Neutral-Fermion constructions of factorial $gp$-and $gq$-Functions

NEUTRAL-FERMION CONSTR UCTIONS OF F A CTORIAL g p -AND g q -FUNCTIONS KOUSHIK BRAHMA 1 , T AKESHI IKEDA 1 , SHINSUKE IW A O 2 , YI Y ANG 3 1 Wase da University, F aculty of Scienc e and Engine ering 3-4-1 Okubo, Shinjuku-ku T okyo 169-8555, Jap an 2 Keio University, F aculty of Business and Commer c e 4-1-1 Hiyoshi, Kohoku-ku Y okohama, Kanagawa 223-8521, Jap an 3 Sun Y at-sen University, Scho ol of Mathematics 135 Xin ’gang West R o ad, Haizhu, Guangzhou 510275, China Email addresses: koushikbr ahma95@gmail.c om, gakuike da@wase da.jp, iwao-s@fb c.keio.ac.jp, yangy875@mail2.sysu.e du.cn Abstract. W e develop neutral-fermionic constructions for the factorial g p - and gq -functions introduced by Nak agaw a and Naruse, whic h are resp ectively dual to the factorial GQ - and GP -functions of Ikeda and Naruse. In particu- lar, we realize the factorial GP -, GQ - and g q -functions as v acuum exp ectation v alues. As applications, we obtain, Jacobi–T rudi t ype determinantal formulas for the transition co efficien ts b et ween functions with differen t equiv ariant pa- rameters for g q and its dual GP , as well as a Pfaffian formula for the factorial g q -functions. W e further pro v e a remark able coincidence among the transition coefficients for parameter changes for gp , g q , GQ , and GP . These co efficien ts admit a description in terms of factorial Grothendiec k polynomials of t ype A. Contents 1. In tro duction 2 Ac knowledgemen ts 3 2. F actorial GP , GQ and g p, g q -functions 3 2.1. Preliminaries on root systems and W eyl groups, and strict partitions 3 2.2. F actorial GP - and GQ -functions 5 2.3. F actorial g p - and g q -functions 7 3. F ree-F ermion formalism 8 3.1. Neutral F ermions and F o ck spaces 8 3.2. β deformed F ermions 9 3.3. β -deformed Boson-F ermion correspondence 10 3.4. F ree-F ermion formalism of g q , g p, GP , GQ — Non-equiv arian t case 11 3.5. GQ λ ( x | b ) and GP λ ( x | b ) as a v acuum exp ectation v alue 13 3.6. g q λ ( x | b ) as a v acuum exp ectation v alue 14 3.7. Pfaffian formula for g q λ ( y | b ) 15 3.8. Jacobi-T rudi t yp e formula of the expansion co effician ts 16 Date : March 31, 2026. 2020 Mathematics Subje ct Classification. 05E05, 14M15, 19L47. Key wor ds and phr ases. Symmetric functions, equiv ariant K -theory , b oson-fermion correspon- dence. Pfaffian formulae, factorial Grothendieck polynomial. 1 2 KOUSHIK BRAHMA, T AKESHI IKEDA, SHINSUKE IW AO, YI Y ANG 3.9. A formula on contour in tegral 17 3.10. Pro of on General expansion formulas 18 3.11. Pro of of the Duality Lemma 3.10 21 3.12. Conjecture on factorial g p -functions 22 4. Co efficien ts in terms of double Grothendieck polynomials 22 4.1. Double Grothendieck p olynomials 22 4.2. Expansion of factorial g p -function and factorial GQ -function 24 References 27 1. Introduction Symmetric functions originating from K -theoretic Sch ub ert calculus provide a ric h algebraic framework connecting geometry , combinatorics, and represen tation theory . Among these, the factorial g p -functions, introduced by Nak agaw a and Naruse [16], form a family of inhomogeneous symmetric functions in y = ( y 1 , y 2 , . . . ) dep ending on an infinite sequence of parameters b = ( b 1 , b 2 , . . . ). Eac h g p λ ( y | b ), indexed b y a strict partition λ , is defined as the dual of the K -theoretic factorial Sc hur Q -function GQ λ ( x | b ) introduced by Ikeda and Naruse [5], whic h represents the Sc h ub ert basis in the torus-equiv ariant K -theory of the Lagrangian Grassman- nian. An affine counterpart of these functions was recen tly introduced by Ik eda, Shi- mozono, and Y amaguchi [6] in connection with the affine Grassmannian of the symplectic group Sp 2 n ( C ). The resulting affine g p -functions , indexed by a certain set of partitions λ , form the ideal-sheaf basis of the corresp onding equiv arian t K - homology ring. The g p function can b e regarded as the limit, as n → ∞ , of affine g p functions. More precisely , when | λ | ≤ 2 n , they coincide with g p λ ( y | b ) under a suitable sp ecialization of the equiv ariant parameters b i . This affine v ersion has motiv ated further study of the underlying combinatorial and algebraic structures of g p -functions. In this paper w e also study the factorial g q -functions, defined as the duals to the factorial GP -functions that represent the Sch ubert basis in the K -theory of the maximal orthogonal Grassmannian. Our aim is to provide a unified framework for these families of functions through the free–F ermion formalism, which generalizes the non-equiv arian t results of Iwao [9]. W e express the functions GP λ ( x | b ), GQ λ ( x | b ), and g q λ ( y | b ) as v acuum exp ec- tation v alues (Theorem 3.8, Theorem 3.10). W e provide a similar expression for g p λ ( y | b ) as a conjecture (Conjecture 3.22). As a corollary , we obtain a Sch ur-type