Equivariant log concavity and the $\operatorname{FI^\sharp}$-module structure on $H^i(\operatorname{Conf}(n,\mathbb{R}^d))$
Previous work has conjectured that the graded $\mathfrak{S}_n$-representations $H^\bullet(\operatorname{Conf}(n,\mathbb{R}^d);\mathbb{Q})$ are strongly equivariantly log concave, and has proven this conjecture in low degrees. By leveraging the theory…
Authors: Benjamin Homan
Equiv arian t log conca vit y and the FI ♯ -mo dule structure on H i (Conf ( n, R d )) Benjamin Homan 1 Departmen t of Mathematics, Univ ersity of Oregon, Eugene, OR A bstr act. Previous w ork has conjectured that the graded S n -represen tations H • (Conf ( n, R d ); Q ) are strongly equiv arian tly log concav e, and has pro ven this conjecture in lo w degrees. By lev eraging the theory of represen tation stabilit y , we are able instead prov e a stronger statement ab out the FI ♯ -mo dule structure on H i (Conf ( n, R d ); Q ) which implies the original conjecture up to degree 19. W e conjecture that this equiv ariant log conca vit y-like property holds in all degrees for the FI ♯ -mo dules H i (Conf ( n, R d ); Q ). 1 In tro duction Giv en a finite group G , w e sa y that a graded representation V • of G is strongly equiv arian tly log conca ve if, for all i < j ≤ k < with i + = j + k , there exists a G -equiv arian t inclusion: V i ⊗ V ℓ → V j ⊗ V k . (1) Previous w ork has conjectured that particular representations of the symmetric group S n are strongly equiv arian tly log concav e [4, Conjecture 5.3] [6, Conjecture 1.6]. They are the cohomology rings: • A • n := H • (Conf ( n, C ); Q ), • C • n := H 2 • (Conf ( n, R 3 ); Q ), where Conf ( n, X ) is the space of ordered n -tuples of distinct p oin ts in a top ological space X . 2 W e sa y that a graded representation is strongly equiv ariantly log conca ve in degree m if (1) holds for all i < j ≤ k < with i + = j + k = m . Both A n and C n are strongly equiv arian tly log conca ve in degree m for all m ≤ 14 [6, Theorem 1.7]. In addition to the structure of graded S n -represen tation, we ha ve natural maps A • n → A • n +1 and C • n → C • n +1 induced by maps Conf ( n + 1 , R d ) → Conf ( n, R d ) that remo ve p oin ts from a configuration. In the language of represen tation stability , this gives b oth A • and C • the structure of an FI-module. Previous w ork leveraged this extra FI-mo dule structure when proving that the graded S n -represen tations A • n and C • n are strongly equiv ariantly log concav e in lo w degrees [6, Theorem 1.7]. 1 Supp orted by NSF grant DMS-2039316. 2 One could consider H ∗ (Conf ( n, R d ); Q ) for any d , how ever when d is even this is isomorphic to A • n up to a change in grading. Likewise, when d is odd w e ha ve an isomorphism with C • n . 1 In fact, A • and C • carry additional structure arising from maps A • n → A • n − 1 and C • n → C • n − 1 . These maps are not well-defined on the lev el of spaces, as there is no canonical wa y to define a map Conf ( n − 1 , R d ) → Conf ( n, R d ) that adds a p oin t to a given configuration. Ho wev er, one can lo osely imagine these maps as “adding a p oint near infinity ,” and they giv e A • and C • a m uch stronger structure of an FI ♯ -mo dule. Ultimately , w e find that strong equiv arian t log conca vity not only app ears when considering A • and C • as graded represen tations, but a similar prop ert y also holds at the level of FI ♯ -mo dules. W e conjecture the follo wing: Conjecture 1.1. Given p ositive inte gers with i < j ≤ k < and i + = j + k , ther e is an inclusion of FI ♯ -mo dules: A i ⊗ A ℓ → A j ⊗ A k , and C i ⊗ C ℓ → C j ⊗ C k . F urther, we prov e this conjecture holds in low degree. Theorem 1.2. Given p ositive inte gers i < j ≤ k < with i + = j + k = m , for m ≤ 19 we have an inclusion of FI ♯ -mo dules A i ⊗ A ℓ → A j ⊗ A k , and C i ⊗ C ℓ → C j ⊗ C k . Corollary 1.3. F or al l n , the gr ade d r epr esentations A • n and C • n ar e str ongly e quivariantly lo g c onc ave up to de gr e e 19. While [6, Theorem 1.7] prov es that these inclusions are S n -equiv arian t, we show that they also