Kazhdan-Lusztig bases of parabolic Hecke algebras and applications to Schur-Weyl duality

Kazhdan–Lusztig bases of parab olic Hec k e algebras and applications to Sc h ur–W eyl dualit y J. Guilhot ∗ † , L. P oulain d’Andecy ‡ Abstract With an ey e to applications to type A and Sch ur–W eyl dualit y , w e study Kazhdan–Lusztig bases for a general parab olic Hec k e algebra. P arab olic Hec ke algebras are idempotent subalgebras of Hec k e algebras corresponding to parab olic subgroups, and for type A they coincide with the fused Heck e algebras app earing in a generalisation of the Sch ur–W eyl duality with the quantum group of GL ( N ). In this pap er we inv estigate tw o diﬀerent Kazhdan–Lusztig bases for parab olic Hec ke algebras, together with the asso ciated cells and the corresponding represen tations. W e quic kly sp ecialise to t yp e A, for which we describ e the cells in terms of the RSK corresp ondence generalising thus the well-kno wn description for the symmetric group. As a ﬁrst application w e reco ver the classiﬁcation of irreducible representations of parab olic Heck e algebras of type A and pro vide a new construction of these representations. Next we turn to the Sch ur–W eyl dualit y and describ e the k ernel in terms of one the basis studied preceden tly . Moreo ver, we formulate some conjectures ab out a generator of these kernels in terms of Kazhdan–Lusztig basis elements, giv e some evidence and pro ve these conjectures in some sp ecial cases. 1 In tro duction P arab olic Heck e algebras w ere introduced in [CIK71] as double-cosets algebras relative to parab olic subgroups of reductive groups ov er ﬁnite ﬁelds. In general, they received far less atten tion than their w ell-known particular case: the usual Iwahori–Hec k e algebra (corresp onding to a Borel subgroup). P arab olic Heck e algebras can also b e deﬁned generically (with a generic parameter q ) as idemp otent subalgebras of usual Heck e algebras asso ciated to Coxeter groups. In formulas, the deﬁnition is H J ( W ) = e j H ( W ) e j , where H ( W ) is the usual Heck e algebra asso ciated to a Coxeter group W and e J is the q -symmetriser corresp onding to a parab olic subgroup W J of W . The generic parab olic Heck e algebras H J ( W ) are the main ob jects of study in this pap er. In [Cur85], Curt is extended to the parab olic case the Lusztig’s isomorphism b etw een a group algebra and the Hec ke algebra. T o do so, he considered a certain basis of the parab olic Heck e algebra, made of a certain subset of Kazhdan–Lusztig elements of the usual Heck e algebra. This can b e seen as one Kazhdan–Lusztig basis for the parab olic Heck e algebra. P arab olic Heck e algebras hav e also b een studied more recently [APV13], but not from the p oin t of view of Kazhdan–Lusztig theory (see also [Gom98]). Our in terest for parab olic Heck e algebra stems from the Sch ur–W eyl dualit y studied in [CP23]. It was shown there that the parab olic Heck e algebra of type A (that was called fuse d He cke algebr as ∗ J ´ er ´ emie Guilhot passed aw ay on 27th July 2025 when most of the pap er w as already written. The second author dedicates this paper to his memory . T ribute to J´ er ´ emie can be found here: https://www.idpoisson.fr/hommage-guilhot/ † Institut Denis P oisson, UMR CNRS 7013, Universit ´ e de T ours, 37200 T ours, F rance ‡ Lab oratoire de math´ ematiques de Reims, UMR CNRS 9008, Universit ´ e de Reims Champagne-Ardenne, 51100 Reims, F rance. email adr ess : loic.p oulain-dandecy@univ-reims.fr 1 in [CP21, CP23]) allows to obtain the centralisers of tensor pro ducts of some represen tations of the quan tum group U q ( g l N ), namely w e hav e a surjective morphism π J : H J ( S n ) → End U q ( g l N )  S µ 1 q V ⊗ · · · ⊗ S µ d q V  , (1) where the represen tations app earing are q -symmetrised p ow ers of the vector representation V of U q ( g l N ). The relev ant parab olic subgroup of S n corresp onds to the c hoice of µ 1 , . . . , µ d . This gener- alises the usual quantum Sc hur–W eyl dualit y inv olving the usual Heck e algebra π : H ( S n ) → End U q ( g l N )  V ⊗ n ) . (2) The surjectivity of the morphisms π and π J is seen as the ﬁrst fundamen tal theorem of in v arian t theory , while the second fundamen tal theorem w ould b e the description of the k ernel of these morphisms. It is w ell-known that the map π is not injectiv e as so on as n > N . Similarly , the map π J is not injectiv e as so on as d > N [CP23] and it remains to understand its kernel. F or the usual quantum Sch ur–W eyl duality in (2), the quotient of the Heck e algebra app earing is w ell-understo o d, see for example [BEG20, EMTW20, GW93, Har99, Jim86, Mat99, Mur95, RSS12, Res87]. F or example for N = 2, it is the T emp erley–Lieb algebra. An explicit generator of the corresp onding ideal of the Hec ke algebra is kno wn, as w ell as a linear basis. It turns out that all this can b e describ ed en tirely and quite naturally in terms of Kazhdan–Lusztig basis elements. Recall that w e hav e tw o Kazhdan–Lusztig bases for a Hec ke algebra { C w } w ∈ W and { C † w } w ∈ W . With our con ven tions, the kernel of the map π is generated by the q -an tisymmetriser on N + 1 letters, whic h turns out to b e the elemen t C † w N +1 corresp onding to the longest elemen t of the symmetric group S N +1 (prop erly embedded into S n ). In contrast, outside of some particular cases [CP23, Dem25, LZ10, PZ24], the kernel of π J in the parab olic Sch ur–W eyl duality (1) is not w ell-understo o d. Building on the example of the usual Hec ke algebra, one could exp ect again the Kazhdan–Lusztig theory to b e useful here, and this is one of the motiv ations for this work. Note ho wev er that this is not going to b e as simple as in the usual case. One ﬁrst reason is that the q -antisymmetriser C † w N +1 b ecomes trivial in the parab olic Heck e algebra: w e hav e e J C † w N +1 e J = 0. So obviously the kernel has to b e describ ed diﬀerently . A second more serious reason for the increase of the diﬃculty is the following. In the usual Sch ur– W eyl duality , roughly sp eaking, we basically ha ve to forget a single irreducible represen tation of the Hec ke algebra, the one corresp onding to the one-column partition of N + 1 b o xes. This b ecomes diﬀeren t fo the parab olic Sc hur–W eyl dualit y where the kernel of π J in (1) contains in general more than one irreducible representation, even at the ﬁrst level where it is non-trivial. It is b est illustrated with a simple example. If we take d = 3 and µ 1 = µ 2 = µ 3 = 2 in (1) then here are the irreducible represen tations (with their dimensions) of the parab olic Heck e algebras: 1 2 3 1 1 2 1 F or N = 2, the k ernel contains the three irreducible represen tations in the shaded area. Therefore, when we had a single canonical c hoice to obtain the ideal in the usual Sc hur–W eyl duality , now we are left with more freedom and no clear indication on what will b e a (natural) generator of the ideal. In [CP23] a conjectural generator of the ideal was giv en relying on diagrammatical considerations. In the presen t pap er we will build on Kazhdan–Lusztig theory (that we need to dev elop a little for parab olic Hec ke algebras) to try to obtain the ideal in a diﬀeren t wa y . 2 T o describ e our strategy in a few words, we notice that the sough t-for ideal, even if consisting of sev eral represen tations, is made up of all those representations which are smaller in the dominance order than a certain hook shape (see the example ab o v e), and moreo v er this hook shape is of dimension 1. Therefore, building on the ideas of cellular algebras (or Kazhdan–Lusztig cells in our case), we lo ok for the unique element in the cell corresp onding to this hook shap e and promote it as our b est candidate for generating the ideal. Remark ably , we conjecture and prov e in some cases that this was exactly the elemen t found diagrammatically in [CP23]. Con tent of the pap er. W e describe in more details the con tent of the pap er. T o follo w the program sk etched ab o ve, we need to dev elop a little bit a theory of Kazhdan–Lusztig cells for parab olic Heck e algebras in order to subsequently apply it to the Sch ur–W eyl duality . So w e start with Kazhdan–Lusztig bases, cells and representations for parab olic Heck e algebras in general, keeping in mind our goal to deal sp eciﬁcally with the t yp e A, where the Kazhdan–Lusztig theory w orks especially well. W e consider in this paper tw o diﬀerent Kazhdan–Lusztig bases for a general parab olic Heck e algebra H J ( W ): { C r + ( D ) } and { e J C † r − ( D ) e J } , indexed b y D ∈ W J \ W /W J . (3) They are indexed b y the double cosets of the parab olic subgroup W J in W , and the ﬁrst one inv olv es the maximal-length represen tatives r + ( D ) of suc h cosets, while the second one in volv es the minimal-length represen tatives r − ( D ). The ﬁrst basis will certainly b e considered as the natural Kazhdan–Lusztig basis for parab olic Heck e algebra and indeed it is the one app earing in [Cur85]. The second one seems to b e new and ma y app ear at ﬁrst to b e b oth less practical due to the presence of the idemp oten t e J . How ever, note ﬁrst that the elements r − ( D ) of the Coxeter group W in volv ed are smaller than their coun terparts r + ( D ). More imp ortan tly , we stress that the second basis is the only one that will b e relev ant in the Sch ur–W eyl duality con text. Maybe it is a go o d place to emphasise that the symmetry b et w een the tw o bases { C w } and { C † w } whic h holds in the usual Heck e algebra is broken in the parab olic setting, due to the presence of the idemp oten t e J . In particular the tw o bases in (3) b eha v e quite diﬀerently . As far as general theory is concerned, we show that the tw o bases (3) hav e indeed the exp ected prop ert y , namely , they are uniquely c haracterised by the bar-inv ariance and a unitriangular decom- p osition with resp ect to a standard basis with co eﬃcien ts having the required p olynomial prop ert y . Ha ving those tw o bases, we can pro ceed with the usual notions of (left, right, t wo-sided) cells and asso ciated cell representations for a general parab olic Heck e algebra. W e pro ve a general result, namely that with this theory the cell mo dules for the parab olic Hec ke algebras are the pro jections (with e J ) of the cell mo dules of the usual Heck e algebras. Here app ears the fact that the second basis is more delicate to handle, and we need an assumption (irreducibility of the cell mo dules) which is going to b e satisﬁed in type A. A t this p oin t, we sp ecialise in the rest of the pap er to the parab olic Heck e algebra of type A. First w e build on the general theory to study the cell structure. Our main results are the following: • W e completely describ e the cells corresp onding to our tw o diﬀeren t bases. F or b oth bases, there is a nice and clean description using tw o diﬀeren t Robinson–Sc hensted–Knuth (RSK) cor- resp ondences inv olving semistandard Y oung tableaux. This we see as the generalisation of the w ell-known description of cells for the usual symmetric group, and w e see the parabolic Hec ke al- gebra (and its Kazhdan–Lusztig theory) as the algebraic incarnation of the RSK corresp ondence b et w een pairs of semistandard Y oung tableaux and double cosets in the symmetric group. • W e recov er the classiﬁcation of [CP23] of the irreducible represen tations in the semisimple setting. This is done from the p oin t of view of the cell represen tations asso ciated to the Kazhdan– Lusztig bases and in particular provides an alternativ e construction, compared to [CP23], of the represen tations. 3 • Using the tw o bases in (3) and the RSK corresp ondence, we make explicit tw o cellular bases in the sense of Graham–Lehrer [GL96]. Finally , we turn to our initial goal, the study of the kernel of the Sch ur–W eyl duality in (1). W e w ork in the semisimple situation in this part. Our ﬁrst main result is that the detailed study of cells and asso ciated represen tations leads very quic kly to a natural description of a linear basis of the ideal in the Sch ur–W eyl dualit y . W e stress again that this is all in terms of the second basis in (3). This could b e the end of the story but we w ould like also to hav e an algebraic generator of this ideal. The second basis provides a natural candidate for such a generator. W e introduce explicitly this natural candidate and formulate tw o conjectures: the ﬁrst one is that it do es provide a generator of the ideal; the second one is that (somewhat miracuously) this generator coincides with the elemen t in tro duced diagrammatically in [CP23]. W e provide some evidence in general and we fully prov e these t wo conjectures in the following cases: • in general for N = 2; therefore the centraliser of an y tensor pro duct of U q ( g l 2 )-represen tations is describ ed in this wa y . • for an y N ≥ 1 when µ is of the form ( µ 1 , 1 , 1 , . . . , 1) (the one-b oundary case). Organisation. Necessary notations and kno wn results on the Kazhdan–Lusztig cells for the sym- metric group are collected in Section 2. The general theory of parab olic Heck e algebras and its Kazhdan-Lusztig bases is developed in Sections 3 and 4. This general theory is applied to type A in Section 5, while the applications to Sch ur–W eyl duality are developed in Section 6. Ac knowledgemen ts. Both authors w ere supp orted b y Agence National de la Recherc he Pro jet AHA ANR-18-CE40-0001 in the course of this inv estigation. Con ten ts 1 In tro duction 1 2 Preliminaries and notations on Heck e algebras 5 2.1 Deﬁnition and standard basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Kazhdan–Lusztig bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Cells and representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 The particular case of type A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 P arab olic Heck e algebras 8 3.1 Double cosets of parab olic subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Deﬁnition of parab olic Heck e algebras and standard basis . . . . . . . . . . . . . . . . 