Pfaffian formula for g q λ ( y | b ) (Corollary 3.14). Note that suc h a form ula has previ- ously b een known for GQ λ ( x | b ) and GP λ ( x | b ) due to the work of Hudson, Ikeda, Matsum ura, and Naruse. Based on the free-F ermion formalism, w e further study the explicit expansion form ula of these functions. W e define GP λ ( x | b ) = X µ C GP λµ ( b, c ) GP µ ( x | c ) , GQ λ ( x | b ) = X µ C GQ λµ ( b, c ) GQ µ ( x | c ) , g p λ ( x | b ) = X µ C g p λµ ( b, c ) g p µ ( x, c ) , g q λ ( x | b ) = X µ C g q λµ ( b, c ) g q µ ( x | c ) . NEUTRAL-FERMION CONSTRUCTIONS OF F ACTORIAL gp -AND g q -FUNCTIONS 3 W e obtain a Jacobi–T rudi type expression for these coefficients (Theorem 3.15), together with an alternative formula in terms of type A Grothendieck p olynomi- als (Theorem 4.10). As a byproduct of the argument, we obtained the follo wing unexp ected coincidence of co efficien ts: (1.1) C g q λµ ( b, c ) = C g p λµ ( b, c ) = C GP µλ ( c, b ) = C GQ µλ ( c, b ) . The pap er organized as follows. In § 2, we fix notation for ro ot systems, W eyl groups and strict partitions. W e explain the definition and some properties of the factorial GP - and GQ -functions. W e introduce the factorial g p - and g q -functions as the dual to GQ and GP resp ectiv ely . W e also provide op erator constructions for these functions. In § 3, w e introduce the neutral-free F ermion, and some of its deformations. The goal of this section is to express GP λ ( x | b ) and g q λ ( y | b ) as v acuum exp ectation v alues for the F o ck space of neutral-free-F ermion. As a corollary , we obtain a Pfaffian form ula for g q λ ( y | b ) . In § 3.8, w e inv astigate the co efficien ts C g q λµ and C GP λµ within the framew ork of F ermion formalism. W e obtain a Jacobi-T rudi type form ula for these co efficinets. In § 4, we will pro v e another expressions for the coefficients C g p λµ and C GQ λµ in terms of t ype A double Grothendieck p olynomials. As a byproduct of this computation, we prov e the coincidence of co efficien ts (1.1) and give a Jacobi-T rudi type formulae of factorial Grothendieck p olynomials. Ac kno wledgemen ts. This work was supported b y JSPS KAKENHI Gran t Num- b ers 24KF0258, 25KF0074, 23K25772, 22K03239, 23K03056. W e are grateful to Professor Hiroshi Naruse for kindly sharing the slides from his talk in the work- shop on Algebraic and Enumerativ e Combinatorics at Shinshu Universit y , Japan. The first author was supp orted by a JSPS P ostdo ctoral F ellowship for Researc h in Japan, during which this research was carried out. The main part of this w ork was carried out while the fourth author was visiting W aseda Universit y . The authors w ould like to thank W aseda Univ ersity for its hospitality . 2. F actorial GP , GQ and g p, g q -functions In this section, we review some results on factorial GP , GQ functions following [5] as w ell as their duals factorial g q , g p functions following [15]. 2.1. Preliminaries on root systems and W eyl groups, and strict parti- tions. 2.1.1. R o ot systems and Weyl gr oups. Let B ∞ and C ∞ b e the following Dynkin diagrams of infinite rank respectively: 0 1 2 · · · < 0 1 2 · · · > F or b oth types, the set of vertex is I = { 0 , 1 , 2 , . . . } . The W eyl group is the infinite h yp er-octhedral group W ∞ generated by s i ( i ∈ I ) satisfying the relations s 2 i = 1 ( i ∈ I ) , s 0 s 1 s 0 s 1 = s 1 s 0 s 1 s 0 , s i s i +1 s i = s i +1 s i s i +1 for i ≥ 1. 4 KOUSHIK BRAHMA, T AKESHI IKEDA, SHINSUKE IW AO, YI Y ANG The group W ∞ acts on the free abelian group L := L ∞ i =1 Z a i b y s 0 ( a j ) = ( − a 1 if j = 1, a j otherwise, , s i ( a j ) =      a i +1 if j = i , a i if j = i + 1, a j otherwise. The simple ro ots { α i } i ∈ I of type B ∞ are defined b y α 0 = − a 1 , α i = a i − a i +1 ( i ≥ 1) , while the simple ro ots { α i } i ∈ I of type C ∞ are defined b y α 0 = − 2 a 1 , α i = a i − a i +1 ( i ≥ 1) . After this p oint, results for t ypes B and C will often b e presented in parallel. Throughout this pap er, we delib erately use the same notation for b oth types with- out changing sym b ols. The interpretation should b e understo o d from the context. 2.1.2. Strict Partitions and Shifte d Diagr ams. A strict partition λ is a strictly de- creasing sequence of p ositive in tegers λ = ( λ 1 > λ 2 > · · · > λ ℓ > 0) . Let SP denote the set of all strict partitions. W e iden tify λ = ( λ 1 > · · · > λ ℓ ) ∈ SP with its shifted Y oung diagram consisting of the b o xes ( i, j ) with 1 ≤ i ≤ ℓ, i ≤ j ≤ i + λ i − 1 , dra wn in matrix-style co ordinates (row i , column j ). The size of λ is defined by | λ | = P ℓ i =1 λ i , which is equal to the total num ber of boxes in the corresp onding shifted Y oung diagram. The length of λ is ℓ , which is the num ber of rows of the diagram. F or each b ox ( i, j ) in the shifted Y oung diagram, we define its c ontent by c ( i, j ) = j − i. Th us, b oxes lying on the same