resp ect the rigid FI ♯ -mo dule structures on A i and C i . As we will see in Remark 2.27, the stronger FI ♯ -mo dule structure can be completely determined with few er computations. Hence, w e are able to prov e Conjecture 1.1 up to degree 19 and obtain strong equiv ariant log conca vity of the graded represen tations as a corollary . First, we review the necessary results of represen tation theory that mak e these calculations p ossible. Then, b y using the equiv alence of the categories FB-mod and FI ♯ -mo d, w e observ e that the determining the FI ♯ -mo dule structures of A i ⊗ A j and C i ⊗ C j requires few er computations than explicitly calculating them as FB-mo dules. Finally , we describ e the algorithm that calculates the FI ♯ -mo dule structures of A i ⊗ A j and C i ⊗ C j and prov es Theorem 1.2. W e run this algorithm using the soft ware pac k age SageMath [3]. A cknow le dgments: W e are grateful to Nic holas Proudfo ot for his mentorship, guidance, and com- men ts throughout the preparation of this manuscript. W e would also like to thank Eric Ramos for the insights in to determining the structure of a tensor pro duct of FI ♯ -mo dules. Finally , we thank Galen Dorpalen-Barry and the T exas A&M compute cluster Whistler for pro viding the computing p o w er necessary to pro ve Theorem 1.2. 2 2 Represen tation Stabilit y In this section we review the definitions and prop erties of FB, FI, and FI ♯ -mo dules. F or a more thorough treatment see [2]. Giv en a partition λ of a p ositive in teger n , we denote by V λ the corresp onding irreducible S n -represen tation. F or the sake of this pap er, all representations will b e o ver the field Q . 2.1 Stabilization of FB-Mo dules F or an y fixed integer i ≥ 0, we can view A i and C i as a sequence of S n -represen tations ( A i n ) and ( C i n ) by v arying n . A categorical description of this data is an FB-mo dule : Definition 2.1. Let FB b e the category whose ob jects are finite sets whose morphisms are bijec- tions b et ween the sets. An FB-mo dule is an y functor from FB to k -Mo d for some commutativ e ring k . 3 Giv en an FB-mo dule V , w e will denote V n := V [ n ] where [ n ] = { 1 , 2 , . . . n } . Example 2.2. Given a sequence { W n } of S n -represen tations, we can construct an FB-mo dule W b y declaring W [ n ] = W n . Then, for an y bijection σ : [ n ] → [ n ], σ acting on W n defines the map W ( σ ) : W n → W n . In particular, this gives A i and C i the structure of an FB-mo dule for an y fixed in teger i ≥ 0, as A i n and C i n ha ve an S n -action induced by p erm uting the p oin ts in a configuration. Some FB-mo dules exhibit an interesting stabilization phenomenon. As an example, consider A 1 = H 1 (Conf ( − , R 2 ) , Q ): Example 2.3. F or this example, let the Y oung diagram of a partition represen t the corresp onding irreducible S n -represen tation. 3 F or this paper w e will only consider Q -mo dules. 3 n A 1 1 0 2 3 L 4 L L 5 L L 6 L L The table con tin ues for all n , but notice that for n > 4, the tableaux indexing the decomposition of A 1 n come from adding a b o x to the first ro w of those in the decomposition of A 1 n − 1 . W e call this phenomenon stabilization . Definition 2.4. Suppose V is an FB-mo dule such that the irreducible comp osition of V m is given b y: V m = M λ ⊢ m ( V λ ) ⊕ c λ . W e say that V stabilizes at m if for all n > m : V n = M λ ⊢ m ( V ( λ 1 +( n − m ) ,λ 2 ,...,λ ℓ ) ) ⊕ c λ . Example 2.3 is no coincidence; for all i ≥ 0, A i and C i will stabilize [1, Theorem 1], and this stabilization w as key to the pro of of [6, Theorem 1.7]. T o understand this b ehavior, we m ust add additional structure to our FB-mo dules. Definition 2.5. Let FI b e the category of finite sets with injections. An FI-mo dule is a functor FI → k -mo d for some commutativ e ring k . Example 2.6. F or any m ≥ 0, we define an FI-mo dule M ( m ) that assigns to any finite set S the Q -mo dule with basis giv en b y injections [ m ] → S . Then, any injection of finite sets ϕ : S → T induces a map ϕ ∗ : M ( m )( S ) → M ( m )( T ). When m = 0, there is only one injection ∅ → S . Th us, M (0) is a constant FI-mo dule assigning to each finite set the trivial represen tation of S n . 