10 4 Kazhdan–Lusztig bases for parab olic Hec ke algebras 11 4.1 A ﬁrst Kazhdan–Lusztig basis for H J ( W ) and its cells . . . . . . . . . . . . . . . . . . 11 4.2 A second Kazhdan–Lusztig basis for H J ( W ) and its cells . . . . . . . . . . . . . . . . 14 5 P arab olic Heck e algebras in type A 18 5.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.2 Cells in S µ \ S n /S µ and RSK corresp ondences . . . . . . . . . . . . . . . . . . . . . . . 20 5.3 Cells and representations of H µ ( S n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.4 Cellular bases of H µ ( S n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4 6 Application to Sch ur–W eyl duality 27 6.1 The Sch ur–W eyl duality for H µ ( S n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6.2 A linear basis for the ideal I µ N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6.3 Conjectures for a generator of the ideal I µ N . . . . . . . . . . . . . . . . . . . . . . . . . 29 2 Preliminaries and notations on Hec k e algebras Let ( W , S ) b e a Coxeter system, for whic h w e denote by ℓ the length function. The left descent L ( x ) of an element x ∈ W consists of the simple transp ositions s ∈ S such that ℓ ( sx ) < ℓ ( x ), and similarly for the righ t descent R ( x ). W e denote by ≤ the strong Bruhat order b et ween elemen ts in W , which means that x ≤ y if a reduced expression for y contains a reduced expression for x (see for example [BB05]). 2.1 Deﬁnition and standard basis W e w ork with an indeterminate q . The Heck e algebra H ( W ) is the Z [ q , q − 1 ]-algebra with basis { T w } w ∈ W , and with multiplication giv en by: T s T w =  T sw if ℓ ( sw ) > ℓ ( w ), ( q − q − 1 ) T w + T sw if ℓ ( sw ) < ℓ ( w ). (4) It is well-kno wn that such an algebra exists, see for example [GP00]. W e denote 1 = T e W . As consequences of the deﬁnition, we hav e: T w T w ′ = T ww ′ if ℓ ( w w ′ ) = ℓ ( w ) + ℓ ( w ′ ), T 2 s = ( q − q − 1 ) T s + 1 or equiv alen tly , ( T s − q )( T s + q − 1 ) = 0 . (5) F or w = s i 1 . . . s i r ∈ W written as a reduced expression, w e hav e T w = T s i 1 . . . T s i r . The basis { T w } w ∈ W is the standar d b asis of H ( W ). 2.2 Kazhdan–Lusztig bases W e recall the construction of Kazhdan–Lusztig bases from [KL79]. W e hav e the follo wing t wo in volu- tiv e ring automorphisms of the Heck e algebra H ( W ) giv en b y their action on the generators and the indeterminate q : · : T i 7→ T − 1 i , q 7→ q − 1 and · † : T i 7→ − T i , q 7→ q − 1 . (6) The ﬁrst Kazhdan–Lusztig basis { C w } w ∈ W is the unique basis satisfying C w = C w and C w = T w + X x 1, T i T i +1 T i = T i +1 T i T i +1 , T 2 i = ( q − q − 1 ) T i + 1 . The notation T i is a shorthand notations for T ( i,i +1) where ( i, i + 1) is the transp osition of S n sw apping i and i + 1. The Robinson–Sc hensted correspondence for p erm utations in S n . The Robinson–Sc hensted corresp ondence (RS for short) is a bijection b etw een the set of p erm utations in S n and the se t of pairs of standard Y oung tableaux of size n and of the same shap e. Giv en a partition λ ⊢ n , we denote b y ST ab( λ ) the set of standard Y oung tableaux of shap e λ . W e denote the RS corresp ondence as follows: S n ↔ G λ ⊢ n ST ab( λ ) 2 w ↔  P ( w ) , Q ( w )  (12) where w e use the same conv en tion as in [F ul97, Knu70]. Given w ∈ S n , the partition λ whic h is the shap e of P ( w ) and Q ( w ) is denoted sh ( w ). T o av oid ambiguit y , we indicate that p erm utations are comp osed from righ t to left, so that s 1 s 2 is the p ermutation  1 2 3 2 3 1  . Applying the usual insertion algorithm to the sequence 231, we hav e the follo wing example of the RS correspondence, which should b e enough to illustrate our con v entions: s 1 s 2 ↔  1 3 2 , 1 2 3  W e see in this example an illustration of the general prop ert y that the p erm utation w has a generator s i in its left descen t (in the example, s 1 ) if and only if the tableau P ( w ) has i in its descent, which means that i + 1 is in a low er row than i . Similarly , w has a generator s i in its right descent (in the example, s 2 ) if and only if the tableau Q ( w ) has i in its descent. Cells and RS corresp ondence. The cell structure of S n turns out to b e in timately related to the RS corresp ondence. Using either the basis { C w } w ∈ S n or the basis { C † w } w ∈ S n , the cells are the same and are describ ed as follows: • w and w ′ are in the same      left righ t 2-sided cell in S n if and only if      Q ( w ) = Q ( w ′ ) P ( w ) = P ( w ′ ) sh ( w ) = sh ( w ′ ) ; • w ⪯ LR w ′ if and only if sh ( w ) ≤ sh ( w ′ ); where in the last item, we use the dominance order on partitions. Cells and representations. Let Γ b e a left cell in S n . As set up in Section 2.3, w e denote V Γ the corresp onding cell representation of H ( S n ) when we use the basis { C w } and we denote V † Γ the represen tation obtained when we use the basis { C † w } . In this paragraph, w e consider the semisimple situation, namely we work ov er the ﬁeld C ( q ) or with a non-zero complex num b er q such that q 2 is not a ro ot of unity whose order is b et w een 2 and 7 n . W e denote by { V λ } λ ⊢ n the set of irreducible representations of the Hec ke algebra H ( S n ), using the standard indexation b y partitions of n . In particular the one-dimensional representation T w 7→ q ℓ ( w ) corresp onds to the single-line partition. The cell represen tations of H ( S n ) are describ ed as follows. Let λ ⊢ n and take Γ an y left cell of S n con taining elemen ts w with sh ( w ) = λ . The isomorphism class of the cell representation of H ( S n ) corresp onding to Γ dep ends on which basis we use. W e hav e: • The cell represen tation V Γ is isomorphic to V λ t . • The cell represen tation V † Γ is isomorphic to V λ . where we use λ t to denote the transp ose of the partition λ (the partition obtained by exchanging the lines and columns of the Y oung diagram of λ ). 3 P arab olic Heck e algebras W e keep ( W, S ) an arbitrary Co xeter system. W e let J b e a non-empty subset of S and W J the corresp onding parab olic subgroup of W . W e assume that W J is ﬁnite. 3.1 Double cosets of parab olic subgroups Minimal-length represen tatives. The following classical facts can b e found in [GP00, chap. 2]. W e denote X J the set of distinguished representativ es for the left cosets of W J in W . An element x ∈ X J is characterised by b eing the unique elemen t of minimal length in its left coset xW J (or equiv alently , the unique minimal elemen t for the Bruhat order in xW J ). It satisﬁes that ℓ ( xu ) = ℓ ( x ) + ℓ ( u ) for any u ∈ W J , and moreo ver (Deo dhar Lemma) we hav e: for an y s ∈ J , either sx ∈ X J , or sx = xt for some t ∈ J . Similarly for righ t cosets, the set of minimal-length representativ e is X − 1 J . In each double cos et in W J \ W /W J , there is also a unique element of minimal length (or equiv a- len tly , minimal for the Bruhat order). W e denote X J J the set of minimal-length represen tatives for the double cosets of W J in W . W e hav e X J J = X J ∩ X − 1 J . Minimal-length represen tatives are characterised in terms of their descents as follows: x ∈ X J ⇔ R ( x ) ∩ J = ∅ . x ∈ X J J ⇔ L ( x ) ∩ J = R ( x ) ∩ J = ∅ . Maximal length represen tatives. Here we use that W J is ﬁnite, and thus that left, righ t or double cosets are ﬁnite. The following classical facts can b e found in [Cur85, Theorem 1.2] or [BKP+18]. There is a unique element of maximal length in each left coset (or equiv alently , a unique maximal elemen t in the Bruhat order). W e denote b y e X J these maximal-length represen tatives. If we denote w J the longest elemen t of W J , then w e hav e: e X J := { xw J , x ∈ X J } . Similarly for righ t cosets, the set of maximal-length representativ e is e X − 1 J . In each double coset in W J \ W /W J , there is also a unique element of maximal length (or equiv a- len tly , maximal for the Bruhat order). W e denote e X J J the set of maximal-length representativ es for the double cosets of W J in W . W e hav e e X J J = e X J ∩ e X − 1 J . In terms of descents, maximal-length representativ es are c haracterised as follows x ∈ e X J ⇔ J ⊂ R ( x ) . x ∈ e X J J ⇔ J ⊂ R ( x ) ∩ L ( x ) . 8 Canonical expressions. Given a double coset D ∈ W J \ W /W J , we introduce a notation for its minimal-length elemen t, and its m aximal-length element: r − ( D ) ∈ D ∩ X J J and r + ( D ) ∈ D ∩ e X J J . Let x = r − ( D ) for some double coset D . The subset J ∩ xJ x − 1 of S gives rise to a parab olic subgroup W J ∩ xJ x − 1 , whic h is a parabolic subgroup of W J . As suc h, there is a set of distinguished representativ es for left cosets of W J ∩ xJ x − 1 in W J , that we denote X J J ∩ xJ x − 1 . With these notations, we ha ve that any elemen t w ∈ D can b e written uniquely as: w = w 1 xw 2 , w 2 ∈ W J and w 1 ∈ X J J ∩ xJ x − 1 . (13) and in this situation, we hav e ℓ ( w ) = ℓ ( w 1 ) + ℓ ( x ) + ℓ ( w 2 ). F urthermore, the maximal-length element r + ( D ) of D can b e written as follows: r + ( D ) = w J w L xw J , where L = J ∩ xJ x − 1 , (14) where w L , w J denote the longest elements of the corresp onding parab olic subgroups of W . Finally , the double coset D consists of the full interv al in the Bruhat order: D = [ r − ( D ) , r + ( D )] , that is, the double coset D coincides with the set of all elemen ts y ∈ W such that r − ( D ) ≤ y ≤ r + ( D ). Example 3.1. T ake W = S 6 the symmetric gr oup gener ate d by s 1 , . . . , s 5 (se e Se ction 5 for the notations) and W J = S 2 × S 2 × S 2 c orr esp onding to J = { s 1 , s 3 , s 5 } . We have w J = s 1 s 3 s 5 . The element x = s 2 s 1 s 4 s 3 s 2 is minimal in its double c oset. Her e we have J ∩ xJ x − 1 = { 1 } (b e c ause xs 3 = s 1 x ). The c orr esp onding maximal element is y = s 3 s 5 .x.s 1 s 3 s 5 . Bruhat order for double cosets. Let D and D ′ b e tw o double cosets in W J \ W /W J . The Bruhat order on W extends naturally to the double cosets W J \ W /W J , through their distinguished represen- tativ es. That is, we set: D ≤ D ′ ⇔ r − ( D ) ≤ r − ( D ′ ) . In our situation ( W J ﬁnite) w e hav e another set of distinguished represen tatives, the maximal-length represen tatives, and they seem to provide an alternativ e c hoice for extending the Bruhat order to W J \ W /W J . W e will use the following result, asserting that the tw o choices are equiv alent. Prop osition 3.2. L et D , D ′ ∈ W J \ W /W J . We have: r − ( D ) ≤ r − ( D ′ ) ⇔ r + ( D ) ≤ r + ( D ′ ) . Pr o of. Let D and D ′ b e distinct cosets. First let w ∈ D and w ′ ∈ D ′ suc h that w < w ′ . T ake s ∈ J \L ( w ). W e hav e that: ∃ w ′′ ∈ D ′ suc h that sw < w ′′ . Indeed, if s / ∈ L ( w ′ ) then we can tak e w ′′ = sw ′ . Whereas if s ∈ L ( w ′ ) then the lifting prop ert y of the Bruhat order [BB05, Prop. 2.2.7] ensures that sw ≤ w ′ . The strict inequality follo ws since sw and w ′ b elong resp ectiv ely to D and D ′ whic h are distinct. No w let x = r − ( D ) and y = r − ( D ′ ) and assume that x < y . W e hav e xw J < y w J since x and y are in particular distinguished left representativ e. F rom (14), a reduced expression for r + ( D ) is r + ( D ) = s 1 . . . s k xw J for some s 1 , . . . , s k in J . Since the expression is reduced, s i is not in the left descen t of s i +1 . . . s k xw J . So we apply the ab o ve observ ation k times and get an elemen t y ′ ∈ D ′ suc h that r + ( D ) < y ′ . Since y ′ ≤ r + ( D ′ ), w e conclude that r + ( D ) ≤ r + ( D ′ ). Finally assume that x and y are not comparable, and assume that r + ( D ) < r + ( D ′ ). This implies b y transitivity that x < r + ( D ′ ). Again by (14) there is a reduced expression for r + ( D ′ ) of the form r + ( D ′ ) = s 1 . . . s k y t 1 . . . , t l , for some s 1 , . . . , s k , t 1 , . . . , t l in J . So a reduced expression of x must b e a sub expression of this, while not a sub expression of y . This implies that x must hav e a reduced expression with some elements of J on the left or on the right. This con tradicts the fact that x is a minimal-length represen tative. 9 P oincar´ e p olynomials. The Poincar ´ e p olynomial of a ﬁnite Coxeter group W ′ is: W ′ ( q 2 ) = X w ∈ W ′ q 2 ℓ ( w ) . (15) No w tak e a parab olic subgroup W K of the ﬁnite Coxeter group W J . Denote X K J the set of minimal- length representativ es of left cosets of W K in W J . W e ha ve that any x ∈ W J is uniquely written as x = du where d ∈ X K J and u ∈ W K , and suc h that ℓ ( x ) = ℓ ( d ) + ℓ ( u ). Therefore, we ﬁnd: W J ( q 2 ) = W K ( q 2 ) X x ∈ X J K q 2 ℓ ( x ) . (16) So W K ( q 2 ) divides W J ( q 2 ) for an y subset K ⊂ J (in particular, (1 + q 2 ) alw ays divides W J ( q 2 )). 3.2 Deﬁnition of parab olic Heck e algebras and standard basis F rom now on, we extend the algebra H ( W ) ov er the following lo calization A := Z [ q , q − 1 , W J ( q 2 ) − 1 ] , (17) where W J ( q 2 ) is the Poincar ´ e p olynomial given in (15). Deﬁnition of the parab olic Heck e algebra. W e deﬁne: 1 J := X w ∈ W J q ℓ ( w ) T w . The element 1 J is a quasi-idemp oten t, and, o ver the base ring A , we renormalise it to get an idemp o- ten t: e J = 1 W J ( q 2 ) 1 J = 1 W J ( q 2 ) X w ∈ W J q ℓ ( w ) T w . (18) This is sometimes called the q -symmetriser asso ciated to the subalgebra of H ( W ) generated b y T s with s ∈ J , and with basis { T w } w ∈ W J . The main prop ert y of e J , implying that e 2 J = e J , is: T w e J = e J T w = q ℓ ( w ) e J for all w ∈ W J . Deﬁnition 3.3. The p ar ab olic He cke algebr a H J ( W ) is the algebr a over A deﬁne d by: H J ( W ) = e J H ( W ) e J . Remark 3.4. Mor e gener al ly, one c an deﬁne, as in [APV13], the p ar ab olic He cke algebr a as the algebr a 1 J H ( W ) 1 J dir e ctly over Z [ q , q − 1 ] . This algebr a is not unital if W J ( q 2 ) is not invertible. Over the extende d b ase ring A , the two deﬁnitions c oincide, and the algebr a H J ( W ) is unital with unit e J . The idempotent e J is almost a Kazhdan–Lusztig basis elemen t. Indeed, let w J b e the longest elemen t of the parab olic subgroup W J . Then we hav e C w J = X w ∈ W J q ℓ ( w ) − ℓ ( w J ) T w = q − ℓ ( w J ) 1 J = q − ℓ ( w J ) W J ( q 2 ) e J . (19) Note that F ormula (19) shows in particular that the idemp otent e J is bar in v ariant: e J = e J , (20) where the bar inv olution was deﬁned in (6). 