diagonal hav e the same con tent. Example 2.1 . The shifted diagram of λ = (5 , 3 , 1) ∈ SP is depicted b elo w, with eac h b ox labeled by its conten ts: 0 1 2 3 4 0 1 2 0 Fix an integer i ≥ 0. An i -addable b ox of λ ∈ SP is a b o x of conten t i that can b e added to the shifted Y oung diagram of λ so that the result is again a shifted Y oung diagram of a strict partition. Similarly , an i -r emovable b o x of λ ∈ SP is a b o x of conten t i that can b e remov ed from the diagram of λ so that the remaining diagram is still a shifted Y oung diagram of a strict partition. If λ has a i -addable b o x, define s i λ to b e the strict partition obtained b y adding that b o x. If λ has a i -remo v able b o x, define s i λ to be the strict partition obtained b y removing that b o x. Otherwise, set s i λ = λ . This defines an action of W ∞ on SP . Let S ∞ b e the subgroup of W ∞ generated by the simple reflections s i ( i ≥ 1) . Let W 0 ∞ denote the set of minimal-length coset representativ es of W ∞ /S ∞ , namely W 0 ∞ = { w ∈ W ∞ | ℓ ( ws i ) = ℓ ( w ) + 1 for all i ≥ 1 } . NEUTRAL-FERMION CONSTRUCTIONS OF F ACTORIAL gp -AND g q -FUNCTIONS 5 There is an isomorphism b et w een W 0 ∞ and SP . F or λ = ( λ 1 , . . . , λ r ) ∈ SP and i ≥ 1, define ρ i := s i − 1 · · · s 1 s 0 . Then set (2.1) w λ := ρ λ r · · · ρ λ 1 . The elemen t w λ is the minimal-length coset representativ e in W 0 ∞ corresp onding to λ ∈ SP . 2.2. F actorial GP - and GQ -functions. This subsection reviews the definition and basic properties of the factorial GP - and GQ -functions. 2.2.1. Notation in the K -the or etic Setting. Let β b e an indeterminate. W e use the follo wing notation x ⊕ y = x + y + β xy , x ⊖ y = x − y 1 + β y , ⊖ x = 0 ⊖ x = − x 1 + β x . Let F = Q ( β )( b 1 , b 2 , . . . ) be the field of rational functions generated b y b 1 , b 2 , . . . and β . Let R b e the Z [ β ]- subalgebra of F generated b y b i and ⊖ b i for i = 1 , 2 , . . . . Define, for t yp e B ∞ (2.2) c ( α 0 ) = b 1 , c ( α i ) = b i +1 ⊖ b i for i ≥ 1. and, for t yp e C ∞ (2.3) c ( α 0 ) = b 1 ⊕ b 1 , c ( α i ) = b i +1 ⊖ b i for i ≥ 1. The ring R is a deformation of the group ring Z [ L ] = Z [ e ± a i ( i ≥ 1)] of L . In fact, via the sp ecialization map Z [ β ] → Z given by β 7→ − 1, Z ⊗ Z [ β ] R is isomorphic to the group ring Z [ L ] of L . More precisely , w e identify the generators b y b i 7→ 1 − e − a i , ⊖ b i 7→ 1 − e a i . Then for eac h i ∈ I , the element c ( α i ) ∈ R sp ecializes to 1 − e α i ∈ Z [ L ] . F or k ≥ 1, define [ [ x | b ] ] k = ( x ⊕ x )( x ⊖ b 1 ) · · · ( x ⊖ b k − 1 ) , (2.4) [ x | b ] k = ( x ⊖ b 1 ) · · · ( x ⊖ b k ) . (2.5) The conv en tion for equiv arian t parameters adopted in this paper differs from that in [5] only in that the parameter b i in [5] corresponds to ⊖ b i in the presen t work. 2.2.2. Definition of GP and GQ . Let λ ∈ SP . Let r = ℓ ( λ ) . F or n ≥ r , we define GQ λ ( x 1 , . . . , x n | b ) = 1 ( n − r )! X w ∈ S n w   r Y i =1 [ [ x i | b ] ] λ i r Y i =1 n Y j = i +1 x i ⊕ x j x i ⊖ x j   , (2.6) GP λ ( x 1 , . . . , x n | b ) = 1 ( n − r )! X w ∈ S n w   r Y i =1 x i [ x i | b ] λ i − 1 r Y i =1 n Y j = i +1 x i ⊕ x j x i ⊖ x j   , (2.7) where w permutes x 1 , . . . , x n . These are symmetric p olynomials in x = ( x 1 , . . . , x n ) with co efficients in R . W e hav e the stabilit y prop erty GQ λ ( x 1 , . . . , x n , 0 | b ) = GQ λ ( x 1 , . . . , x n | b ) , GP λ ( x 1 , . . . , x n , 0 | b ) = GP λ ( x 1 , . . . , x n | b ) , and so the infinite v ariable version GQ λ ( x 1 , x 2 , . . . | b ) and GP λ ( x 1 , x 2 , . . . | b ) makes sense, which w e simply denote by GQ λ ( x | b ) and GP λ ( x | b ) resp ectiv ely . 6 KOUSHIK BRAHMA, T AKESHI IKEDA, SHINSUKE IW AO, YI Y ANG By setting all parameters b i to zero, w e obtain the non-equiv arian t versions GQ λ ( x ) := GQ λ ( x | 0) , GP λ ( x ) := GP λ ( x | 0) . Define (2.8) G Γ R ( x ) := M λ ∈ SP R GP λ ( x ) , G Γ + R ( x ) := M λ ∈ SP R GQ λ ( x ) . Then { GP λ ( x | b ) | λ ∈ SP } (resp. { GQ λ ( x | b ) | λ ∈ SP } ) forms an R -basis of G Γ R ( x ) (resp. G Γ + R ( x )). It is kno wn that both are R -algebras, and that G Γ + R ( x ) is a subalgebra of G Γ R ( x ); see [5] and [2] for details. R emark 2.2 . The family of functions GP λ ( x | b ) defined here is of t ype B, in the sense that they corresp ond to the Sch ub ert basis of the torus equiv ariant K -theory for the maximal isotropic Grassmannian of t yp e B (see [5, § 8]). More precisely , the function defined in (2.7) is GB ( n ) λ ( x | ⊖ b ) = GP λ ( x 1 , . . . , x n | 0 , ⊖ b ) in the notation of [5, § 6.1]. 2.2.3. V anishing pr op erty. F or each strict partition µ = ( µ 1 , · · · , µ r ) of length r define a sequence b µ b y b µ = ( b µ 1 , · · · , b µ r , 0 , · · · ) Prop osition 2.3 (V anishing prop ert y , [5]) . L et λ and µ b e strict p artitions. If λ ⊆ µ then GQ λ ( b µ | b ) = 0 , and GP λ ( b µ | b ) = 0 . Mor e over, we have GQ λ ( b λ | b )  = 0 , GP λ ( b λ | b )  = 0 . 