4 Additionally , M (1) n has a basis indexed b y elemen ts of [ n ], since injections { 1 } → [ n ] are completely determined b y the image of 1. The S n -action on M (1) n is giv en b y p erm uting the index set [ n ], and thus M (1) n is the p erm utation representation of S n . Example 2.7. F or an y fixed i , A i and C i ha ve the structure of an FI-mo dule. Giv en an injection f : [ m ] → [ n ], we define a map Conf ( n, R d ) → Conf ( m, R d ) by ( p 1 , . . . , p n ) 7→ ( p f (1) , . . . , p f ( m ) ). The induced maps on cohomology are Q -mo dule homomorphisms A i ( f ) : A i m → A i n and C i ( f ) : C i m → C i n . When f : [ n ] → [ n + 1] this corresp onds to removing a p oin t from the configuration as discussed in the introduction. Remark 2.8. There is a forgetful functor is induced by the inclusion FB → FI. W e denote this functor π : FI-mo d → FB-mod and it forgets all non-bijectiv e morphisms. The additional structure of an FI-mo dule allo ws for a c haracterization of the FB-modules which stabilize. Definition 2.9. [2, Prop osition 2.3.5] An FI-mo dule V is finitely generated 4 if there exists a finite sequence of integers { m i } and a surjection: M i M ( m i ) ↠ V , where M ( m i ) is the FI-mo dule defined in Example 2.6. Theorem 2.10. [2, Theorem 1.13] If V is an FB -mo dule over a field of char acteristic 0 , then V stabilizes if and only if ther e exists a finitely gener ate d FI -mo dule W such that V = π ( W ) . Imp ortan tly , b oth A i and C i are finitely generated FI-mo dules and thus the FB-mo dules π ( A i ) and π ( C i ) stabilize [1, Theorem 1]. In fact, there are sharp b ounds for this stabilization: Theorem 2.11. [5, Theorem 1.1] F or al l inte gers i ≥ 0 , the FB -mo dules A i and C i stabilize sharply at n = 3 i + 1 and n = 3 i , r esp e ctively. The formulation of a strong equiv arian t log concavit y prop ert y akin to (1) requires a notion of the tensor pro duct of FB-mo dules: Definition 2.12. Giv en tw o FB-mo dules V and W , their tensor pro duct V ⊗ W is the FB-mo dule assigning: ( V ⊗ W ) n = V n ⊗ W n . If V and W are FI-modules, then V ⊗ W also has an FI-module structure; an injection f : [ m ] → [ n ] defines a map V ⊗ W ( f ) := V ( f ) ⊗ W ( f ) : V m ⊗ W m → V n ⊗ W n . If tw o FB-mo dules stabilize, it follows that their tensor product will stabilize: 4 This is not the definition used in [2, Definition 2.3.4]. How ever, [2, Prop osition 2.3.5] giv es that this characteri- zation is equiv alent. 5 Theorem 2.13. [6, Theorem 3.3] If FB -mo dules V and W stabilize at n and m , r esp e ctively, then V ⊗ W stabilizes at n + m . Corollary 2.14. F or any inte ger i ≥ 0 , the FB -mo dules A i ⊗ A j and C i ⊗ C j stabilize at 3( i + j ) + 2 and 3( i + j ) , r esp e ctively. Pr o of. Theorem 2.11 and Theorem 2.13 immediately imply the corollary . Corollary 2.14 gives a finite bound on the computations necessary to pro ve equiv arian t log conca vity in low degrees. W e need only compute the irreducible decompositions of ( A i ⊗ A j ) n and ( C i ⊗ C j ) n up to the stabilization b ounds n = 3( i + j ) + 2 and n = 3( i + j ) and verify (1) for these represen tations. 2.2 FI ♯ -Mo dules Determining the structure of A i ⊗ A j and C i ⊗ C j b y computing their underlying FB-mo dules up to stabilization is a computation in tensive pro cess that requires determining Kroneck er co efficients for large S n -represen tations. W e improv e this approach by observing that b oth A i and C i ha ve a stronger FI ♯ -mo dule structure: Definition 2.15. [2, Definition 4.1.1] Let FI ♯ b e the category of finite sets with partially-defined injections . A partially-defined injection b et ween finite sets S and T are subsets A ⊆ S and B ⊆ T together with a bijection φ : A → B . An FI ♯ -mo dule is a functor FI ♯ → k -mo d for a commutativ e ring k . Remark 2.16. Let co-FI b e the category FI op . One can think of FI ♯ -mo dules as ha ving both an FI-mo dule and a co-FI-mo dule structure that