10 Standard basis of H J ( W ) . F or a double coset D ∈ W J \ W /W J , deﬁne: T D := X w ∈D q ℓ ( w ) − ℓ ( r + ( D )) T w . The substraction of ℓ ( r + ( D )) is a normalisation choice, and is such that the co eﬃcien t of the longest elemen t T r + ( D ) in T D is equal to 1 (note that all other co eﬃcien ts are negative p ow ers of q ). The standard basis of H J ( W ) is [APV13, Cur85, CIK71]: { T D } D∈ W J \ W/W J . (21) It might not b e immediately clear that the elements T D b elong to the algebra H J ( W ). In fact, if w e denote x = r − ( D ) the minimal-length representativ e of D , from (13) w e deduce (see [APV13]) that: T D = W J ( q 2 ) 2 W J ∩ xJ x − 1 ( q 2 ) q ℓ ( r − ( D )) − ℓ ( r + ( D )) e J T x e J . (22) As recalled in (16), W J ∩ xJ x − 1 ( q 2 ) divides W J ( q 2 ), and moreov er, in the ring A , W J ( q 2 ) and all its factors are inv ertible. So ov er the ring A , renormalising the basis elemen ts ab ov e, w e also ha ve bases of H J ( W ) of the form: { e J T w e J } w ∈ R J for an y set R J of represen tatives of W J \ W /W J . (23) Indeed we hav e that e J T w e J = q ℓ ( w ) − ℓ ( x ) e J T x e J whenev er w ∈ W J xW J . This follows from the prop ert y (13) of double cosets and from the basic prop ert y (18) of e J . Two natural choices for R J are of course R J = X J J and R J = e X J J , the set of, respectively , minimal-length and maximal-length represen tatives. 4 Kazhdan–Lusztig bases for parab olic Hec k e algebras 4.1 A ﬁrst Kazhdan–Lusztig basis for H J ( W ) and its cells 4.1.1 The basis Let D a double coset in W J \ W /W J . Recall that r + ( D ) is the unique element of maximal-length in D . Since any s ∈ J is in the left descent and in the righ t descent of r + ( D ), from prop ert y (9), w e ha ve immediately that: e J C r + ( D ) e J = C r + ( D ) , (24) so that the elemen ts C r + ( D ) b elong to H J ( W ). The next result shows that the ab o ve set of elements forms a basis and compares it with the standard basis { T D } of H J ( W ) deﬁned in (21). Prop osition 4.1. The set { C r + ( D ) } D∈ W J \ W/W J is a b asis of H J ( W ) , and we have: C r + ( D ) = X D ′ ≤D p r + ( D ′ ) ,r + ( D ) T D ′ . (25) Mor e over, C r + ( D ) is the unique element B D of H J ( W ) satisfying B D = B D and B D = T D + X D ′ < D a D ′ D T ′ D with a D ′ D ∈ q − 1 Z [ q − 1 ] . Pr o of. The fact that { C r + ( D ) } D∈ W J \ W/W J forms a basis of H J ( W ) can b e found in [Cur85, Theo. 1.10]. It follows from (25) whic h can b e chec ked by a short direct calculation that we provide here. Let x = r + ( D ) and let D ′ ∈ W J \ W /W J . Recall from (13) that any element y of D ′ can b e written as a reduced expression of the form: y = s 1 . . . s k r − ( D ) t 1 . . . t l , with s i , t i ∈ J . 11 Moreo ver any s i , t i ∈ J are in the left and right descen ts of x (maximal-length represen tative). There- fore w e can use the prop erty (10) of Kazhdan–Lusztig p olynomials and obtain: p y ,x = q ℓ ( y ) − ℓ ( r − ( D ′ )) p r − ( D ′ ) ,x = q ℓ ( y ) − ℓ ( r + ( D ′ )) p r + ( D ′ ) ,x for an y y ∈ D ′ . (26) The last equality uses p r + ( D ′ ) ,x = q ℓ ( r + ( D ′ )) − ℓ ( r − ( D ′ )) p r − ( D ′ ) ,x , which is simply the ﬁrst equalit y for y = r + ( D ′ ). No w we can calculate as follows: C x = X y ≤ x p y ,x T y = X D ′ ∈ W J \ W/W J X y ∈D ′ p y ,x T y = X D ′ ∈ W J \ W/W J  p r + ( D ′ ) ,x X y ∈D ′ q ℓ ( y ) − ℓ ( r + ( D ′ )) T y  = X D ′ ∈ W J \ W/W J p r + ( D ′ ) ,x T D ′ . In the last sum ab o ve, if T D ′ app ears with a non-zero co eﬃcien t, this means that T r + ( D ′ ) app eared with a non-zero co eﬃcien t in C x , and this means that r + ( D ′ ) ≤ r + ( D ). According to Prop osition 3.2, this is equiv alent to D ′ ≤ D . The form ula in the prop osition expresses then a unitriangular change of basis, since the co eﬃcien t in front of T D is ob viously 1. The element C r + ( D ) is indeed stable under the bar inv olution, by property of the Kazhdan–Lusztig basis of H ( W ). It is immediate that the ﬁrst co eﬃcien t in the decomp osition is 1 while the others are in q − 1 Z [ q − 1 ], from the similar prop erties of the p olynomials p x,y . It remains to prov e the unicity statemen t. It is easily c heck ed [Lus03, Theorem 5.2] that for an elemen t h = P w ∈ W a w T w in the Hec ke algebra with co eﬃcien ts a w ∈ q − 1 Z [ q − 1 ], we hav e that if h = h then h = 0. If there is another elemen t X D satisfying the required prop erties, then X D − C r + ( D ) is such an elemen t h and therefore is 0. Example 4.2. T ake W = S 4 the symmetric gr oup gener ate d by s 1 , s 2 , s 3 and W J = S 2 × S 2 c orr e- sp onding to J = { s 1 , s 3 } . Ther e ar e thr e e double c osets: [13] , [12321] , [121321] , whose names r eﬂe ct how their longest r epr esentatives write in terms of the gener ators s 1 , s 2 , s 3 . The formulas il lustr ating the pr op osition ar e: C 13 = T [13] , C 12321 = T [12321] + ( q − 1 + q − 3 ) T [13] , C 121321 = T [121321] + q − 1 T [12321] + q − 4 T [13] . F or example, q − 1 + q − 3 is the Kazhdan–Lusztig p olynomial p s 1 s 3 ,s 1 s 2 s 3 s 2 s 1 giving the c o eﬃcients of T s 1 s 3 in the exp ansion of C s 1 s 2 s 3 s 2 s 1 . 4.1.2 Cells in W J \ W /W J and represen tations of H J ( W ) In this subsection, the orders, the cells and the asso ciated representations of H ( W ) are those con- structed from the basis { C w } of H ( W ). The orders and the cells on W J \ W /W J as well as the asso ciated represen tations of H J ( W ) are deﬁned similarly , using the basis { C r + ( D ) } of H J ( W ) just obtained. Prop osition 4.3. • L et X stands for L or R or LR . We have D ⪯ X D ′ in W J \ W /W J ⇐ ⇒ r + ( D ) ⪯ X r + ( D ′ ) in W . • In p articular, the left c el ls in W J \ W /W J (and similarly for right and double-side d c el ls) ar e the non-empty sets of the form: Γ ∩ e X J J , for a left c el l Γ in W , 12 wher e we have identiﬁe d double c osets in W J \ W /W J with their maximal-length r epr esentatives in e X J J . Pr o of. Let D , D ′ ∈ W J \ W /W J . If there is h ∈ H J ( W ) such that C r + ( D ) → h C r + ( D ′ ) in H J ( W ) then seeing h as an element of H ( W ), it trivially implies the same prop ert y in H ( W ). Recipro cally , assume that there is h ∈ H ( W ) such that C r + ( D ) → h C r + ( D ′ ) . Recall that the ele- men ts C r + ( D ) and C r + ( D ′ ) are in v arian t b y left or righ t m ultiplication b y e J . So w e ha ve e J he J C r + ( D ) = e J hC r + ( D ) and thus C r + ( D ) → e J he J e J C r + ( D ′ ) = C r + ( D ′ ) in H J ( W ). This sho ws the equiv alence for the left order. A similar reasoning shows the desired result for the right m ultiplication, and the ﬁrst item follo ws. The second item is an immediate consequence of the ﬁrst one. Example. W e take W = S 6 the symmetric group generated b y s 1 , . . . , s 5 and W J = S 2 × S 2 × S 2 generated b y s 1 , s 3 , s 5 . Using the one-line notation for a p erm utation, here is a left cell in S 6 : Γ = { 615432 , 625431 , 635421 , 645321 , 546321 } . Among these 5 elemen ts, only tw o are in e X J J (one has to lo ok for those elements with left and right descen ts con taining 1 , 3 , 5) and the resulting cell in W J \ W /W J consists of the tw o double cosets of the follo wing elements Γ ∩ e X J J = { 625431 , 645321 } . In this example, the left cell Γ corresp onds to p ermutations having their right tableau in the RS corresp ondence equal to 1 3 2 4 5 6 . The in tersection with e X J J is not empty b ecause we ha ve chosen a tableau containing 1 , 3 , 5 in its descent. Actually , the intersection is made of those p erm utations ha ving a left tableau whic h also contains 1 , 3 , 5 in its desce n t. In t yp e A, the description of the cells of W J \ W /W J in terms of the RS corresp ondence is simple and will b e explicited in the next section. Represen tations of H J ( W ) . Then we discuss the represen tations of H J ( W ) induced by its left cells and relate them to the cell representations of H ( W ). W e note that for any representation V of H ( W ), the vector space e J ( V ) (which ma y b e { 0 } ) carries naturally a representation of H J ( W ). Let Γ b e a left cell for W and V Γ the asso ciated representation of H ( W ), with basis { C w + I ≺ L Γ } w ∈ Γ . If non-empt y , the subset Γ ∩ e X J J indexes the elements in a left cell of W J \ W /W J , thanks to the prop osition ab o ve. W e denote V Γ ∩ e X J J the asso ciated representation of H J ( W ). Prop osition 4.4. L et Γ b e a left c el l of W such that Γ ∩ e X J J  = ∅ . As r epr esentations of H J ( W ) , we have V Γ ∩ e X J J = e J ( V Γ ) . Pr o of. Recall that e X − 1 J denotes the set of maximal-length representativ es of righ t cosets in W J \ W , or equiv alen tly , the set of elemen ts con taining J in their left descent. First we note that the set of elemen ts: { C x } x ∈ e X − 1 J is a basis of the subspace e J H ( W ) (the image of H ( W ) b y left m ultiplication by e J ). The pro of is similar to the pro of of Prop osition 4.1. Namely , we take x ∈ e X − 1 J and we write, using the same argumen ts, that: C x = X x ′ ∈ W J \ W X y ∈ W J x ′ p y ,x T y = X x ′ ∈ W J \ W p x ′ ,x  X y ∈ W J x ′ q ℓ ( y ) − ℓ ( x ′ ) T y  . The sum in parenthesis, deﬁned for any coset in W J \ W forms a basis of e J H ( W ) (since e J T z is prop ortional to it for any z ∈ W J x ′ ) and the ab o ve form ula th us expresses a triangular c hange of basis. 13 Then we take Γ an arbitrary left cell of W , and w e prov e that the following subset is a basis of the subspace e J ( V Γ ): { C w + I ≺ L Γ } w ∈ Γ ∩ e X − 1 J . (27) F rom prop ert y (9), we ha ve immediately that e J C w = C w if w ∈ e X − 1 J , so that the elements in (27) indeed b elong to e J ( V Γ ). They are obviously linearly indep enden t, as a subset of the basis of V Γ . Then take w ∈ Γ which is not in e X − 1 J . Inside e J H ( W ), the elemen t e J C w decomp oses in the basis { C x } x ∈ e X − 1 J , and moreov er uses only elemen ts in Γ or strictly less than Γ (for ≺ L ). Thus, e J C w + I ≺ L Γ can b e written in terms of the elemen ts in (27), which is therefore a generating set of e J ( V Γ ). No w we assume that Γ is such that Γ ∩ e X J J  = ∅ . It means that there is an element in Γ whic h con tains J in its righ t descen t. Since the righ t descen ts of all elements in the same left cell coincide (see (11), this means that all elements in Γ con tain J in their righ t descent. In this case, the basis ab o v e of e J ( V Γ ) b ecomes: { C w + I ≺ L Γ } w ∈ Γ ∩ e X J J . This iden tiﬁes immediately with the basis of the cell represen tation V Γ ∩ e X J J of H J ( W ) and the H J ( W )- mo dule structures (by left multiplication) are the same. 4.2 A second Kazhdan–Lusztig basis for H J ( W ) and its cells In the previous subsection, we ha v e studied a Kazhdan–Lusztig basis of H J ( W ) coming from the basis { C w } of H ( W ). In this section w e wan t to use the other basis { C † w } of H ( W ). W e will see that it b eha v es quite diﬀerently with resp ect to the idemp oten t e J . In t yp e A, this basis will b e relev ant for the applications to Sch ur–W eyl duality in Section 6. 4.2.1 The basis The basis { C r + ( D ) } of the previous section has the property that e J C r + ( D ) e J = C r + ( D ) , so that it answ ers nicely the question of describing the image of the tw o-sided multiplication by e J on H ( W ). W e can also wonder ab out the kernel of the tw o-sided m ultiplication by e J on H ( W ). It turns out that there is also a nice description, this time in terms of the other Kazhdan–Lusztig basis { C † w } of H ( W ). Indeed, w e hav e: e J C † w e J = 0 for all w / ∈ X J J . T o see this, let w / ∈ X J J . This means that there is some s ∈ J suc h that s ∈ R ( w ) or s ∈ L ( w ). Sa y s ∈ R ( w ). So we ha ve: C † w e J = C † w 1 + q T s 1 + q 2 e J = 0 . The ﬁrst equalit y uses T s e J = q e J if s ∈ J , while the second equalit y uses C † w T s = − q − 1 C † w if s ∈ R ( w ). Recall also that (1 + q 2 ) is in vertible in A . A similar pro of works if s ∈ L ( w ). In fact the elemen ts C † w with w / ∈ X J J form a basis of the kernel of the tw o-sided multiplication b y e J , and w e obtain a basis of H J ( W ) from the remaining elements, as w e show in the follo wing prop osition. Note that for this second basis, it is more natural to w ork with the renormalised standard basis { e J T r − ( D ) e J } of H J ( W ), see (22) and (23). Prop osition 4.5. The set { e J C † r − ( D ) e J } D∈ W J \ W/W J is a b asis of H J ( W ) and we have: e J C † r − ( D ) e J = X D ′ ≤D a D ′ , D e J T r − ( D ′ ) e J , wher e a D ′ , D = X y ∈D ′ ( − 1) ℓ ( y ) q ℓ ( y ) − ℓ ( r − ( D ′ )) p y ,r − ( D ) . (28) Mor e over, e J C † r − ( D ) e J is the unique element B D of H J ( W ) satisfying B D = B D and B D = ( − 1) ℓ ( r − ( D )) e J T r − ( D ) e J + X D ′ < D α D ′ D e J T r − ( D ′ ) e J with α D ′ D ∈ q Z [ q ] . 