2.2.4. F actorization pr op erty. F or a partition λ of length less than or equal to r , define the factorial Gr othendie ck p olynomial b y (2.9) G λ ( x 1 , . . . , x r | b ) = X w ∈ S r w Q r i =1 [ x i | b ] λ i + r − i Q 1 ≤ i w λ , 2.3. F actorial g p - and g q -functions. Let ˆ Λ( y ) b e the completed ring of symmet- ric functions in y = ( y 1 , y 2 , . . . ); an elemen t of ˆ Λ( y ) is a formal sum P ∞ n =0 f n , where f n is a symmetric function of degree n . Define ˆ Λ R ( y ) := R ⊗ Z ˆ Λ( y ) . There exist unique families { g p λ ( y | b ) } λ ∈ SP and { g q λ ( y | b ) } λ ∈ SP in ˆ Λ R ( y ) such that ∞ Y i,j =1 1 − ¯ x i y j 1 − x i y j = X λ ∈ SP GQ λ ( x | b ) g p λ ( y | b ) , (2.19) ∞ Y i,j =1 1 − ¯ x i y j 1 − x i y j = X λ ∈ SP GP λ ( x | b ) g q λ ( y | b ) . (2.20) Define the follo wing element in ˆ Λ R ( y ) (2.21) Ω( b i | y ) := ∞ Y j =1 1 − b i y j 1 − b i y j . 8 KOUSHIK BRAHMA, T AKESHI IKEDA, SHINSUKE IW AO, YI Y ANG The group W ∞ acts on ˆ Λ R ( y ) by s 0 f ( x | b ) = Ω( b 1 | y ) f ( x | ⊖ b 1 , b 2 , b 3 , . . . ) , s i f ( x | b ) = f ( x | . . . , b i +1 , b i , . . . ) for i ≥ 1 . F or i ∈ I , the op erator T i on ˆ Λ R ( y ) is defined by (2.22) T i f ( y | b ) = c ( α i ) − 1  s i f ( y | b ) − f ( y | b )  , where c ( α i ) is defined by (2.3) in type C ∞ and by (2.2) in t yp e B ∞ . These operators satisfy (2.23) T 2 i = β T i for i ∈ I , and T 0 T 1 T 0 T 1 = T 1 T 0 T 1 T 0 , T i T i +1 T i = T i +1 T i T i +1 for i ≥ 1 . (2.24) Theorem 2.6 ([16]) . L et x b e either p or q , and write g x λ for g p λ or g q λ ac c or d- ingly. Then for i ∈ I we have T i g x λ ( y | b ) =      g x s i λ ( y | b ) if s i w λ > w λ and s i w λ ∈ W 0 ∞ , 0 if s i w λ > w λ and s i w λ / ∈ W 0 ∞ , β g x λ ( y | b ) if s i w λ < w λ . Mor e over, for λ ∈ SP , we have g x λ ( y | b ) = T w λ (1) . Pr o of. The result is due to [16, Prop. 5.1] (see [6, App endix A] for a pro of ). □ 3. Free-Fermion formalism In this section, after recalling neutral fermions and the F o c k space, we review the results of Iwao [9]. In § 3.5, we express the factorial GP - and GQ -functions as v acuum exp ectation v alues. In § 3.6, we give a v acuum exp ectation v alue formula (Theorem 3.10) for the factorial g q -functions. As consequences, we obtain their generating function formula (Corollary 3.11) and Pfaffian formula in § 3.7. In § 3.11, we present a general expansion theorem (Theorem 3.15), whose pro of is giv en in § 3.18. In § 3.12, we state conjectures on the factorial g p -functions. 3.1. Neutral F ermions and F o ck spaces. In this section, we give a brief sum- mery of neutral fermions, F o c k mo dule, and v acuum exp ectation v alue. Let C b e the acco ciativ e Q ( β ) − algebra defined by the generators { ϕ n } n ∈ Z and the relations: [ ϕ m , ϕ n ] + = 2( − 1) m δ m + n, 0 , (3.1) where [ A, B ] + := AB + B A is the anti-c ommutator . Let F denote the left C –mo dule generated by the vector | v ac ⟩ suc h that (3.2) ϕ n | v ac ⟩ = 0 for n < 0 . The vector | v ac ⟩ is called the vacuum ve ctor . Let F † denote the right C –mo dule generated by the vector ⟨ 0 | , the dual vacuum ve ctor , such that (3.3) ⟨ v ac | ϕ n = 0 for n > 0 . F (resp. F † ) is called the F o ck mo dule (resp. dual F o ck mo dule ) of C . The F o ck space F decomp oses in to tw o subspaces F even and F odd , where F even (resp. F odd ) is generated by all v ectors obtained from | v ac ⟩ b y applying even (resp. o dd) num b ers NEUTRAL-FERMION CONSTRUCTIONS OF F ACTORIAL gp -AND g q -FUNCTIONS 9 of ϕ n ( n ≥ 0). There exists an an ti-algebra automorphism x 7→ x ∗ of C as Q ( β )- algebra such that ϕ ∗ n = ( − 1) n ϕ − n for n ∈ Z and ( xy ) ∗ = y ∗ x ∗ . There is a unique non-degenerate bilinear form ⟨ · | · ⟩ : F † × F → Q ( β ) suc h that ⟨ ua | v ⟩ = ⟨ u | av ⟩ for u ∈ F † , v ∈ F , a ∈ C , (3.4) ⟨ v ac | v ac ⟩ = 1 . (3.5) F or u ∈ F † and v ∈ F , we denote u , v by ⟨ u | , | v ⟩ resp ectiv ely . Moreov er, for a ∈ C , w e denote the v alue ⟨ ua | v ⟩ = ⟨ u | av ⟩ by ⟨ u | a | v ⟩ . In particular, ⟨ v ac | a | v ac ⟩ is called the vacuum exp e ctation value of a ∈ C . F or example, we ha v e (3.6) ⟨ v ac | ϕ m ϕ n | v ac ⟩ =      2( − 1) m δ m + n, 0 ( n > 0) δ m, 0 ( n = 0) 0 ( n < 0) If A is a sk ew-symmetric matrix of even size, we denote its Pfaffian by Pf ( A ). Prop osition 3.1 (Wick’s theorem, [14], [1]) . F or v 1 , . . . , v r ∈ L n ∈ Z Q ( β ) ψ n , ⟨ v ac | v 1 · · · v r | v ac ⟩ = ( Pf ( ⟨ v ac | v i v j | v ac ⟩ ) 1 ≤ i,j ≤ r if r is even, 0 if r is o dd. (3.7) The neutr al fermion field is defined as ϕ ( z ) = P n ∈ Z ϕ n z n , where z is a formal v ariable. W e hav e from (3.6) ⟨ v ac | ϕ ( z ) ϕ ( w ) | v ac ⟩ = X m,n ∈ Z ⟨ v ac | ϕ m ϕ n | v ac ⟩ z m w n = 1 + ∞ X n =1 2( − 1) n z − n w n = 1 − w /z 1 + w /z . 