are compatible with eac h other. An FI ♯ -mo dule structure is very rigid, but w e will see in Remark 2.27 that it is exactly this rigidit y that allows us to completely determine the FI ♯ -mo dule structure of A i ⊗ A j and C i ⊗ C j more efficiently . Example 2.17. Both A i and C i are FI ♯ -mo dules b ecause Conf ( n, R d ) is a homotopy FI ♯ -space [2, Prop osition 6.4.2]. That is, it defines a functor FI ♯ → hT op, the category of top ological spaces with homotop y classes of maps as morphisms. As discussed in E xample 2.7, the FI-mo dule structure on cohomology is induced b y deleting p oin ts from a configuration. Additionally , while “adding a p oin t at infinity” is not w ell defined at the level of spaces, it is up to homotop y . This induces a co-FI structure on cohomology . Remark 2.18. Just lik e in Remark 2.8 there is a map FI → FI ♯ taking only the morphisms where the bijectiv e comp onen t is defined on the whole domain. This induces another forgetful functor FI ♯ -mo d → FI-mod. Ultimately , the categories FI ♯ -mo d and FB-mo d are equiv alen t [2, Theorem 4.1.5]. T o define this equiv alence, we first consider a functor M : FB-mo d → FI-mo d. Recall from Remark 2.8, that π : FI-mo d → FB-mod is the forgetful functor induced by the inclusion of FB → FI. 6 Definition 2.19. [2, Definition 2.2.2] The functor M : FB-mod → FI-mo d is the left-adjoint of the map π : FI-mo d → FB-mo d. Explicitly , given an FB-mo dule W , the S n -represen tations M ( W ) n are given b y: M ( W ) n = M a ≤ n Ind S n S a × S n − a W a ⊠ Q , where S n − a acts on Q trivially . Example 2.20. In Example 2.6 w e defined the FI-mo dule M ( m ). This FI-mo dule is the image of the regular represen tation Q [ S m ] under the functor M . Remark 2.21. By using [5, Corollary 2.10], one can confirm that in Example 2.3 A 1 ∼ = M ( V (2) ) where we consider the represen tation V (2) as the follo wing FB-mo dule: V (2) n = V (2) if n = 2; 0 else. In fact, [5, Corollary 2.10] allows us to compute the FB-mo dules that map via M to A i and C i for an y i . It happ ens that FI-mo dules in the image of M alwa ys ha ve a stronger FI ♯ -mo dule structure [2, Example 4.1.4], so we can take M to b e a functor FB-mo d → FI ♯ -mo d. An explicit description of the other half of the equiv alence requires that w e define the span of elemen ts in an FI-mo dule: Definition 2.22. [2, Definition 2.3.1] Let V b e an FI-mo dule and A b e some collection of elements in F n V n . W e sa y span V ( A ) is the minimal sub-FI-mo dule of V that has ev ery elemen t of A . W e now define a functor H 0 : FI-mo d → FB-mo d. F or any p ositiv e integer n define V 2( k + j ) . Then H 0 ( A j ⊗ A k ) m = 0 and H 0 ( C j ⊗ C k ) m = 0 . Pr o of. By [5, Corollary 2.10] w e ha ve that H 0 ( A j ) and H 0 ( C j ) are non-zero in degrees j + 1 to 2 j . Supp ose V and W are finitely-generated FI-mo dules such that H 0 ( V ) and H 0 ( W ) are zero in all degrees strictly greater than n and m , respectively . Then [2, Prop osition 2.3.6, Remark 2.3.8] giv es that H 0 ( V ⊗ W ) is zero in degrees strictly greater than n + m . Hence, b oth H 0 ( A j ⊗ A k ) and H 0 ( C j ⊗ C k ) are zero in degrees strictly greater that 2( i + j ) as desired. Remark 2.27. This b ound is low er than the stabilization degree for these FB-mo dules, whic h are 3( j + k ) + 2 for A j ⊗ A k and 3( j + k ) for C j ⊗ C k b y Corollary 2.14. Example 2.28. W e will compute H 0 ( A 1 ⊗ A 1 ). W e noted in Remark 2.21 that A 1 = M ( V (2) ) so ( A 1 ⊗ A 1 ) i will first b e non-zero in degree 2. In general, if the minimal degree of a generator for V is n and for W the minimal degree is m , w e know that H 0 ( V ⊗ W ) i = 0 for all i < max( n, m ). W e also know from Lemma 2.26 that H 0 ( A 1 ⊗ A 1 ) i = 0 for an y i > 4 so w e ha ve a finite num ber of computations that we can p erform in SageMath [3]: i 2 3 4 ( A 1 ⊗ A 1 ) i V (2) V (1 , 1 , 1) ⊕ V ⊕ 3 (2 , 1) ⊕ V ⊕ 2 (3) V (1 , 1 , 1 , 1) ⊕ V ⊕ 3 (2 , 1 , 1) ⊕ V ⊕ 4 (2 , 2) ⊕ V ⊕ 5 (3 , 1) ⊕ V ⊕ 3 (4) M ( H 0 ( A 1 ⊗ A 1 )
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