14 Pr o of. The pro of is a calculation similar to the pro of of Prop osition 4.1. W e use that e J T y e J = q ℓ ( y ) − ℓ ( r − ( D ′ )) e J T r − ( D ′ ) e J , if y is in the double coset D ′ . T o give details, we let x = r − ( D ) and we write: e J C † x e J = X y ≤ x ( − 1) ℓ ( y ) p y ,x e J T y e J = X D ′ ∈ W J \ W/W J X y ∈D ′ ( − 1) ℓ ( y ) p y ,x e J T y e J = X D ′ ∈ W J \ W/W J  X y ∈D ′ ( − 1) ℓ ( y ) p y ,x q ℓ ( y ) − ℓ ( r − ( D ′ ))  e J T r − ( D ′ ) e J . The double cosets D ′ app earing with non-zero co eﬃcien t must con tain an element y ∈ D ′ suc h that y ≤ x . This implies that r − ( D ′ ) ≤ x . So the sum can b e restricted to D ′ ≤ D . Moreo v er it is immediate that a D , D = ( − 1) ℓ ( r − ( D )) since the only element y in D satisfying y ≤ x is y = x , due to the minimalit y of x = r − ( D ). Thus the formula expresses a unitriangular (up to a sign) c hange of basis. The stability of e J C † r − ( D ) e J under the bar inv olution is immediate since each factor is stable, see (20). The ﬁrst co eﬃcient was calculated b efore and the fact that the other co eﬃcien ts a D ′ , D are in q Z [ q ] is immediate since the p olynomials p y ,r − ( D ) are in q Z [ q ] for y ∈ D ′ . The unicit y statement is pro ved exactly as in the end of the pro of of Prop osition 4.1 (with q instead of q − 1 ). Remark 4.6. The c o eﬃcients in (28) ar e p olynomials in q with inte ger c o eﬃcients, but they do not have to b e in Z ≥ 0 [ q ] or in Z ≤ 0 [ q ] as shown in the se c ond example b elow. Example 4.7. • T ake W = S 4 and W J = S 2 × S 2 as in Example 4.2. The thr e e double c osets have as minimal r epr esentatives: e, s 2 , s 2 s 1 s 3 s 2 . The formulas il lustr ating the pr op osition ar e: e J C † e e J = e J e J C † s 2 e J = − e J T s 2 e J + q e J e J C † s 2 s 1 s 3 s 2 e J = e J T s 2 s 1 s 3 s 2 e J − ( q + q 3 ) e J T s 2 e J + q 2 e J F or example, in C † s 2 s 1 s 3 s 2 , the terms c orr esp onding to the trivial double c oset ar e q 2 T s 1 s 3 − q 3 T s 1 − q 3 T s 3 + ( q 2 + q 4 ) T e , so that the q 2 in fr ont of e J ab ove is obtaine d as q 4 − q 4 − q 4 + ( q 2 + q 4 ) . • T ake W = S 4 and W J = S 1 × S 2 × S 1 . The p ar ab olic sub gr oup is gener ate d by s 2 . Ther e ar e 7 double c osets and one of the minimal-length r epr esentatives is s 1 s 2 s 3 s 2 s 1 . The formula in the pr op osition is: eC † 12321 e = − eT 12321 e + q 2 eT 123 e + q 2 eT 321 e + ( q − q 3 ) eT 13 e − q 2 eT 1 e − q 2 eT 3 e + q 3 e , wher e we have abbr eviate d e J by e , and the gener ators s i by their letters i . We note the c o eﬃcient ( q − q 3 ) in fr ont of eT 13 e c ontaining b oth signs ± 1 . It is obtaine d by lo oking at the fol lowing c o eﬃcients in C † 12321 : C † 12321 = ... + ( q + q 3 ) T 13 − q 2 T 213 − q 2 T 132 + 0 T 2132 , which ar e the c oﬃcients in fr ont of the elements in the double c oset of s 1 s 3 . The c oﬃcient ( q − q 3 ) is obtaine d as ( q + q 3 ) − q 3 − q 3 . Remark 4.8. L o oking at the b asis in the pr evious pr op osition, we may wonder why not c onsidering the set of elements { e J C r − ( D ) e J } D∈ W J \ W/W J . In fact, one c an pr ove exactly as ab ove that this is inde e d a b asis of H J ( W ) and that we have: e J C r − ( D ) e J = X D ′ ≤D b D ′ , D e J T r + ( D ′ ) e J , (29) 15 wher e b D ′ , D = P y ∈D ′ q ℓ ( y ) − ℓ ( r + ( D ′ )) p y ,x . These c o eﬃcients ar e p olynomials in q − 1 with inte ger c o eﬃ- cients, thanks to the use of the b asis elements e J T r + ( D ′ ) e J inste ad of e J T r − ( D ′ ) e J . Obviously, if the p olynomials p y ,x ar e in Z ≥ 0 [ q − 1 ] then so ar e the c o eﬃcients b D ′ , D . Stil l, this b asis is less natur al and less e asy to hand le than the b asis in Pr op osition 4.1, and unlike the b asis in Pr op osition 4.5, it do es not play any r ole in our study of the Schur–Weyl duality in Se ction 6. So we wil l not c onsider it further. 4.2.2 Cells in W J \ W /W J and represen tations of H J ( W ) In this subsection, the orders, the cells and the asso ciated representations of H ( W ) are those con- structed from the basis { C † w } w ∈ W of H ( W ). The orders, the cells and the asso ciated representations of H J ( W ) are deﬁned similarly , using the basis { e J C † r − ( D ) e J } D∈ W J \ W/W J of H J ( W ) just obtained. Cells in W J \ W /W J . Here is the statemen t that is v alid in general for the t yp e of cells in W J \ W /W J considered in this section. Prop osition 4.9. L et X stands for L or R or LR . We have D ⪯ X D ′ in W J \ W /W J = ⇒ r − ( D ) ⪯ X r − ( D ′ ) in W . Pr o of. Assume that there exists h ∈ H J ( W ) such that e J C † r − ( D ) e J app ears in he J C † r − ( D ′ ) e J , which is equal to e J hC † r − ( D ′ ) e J since h ∈ H J ( W ). No w expand hC † r − ( D ′ ) in the basis { C † w } in H ( W ) and then m ultiply on b oth sides b y e J . All terms C † w with w / ∈ X J J giv e 0. This shows that C † r − ( D ) app ears in hC † r − ( D ′ ) and this sho ws the implication for the left order. The veriﬁcation for the right order is the same and this implies the implication for the tw o-sided order. Note that w e do not pro ve the equiv alence (in contrast with Prop osition 4.3). Nevertheless, we pro ve b elo w a description of the cells under an irreducibility assumption for the cell mo dules. Due to this irreducibilit y assumption, the description is less complete than for the previous basis. Ho wev er, this will b e enough for type A, where all cell represen tations are irreducible ov er C ( q ). F or brevity , w e treat only the left cells. Prop osition 4.10. L et D ∈ W J \ W /W J and Γ the left c el l in W c ontaining r − ( D ) . Assume that the c orr esp onding r epr esentation V † Γ is irr e ducible for H ( W ) over C ( q ) . We have D ∼ L D ′ in W J \ W /W J ⇐ ⇒ r − ( D ) ∼ L r − ( D ′ ) in W . (30) Assume in p articular that al l c el l r epr esentations V † Γ ar e irr e ducible for H ( W ) over C ( q ) . Then the left c el ls in W J \ W /W J ar e the non-empty sets of the form: Γ ∩ X J J , for a left c el l Γ in W , wher e we have identiﬁe d double c osets in W J \ W /W J with their minimal-length r epr esentatives in X J J . Pr o of. F rom Prop osition 4.9, we already hav e the direct implication of (30). F or the rev erse impli- cation, assume that r − ( D ) ∼ L r − ( D ′ ) in W , so that b oth r − ( D ) , r − ( D ′ ) are in the same cell Γ. W e need to sho w that D ⪯ L D ′ in W J \ W /W J . First w e note that w e can write: e J C † r − ( D ′ ) = C † r − ( D ′ ) + X x  = r − ( D ′ ) α x C † x . Indeed ﬁrst w e decompose the elemen t e J C † r − ( D ′ ) in the basis { C † w } of H ( W ) as e J C † r − ( D ′ ) = P w α w C † w and we multiply from left and right by e J . W e ﬁnd that the co eﬃcien t in front of C † r − ( D ′ ) m ust b e 16 equal to 1, and for that we used that e J C † r − ( D ′ ) e J is a basis elemen t of H J ( W ) and all other terms e J C † w e J are either 0 or diﬀerent basis elements. Since r − ( D ′ ) is in the cell Γ, the elemen t e J C † r − ( D ′ ) is in the ideal I † ⪯ L Γ , and moreo v er the preceding discussion shows that it is not in I † ≺ L Γ (b ecause of the non-zero co eﬃcien t in fron t of C † r − ( D ′ ) ). Recalling that V † Γ is deﬁned as the quotient of the left ideal I † ⪯ L Γ b y the left ideal I † ≺ L Γ , this means that e J C † r − ( D ′ ) + I ≺ L Γ  = 0 V † Γ . (31) No w, given another elemen t y ∈ V † Γ , from the irreducibility assumption on V † Γ it is alwa ys p ossible to ﬁnd an element h ∈ C ( q ) H ( q ) sending e J C † r − ( D ′ ) to this element y . W e choose for y the element C † r − ( D ) + I † ≺ L Γ . It is p ossible to do so since r − ( D ) is also in the cell Γ. W e obtain that he J C † r − ( D ′ ) = C † r − ( D ) + x , with x ∈ I ≺ L Γ . for some h ∈ C ( q ) H ( q ). Multiplying from left and righ t by e J , w e ﬁnd e J he J C † r − ( D ′ ) e J = e J C † r − ( D ) e J + e J xe J . (32) The elemen t x ∈ I ≺ L Γ writes in terms of basis elements C † w with w  = r − ( D ) and therefore e J xe J writes in terms of basis elemen ts e J C † w e J diﬀeren t from e J C † r − ( D ) e J (here w e used again that e J C † w e J is either 0 or directly a basis element). Therefore, when the rhs of (32) is written in the basis { e J C † w e J } , the elemen t e J C † r − ( D ) e J app ears with co eﬃcien t 1. So far, the element e J he J b y whic h w e m ultiply b elongs to C ( q ) H J ( W ). But we can multiply b y a suitable element of the base ring A to pro duce an equalit y in H J ( W ). And we ha ve found e J C † r − ( D ) e J with a non-zero co eﬃcien t, so we conclude that D ⪯ L D ′ in W J \ W /W J , as required. Example. W e take W = S 6 the symmetric group generated b y s 1 , . . . , s 5 and W J = S 2 × S 2 × S 2 generated b y s 1 , s 3 , s 5 . Using the one-line notation for a p erm utation, here is a left cell in S 6 : Γ = { 231456 , 132456 , 142356 , 152346 , 162345 } . Among these 5 elemen ts, only tw o are in X J J (one has to lo ok for those elements with left and right descen ts disjoints from { 1 , 3 , 5 } ) and the resulting cell in W J \ W /W J consists of the tw o double cosets of the follo wing elements Γ ∩ X J J = { 132456 , 152346 } . In this example, the left cell Γ corresp onds to p ermutations having their right tableau in the RS corresp ondence equal to 1 2 4 5 6 3 . The in tersection with X J J is not empty b ecause w e hav e c hosen a tableau not con taining 1 , 3 , 5 in its descent. Actually , the in tersection is made of those p erm utations having a left tableau which also do es not con tain 1 , 3 , 5 in its descent. In type A, the description of the cells of W J \ W /W J in terms of the RS corresp ondence is simple and will b e explicited in the next section. Represen tations of H J ( W ) . W e recall again that for any representation V of H ( W ), the vector space e J ( V ) carries naturally a representation of H J ( W ), and moreov er if V is an irreducible H ( W )- mo dule then e J ( V ) (if non-zero) is an irreducible H J ( W )-mo dule, see for example [Gre80, § 6.2]. Let Γ be a left cell for W and V † Γ the asso ciated representation of H ( W ), with basis { C † w + I ≺ L Γ } w ∈ Γ . If non-empt y , the subset Γ ∩ X J J indexes the elements in a left cell of W J \ W /W J , thanks to the prop osition ab o ve. W e denote V † Γ ∩ X J J the asso ciated representation of H J ( W ). 17 Prop osition 4.11. L et Γ b e a left c el l of W such that Γ ∩ X J J  = ∅ . As r epr esentations of H J ( W ) , we have V † Γ ∩ X J J = e J ( V † Γ ) . Pr o of. Recall that X − 1 J denotes the set of minimal-length representativ es of right cosets in W J \ W , or equiv alently , the set of elemen ts with left descen t disjoin t from J . First we note that the set of elemen ts: { e J C † x } x ∈ X − 1 J (33) is a basis of the subspace e J H ( W ) (the image of H ( W ) b y left m ultiplication b y e J ). Indeed, using the same argumen ts than b efore Prop osition 4.5, we see that e J C † x = 0 if x is not in X − 1 J . So the ab o ve set is a spanning set. The linear indep endence easily follo ws from the fact that e J C † x decomp oses in the standard basis using only elements T y with y ≤ w J x , with an inv ertible coeﬃcients in front of T w J x . Next w e deduce that the following subset is a basis of the subspace e J ( V † Γ ): { e J C † w + I † ≺ L Γ } w ∈ Γ ∩ X − 1 J . (34) Again we hav e that e J C † w = 0 if w / ∈ X − 1 J , so that the elemen ts in (34) span e J ( V † Γ ). Besides, a relation of linear dep endency b et ween these elements w ould contradict the linear indep endence of the basis elemen ts of e J H ( W ) describ ed ab ov e. Since Γ ∩ X J J  = ∅ (the elemen t r − ( D ) is in here), it means that there is at least one element in Γ with right descent disjoin t from J . Since the right descen ts of all elements in the same left cell coincide, this means that all elements in Γ hav e their righ t descents disjoint from J . In this case, the basis ab o v e of e J ( V † Γ ) b ecomes: { e J C † w + I † ≺ L Γ } w ∈ Γ ∩ X J J . (35) This identiﬁes naturally with the deﬁning basis { e J C † w e J + I † ≺ L Γ ∩ X J J } w ∈ Γ ∩ X J J of the cell represen tation V † Γ ∩ X J J of H J ( W ). It remains to c heck that the actions of H J ( W ) are the same. Let h ∈ H J ( W ). W e write, for w ∈ Γ ∩ X J J , he J C † w = hC † w = X w ′ ∈ Γ α w ′ C † w ′ + X y ≺ L Γ α y C † y . The second sum is in I † ≺ L Γ and therefore the co eﬃcien ts α w ′ giv e the action of h on the basis (35). No w multiply this equality by e J from left and right to get he J C † w e J = X w ′ ∈ Γ ∩ X J J α w ′ e J C † w ′ e J + X y ≺ L Γ y ∈ X J J α y e J C † y e J , where we ha v e b een using again that e J C † x e J = 0 when x / ∈ X J J . The second sum is clearly in I † ≺ L Γ ∩ X J J b y construction of this ideal, and therefore the co eﬃcien ts α w ′ also giv e the action of h in the represen tation V † Γ ∩ X J J of H J ( W ). 5 P arab olic Heck e algebras in t yp e A 5.1 Notations F rom no w on, using Notations from Section 2.4, we take W = S n the symmetric group on n letters with generators s i = ( i, i + 1), i = 1 , . . . , n − 1. The asso ciated Hec ke algebra is denoted H ( S n ) and its generators are T 1 , . . . , T n − 1 . 18 P arab olic subgroups S µ . All parab olic subgroups of S n are obtained as W J = S µ = S µ 1 × · · · × S µ d , where d > 1 and µ = ( µ 1 , . . . , µ d ) ∈ Z > 0 suc h that µ 1 + · · · + µ d = n . F rom now on we ﬁx suc h an integer d and such a comp osition µ (with d non-zero parts). The parab olic subgroup S µ is naturally embedded in S n and is the parab olic subgroup corresp onding to the follo wing subset of simple transp ositions: J = { 1 , . . . , µ 1 − 1 , µ 1 + 1 , . . . , µ 1 + µ 2 − 1 , . . . . . . } , (36) where we ha ve iden tiﬁed a generator s i with its index. In other words, to get J , we remo ve from { 1 , . . . , n − 1 } the indices µ 1 , µ 1 + µ 2 , . . . , µ 1 + · · · + µ d − 1 . P arab olic Heck e algebra H µ ( S n ) . In the Heck e algebra H ( S n ), the subalgebra generated by the subset T s , with s ∈ J , is isomorphic in this case to H ( S µ 1 ) ⊗ · · · ⊗ H ( S µ d ). In each subalgebra H ( S k ), w e hav e the q -symmetriser (normalised to b e an idemp oten t) whic h is: e k = 1 P w ∈ S k q 2 ℓ ( w ) X w ∈ S k q ℓ ( w ) T w . The idemp oten t in H ( S n ) corresp onding to the choice of the comp osition µ is: e µ = e µ 1 ⊗ · · · ⊗ e µ d = 1 P w ∈ S µ q 2 ℓ ( w ) X w ∈ S µ q ℓ ( w ) T w . (37) The Poincar ´ e p olynomial of S n satisﬁes : P w ∈ S n q 2 ℓ ( w ) = Q n a =1 (1 + q 2 + · · · + q 2( a − 1) ). So here the normalizing factor in e µ is d Y i =1  X w ∈ S k i q 2 ℓ ( w )  = d Y i =1 [ k i ] q ! where we ha ve set [ n ] q ! = [2] q [3] q . . . [ n ] q and [ m ] q = 1 + q 2 m 1 + q 2 for any in teger m . So the ground ring for the parab olic Heck e algebra is in this case: A = Z [ q , q − 1 , ([ K ] q !) − 1 ] , where K = Max { µ 1 , . . . , µ d } . (38) W e rep eat the deﬁnition, for ﬁxing the notations, of the parab olic Heck e algebra of t yp e A. Deﬁnition 5.1. We denote H µ ( S n ) the p ar ab olic He cke algebr a asso ciate d to W = S n and J as ab ove. It is the fol lowing sub algebr a of H ( S n ) , deﬁne d over A in (38), H µ ( S n ) = e µ H ( S n ) e µ , wher e e µ is the idemp otent deﬁne d in (37). A diagrammatic, braid-lik e, description of the algebra H µ ( S n ) was describ ed in [CP23] and the algebra w as called the “fused Heck e algebra”. Remark 5.2. In A ar e invertible al l p olynomials of the form (1 + q 2 + · · · + q 2( k − 1) ) , with k smal ler or e qual to some µ i . The p ossible sp e cializations of q to, say, a c omplex numb er ar e those such that q 2 is not a r o ot of unity whose or der is b etwe en 2 and K = Max { µ 1 , . . . , µ d } . 