3.2. β deformed F ermions. W e define t wo β - deforme d fermion fields ϕ ( β ) ( z ) = P n ∈ Z ϕ ( β ) n z n and ϕ [ β ] ( z ) = P n ∈ Z ϕ [ β ] n z n via the follo wing expansions: ∞ X n =0 ϕ ( β ) n z n = ∞ X n =0 ϕ n  z + β 2  n , ∞ X n =1 ϕ ( β ) − n z − n = ∞ X n =1 ϕ − n z − 1 1 + β 2 z − 1 ! n , ∞ X n =1 ϕ [ β ] n z n = ∞ X n =1 ϕ n z 1 + β 2 z ! n , ∞ X n =0 ϕ [ β ] − n z − n = ∞ X n =0 ϕ − n  z − 1 + β 2  n . F rom the v acuum conditions (3.2), (3.3), ϕ [ β ] − n | v ac ⟩ = ϕ ( β ) − n | v ac ⟩ = 0 , ⟨ v ac | ϕ [ β ] n = ⟨ v ac | ϕ ( β ) n = 0 , for n > 0. (3.8) Then it follows from equation (3.1) that these op erators satisfy the anticomm utation relation [ ϕ ( β ) ∗ m , ϕ [ β ] n ] + = 2 δ m,n + β δ m,n − 1 . (3.9) 10 KOUSHIK BRAHMA, T AKESHI IKEDA, SHINSUKE IW AO, YI Y ANG In addition, ⟨ v ac | ϕ ( β ) ( u ) ϕ ( β ) ( w ) | v ac ⟩ = w − 1 − u − 1 w − 1 ⊕ u − 1 , (3.10) ⟨ v ac | ϕ [ β ] ∗ ( u ) ϕ [ β ] ∗ ( w ) | v ac ⟩ = w − u w ⊕ u , (3.11) where w − 1 − u − 1 w − 1 ⊕ u − 1 and w − u w ⊕ u are resp ectively expanded as 1 − w u − 1 1 − w u − 1 + β u − 1 = 1 − ( β + 2 w ) u − 1 + (2 w 2 + 3 β w + β 2 ) u − 2 + · · · , and 1 − uw − 1 1 + uw − 1 + β u = 1 − ( β + 2 w − 1 ) u + (2 w − 2 + 3 β w − 1 + β 2 ) u 2 + · · · . Define another t w o β - deforme d fermion fields as Φ ( β ) ( z ) := X n ∈ Z Φ ( β ) n z n = 1 2 + β z − 1 ϕ ( β ) ( z ) , (3.12) Φ [ β ] ( z ) := X n ∈ Z Φ [ β ] n z n = 1 2 + β z ϕ [ β ] ( z ) . (3.13) F or an y n ∈ Z , these β -deformed fermions are related to the neutral fermion through formal series expansions: Φ ( β ) n = 1 2 ϕ ( β ) n − β 4 ϕ ( β ) n +1 + β 2 8 ϕ ( β ) n +2 + · · · , Φ [ β ] n = 1 2 ϕ [ β ] n − β 4 ϕ [ β ] n − 1 + β 2 8 ϕ [ β ] n − 2 + · · · , Using (3.8), we ha v e the following annihilation rule Φ ( β ) ∗ n | v ac ⟩ = Φ [ β ] − n | v ac ⟩ = 0 , ⟨ v ac | Φ ( β ) n = 0 for n > 0. (3.14) These β -deformed fermions satisfies the follo wing commutation relations [Φ ( β ) ∗ n , ϕ [ β ] m ] + = [Φ [ β ] ∗ n , ϕ ( β ) m ] + = δ m,n . (3.15) 3.3. β -deformed Boson-F ermion corresp ondence. Let p n ( x ) = P k> 0 x n k b e the n -th p o w er sum in x = ( x 1 , x 2 , · · · ). The corresp onding β -deformed p o wer sums p ( β ) n and p [ β ] n are defined b y: p ( β ) n ( x ) = p n  x 1 + β / 2  = ∞ X i =0  − n i  ( β / 2) i p n + i ( x ) , p [ β ] n ( y ) = p n ( y + β / 2) − p n ( β / 2) = n X i =0  n i  ( β / 2) i p i ( y ) . F or eac h o dd integer n , we define the following operators acting on F and F † (3.16) b n = 1 4 X i ∈ Z ( − 1) i ϕ − i − m ϕ i . NEUTRAL-FERMION CONSTRUCTIONS OF F ACTORIAL gp -AND g q -FUNCTIONS 11 They satisfy following the Heisenberg relation, and the commutation relation with neutral F ermion: (3.17) [ b m , b n ] = m 2 δ m + n, 0 , [ b m , ϕ n ] = ϕ n − m . The β - deforme d Hamiltonian op er ators are defined as H ( β ) ( x ) = 2 X n =1 , 3 , 5 , ··· p ( β ) n ( x ) n b n , H [ β ] ( y ) = 2 X n =1 , 3 , 5 , ··· p [ β ] n ( y ) n b n . Define c G Γ( x ) := Q ( β )[ [ p ( β ) 1 ( x ) , p ( β ) 3 ( x ) , · · · ] ] , (3.18) g Γ( y ) := Q ( β )[ p [ β ] 1 ( y ) , p [ β ] 3 ( y ) , · · · ] . (3.19) There exists a p erfect pairing c G Γ( x ) × g Γ( y ) − → Q ( β ) suc h that for all o dd partitions λ and µ (3.20) ⟨ p ( β ) λ ( x ) , p [ β ] µ ( y ) ⟩ = 2 − ℓ ( λ ) δ λµ , F or odd partition λ , (3.21) p ( β ) λ ( x ) := p ( β ) λ 1 ( x ) p ( β ) λ 2 ( x ) · · · , p [ β ] λ ( y ) := p [ β ] λ 1 ( y ) p [ β ] λ 2 ( y ) · · · . W e define (3.22) ˆ F even = Hom Q ( β ) ( F † even , Q ( β )) . There is a linear isomorphism F † even → F even , ⟨ u | 7→ | u ∗ ⟩ given b y (3.23) | u ∗ ⟩ := a ∗ | v ac ⟩ whenever ⟨ u | = ⟨ v ac | a, a ∈ C . Lemma 3.2 (Boson-F ermion corresp ondence, [9], § 3.4) . Ther e exist Q ( β ) -line ar isomorphisms such that ˆ F even − → c G Γ( x ) , ⟨ u | 7→ ⟨ v ac | e H ( β ) ( x ) | u ∗ ⟩ , F even − → g Γ( y ) , | v ⟩ 7→ ⟨ v ac | e H [ β ] ( y ) | v ⟩ . Mor e over, via these isomorphisms, the natur al p airing ˆ F even × F even → Q ( β ) c or- r esp onds to the p airing (3.20) on c G Γ( x ) × g Γ( y ) → Q ( β ) . 3.4. F ree-F ermion formalism of gq , g p, GP , GQ — Non-equiv ariant case. W e correct the results of free-F erminon formalism for the non-equiv ariant case. Let θ = 2 X n =1 , 3 , 5 , ···  β 2  n b n n , θ ∗ = 2 X n =1 , 3 , 5 , ···  β 2  n b − n n . W e ha v e (3.24) e θ | v ac ⟩ = | v ac ⟩ , ⟨ v ac | e θ ∗ = ⟨ v ac | . Lemma 3.3. We have e θ ϕ [ β ] ( z ) e − θ = (1 + β z ) ϕ [ β ] ( z ) , (3.25) e θ ∗ ϕ ( β ) ( z ) e − θ ∗ = (1 + β z − 1 ) ϕ ( β ) ( z ) . (3.26) 12 KOUSHIK BRAHMA, T AKESHI IKEDA, SHINSUKE IW AO, YI Y ANG Pr o of. Equation (3.25) is obtained from [9, Prop osition 9.2] b y applying the anti- automorphism ∗ . Equation (3.26) is giv en in [9, Prop osition 4.5]. □ F or a strict partition λ of length r , w e define the following v ectors | λ ⟩ g q = ( ϕ [ β ] λ 1 e − θ ϕ [ β ] λ 2 e − θ · · · ϕ [ β ] λ r e − θ | v ac ⟩ , r : even ϕ [ β ] λ 1 e − θ ϕ [ β ] λ 2 e − θ · · · ϕ [ β ] λ r e − θ ϕ [ β ] 0 e − θ | v ac ⟩ , r : o dd | λ ⟩ GP = ( Φ ( β ) λ 1 