19 Standard basis of H µ ( S n ) . The standard basis { T D } of the parab olic Hec ke algebra H µ ( S n ) is indexed b y double cosets of S n b y the Y oung subgroup S µ , see Section 3. These double cosets are naturally in bijection with the following sets of ob jects: • Diagrams generalising the usual diagrams for p erm utations. They connect tw o lines of n dots. The i -th dot of each line has µ i edges attac hed to it. • n × n matrices with non-negativ e integer entries such that the sum of the entries in the i -th row is equal to the sum of the entries in the i -th column and is equal to µ i . W e refer to [CP23, PdA20] for more details. 5.2 Cells in S µ \ S n /S µ and RSK corresp ondences 5.2.1 Tw o RSK corresp ondences W e will use the following map from { 1 , . . . , n } to { 1 , . . . , d } :      1 . . . µ 1 7→ 1 ,      µ 1 + 1 . . . µ 1 + µ 2 7→ 2 , . . .      µ 1 + · · · + µ d − 1 + 1 . . . n 7→ d , (39) so that the ﬁrst µ 1 in tegers are replaced by 1, the next µ 2 in tegers are replaced by 2, and so on. Recall that a semistandard Y oung tableau of shap e λ and of w eight µ is a ﬁlling of the Y oung diagram of λ with in tegers, suc h that the in teger i app ears µ i times, and the ﬁlling is w eakly increasing along the ro ws and strictly increasing along the columns. W e denote: SST ab( λ, µ ) = { semistandard Y oung tableaux of shap e λ and weigh t µ } . Deﬁnition 5.3. F or a standar d Y oung table au t ∈ ST ab( λ ) , we denote by t the table au obtaine d fr om t by applying the map (39) to al l the entries. The resulting tableau t is sometimes a semistandard Y oung tableau in SST ab( λ, µ ). In fact, it happ ens exactly when the standard Y oung tableau t do es not contain in its descent an y element of J (the subset in (36) deﬁning the subgroup S µ ). W e recall that: SST ab( λ, µ )  = ∅ ⇔ λ ≥ µ ord , (40) where µ ord is the partition obtained from µ by ordering the parts in decreasing order. This can b e pro ved combinatorially (see [Sta99]). F or what follo ws, recall from Section 2 that the usual Robinson–Schensted correspondence for p erm utations in S n is denoted: S n ← → G λ ⊢ n ST ab( λ ) 2 w ← →  P ( w ) , Q ( w )  . (41) The Robinson–Schensted–Kn uth corresp ondence [Kn u70]. W e are ready to describ e the pro cedure giving a pair of elements in SST ab( λ, µ ), for some λ , starting from a double coset D ∈ S µ \ S n /S µ . Sc hematically , w e apply the following pro cedure: D ↔ r − ( D ) = w ← →  P ( w ) , Q ( w )  ↔  P ( w ) , Q ( w )  . Since w = r − ( D ) is a minimal-length coset representativ e, it do es not contain any element of J in its left or its right descent. So b oth P ( w ) and Q ( w ) are suc h that their images b y the map · are semistandard Y oung tableaux. W e denote the resulting bijection b y: S µ \ S n /S µ ↔ G λ ⊢ n SST ab( λ, µ ) 2 D ↔  P ( D ) , Q ( D )  (42) 20 As recalled in (40), only shap es λ suc h that λ ≥ µ ord giv e non-empty sets of semistandard tableaux. Remark 5.4. T ake a double c oset D ∈ S µ \ S n /S µ and take its minimal r epr esentative r − ( D ) . Write it with the two-line notation for p ermutations and then apply the map (39) to al l the entries. This r esults in a two-line arr ay of inte gers, arr ange d in incr e asing lexic o gr aphic or der, with µ i entries e qual to i in e ach line. In [Knu70], the bije ction (42) is describ e d dir e ctly fr om these arr ays of inte gers. It is e asy to se e that the two descriptions ar e e quivalent. Another corresp ondence. W e will use another natural bijection alternative to (42). Schematically , it go es as follows: D ↔ r + ( D ) = w ← →  P ( w ) , Q ( w )  ↔  P ( w ) t , Q ( w ) t  . Since w = r + ( D ) is a maximal-length coset representativ e, it contains every elemen t of J in its left and its right descent. So b oth P ( w ) and Q ( w ) also contain J in their descent. This means that their transp osed tableaux (exc hanging lines and columns) are such that their images by · are semistandard Y oung tableaux. W e denote the resulting bijection b y: S µ \ S n /S µ ↔ G λ ⊢ n SST ab( λ, µ ) 2 D ↔  e P ( D ) , e Q ( D )  (43) Again, see (40), only shap es λ such that λ ≥ µ ord giv e non-empty sets of semistandard tableaux. Example 5.5. We c onsider the p ar ab olic sub gr oup S 2 × S 1 × S 1 inside S 4 . Ther e ar e 7 double c osets. Her e we give for e ach double c oset the minimal r epr esentative and the c orr esp onding p air of semistandar d table aux for the RSK c orr esp ondenc e (42). We do not r ep e at the se c ond table au when it c oincides with the ﬁrst: D 1 D 2 D 3 D 4 D 5 D 6 D 7 r − ( D ) e s 2 s 3 s 2 s 3 s 3 s 2 s 2 s 3 s 2 s 2 s 1 s 3 s 2  P ( D ) , Q ( D )  ( 1123 , · ) ( 113 2 , · ) ( 112 3 , · ) ( 113 2 , 112 3 ) ( 112 3 , 113 2 ) ( 11 2 3 , · ) ( 11 23 , · ) Then, with the same c onvention and numb ering of the c osets, we give the maximal r epr esentative and the c orr esp onding p air of semistandar d table aux for the other c orr esp ondenc e (43): D 1 D 2 D 3 D 4 D 5 D 6 D 7 r + ( D ) s 1 s 1 s 2 s 1 s 1 s 3 s 1 s 2 s 3 s 1 s 1 s 3 s 2 s 1 s 1 s 2 s 3 s 2 s 1 s 1 s 2 s 1 s 3 s 2 s 1  P ( D ) , Q ( D )  ( 11 2 3 , · ) ( 112 3 , · ) ( 11 23 , · ) ( 112 3 , 113 2 ) ( 113 2 , 112 3 ) ( 113 2 , · ) ( 1123 , · ) 5.2.2 Cells in S µ \ S n /S µ and RSK corresp ondences The ﬁrst basis of H µ ( S n ) . In this paragraph, we use the ﬁrst Kazhdan–Lusztig basis { C r + ( D ) } D∈ S µ \ S n /S µ of H µ ( S n ) studied in Section 4.1 to deﬁne orders and cells on S µ \ S n /S µ . F rom the results prov ed in Section 4, in particular Prop osition 4.3, we can immediately describ e the cells in S µ \ S n /S µ asso ciated to this basis { C r + ( D ) } D∈ S µ \ S n /S µ , since they are simply the intersection of the cells of S n with the subset of maximal-length representativ es. Almost by construction, the relev an t RSK corresp ondence here is the second one (43), which uses the maximal-length represen tatives. The notation is that to a double coset D is asso ciated a pair  e P ( D ) , e Q ( D )  of semistandard Y oung tableaux of the same shap e. W e will denote f sh ( D ) this common shap e. 21 Corollary 5.6. We have: • D and D ′ ar e in the same      left right 2-side d c el l of S µ \ S n /S µ if and only if        e Q ( D ) = e Q ( D ′ ) e P ( D ) = e P ( D ′ ) f sh ( D ) = f sh ( D ′ ) ; • D ⪯ LR D ′ if and only if f sh ( D ) ≥ f sh ( D ′ ) . Note that the tw o-sided cells in S µ \ S n /S µ are indexed by partitions λ suc h that λ ≥ µ ord , see the sen tence after (43). The tw o-sided order ⪯ LR is given in the second item by the rev erse dominance order since w e needed to transp ose the tableaux in (43) in order to deﬁne e P ( D ) and e Q ( D ), and the dominance order is reversed b y transp osition. The second basis of H µ ( S n ) . In this paragraph, we use the second basis { e J C † r − ( D ) e J } D∈ S µ \ S n /S µ of H µ ( S n ) studied in Section 4.2 to deﬁne orders and cells on S µ \ S n /S µ . As already recalled in Section 2, the left cell mo dules of H ( S n ) are all irreducible ov er C ( q ). So we can use Prop osition 4.10 from Section 4 (and ev erything is similar for righ t cell mo dules). Therefore we can immediately describ e the cells associated to this basis since they are simply the in tersection of the cells of S n with the subset of minimal-length represen tatives. Almost by construction, the relev an t RSK corresp ondence here is the ﬁrst one (42), which uses the minimal-length representativ es. The notation is that to a double coset D is asso ciated a pair  P ( D ) , Q ( D )  of semistandard Y oung tableaux of the same shap e. W e will denote sh ( D ) this common shap e. Prop osition 5.7. We have • D and D ′ ar e in the same      left right 2-side d c el l of S µ \ S n /S µ if and only if      Q ( D ) = Q ( D ′ ) P ( D ) = P ( D ′ ) sh ( D ) = sh ( D ′ ) ; • If D ⪯ LR D ′ then sh ( D ) ≤ sh ( D ′ ) ; Note again that the t w o-sided cells are indexed b y partitions λ suc h that λ ≥ µ ord , see the sen tence after (42). This time, since w e do not hav e a general res ult for the cell orders (see Prop osition 4.9), w e only know that the LR -order is w eaker than the dominance order on partitions. Example 5.8. The p artition of the set S µ \ S n /S µ into c el ls is diﬀer ent dep ending on which of the two b ases we c onsider. F or example, for n = 4 and µ = (2 , 1 , 1) , ther e ar e 7 c osets denote d D 1 , . . . , D 7 . Ther e ar e 4 two-side d c el ls indexe d by p artitions λ ≥ µ . Builiding on the c alculation of the RSK c orr esp ondenc es in Example 5.5, we show her e the p artition into c el ls: First b asis Se c ond b asis : {D 7 } {D 1 } : {{D 2 , D 5 } , {D 4 , D 6 }} {{D 2 , D 5 } , {D 3 , D 4 }} : {D 3 } {D 7 } : {D 1 } {D 6 } On e ach line, we show the c orr esp onding two-side d c el ls, with its de c omp osition into left c el ls. 22 5.3 Cells and represen tations of H µ ( S n ) Using the Kazhdan–Lusztig bases of H ( S n ), we discuss when the cell representations of H ( S n ) are killed or not b y the idemp oten t e µ . Restricting to the semisimple case, w e th us reco ver, with a diﬀeren t pro of, the classiﬁcation of irreducible representations of H µ ( S n ) obtained in [CP23]. 5.3.1 Classiﬁcation Recall that w e hav e denoted { V Γ } the cell represen tations of H ( S n ) using the basis { C w } , and { V † Γ } the cell represen tations of H ( S n ) using the basis { C † w } . Theorem 5.9. L et λ ⊢ n and let Γ b e a left c el l of S n c ontaining elements w with sh ( w ) = λ . We have e µ ( V Γ )  = 0 ⇔ λ t ≥ µ ord , e µ ( V † Γ )  = 0 ⇔ λ ≥ µ ord , wher e µ ord is the p artition obtaine d fr om µ by or dering the p arts in de cr e asing or der. Pr o of. W e consider ﬁrst the representation V Γ . W e pro ve that e µ ( V Γ ) = 0 if λ t ≱ µ ord . Indeed, the idemp oten t e µ is prop ortional to the elemen t C w µ , where w µ is the longest element of the parab olic subgroup S µ . It is easy to see that under the RS corresp ondence, w µ is of shap e ( µ ord ) t . When m ultiplying C w µ with any elemen t and expanding in the { C w } basis, we ﬁnd only elements C x with sh ( x ) ≤ ( µ ord ) t . If λ t ≱ µ ord then λ ≰ ( µ ord ) t , and w e can nev er ﬁnd elements C w with sh ( w ) = λ . F or the rest of the pro of, we recall from (40) that λ ≥ µ ord ⇔ SST ab( λ, µ )  = ∅ . (44) Note then that the condition SST ab( λ, µ )  = ∅ is equiv alent to the existence of a standard tableau of shap e λ with descent disjoin t from J . Indeed the map (39) applied to all entries provides a bijection b et w een such standard tableaux and SST ab( λ, µ ). W e must only c heck that from T ∈ SST ab( λ, µ ), w e can c ho ose a preimage of T for the map (39) with descent disjoint from J . Since eac h set of b o xes in T with the same entry , sa y a ∈ { 1 , . . . , d } , contains at most one b o x in each column, we can choose a preimage suc h that the num b ers µ 1 + · · · + µ a − 1 + 1 , . . . , µ 1 + · · · + µ a are placed strictly from left to righ t. F rom what we just said, if λ t ≥ µ ord then we can take a standard tableau of shap e λ t with descent disjoin t from J . T ransp osing, we obtain a standard tableau t of shap e λ with descent con taining J . Since Γ is a left cell of S n con taining elemen ts of shap e λ , there is an elemen t w in Γ with t as its left tableau under the RS corresp ondence. Therefore, J is included in the left descen t of w , and w e thus ha ve e µ C w = C w . This shows that e µ ( V Γ )  = 0. The reasoning is quite similar for the representation V † Γ . F rom the discussion ab ov e, the condition λ ≥ µ ord is equiv alent to the existence of a standard tableau of shap e λ with descent disjoint from J , whic h is in turn equiv alent to the existence of an element in Γ with left descent disjoin t from J . These elemen ts in Γ with left descent disjoint from J index a basis of e J ( V † Γ ), as was shown during the pro of of Prop osition 4.11, see (34). Th us if λ ≱ µ ord , there is no such elemen t in Γ and w e hav e e J ( V † Γ ) = 0 while otherwise there is such an element and we hav e e J ( V † Γ )  = 0. The semisimple situation. In this paragraph, we consider the semisimple situation, namely the ﬁeld C ( q ) or a non-zero complex num b er q such that q 2 is not a ro ot of unit y whose order is b et ween 2 and n . F rom the general results for idemp oten t subalgebras, (see e.g. [Gre80, § 6.2]) a complete set of pairwise distinct irreducible represen tations of H µ ( S n ) are all the non-zero e µ ( V λ ), where the V λ ’s are the irreducible represen tations of H ( S n ). The identiﬁcation b et ween the cell representations and the V λ ’s was recalled in Subsection 2.4. Using the previous theorem, we conclude immediately with the follo wing corollary , which w as prov en in [CP23] with other metho ds. 23 Corollary 5.10. A c omplete set of non-zer o p airwise distinct irr e ducible r epr esentations of H µ ( S n ) is: { e µ ( V λ ) such that λ ≥ µ ord } , wher e µ ord is the p artition obtaine d fr om µ by or dering the p arts in de cr e asing or der. Remark 5.11. In the p articular situation µ = ( k , . . . , k ) (a p ositive inte ger k r ep e ate d d times), then the c ondition λ ≥ µ ord is e asily se en to b e e quivalent to the c ondition that the numb er of non-zer o p arts of λ is less or e qual to d (which is only a ne c essary c ondition for gener al µ ). We r efer to [CP23] for mor e details. The semisimple represen tation theory of the algebra H µ ( S n ) is concisely summarised b y its Bratteli diagram. W e refer to [CP23]. F or example, if n = 6 and µ = (2 , 2 , 2), the Bratteli diagram is: ∅ 1 1 1 1 1 2 3 1 1 2 1 n = 1 n = 2 n = 3 The rule is that w e add t wo b o xes at eac h step (since µ = (2 , 2 , 2)) and t wo partitions λ ′ ⊢ n and λ ⊢ n + 2 are connected if λ ′ ⊂ λ and moreov er λ/λ ′ do es not con tain tw o b o xes in the same column. F or example λ = (2 , 2) is not connected to λ ′ = (3 , 3). The dimension of e µ ( V λ ) is a Kostk a num b er, that is, the following n umber of semistandard Y oung tableaux: dim e µ ( V λ ) = | SST ab( λ, µ ) | . It follo ws at once from the description of the cells of H µ ( S n ) (see also [CP23]). 5.3.2 Examples Example: λ = ( n ) . The representation is obviously of dimension 1. It is the restriction, or pro jec- tion, to H µ ( S n ) of the one-dimensional representation T w 7→ q ℓ ( w ) of the Hec ke algebra H ( S n ). In terms of the ﬁrst basis of H µ ( S n ), this partition is at the bottom of the cell order (rev erse dominance ordering). The basis elemen t corresp onding to this cell is C w n , where w n is the longest element of S n . It is indeed a simple matter to see that elements of H µ ( S n ) acts as follows: e µ T w e µ C w n = q ℓ ( w ) C w n . In terms of the second basis of H µ ( S n ), the partition ( n ) is at the top of the cell order. The corresp onding cell consists only of the trivial double coset and the corresp onding basis element is e µ C † e e µ = e µ . The action of H µ ( S n ) is b y left multiplication mo dulo all the basis elemen ts diﬀeren t from e µ , and th us, in this picture, the action of the basis elements of H µ ( S n ) is giv en as follo ws e µ C † w e µ 7→ ( 1 if w = e , 0 otherwise . 24 Remark 5.12. Note that the other one-dimensional r epr esentation T w 7→ ( − q − 1 ) ℓ ( w ) of H ( S n ) do es not give a r epr esentation of H µ ( S n ) sinc e the pr oje ctor e µ is 0 in this r epr esentation. However, ther e ar e other one-dimensional r epr esentations of H µ ( S n ) . One of them for λ = µ ord is discusse d b elow, and the other one (pr ob ably the most inter esting b e c ause not at one extr emity of the c el l or dering) is for λ a ho ok p artition of maximal-length and wil l play a pr ominent r ole in the next se ction. Example: λ = µ ord . This partition also corresp onds to a one-dimensional representation of H µ ( S n ) (ev en if the corresp onding representation of H ( S n ) is not one-dimensional). T o see this, recall that the dimension is the cardinalit y of SST ab( λ, µ ) and that this num b er do es not dep end on the ordering of the comp osition µ (see [Sta99, Theorem 7.10.2]). So for simplicity w e assume that µ = µ ord , that is µ = ( µ 1 , . . . , µ d ) with µ 1 ≥ · · · ≥ µ d . In this case, one easily sees that there is a single semistandard tableau in SST ab( λ, µ ) for λ = µ . It is obviously obtained by ﬁlling the ﬁrst line with 1’s, the second line with 2’s and so on. F or the ﬁrst basis of H µ ( S n ), the unique basis element corresp onding to this cell is C w µ = e µ , where w µ is the longest elemen t of the subgroup S µ inside S n . In this case, the partition µ is at the top of the cell order and the action of H µ ( S n ) is b y left multiplication mo dulo all the basis elements diﬀeren t from e µ . Th us, in this picture, the action of the basis elements of H µ ( S n ) is given as follows C w 7→ ( 1 if w = w µ , 0 otherwise . F or the second basis of H µ ( S n ), the partition λ = µ is at the b ottom of the cell order. The corrrsp onding basis element is e µ C † ˜ w µ e µ , for some p erm utation ˜ w µ , whic h is a minimal-length representativ e corresp onding through the RSK corresp ondence to the unique semistandard Y oung tableau in SST ab( µ, µ ). It is straigh tforward to c heck that the p erm utation ˜ w µ is giv en, visually , as follows: µ 1 . . . µ 2 . . . . . . . . . . . . . . . µ d − 1 . . . µ d . . . . It is the in volution sending the last µ d in tegers (preserving their order) to the last µ d in tegers in the subset { 1 , . . . , µ 1 } , and then rep eating the pro cedure to the remaining subsets of sizes µ 2 , . . . , µ d − 1 as long as there remains at least tw o subsets. F or example, if µ = (3 , 2 , 2) then ˜ w µ = 1674523. Since µ is at the b ottom of the cell order, we know that left m ultiplication of the basis elemen t e µ C † ˜ w µ e µ b y any h ∈ H µ ( S n ) is prop ortional to the basis element: he µ C † ˜ w µ e µ = α ( h ) e µ C † ˜ w µ e µ . The co eﬃcien t α ( h ) is the v alue of the element h in this one-dimensional representation. 5.4 Cellular bases of H µ ( S n ) 5.4.1 Cellularit y of the Heck e algebra H ( S n ) It is well-kno wn [Gec07, KL79] that com bined with the RS corresp ondence, the Kazhdan–Lusztig bases { C w } w ∈ S n and { C † w } w ∈ S n b ecome cellular bases of H ( S n ), thus providing the Heck e algebra H ( S n ) with a structure of a cellular algebra in the sense of [GL96]. 25 W e reindex the tw o Kazhdan–Lusztig bases, using the RS correspondence w ↔  P ( w ) , Q ( w )  , deﬁning C P ( w ) ,Q ( w ) = C w and C † P ( w ) ,Q ( w ) = C † w . W e consider the antiautomorphism ι of order 2 of H ( S n ), which sends an y generator T i to itself, which means that it is given on the standard basis by: ι : T w 7→ T w − 1 . Then the t wo sets of elements: { C s , t } s , t , { C † s , t } s , t where ( s , t ) runs ov er G λ ⊢ n ST ab( λ ) 2 , form t wo cellular bases of H ( S n ) with resp ect to the in volution ι and to the p oset of partitions of n with the dominance order. T o b e precise, this means that: • ι ( C s , t ) = C t , s for an y s , t ; • for all h ∈ H ( S n ) and s , t ∈ ST ab( λ ), we ha ve hC s , t = X s ′ ∈ ST ab( λ ) r h ( s , s ′ ) C s ′ , t mo d I <λ and hC † s , t = X s ′ ∈ ST ab( λ ) ˜ r h ( s , s ′ ) C † s ′ , t mo d I † <λ . where r h ( s , s ′ ) and ˜ r h ( s , s ′ ) are co eﬃcien ts indep enden t of t . Ab ov e, the ideal I <λ is the span of all elemen ts C s , t for standard tableaux s , t of shap e strictly smaller than λ in the dominance order. The ideal I † <λ is deﬁned similarly using elements C † s , t . 5.4.2 Tw o cellular bases of H µ ( S n ) It follows immediately from its deﬁnition (37) that the pro jector e µ is stable by the in volution ι . F rom a general prop ert y of cellular algebras (see [KX98, Prop ostion 4.3]), the stability of e µ b y ι ensures that the algebra H µ ( S n ) = e µ H n e µ is also cellular with resp ect to the same inv olution. Our goal to conclude this section is to mak e explicit the t wo cellular bases of H µ ( S n ) that w e obtain using the bases previously considered. The p oset. W e recall that a basis of H µ ( S n ) is indexed by the double cosets S µ \ S n /S µ , and that these double cosets are in bijection with a certain set of pairs of semistandard tableaux: S µ \ S n /S µ ↔ G λ ⊢ n SST ab( λ, µ ) 2 , via either one of the tw o RSK corresp ondences from Section 5.2. Recall also that the set SST ab( λ, µ ) is not empt y if and only if λ ≥ µ ord . Th us the poset inv olv ed in the cellular datum for H µ ( S n ) is going to b e the set of partitions λ of n satisfying λ ≥ µ ord , endo wed with either the dominance ordering on partitions or the reverse dominance ordering. Cellular bases. Let ( S , T ) ∈ SST ab( λ, µ ) 2 for some λ ≥ µ ord . There is a unique standard tableau s in ST ab( λ ) whic h do es not contain J in its descent and such that s = S , where · is the map (39). Similarly , there is a unique standard tableau t in ST ab( λ ) which do es not con tain J in its descent and suc h that t = T . Example 5.13. If S = 1 1 2 3 2 3 then s = 1 2 3 5 4 6 . 26 Unra veling the deﬁnition of the RSK corresp ondence (43), w e see that if the pair ( S , T ) corresp onds to the coset D , then the pair ( s t , t t ) corresponds to r + ( D ) through the usual RS correspondence. Th us w e set: C S , T := C r + ( D ) = C s t , t t . (45) F or the second basis, unrav eling the deﬁnition of the RSK corresp ondence (42), we see that if the pair ( S , T ) corresp onds to the coset D , then the pair ( s , t ) corresp onds to r − ( D ) through the usual RS corresp ondence. Thus we set: C † S , T := e µ C † r − ( D ) e µ = e µ C † s , t e µ . (46) With these notations, we ha ve that the tw o following sets form tw o cellular bases of H µ ( S n ): { C S , T } , { C † S , T } where ( S , T ) runs ov er G λ ⊢ n λ ≥ µ ord SST ab( λ, µ ) 2 . Indeed we already kno w that these are bases of H µ ( S n ). Moreov er, F ormulas (45) and (46), only in terms of tableaux, give immediately that ι ( C S , T ) = C T , S and ι ( C † S , T ) = C † T , S . Regarding the prop ert y with resp ect to the left multiplication, let us consider the ﬁrst basis. F or h ∈ H ( S n ), w e hav e explicitly e µ he µ C S , T = e µ he µ C s t , t t = X u , v ∈ ST ab( λ ) r e µ ( u t , v t ) r h ( s t , u t ) C v t , t t mo d I <λ t = X V ∈ SST ab( λ,µ )  X u ∈ ST ab( λ ) r e µ ( u t , v t ) r h ( s t , u t )  C V , T mo d I <λ t ∩ H µ ( S n ) . The ﬁrst multiplication by e µ do es nothing since C s t , t t is already in H µ ( S n ). Since this calculation ends up in H µ ( S n ), the sum ov er v can b e tak en ov er the standard tableaux in ST ab( λ ) whic h do es not con tain J in their descent. T o each such tableau v corresponds a unique V ∈ SST ab( λ, µ ). In the resulting expression, the co eﬃcien t in front of C V , T do es not dep end on T . Finally , note that, due to the transp osition of tableaux app earing in (45), the ideal I <λ t ∩ H µ ( S n ) is spanned by elements C U , U’ asso ciated to shap es ν such that ν t < λ t . This is equiv alent to ν > λ and therefore the order on partitions that we hav e to use is the reverse dominance ordering. F or the second basis, still with h ∈ H ( S n ), w e hav e explicitly: e µ he µ C † S , T = e µ he µ C † s , t e µ = X u , v ∈ ST ab( λ ) ˜ r h ( u , v ) ˜ r e µ ( s , u ) e µ C † v , t e µ mo d I <λ = X V ∈ SST ab( λ,µ )  X u ∈ ST ab( λ ) ˜ r h ( u , v ) ˜ r e µ ( s , u )  C † V , T mo d I <λ ∩ H µ ( S n ) . The sum ov er v can b e taken ov er the standard tableaux in ST ab( λ ) which do es not con tain J in their descent since otherwise e µ C † v , t w ould b e 0. Again to each such tableau v corresp onds a unique V ∈ SST ab( λ, µ ). In the resulting expression, the co eﬃcient in front of C † V , T do es not dep end on T . In this case, the order on partitions that we hav e to use is the usual dominance ordering. 6 Application to Sc h ur–W eyl dualit y Unless otherwise sp eciﬁed, we w ork in this section in the semisimple situation, namely ov er the ﬁeld C ( q ) or with a non-zero complex num b er q such that q 2 is not a ro ot of unity whose order is b et ween 2 and n ( q 2 = 1 is allo wed). In this section N is an integer suc h that N ≥ 1 and w e will assume that d (the length of the comp osition µ = ( µ 1 , . . . , µ d )) satisﬁes d > N , b ecause otherwise the ideal I µ N studied in this section is { 0 } . 27 6.1 The Sc h ur–W eyl dualit y for H µ ( S n ) F rom [CP23], the algebra H µ ( S n ) app ears in the Sch ur–W eyl duality through a representation: π N : H µ ( S n ) → End  S µ 1 q V ⊗ · · · ⊗ S µ d q V  , (47) where V is of dimension N and S k q V is the k -th q -symmetrized p o wer of V , which is an irreducible represen tation of the quantum group U q ( g l N ). The map π n is surjective on to the U q ( g l N )-cen traliser, but it is not injective as so on as d > N and it remains to understand its kernel. W e set: I µ N = Ker π N , so that the quotient of H µ ( S n ) by its ideal I µ N is isomorphic to the centraliser. W e are interested in ﬁnding a linear basis of the ideal I µ N and a set (as simple as p ossible) of generators of it. W e will need the description of the ideal I µ N in terms of the representations of H µ ( S n ) obtained in [CP23]. F rom the classiﬁcation of its irreducible representations (Section 5.3), the algebra H µ ( S n ) has an Artin–W edderburn decomp osition: H µ ( S n ) ∼ = M λ ⊢ n λ ≥ µ ord End  e µ ( V λ )  . It is pro ved in [CP23] that the ideal I µ N is the one made of the summands ab ov e corresp onding to partitions with strictly more than N ro ws. Example 6.1. T ake n = 6 and µ = (2 , 2 , 2) . The irr e ducible r epr esentations of H µ ( S n ) ar e indexe d by the fol lowing p artitions (the p artitions λ ⊢ 6 with λ ≥ (2 , 2 , 2) ): 1 2 3 1 1 2 1 The numb ers ar e the dimensions. When N = 2 , the ide al I µ N c orr esp onds to the thr e e p artitions in the shade d ar e a. 6.2 A linear basis for the ideal I µ N W e are going to use the results prov ed in the preceding sections for the second basis of H µ ( S n ). This basis was in tro duced in Section 4.2 for an arbitrary parab olic Hec ke algebra, and discussed in Sections 5 for the t yp e A. F or simplicit y of notation here, we will identify a double coset D ∈ S µ \ S n /S µ with its minimal represen tative r − ( D ) in X J J . Recall that X J J is the set of elements in S n with left and righ t descents disjoint from J , where J is asso ciated to µ by J = { 1 , . . . , µ 1 − 1 , µ 1 + 1 , . . . , µ 1 + µ 2 − 1 , . . . . . . } ; that is, in order to get J , we remov e from { 1 , . . . , n − 1 } the indices µ 1 , µ 1 + µ 2 , . . . , µ 1 + · · · + µ d − 1 . The basis of H µ ( S n ) we consider is th us { e µ C † w e µ } w ∈ X J J . Recall also that the shap e sh ( w ) of an elemen t w ∈ S n is b y deﬁnition the partition λ which is the shap e of the standard tableaux ( P ( w ) , Q ( w )) corresp onding to w via the RS corresp ondence. Prop osition 6.2. A b asis of I µ N is: { e µ C † w e µ , with w ∈ X J J such that sh ( w ) has strictly mor