e θ ∗ Φ ( β ) λ 2 e θ ∗ · · · Φ ( β ) λ r e θ ∗ | v ac ⟩ , r : even Φ ( β ) λ 1 e θ ∗ Φ ( β ) λ 2 e θ ∗ · · · Φ ( β ) λ r e θ ∗ ϕ ( β ) 0 e θ ∗ | v ac ⟩ . r : o dd | λ ⟩ g p = (Φ [ β ] λ 1 − 1 2 ( − β 2 ) λ 1 ) e − θ (Φ [ β ] λ 2 − 1 2 ( − β 2 ) λ 2 ) e − θ · · · (Φ [ β ] λ r − 1 2 ( − β 2 ) λ r ) e − θ ( ϕ 0 + 1) | v ac ⟩ , | λ ⟩ GQ = ( ϕ ( β ) λ 1 e θ ∗ ϕ ( β ) λ 2 e θ ∗ · · · ϕ ( β ) λ r e θ ∗ | v ac ⟩ r : even ϕ ( β ) λ 1 e θ ∗ ϕ ( β ) λ 2 e θ ∗ · · · ϕ ( β ) λ r e θ ∗ ϕ ( β ) 0 e θ ∗ | v ac ⟩ r : o dd Theorem 3.4. L et λ ∈ SP , we have GP λ ( x ) = ⟨ v ac | e H ( β ) ( x ) | λ ⟩ GP , GQ λ ( x ) = ⟨ v ac | e H [ β ] ( x ) | λ ⟩ GQ , (3.27) g p λ ( y ) = ⟨ v ac | e H [ β ] ( y ) | λ ⟩ g p , g q λ ( y ) = ⟨ v ac | e H [ β ] ( y ) | λ ⟩ g q . (3.28) The functions for the single row partitions can b e computed by the following generating functions. Prop osition 3.5 ([9]) . L et GP ( z ) = P m ∈ Z GP m ( x ) z m , GQ ( z ) = P m ∈ Z GQ m ( x ) z m , and g p ( z ) = ∞ P m =0 g p m ( y ) z m , g q ( z ) = ∞ P m =0 g q m ( y ) z m . Then we have GQ ( u − 1 ) = 1 (1 + β u ) ∞ Y j =1 u ⊕ x j u ⊖ x j , GP ( u − 1 ) = 1 2 + β u GQ ( u − 1 ) , (3.29) g q ( z ) = ∞ Y j =1 1 − y j ¯ z 1 − y j z , g p ( z ) = 1 2 + β z ( g q ( z ) + β z + 1) . (3.30) Note that for m > 0 we ha ve GQ − m ( x ) = ( − β ) m . Corollary 3.6. L et λ ∈ SP . We have GQ λ ( x ) = [ u − λ 1 1 . . . u − λ r r ] r Y i =1 GQ ( u − 1 i ) Y 1 ≤ i 1. W e define that Φ ( β )( k ) ( u ) = ϕ ( β )( k ) ( z ) = ϕ ( β ) ( z ) for all k ≤ 0. In the sp ecific case of k = 1, w e also allo w the notation Φ ( β )(1) ( z ) = Φ ( β ) ( z ) and ϕ ( β )(1) ( z ) = ϕ ( β ) ( z ). It can b e directly calculated that for k ≥ 1 Φ ( β )( k ) i = k − 1 Y j =1 1 1 + β b j k − 1 X j =0 ( − 1) j e j ( b 1 . . . , b k )Φ ( β ) i − j , (3.31) ϕ ( β )( k ) i = k − 1 Y j =1 1 1 + β b j k − 1 X j =0 ( − 1) j e j ( b 1 . . . , b k ) ϕ ( β ) i − j . (3.32) Lemma 3.7 (cf. [3], Lemma 5.19) . The factorial GP -function GP λ ( x | b ) and factorial GQ -function GQ λ ( x | b ) ar e given by the fol lowing formula GP λ ( x | b ) = [ u − λ 1 1 . . . u − λ r r ] r Y i =1 λ i − 1 Y j =1 GP ( u − 1 i ) u i − b j u i (1 + β b j ) Y 1 ≤ i 0, w e make the con v ention that ϕ [ β ]( k ) ( z ) = ϕ [ β ] ( z ) for k ≤ 0. It can b e directly calculated that ϕ [ β ]( k ) i = k − 1 Y ℓ =1 (1 + β b ℓ ) ∞ X j =0 h j ( b 1 . . . , b k ) ϕ [ β ] i + j . (3.39) F or any arbitrary strict partition λ = ( λ 1 . . . , λ r ), we denote r ′ b y the minimum ev en integer that is equal to or greater than r , if r ′  = r , let λ r ′ = 0, define | λ ⟩ ( g q,b ) := ϕ [ β ]( λ 1 ) λ 1 e − θ ϕ [ β ]( λ 2 ) λ 2 e − θ · · · ϕ [ β ]( λ r ′ ) λ r ′ e − θ | v ac ⟩ = ( ϕ [ β ]( λ 1 ) λ 1 e − θ ϕ [ β ]( λ 2 ) λ 2 e − θ · · · ϕ [ β ]( λ r ) λ r e − θ | v ac ⟩ , r : even ϕ [ β ]( λ 1 ) λ 1 e − θ ϕ [ β ]( λ 2 ) λ 2 e − θ · · · ϕ [ β ]( λ r ) λ r e − θ ϕ [ β ] 0 e − θ | v ac ⟩ . r : o dd The pro of of the following key lemma will b e given in the next subsection. Lemma 3.9 (Duality Lemma) . F or any arbitr ary strictly p artitions λ and µ , ( GP,b ) ⟨ µ | λ ⟩ ( g q,b ) = δ µλ , (3.40) wher e ( GP,b ) ⟨ µ | = ( | µ ⟩ ( GP,b ) ) ∗ . NEUTRAL-FERMION CONSTRUCTIONS OF F ACTORIAL gp -AND g q -FUNCTIONS 15 Theorem 3.10. F or λ ∈ SP , we have g q λ ( y | b ) = ⟨ v ac | e H [ β ] ( y ) | λ ⟩ ( g q,b ) . Pr o of. F rom Lemma 3.2, Lemma 3.9, and Theorem 3.8 w e deduce that the family of functions {⟨ v ac | e H [ β ] ( y ) | λ ⟩ ( g q,b ) | λ ∈ SP } is dual to { GP λ ( x | b ) | λ ∈ SP } . □ Corollary 3.11. The factorial g q -functions ar e given by the fol lowing formula (3.41) g q λ ( y | b ) = [ z λ 1 1 . . . z λ r r ] r Y i =1 g q ( λ i ) ( z i ) Y 1 ≤ i · · · > λ r > 0) with r even, we have (3.42) Pf ( G i,j λ i ,λ j ) = [ u − λ 1 1 · · · u − λ r r ] r Y i =1 G i ( u i ) Y 1 ≤ i 0 and n, d ≥ 0 . Then we have (3.52) [ u m + d w n ] (1 + β w ) A (1 + β u − 1 ) B Q m − 1 k =1 (1 − uc k ) Q n k =1 (1 − w − 1 b k ) uw 1 − uw ! = X l ≥ 0  A + B l  β l h l − n + m + d ( b 1 , . . . , b n ; − c 1 , . . . , − c m − 1 ) wher e h p ( b 1 , . . . , b n ; − c 1 , . . . , − c m − 1 ) = 0 if p < 0 and  − n l  = ( − 1) l  n + l − 1 l  . Pr o of. Let I be the expression on the left-hand side in (3.52). Then w e hav e I = 1 (2 π i ) 2 I I | b | , | c | < | w | < | u | − 1 < | β | − 1 (1 + β w ) A (1 + β u − 1 ) B Q m − 1 k =1 (1 − uc k ) Q n k =1 (1 − w − 1 b k ) uw 1 − uw dudw u m + d +1 w n +1 t = u − 1 = − 1 (2 π i ) 2 I I | b | , | c | < | w | < | t | < | β | − 1 (1 + β w ) A (1 + β t ) B Q m − 1 k =1 (1 − t − 1 c k ) Q n k =1 (1 − w − 1 b k ) w t − w d ( t − 1 ) dw t − m − d − 1 w n +1 d ( t − 1 )= − t − 2 dt = 1 (2 π i ) 2 I I | b | , | c | < | w | < | t | < | β | − 1 (1 + β w ) A (1 + β t ) B t d Q m − 1 k =1 ( t − c k ) Q n k =1 ( w − b k ) dtdw t − w . Applying the Residue theorem for the contour integral H dt , we obtain I = 1 2 π i I | b | , | c | < | w | < | β | − 1 (1 + β w ) A + B w d Q m − 1 k =1 ( w − c k ) Q n k =1 ( w − b k ) dw . 