e than N r ows . } (48) 28 In terms of the cellular basis { C † S , T } from Section 5.4, that is when iden tifying an elemen t w ∈ S n with a pair of standard tableaux via the RS corresp ondence, the basis of I µ N reads: { e µ C † s , t e µ , s , t ha ve strictly more than N rows and descents disjoint from J . } Note that from the RSK corresp ondence recalled in (42) and the classical com binatorial result recalled in (40), we know that the shap e λ of an elemen t w ∈ X J J , or equiv alently , the shap e of a standard tableau with descen t disjoint from J , must satisfy λ ≥ µ ord . Pr o of. W e use the description of the ideal I µ N in terms of the representations recalled ab o ve. The ideal I µ N is made of the summands in the Artin–W edderburn decomposition corresp onding to partitions with strictly more than N ro ws (see Example 6.1). Luckily , this condition is simply characterised in terms of the dominance ordering. Indeed, denote b y Ho ok N +1 ,n the ho ok partition with N + 1 rows and of size n . This partition indeed satisﬁes Ho ok N +1 ,n ≥ µ ord due to the condition d > N . Then it is easy to see that: λ ⊢ n has strictly more than N ro ws ⇔ λ ≤ Ho ok N +1 ,n . This remark allo ws to write the set (48) ab o v e as: { e µ C † w e µ , with w ∈ X J J suc h that sh ( w ) ≤ Ho ok N +1 ,n . } This sho ws, using the cell order prop ert y from Prop osition 5.7, that this set indeed spans an ideal of H µ ( S n ). This is in fact the sum of the (t wo-sided) cell ideals asso ciated to the tw o-sided cells corresp onding to shap es λ ≤ Ho ok N +1 ,n . The irreducible representations of H µ ( S n ) corresp onding to this ideal are exactly the ones corresp onding to I µ N , therefore this ideal is I µ N . Remark 6.3. We emphasize that we c an not use dir e ctly the other Kazhdan–Lusztig b asis { C w } w ∈ X J J of H µ ( S n ) sinc e the c el l r epr esentations c orr esp ond to the r epr esentations e µ ( V λ t ) . This me ans that the desir e d r epr esentations c orr esp ond to al l c el ls ab ove the shap e Ho ok t N +1 ,n in the c el l or der, and this do es not form a c el l ide al. Remark 6.4. In the gener al situation, that is over the deﬁning ring A , or for an arbitr ary authorise d sp e cialisation of q (se e R emark 5.2), the r epr esentation π N in (47) is stil l deﬁne d, and thus one c an wonder if the kernel stil l admits the description of the pr evious pr op osition. It turns out to b e true for the usual He cke algebr a; se e [GW93] or [Mar92]. We le ave this question op en for the algebr a H µ ( S n ) . 6.3 Conjectures for a generator of the ideal I µ N As detailed in [CP23], in the semisimple regime, it is someho w enough to understand the ideal I µ N at the ﬁrst lev el where it is not trivial, namely when d = N + 1. Indeed, say we hav e an elemen t X of H µ ( S n ), where µ = ( µ 1 , . . . , µ N , µ N +1 ), whic h generates the ideal I µ N . If we increase the length of µ to µ + = ( µ 1 , . . . , µ N , µ N +1 , µ N +2 ) then w e can see naturally the algebra H µ ( S n ) as a subalgebra of H µ + ( S n ′ ). Then the ideal I µ + N of H µ + n ′ will simply b e generated by X , no w seen as an element of H µ + ( S n ′ ) by the natural embedding. This is prov ed in [CP23] under the (necessary) assumption that µ and µ + are partitions, instead of general comp ositions. F or simplicity and due to the preceding short discussion, w e will now b e considering that d = N + 1 , that is, µ = ( µ 1 , . . . , µ N , µ N +1 ) . As alwa ys, w e hav e n = µ 1 + · · · + µ N +1 . Note that for what follows we do not need to assume that µ is a partition since we will only b e sp eaking of the ideal I µ N at lev el d = N + 1. 29 6.3.1 A ﬁrst ten tative generator Here w e study the p ossibilit y to ﬁnd a generator of the id eal I µ N b y looking at the basis { e µ C † w e µ } w ∈ X J J of H µ ( S n ). As men tioned b efore, one diﬃculty is that the ideal I µ N con tains more than one irreducible represen tation. How ev er, we still hav e a bit of luck, in the sense that the subset of partitions that we need to consider has a simple description in terms of the dominance order, that is, in terms of the cell order corresp onding to the basis { e µ C † w e µ } w ∈ X J J of H µ ( S n ). More precisely , recall that Ho ok N +1 ,n is the ho ok shap e partition with a ﬁrst column of size N + 1 (and total size n ). As used in the preceding subsection, it is easy to see that { λ ⊢ n with strictly more than N ro ws } = { λ ⊢ n suc h that λ ≤ Ho ok N +1 ,n } . Therefore, to generate the ideal I µ N , the only choice for the basis { e µ C † w e µ } is to take elements corre- sp onding to the cell asso ciated to the ho ok shap e Ho ok N +1 ,n . Moreo ver, the irreducible represen tation of H µ ( S n ) corresp onding to Ho ok N +1 ,n is one-dimensional. Indeed there is a unique semistandard tableau in SST ab(Ho ok N +1 ,n , µ ) corresp onsing to the unique standard tableau in ST ab(Ho ok N +1 ,n ) with descent disjoint from J . This standard tableau of shap e Ho ok N +1 ,n , denoted ˜ t N +1 , has the following en tries in the ﬁrst column: 1 , µ 1 + 1 , µ 1 + µ 2 + 1 , . . . , µ 1 + · · · + µ N + 1 . Its descen t, whic h is { µ 1 , µ 1 + µ 2 , . . . , µ 1 + · · · + µ N } , is indeed disjoint from J , whic h we recall is exactly { 1 , . . . , n } minus the preceding subset. Example 6.5. T ake again n = 6 and µ = (2 , 2 , 2) . The irr e ducible r epr esentations of H µ ( S n ) ar e: 1 2 3 1 1 2 1 and, for N = 2 , the ide al I µ N c orr esp onds to the thr e e p artitions in the shade d ar e a. The ho ok shap e (4 , 1 , 1) dominates them al l in the dominanc e or der and c orr esp onds to a one-dimensional r epr esen- tation. The unique semistandar d table au of this shap e is 1 1 2 3 2 3 c orr esp onding to the unique standar d table au ˜ t N +1 = 1 2 4 6 3 5 not c ontaining 1 , 3 , 5 in its desc ent. Mor e examples ar e given b elow T o summarise, if we wan t to generate the cell ideal I µ N , there is only one choice from the p oin t of view of the basis e µ C † w e µ , whic h we formalise in the following deﬁnition Deﬁnition 6.6. We deﬁne: Y µ N = e µ C † ˜ t N +1 , ˜ t N +1 e µ = e µ C † ˜ w N +1 e µ , wher e ˜ t N +1 is the unique standar d table au of shap e Ho ok N +1 ,n with desc ent disjoint fr om J and ˜ w N +1 is the p ermutation in S n c orr esp onding to ( ˜ t N +1 , ˜ t N +1 ) under the RS c orr esp ondenc e. The p erm utation ˜ w N +1 app earing in the ab o v e deﬁnition can be describ ed explicitly . It is the p erm utation of order 2 doing the following transp ositions: µ 1 + · · · + µ i ↔ µ 1 + · · · + µ N +1 − i + 1 , ∀ i ≤ N + 1 2 . 30 More visually , recall the decomp osition of { 1 , . . . , n } in to consecutive subsets of sizes µ 1 , µ 2 , . . . , µ N +1 . Then send the last letter of the ﬁrst subset to the ﬁrst letter of the last subset, then the last letter of the second subset to the ﬁrst letter of the second to last subset, and so on. If there is an o dd n umber of subsets ( N + 1 is o dd), the subset in the middle is untouc hed. The follo wing diagram should be helpful to visualize ˜ w N +1 (see also the examples b elo w): µ 1 . . . µ 2 . . . . . . . . . . . . . . . µ N . . . µ N +1 . . . ˜ w N +1 . It is straigh tforward to chec k that the RS corresp ondence pro duces the desired pair ( ˜ t N +1 , ˜ t N +1 ). Example 6.7. When µ = (2 , 2 , . . . , 2) , we show for smal l N the standar d table au ˜ t N +1 and the p ermutation ˜ w N +1 in one-line notation: N = 1 and µ = (2 , 2) : ˜ t N +1 = 1 2 4 3 ˜ w N +1 = 1324 , N = 2 and µ = (2 , 2 , 2) : ˜ t N +1 = 1 2 4 6 3 5 ˜ w N +1 = 153426 , N = 3 and µ = (2 , 2 , 2 , 2) : ˜ t N +1 = 1 2 4 6 8 3 5 7 ˜ w N +1 = 17354628 . 6.3.2 A second ten tative generator Here we recall the deﬁnition of another tentativ e generator of the ideal I µ N from [CP23]. This is done in sev eral steps. The elemen t T γ µ . Consider the following element of the Heck e algebra H ( S n ): T γ µ = T µ 1 . . . T 2 · T µ 1 + µ 2 . . . T 3 · . . . . . . · T µ 1 + ··· + µ N . . . T N +1 , (49) where the dots b etw een T µ 1 + ··· + µ a and T a +1 indicate the pro duct of the generators in decreasing order of their indices. By conv ention, this pro duct is 1 when µ 1 + · · · + µ a < a + 1. Note that T γ µ = 1 only if µ = (1 , 1 , . . . , 1). Graphically , for example if N = 2, the elemen t T γ µ is depicted as: T γ µ = µ 1 1 2 3 . . . . . . µ 2 . . . . . . µ 3 . . . . . . As a p erm utation (recall that w e read from b ottom to top, which corresp onds to comp osing p erm u- tations in the usual wa y , from right to left), we ha ve γ µ (1) = 1, γ µ (2) = µ 1 + 1, ..., γ µ ( N + 1) = µ 1 + · · · + µ N + 1, so the p erm utation γ µ sends 1 , . . . , N + 1, p erserving their order, to the ﬁrst num b er in each of the subsets of sizes µ 1 , µ 2 , . . . , µ N +1 . The elements after N + 1 are “pushed to the left”, namely , they are sent, preserving their order, to the remaining a v ailable elemen ts. 31 In the Heck e algebra, the element T γ µ ab o v e is obtained from a reduced expression of γ µ b y using only p ositiv e crossing (left strand ab ov e righ t strand), that is T γ µ in volv es only generators T i ’s and not their in verses. W e note at once the following prop ert y . W e hav e e µ T γ µ X T − 1 γ µ e µ = e µ T γ ′ µ X T − 1 γ ′ µ e µ , (50) where, for example for N = 2, T γ ′ µ is an y element of the form T γ ′ µ = µ 1 i 1 1 2 3 . . . µ 2 i 2 . . . µ 3 i 3 . . . that is, we can instead send 1 to i 1 , 2 to i 2 , ..., for any choice of i 1 in the ﬁrst set of size µ 1 , any c hoice of i 2 in the following subset of size µ 2 and so on. Indeed note that T γ ′ µ is obtained from T γ µ b y precomp osing by elements T i where i ∈ J . The explicit formula is: T γ ′ µ = T i 1 − 1 . . . T 1 · T i 2 − 1 . . . T µ 1 +1 · . . . · T i N +1 − 1 . . . T µ 1 + ··· + µ N +1 · T γ µ . All elements app earing b efore T γ µ ha ve indices in J and therefore satisfy e µ T i = q e µ . The additional p o w ers of q app earing are cancelled b y the presence of T − 1 γ ′ µ next to e µ on the other side. The second ten tative generator X µ N . Now denote w N +1 the longest element of the symmetric group S N +1 and consider the corresp onding Kazhdan–Lusztig element in H ( S N +1 ): C † w N +1 = X w ∈ S N +1 ( − 1) ℓ ( w ) q ℓ ( w N +1 ) − ℓ ( w ) T w . (51) It is called the (unnormalised) q -antisymmetriser of H ( S N +1 ). Then, w e see this elemen t as an element of H ( S n ) b y the natural em b edding of H ( S N +1 ) in to H ( S n ). Finally , we are ready to deﬁne our second candidate for a generator. Deﬁnition 6.8. We deﬁne X µ N = e µ T γ µ C † w N +1 T − 1 γ µ e µ . (52) wher e T γ µ was intr o duc e d ab ove in (49). Note the other p ossible formulas for X µ N from (50). W e will giv e others b elo w, see Remark 6.14. W e concede that the algebraic deﬁnition of the generator lo oks complicated. How ever it has a nice and relativ ely simple diagrammatic interpretation in terms of fused braids as dev elop ed in [CP23]. W e illustrate this with an example. Example 6.9. F or example, take µ = (2 , 2 , 2) and N = 2 . In the He cke algebr a, the element C † w 3 of H ( S 3 ) is depicte d as: q 3 − q 2 − q 2 + q + q − . 32 This is the sum over al l standar d b asis elements of H ( S 3 ) and the c o eﬃcient in fr ont of the b asis element T w is ( − 1) ℓ ( w ) q ℓ ( w 3 ) − ℓ ( w ) wher e w 3 is the longest element (her e of length 3). Her e is the diagr ammatic depiction of the element X µ N deﬁne d ab ove: q 3 − q 2 − q 2 + + q 2 − . Each term is se en as an element of H µ ( S n ) = e µ H ( S 6 ) e µ as fol lows. The str ands r epr esent an element of H ( S 6 ) in the usual way, while the black el lipses r epr esent q -symmetrisers (her e on two str ands sinc e µ = (2 , 2 , 2) ). So the thr e e black el lipses on top and on b ottom r epr esents the left and right multiplic ation by e µ . The diagr ammatic pr o c e dur e is as fol lows: Start with a standar d b asis element of H ( S 3 ) ; pr omote the dots into black el lipses; add to e ach el lipse vertic al str ands so that ther e ar e two str ands attache d to e ach el lipse. We add these str ands on top of the pictur e (with r esp e ct to the she et of p ap er on which it is dr awn). We apply this pr o c e dur e to e ach term in C † w 3 . Note that we have use d the pr op erty T ± 1 i e µ = q ± 1 e µ if i ∈ J to simplify some cr ossings and to have elements which ar e minimal-length r epr esentatives. In doing so, the c o eﬃcients change d a little, but the rule is simple. What we get is that in fr ont of an element c orr esp onding to a minimal-length r epr esentative r − ( D ) , the c oﬃcient is ( − 1) ♯ q ℓ ( w 3 ) − ♯ , wher e ♯ is the numb er of cr ossings c ounte d with signs. Remark 6.10. Ther e is a c onne ction b etwe en the longest element w N +1 and the p ermutation ˜ w N +1 intr o duc e d in Deﬁnition 6.6. This is thr ough the p ermutation γ µ intr o duc e d ab ove in (49) and it go es as fol lows. Under the RS c orr esp ondenc e, the element w N +1 c orr esp onds to ( t N +1 , t N +1 ) , wher e t N +1 is the standar d table au of shap e Ho ok N +1 ,n which has the fol lowing entries in the ﬁrst c olumn: 1 , 2 , 3 , . . . , N + 1 . Note that the desc ent of t N +1 is not disjoint fr om J . The only standar d table au of this shap e with desc ent disjoint fr om J was denote d ˜ t N +1 in Deﬁnition 6.6 and c orr esp onds to the p ermutation ˜ w N +1 . Now it is e asy to che ck that the p ermutation γ µ is exactly the p ermutation r elating the two standar d table aux t N +1 and ˜ t N +1 : ˜ t N +1 = γ µ ( t N +1 ) , wher e the right hand side me ans γ µ applie d to the entry of t N +1 . This c an b e se en as the heuristic b ehind the c onje ctur es of the next se ction. 6.3.3 Conjectures and partial results The conjectures. T o summarise, w e ha ve introduced in Deﬁnitions 6.6 and 6.8 tw o elements of the algebra H µ ( S n ) in the following form: X µ N = e µ T γ µ C † w N +1 T − 1 γ µ e µ and Y µ N = e µ C † ˜ w N +1 e µ . No w we formulate our conjectures regarding these elements and the ideal I µ N w e are interested in. Conjecture 6.11. a) The element X µ N gener ates the ide al I µ N . b) The element Y µ N gener ates the ide al I µ N . c) The two elements X µ N and Y µ N ar e e qual. 33 Statemen t a) ab o v e w as conjectured in [CP23, § 9]. In there, it is shown, ﬁrst, that the element X µ N do es b elong to the ideal I µ N . Moreo ver statemen t a) was prov ed in some particular cases, namely for N = 2 and an y µ , and for any N if µ either con tains only 1’s and 2’s, or con tains only a single part diﬀeren t than 1. W e refer to [CP23] for more details. The no velt y of the present work is to conjecture statemen ts b) and c). Needless to say that if c) is true then statemen ts a) and b) b ecome equiv alent (but of course a) and b) can b oth b e true while c) is not). W e will manage b elo w to give some supp orting evidence for c) in general and to prov e all statemen ts a), b) and c) in some sp ecial cases. Remark 6.12. It is also c onje ctur e d in [CP23, Conje ctur e 9.3] that the (c onje ctur al) gener ator X µ N of the ide al is c entr al in H µ ( S n ) . P artial results. Recall that the element Y µ N = e µ C † ˜ w N +1 e µ has in particular the imp ortant prop ert y of b eing stable b y the bar inv olution · , from the general prop ert y of Kazhdan–Lusztig elemen ts, see Prop osition 4.5. The bar in volution is the morphism sending each T i to its in verse T − 1 i and q to q − 1 . What w e can show in general is that this stabilit y prop erty is satisﬁed by the element X µ N . Prop osition 6.13. We have: X µ N = X µ N , namely, e µ T γ µ C † w N +1 T − 1 γ µ e µ = e µ T γ µ C † w N +1 T γ µ − 1 e µ . Pr o of. W e are going to sho w the following identit y: e µ T γ µ C † w N +1 = e µ T γ µ C † w N +1 . (53) Assume this identit y is veriﬁed and apply to it the antiautomorphism ι of H n sending each generator T i to itself, follo wed by the inv olution · . The map ι leav es in v ariant b oth e µ and C † w N +1 (see (18) and (51)) as do es the inv olution · (see (20)). The comp osition of b oth maps sends T γ µ to T − 1 γ µ . Therefore, w e obtain from (53) C † w N +1 T − 1 γ µ e µ = C † w N +1 T γ µ − 1 e µ . Com bined with (53), this prov es the statement of the prop osition. The pro of of (53) will b e using induction on N . First tak e N = 1, so that µ = ( µ 1 , µ 2 ). In this case, w e hav e: T γ µ = T µ 1 . . . T 2 and C † w N +1 = q − T 1 . Let i ∈ { 2 , . . . , µ 1 } . W e claim that: e µ T ± 1 µ 1 . . . T ± 1 i +1 ˇ T i T i − 1 . . . T 2 ( q − T 1 ) = 0 . (54) T o b e precise, the exp onan ts ± 1 means that we can choose indep endently +1 or − 1 for all generators to the left of T i , and the notation ˇ T i means that T i is omitted. The statemen t follo ws from the fact that T i − 1 . . . T 2 ( q − T 1 ) comm utes to the left and hits e µ where eac h elemen t T 1 , T 2 , . . . , T i − 1 is replaced b y q . In particular, the factor ( q − T 1 ) giv es 0. No w, in e µ T γ µ C † w N +1 , we can replace (from left to right) all the T i ’s in T γ µ b y their inv erses, using the relation T i = T − 1 i + ( q − q − 1 ) and (54). This pro ves (53) for N = 1. Next, let N > 1. Setting µ − = ( µ 1 , . . . , µ N ), w e hav e: T γ µ = T γ µ − · T µ 1 + ··· + µ N . . . T N +1 with T γ µ − ∈ ⟨ T 1 , . . . , T µ 1 + ··· + µ N − 1 ⟩ . Similarly as for N = 1, we start by proving that e µ T γ µ − T ± 1 µ 1 + ··· + µ N . . . T ± 1 i +1 ˇ T i T i − 1 . . . T N +1 C † w N +1 = 0 ∀ i = N + 1 , . . . , µ 1 + · · · + µ N . (55) This is the diﬃcult part of the pro of. Recall that the elemen t T γ µ is as follo ws: 34 µ 1 1 2 . . . . . . a . . . N + 1 µ 2 . . . . . . . . . µ a i a . . . . . . µ N +1 . . . . where we hav e circled the crossing that is remov ed when we remov e T i in (55). This crossing in volv es the strand coming from dot num b er µ 1 + · · · + µ N + 1 and another strand. Denote i a the dot from where this strand is coming. This i a b elongs to one of the subsets, sa y the one of size µ a , and can not b e the ﬁrst element of this subset. The ± 1 crossings in (55) will also b e on the strand coming from dot num b er µ 1 + · · · + µ N + 1, and they will b e to the right of the circled crossing and therefore will not in terfere at all with our reasoning. First, recall from (50) the freedom that we hav e in T γ µ . W e use this freedom to mo ve a little bit the starting p oint of the strands descending on dot num b er a . This do es not change the left-hand-side of (55) up to an irrelev ant factor (a p o w er of q ). Doing so, the element in (55) betw een the idemp oten t e µ and C † w N +1 is: µ 1 1 2 . . . . . . a . . . N + 1 µ 2 . . . . . . . . . µ a i a . . . . . . µ N +1 . . . where we hav e indeed remov ed the crossing circled in the preceding diagram. No w what w e claim is that precomp osing (on top) by the generator T i a − 1 is equal to comp osing (at the b ottom) by T a ...T N − 1 T N T − 1 N − 1 . . . T − 1 a . One has to lo ok at the following picture, where one can see that the added crossing on top is transp orted to the braid added at the b ottom. µ 1 1 2 . . . . . . . . . µ 2 . . . . . . . . . . . . . . . µ N +1 . . . The mo ve is entirely happ ening b elo w the strands descending from left to right, which therefore do not in terfere with the pro cedure. T o summarize, if w e denote X the element in (55) betw een the idemp oten t e µ and C † w N +1 , w e hav e shown in particular that: e µ T i a − 1 X C † w N +1 = e µ X T a ...T N − 1 T N T − 1 N − 1 . . . T − 1 a C † w N +1 . 35 T o conclude w e notice that T i a − 1 is absorbed b y e µ and replaced by q , while T a . . . T N − 1 T N T − 1 N − 1 . . . T − 1 a is absorb ed by C † w N +1 and replaced by − q − 1 (recall that T i C † w N +1 = − q − 1 C † w N +1 for any i = 1 , . . . , N ). Therefore w e get: ( q + q − 1 ) e µ X C † w N +1 = 0 , whic h implies that the elemen t (55) is 0 as desired, since we assumed in this section that we work ed o ver the ﬁeld C ( q ) or that q 2 is not a to o small ro ot of unity . No w that we hav e shown (55), we deduce that, as we did for N = 1, in e µ T γ µ C † w N +1 = e µ T γ µ − T µ 1 + ··· + µ N . . . . . . T N +1 C † w N +1 , w e can replace (from left to right) all app earing T i b y their inv erses, using the relation T i = T − 1 i + ( q − q − 1 ) and (55). W e thus hav e e µ T γ µ C † w N +1 = e µ T γ µ − T µ 1 + ··· + µ N . . . . . . T N +1 C † w N +1 . (56) Finally , to b e able to use the induction hypothesis, we write: e µ = e µ − e µ N +1 with e µ N +1 ∈ ⟨ T µ 1 + ··· + µ N +1 , . . . , T n − 1 ⟩ , and w e note that e µ N +1 comm utes with T γ µ − ∈ ⟨ T 1 , . . . , T µ 1 + ··· + µ N − 1 ⟩ . F urthermore, we write C † w N +1 = C † w N Y for some elemen ts Y ∈ H n . The precise form of Y is not imp ortant here, the imp ortant prop ert y is that C † w N comm utes with all T i with i ≥ N + 1. In particular, it commutes with e µ N +1 . Therefore, we rewrite (56) as e µ T γ µ C † w N +1 = e µ − T γ µ − C † w N e µ N +1 T µ 1 + ··· + µ N . . . . . . T N +1 Y . W e use the induction hypothesis, namely , e µ − T γ µ − C † w N = e µ − T γ µ − C † w N , and we put ev erything back to their original p osition to get ﬁnally e µ T γ µ C † w N +1 = e µ T γ µ − T µ 1 + ··· + µ N . . . . . . T N +1 C † w N +1 . This concludes the pro of of (53) and in turn of the prop osition. Remark 6.14. In the pr e c e ding pr o of, we have actual ly shown that in the expr ession: X µ N = e µ T γ µ C † w N +1 T − 1 γ µ e µ , one c an r eplac e any gener ator T i app e aring in T γ µ by its inverse T − 1 i , and similarly, one c an r eplac e any T − 1 i app e aring in T − 1 γ µ by T i . F or example, we have X µ N = e µ T γ µ C † w N +1 T γ − 1 µ e µ , which is an expr ession explicitly involving only the gener ators T i and never their inverses. W e conclude this pap er b y proving the preceding conjectures in some sp ecial cases. Prop osition 6.15. In the fol lowing situations: • for any N ≥ 1 with µ = ( µ 1 , 1 , 1 , . . . , 1) . • for N = 1 and any µ = ( µ 1 , µ 2 ) ; • for N = 2 and any µ = ( µ 1 , µ 2 , µ 3 ) , 36 we have: X µ N = Y µ N , and this element gener ates the ide al I µ N . Pr o of. In these sp eciﬁc cases, the fact that X µ N generates the ideal I µ N w as prov en in [CP23], so w e only need to prov e the equalit y in the prop osition, which is e µ T γ µ C † w N +1 T − 1 γ µ e µ = e µ C † ˜ w N +1 e µ . (57) Recall from (50) that we can mo dify T γ µ in to some diﬀerent element T γ ′ µ without changing the left hand side of (57). W e will use this freedom b elow. • Let N ≥ 1 and µ = ( µ 1 , 1 , 1 , . . . , 1). Here w e choose to w ork with the following element T γ ′ µ : T γ ′ µ = µ 1 1 2 . . . N + 1 . . . µ 1 + N . . . . With a form ula, we hav e: T γ ′ µ = T µ 1 − 1 . . . T 1 · T γ µ = T µ 1 − 1 . . . T 1 · T µ 1 . . . T 2 · . . . . . . · T µ 1 + N − 1 . . . T N +1 . Using sev eral times the braid relations or lo oking at the braid element ab o ve, it is easy to ﬁnd that: T γ ′ µ T i = T µ 1 + i − 1 T γ ′ µ ∀ i = 1 , . . . , N . No w, w N +1 is the longest p ermutation of the letters { 1 , . . . , N + 1 } , and in this case ˜ w N +1 is the longest p ermutation of the letters { µ 1 , . . . , µ 1 + N } . Therefore, we hav e that C † ˜ w N +1 is obtained from C † w N +1 b y replacing the generators T 1 , . . . , T N b y , resp ectiv ely , T µ 1 , . . . , T µ 1 + N − 1 . F rom the preceding calculations, w e see that: T γ ′ µ C † w N +1 = C † ˜ w N +1 T γ ′ µ , and this pro ves (57). • Let N = 1 and µ = ( µ 1 , µ 2 ). Here we mo dify again the element T γ µ in to T γ ′ µ = T µ 1 − 1 . . . T 1 · T γ µ = T µ 1 − 1 . . . T 1 · T µ 1 . . . T 2 . The p erm utations w N +1 and ˜ w N +1 are, resp ectiv ely , the transp ositions (1 , 2) and ( µ 1 , µ 1 + 1). There- fore, w e hav e: C † w N +1 = q − T 1 and C † ˜ w N +1 = q − T µ 1 . Using the braid relations, it is easy to see that T γ ′ µ T 1 = T µ 1 T γ ′ µ and this pro v es (57) with T γ ′ µ as required. • Let N = 2 and µ = ( µ 1 , µ 2 , µ 3 ). This case is more in volv ed. Here w e hav e: T γ µ = T µ 1 . . . T 2 · T µ 1 + µ 2 . . . T 3 = T µ 1 + µ 2 . . . T µ 1 +2 · T µ 1 . . . T 2 · T µ 1 +1 . . . T 3 . Again, w e are going to w ork with the mo diﬁed element 37 T γ ′ µ = µ 1 1 2 3 . . . . . . µ 2 . . . . . . µ 3 . . . . . . The form ula is: T γ ′ µ = T µ 1 − 1 . . . T 1 · T γ µ = T µ 1 + µ 2 . . . T µ 1 +2 · T µ 1 − 1 . . . T 1 · T µ 1 . . . T 2 · T µ 1 +1 . . . T 3 . Using the braid relations, we ﬁnd that this element satisﬁes: T γ ′ µ T 1 = T µ 1 T γ ′ µ , T γ ′ µ T 2 = ˜ T µ 1 + µ 2 T γ ′ µ , with ˜ T µ 1 + µ 2 = T − 1 µ 1 +1 . . . T − 1 µ 1 + µ 2 − 1 T µ 1 + µ 2 T µ 1 + µ 2 − 1 . . . T µ 1 +1 . The elemen t C † w N +1 b eing the q -antisymmetriser made out of the generators T 1 , T 2 , w e hav e: T γ ′ µ C † w N +1 T − 1 γ ′ µ = q 3 − q 2 ( T µ 1 + ˜ T µ 1 + µ 2 ) + q ( T µ 1 ˜ T µ 1 + µ 2 + ˜ T µ 1 + µ 2 T µ 1 ) − T µ 1 ˜ T µ 1 + µ 2 T µ 1 . (58) It remains to multiply on b oth sides by the idemp oten t e µ . Recall that T i e µ = e µ T i = q e µ for i ∈ { µ 1 , . . . , µ 1 + µ 2 − 1 } . W e hav e the follo wing formulas: e µ ˜ T µ 1 + µ 2 e µ = e µ T µ 1 + µ 2 e µ , e µ ˜ T µ 1 + µ 2 T µ 1 e µ = q 1 − µ 2 e µ T µ 1 + µ 2 . . . T µ 1 e µ , e µ T µ 1 ˜ T µ 1 + µ 2 e µ = q µ 2 − 1 e µ T µ 1 . . . T µ 1 + µ 2 e µ − q µ 2 − 1 ( q µ 2 − 1 − q 1 − µ 2 ) e µ T µ 1 T µ 1 + µ 2 e µ , e µ T µ 1 ˜ T µ 1 + µ 2 T µ 1 e µ = e µ T µ 1 . . . T µ 1 + µ 2 . . . T µ 1 e µ − q ( q µ 2 − 1 − q 1 − µ 2 ) e µ T µ 1 + µ 2 . . . T µ 1 e µ . (59) W e will explain ho w to chec k them at the end of the pro of, and for now we use them in (58) and w e conclude that: e µ T γ ′ µ C † w N +1 T − 1 γ ′ µ e µ = e µ  q 3 − q 2 ( T µ 1 + T µ 1 + µ 2 ) + q 2 − µ 2 T µ 1 + µ 2 . . . T µ 1 + q µ 2 T µ 1 . . . T µ 1 + µ 2 − q µ 2 ( q µ 2 − 1 − q 1 − µ 2 ) T µ 1 T µ 1 + µ 2 − T µ 1 . . . T µ 1 + µ 2 . . . T µ 1 + q ( q µ 2 − 1 − q 1 − µ 2 ) T µ 1 + µ 2 . . . T µ 1  e µ = e µ  q 3 − q 2 ( T µ 1 + T µ 1 + µ 2 ) + q µ 2 ( T µ 1 + µ 2 . . . T µ 1 + T µ 1 . . . T µ 1 + µ 2 ) − ( q 2 µ 2 − 1 − q 1 ) T µ 1 T µ 1 + µ 2 − T µ 1 . . . T µ 1 + µ 2 . . . T µ 1  e µ . (60) Note that in the ﬁnal result, apart from the co eﬃcien t ( − 1) in fron t of the longest element, the co eﬃcien ts are all in q Z [ q ]. T o conclude the pro of, we remark that the p erm utation ˜ w N +1 is the transp osition of the letters µ 1 and µ 1 + µ 2 + 1 and therefore: T ˜ w N +1 = T µ 1 . . . T µ 1 + µ 2 . . . T µ 1 . So the ab o ve calculation resulted in e µ T γ ′ µ C † w N +1 T − 1 γ ′ µ e µ = ( − 1) ℓ ( ˜ w N +1 ) e µ T ˜ w N +1 e µ + X x< ˜ w N +1 α x e µ T x e µ with α x ∈ q Z [ q ]. 38 By the unicit y prop ert y of the Kazhdan–Lusztig basis prov ed in Prop osition 4.5, we conclude that this elemen t is indeed e µ C † ˜ w N +1 e µ as w as required. T o ﬁnish the pro of, w e brieﬂy indicate how to ﬁnd formulas (59). The ﬁrst tw o formulas are immediate. F or the third one, we write e µ T µ 1 ˜ T µ 1 + µ 2 e µ = q µ 2 − 1 e µ T µ 1 T − 1 µ 1 +1 . . . T − 1 µ 1 + µ 2 − 1 T µ 1 + µ 2 e µ , and w e remov e from left to right the inv erses, using the following relations e µ T µ 1 T µ 1 +1 . . . | {z } i − 1 terms ( T − 1 µ 1 + i − T µ 1 + i ) . . . T − 1 µ 1 + µ 2 − 1 | {z } µ 2 − i − 1 terms T µ 1 + µ 2 e µ = − ( q − q − 1 ) q − µ 2 +2 i e µ T µ 1 T µ 1 + µ 2 e µ . This relies on T − 1 i = T i + ( q − q − 1 ) and the absorption prop erty of the idemp oten t e µ . Indeed the i − 1 underbraced factors comm ute to hit the idemp oten t on the righ t and pro duce positive p o wers of q . The remaining µ 2 − i − 1 factors commute and hit the idemp otent on the left pro ducing negative p o wers of q . The sum o ver i ∈ { 1 , . . . , µ 2 − 1 } concludes the v eriﬁcation using ( q − q − 1 )( q − µ 2 +2 + · · · + q µ 2 − 2 ) = q µ 2 − 1 − q 1 − µ 2 . F or the last formulas in (59), a similar reasoning w orks. 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Kazhdan-Lusztig bases of parabolic Hecke algebras and applications to Schur-Weyl duality

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