18 KOUSHIK BRAHMA, T AKESHI IKEDA, SHINSUKE IW AO, YI Y ANG Expanding the in tegrand ov er the domain {| b | , | c | < | w | < | β | − 1 } , we obtain I = 1 2 π i I | b | , | c | < | w | < | β | − 1 " (1 + β w ) A + B w d ( ∞ X i =0 ( − 1) i h i ( b 1 , . . . , b n ; − c 1 , . . . , − c m − 1 ) w − n + m − i − 1 )# dw = X l ≥ 0  A + B l  β l h l − n + m + d ( b 1 , . . . , b n ; − c 1 , . . . , − c m − 1 ) . □ 3.10. Pro of on General expansion form ulas. Lemma 3.18. F or n > 0 and m ∈ Z , we have ⟨ v ac | e − θ ∗ ϕ [ β ] ∗ m Φ ( β ) n e θ ∗ | v ac ⟩ = δ n,m . Pr o of. In fact for n > 0, it follo ws from (3.8), (3.14), (3.15) and Lemma 3.3 that ⟨ v ac | e − θ ∗ ϕ [ β ] ∗ 0 Φ ( β ) n e θ ∗ | v ac ⟩ = ⟨ v ac | e − θ ∗ ϕ [ β ] ∗ 0 e θ ∗ e − θ ∗ Φ ( β ) n e θ ∗ | v ac ⟩ = ⟨ v ac | ( ϕ [ β ] ∗ 0 + β ϕ [ β ] ∗ − 1 )(Φ ( β ) n + β Φ ( β ) n +1 + β 2 Φ ( β ) n +2 + · · · ) | v ac ⟩ = 0 . If we imp ose m  = 0, then ⟨ v ac | e − θ ∗ ϕ [ β ] ∗ m Φ ( β ) n e θ ∗ | v ac ⟩ = ⟨ v ac | ϕ [ β ] ∗ m Φ ( β ) n e θ ∗ | v ac ⟩ δ m> 0 = δ n,m δ m> 0 − ⟨ v ac | Φ ( β ) n ϕ [ β ] ∗ m e θ ∗ | v ac ⟩ δ m> 0 = δ n,m δ m> 0 . □ Lemma 3.19. F or m ∈ Z , we have ⟨ v ac | e − θ ∗ ϕ [ β ] ∗ m ϕ ( β ) 0 e θ ∗ | v ac ⟩ = δ m, 0 . Pr o of. F or m < 0, the lemma can ob viously b e derived. If m > 1, b y using (3.8), (3.9) and Lemma 3.3, w e can obtain ⟨ v ac | e − θ ∗ ϕ [ β ] ∗ m ϕ ( β ) 0 e θ ∗ | v ac ⟩ = −⟨ v ac | ϕ ( β ) 0 ϕ [ β ] ∗ m e θ ∗ | v ac ⟩ = −⟨ v ac | ϕ ( β ) 0 e θ ∗ ( ϕ [ β ] ∗ m + β ϕ [ β ] ∗ m − 1 ) | v ac ⟩ = 0 . If m = 0, ⟨ v ac | e − θ ∗ ϕ [ β ] ∗ 0 ϕ ( β ) 0 e θ ∗ | v ac ⟩ = ⟨ v ac | ( ϕ [ β ] ∗ 0 + β ϕ [ β ] ∗ − 1 )( ϕ ( β ) 0 + β ϕ ( β ) 1 + β 2 ϕ ( β ) 2 + · · · ) | v ac ⟩ = ⟨ v ac | ϕ [ β ] ∗ 0 ϕ ( β ) 0 | v ac ⟩ = ⟨ v ac | ϕ ∗ 0 ϕ 0 | v ac ⟩ = 1 . If m = 1, it follows from (3.8), (3.9) that ⟨ v ac | e − θ ∗ ϕ [ β ] ∗ 1 ϕ ( β ) 0 e θ ∗ | v ac ⟩ = β − ⟨ v ac | e − θ ∗ ϕ ( β ) 0 ϕ [ β ] ∗ 1 e θ ∗ | v ac ⟩ = β − ⟨ v ac | ( ϕ ( β ) 0 + β ϕ ( β ) 1 + β 2 ϕ ( β ) 2 + · · · )( ϕ [ β ] ∗ 1 + β ϕ [ β ] ∗ 0 ) | v ac ⟩ = β − β ⟨ v ac | ϕ ( β ) 0 ϕ [ β ] ∗ 0 | v ac ⟩ = 0 . □ Let λ ∈ SP . Define (3.53) d λ ( b ) := ℓ ( λ ) Y i =1 λ i − 1 Y l =1 (1 + β b l ) . NEUTRAL-FERMION CONSTRUCTIONS OF F ACTORIAL gp -AND g q -FUNCTIONS 19 Prop osition 3.20. L et λ, µ b e strict p artitions. L et c = ( c 1 , c 2 , . . . ) a p ar ameter se quenc e. Then ( GP,c ) ⟨ µ | λ ⟩ ( g q,b ) is non-zer o only if λ, µ have the same length, say r . If this holds we have ( GP,c ) ⟨ µ | λ ⟩ ( g q,b ) = d λ ( b ) d µ ( c ) − 1 det( D ) , (3.54) wher e D is an r × r matrix define d by (3.55) D ij = ∞ X k =0 β k  i − j k  h µ i − λ j + k ( b 1 , . . . , b λ j ; − c 1 , . . . , − c µ i − 1 ) , wher e  − n k  denotes the gener alize d binomial c o efficients given by ( − 1) k  n + k − 1 k  for n > 0 and k ≥ 0 , Pr o of. First we assume that the length r of the strictly partition µ is even. It follo ws from the definition of | µ ⟩ ( GP,b ) that ( GP,c ) ⟨ µ | = ( | µ ⟩ ( GP,c ) ) ∗ = ⟨ v ac | e θ Φ ( β ) ∗ ( µ s ) µ r · · · e θ Φ ( β ) ∗ ( µ 1 ) µ 1 where we simply denote Φ ( β ) ∗ ( µ i ) µ i the equiv ariantly deformed F ermion with equi- v ariant parameter c = ( c 1 , c 2 , . . . ) . Let us denote u µ w λ = u µ 1 1 · · · u µ r r w λ 1 1 · · · w λ s ′ s ′ . W e hav e ( GP,c ) ⟨ µ | λ ⟩ ( g q,b ) = [ u µ w λ ] ⟨ v ac | e θ Φ ( β ) ∗ ( µ r ) ( u r ) · · · e θ Φ ( β ) ∗ ( µ 1 ) ( u 1 ) ϕ [ β ]( λ j ) ( w 1 ) e − θ · · · ϕ [ β ]( λ s ′ ) ( w s ′ ) e − θ | v ac ⟩ = [ u µ w λ ] ⟨ v ac | ← Y 1 ≤ i ≤ r e ( r +1 − i ) θ Φ ( β ) ∗ ( µ i ) ( u i ) e − ( r +1 − i ) θ → Y 1 ≤ j ≤ s ′ e ( s ′ +1 − j ) θ ϕ [ β ]( λ j ) ( w j ) e − ( s ′ +1 − j ) θ | v ac ⟩ = [ u µ w λ ] s ′ Q j =1  1 + β w j  s ′ +1 − j r Q i =1  1 + β u − 1 i  r +1 − i ⟨ v ac | ← Y 1 ≤ i ≤ r Φ ( β ) ∗ ( µ i ) ( u i ) → Y 1 ≤ j ≤ s ′ ϕ [ β ]( λ j ) ( w j ) | v ac ⟩ , (3.56) where we used Lemma 3.3 in the last equality . By Wick’s theorem, the v acuum exp ectation v alue is zero unless s ′ = r . So now we set s ′ = r. Then the v acuum exp ectation v alues in (3.56) is written as d λ ( b ) d µ ( c ) [ u µ w λ ] r Y i =1 Q µ i − 1 k =1 (1 − u i c k ) Q λ j k =1 (1 − w − 1 j b k ) ⟨ v ac | ← Y 1 ≤ i ≤ r Φ ( β ) ∗ ( µ i ) ( u i ) → Y 1 ≤ j ≤ s ′ ϕ [ β ]( λ j ) ( w j ) | v ac ⟩ . Since the rational function in this expression is a pro duct form suc h that all v ariables are separated, the function (3.56) is, up to scaler d λ ( b ) d µ ( c ) − 1 , equal to the Pfaffian of a 2 r × 2 r skew-symmetric matrix such that each entry in volving only a pair of v ariables in u r , . . . , u 1 , w 1 , . . . , w r . Consider the factor inv olving u i and u j : (3.57) [ u µ i i u µ j j ] f i ( u i ) f j ( u j ) ⟨ v ac | Φ ( β ) ∗ ( u i )Φ ( β ) ∗ ( u j ) | v ac ⟩ for 1 ≤ i, j ≤ r , where f i ( u ) = Q µ i − 1 k =1 (1 − uc k ) (1 + β u − 1 ) r +1 − i . 20 KOUSHIK BRAHMA, T AKESHI IKEDA, SHINSUKE IW AO, YI Y ANG Since f i ( u i ) is a series in u i of the highest degree µ i − 1, and the v accum exp ectation v alue in (3.57) do es not contain terms that could increase the p ow er of u i and u j b y (3.14), thus the co efficient extraction [ u µ i i u µ j j ] yields zero. Hence the Pfaffian reduces to a determinant, and we hav e (3.58) ( GP,c ) ⟨ µ | λ ⟩ ( g q,b ) = d λ ( b ) d µ ( c ) − 1 det( D ) , where (3.59) D ij = [ u µ i i w λ j j ] (1 + β w j ) r +1 − j (1 + β u − 1 i ) r +1 − i Q µ i − 1 k =1 (1 − u i c k ) Q λ j k =1 (1 − w − 1 j b k ) ⟨ v ac | Φ ( β ) ∗ ( u i ) ϕ [ β ] ( w j ) | v ac ⟩ . W e compute by using (3.18) ⟨ v ac | Φ ( β ) ∗ ( u i ) ϕ [ β ] ( w j ) | v ac ⟩ = 1 + β u − 1 i 1 + β w j ⟨ v ac | e θ Φ ( β ) ∗ ( u i ) ϕ [ β ] ( w j ) e − θ | v ac ⟩ = 1 + β u − 1 i 1 + β w j u i w j 1 − u i w j . Th us we hav e (3.60) D ij = [ u µ i i w λ j j ] (1 + β w j ) r − j (1 + β u − 1 i ) r − i Q µ i − 1 k =1 (1 − u i c k ) Q λ j k =1 (1 − w − 1 j b k ) u i w j 1 − u i w j . W e show the inner pro duct ( GP,c ) ⟨ µ | λ ⟩ ( g q,b ) v anishes unless s = r . In fact, if the length s of λ is o dd, that is, s ′ = s + 1 = r and λ s ′ = 0. Then we hav e D ir = [ u µ i i w 0 r ] Q µ i − 1 k =1 (1 − u i c k ) (1 + β u − 1 i ) r − i u i w r 1 − u i w r = 0 for 1 ≤ i ≤ r because the constant term of w r in the series is zero, and hence det D = 0 . No w we consider the case when the length r of µ is o dd, so r ′ = r + 1 and µ r +1 = 0. The pro of pro ceeds similarly , but now r + 1 = s ′ b y Wick’s theorem. W e in tro duce a matrix A = ( A ij ) 1 ≤ i,j ≤ r +1 b y A ij = ⟨ v ac | Φ ( β ) ∗ ( µ i ) ( u i ) ϕ [ β ]( λ j ) ( w j ) | v ac ⟩ for 1 ≤ i ≤ r and 1 ≤ j ≤ r + 1 , A r +1 ,j = ⟨ v ac | ϕ ( β ) ∗ ( u r +1 ) ϕ [ β ]( λ j ) ( w j ) | v ac ⟩ for 1 ≤ j ≤ r + 1 . Then we ha ve ( GP,c ) ⟨ µ | λ ⟩ ( g q,b ) = d λ ( b ) d µ ( c ) − 1 det( e D ) , where e D is a ( r + 1) × ( r + 1) matrix with entries e D ij = [ u µ i i w λ j j ] (1 + β w j ) r +2 − j  1 + β u − 1 i  r +2 − i µ i − 1 Q k =1 (1 − u i c k ) λ j Q k =1 (1 − w − 1 j b k ) A ij . F or 1 ≤ j ≤ r , it follows from Lemma 3.18 that e D r +1 ,j = 0. If λ s +1 = λ r ′ > 0, then in this case e D r +1 ,r +1 = 0 again from Lemma 3.18. This leads to NEUTRAL-FERMION CONSTRUCTIONS OF F ACTORIAL gp -AND g q -FUNCTIONS 21 ( GP,c ) ⟨ µ | λ ⟩ ( g q,b ) = 0. Th us if we wan t ( GP,c ) ⟨ µ | λ ⟩ ( g q,b ) to b e non-zero, it is neces- sary to require λ r +1 = λ s ′ = 0. So s is o dd and r = ℓ ( µ ) = ℓ ( λ ) = s . Then we hav e e D r +1 ,r +1 = 1. Now the inner pro duct then simplifies to ( GP,c ) ⟨ µ | λ ⟩ ( g q,b ) = d λ ( b ) d µ ( c ) − 1 det( D ) , where D is the r × r submatrix of ˜ D with 1 ≤ i, j ≤ r . F rom Lemma 3.17, one sees that the entries of D is given by the same formula (3.55) as the case of r is ev en. □ 3.11. Pro of of the Duality Lemma 3.10. Lemma 3.21. We have ( GP,c ) ⟨ µ | λ ⟩ ( g q,b ) = 0 unless λ ⊂ µ . Pr o of. W e know equations (3.54), (3.55) hold. Suppose there exists an index a with 1 ≤ a ≤ r such that λ a > µ a . F or p = 0 , . . . , r − a and q = 0 , . . . , a − 1, we hav e the inequalities: µ a + p ≤ µ a − p < λ a − p ≤ λ a − q − p − q . No w let us consider the summation index k in matrix entry D a + p,a − q giv en by (3.55). Giv en that a + p ≥ a − q , the binomial co efficients in (3.55) v anish unless k ≤ ( a + p ) − ( a − q ) = p + q , and thus µ a + p − λ a − q + k ≤ µ a + p − λ a − q + p + q < 0 . This implies h µ a + p − λ a − q + k = 0 for all admissible k , so D a + p,a − q = 0. These v anishing entries form a zero blo ck in rows a to r and columns 1 to a , rendering D singular. Thus det( D ) = 0 and ( GP,c ) ⟨ µ | λ ⟩ ( g q,b ) = 0, completing the pro of. □ Pr o of of L emma 3.9. By using Theorem 3.20, w e can assume r = s and then ( GP,b ) ⟨ µ | λ ⟩ ( g q,b ) = det( D | c = b ) , where D | c = b is an r × r matrix giv en by D ij | c = b = ∞ X k =0 β k  i − j k  h µ i − λ j + k ( b 1 , . . . , b λ j ; − b 1 , . . . , − b µ i − 1 ) =      P ∞ k =0 β k  i − j k  h µ i − λ j + k ( b µ i , · · · , b λ j ) if µ i ≤ λ j 0 if µ i = λ j + 1 , P ∞ k =0 β k  i − j k  e µ i − λ j + k ( − b λ j +1 , · · · , − b µ i − 1 ) , if µ i > λ j + 1 . (3.61) By Lemma 3.21, we can assume λ i ≤ µ i for 1 ≤ i ≤ r. W e will show D ii | c = b = δ λ i µ i for 1 ≤ i ≤ r. If µ i = λ i w e ha ve D ii | c = b = 1 from (3.61). Suppose λ i < µ i . If µ i = λ i + 1 then D ii = 0 again from (3.61). If µ i > λ i + 1 we hav e from (3.61) D ii | c = b = e µ i − λ i ( − b λ i +1 , · · · , − b µ i − 1 ) = 0 b ecause the num ber of v ariables is µ i − λ i − 1 . Next we will show that D | c = b is upp er triangular. In fact, if i < j w e hav e µ i > µ j ≥ λ j so D ij | c = b = ∞ X k =0 β k  i − j k  e µ i − λ j + k ( − b λ j +1 , · · · , − b µ i − 1 ) = 0 b ecause the num ber of v ariables is µ i − λ j − 1 . Therefore w e hav e det D | c = b = δ λ 1 µ 1 · · · δ λ r µ r = δ λµ , as desired. □ 22 KOUSHIK BRAHMA, T AKESHI IKEDA, SHINSUKE IW AO, YI Y ANG 3.12. Conjecture on factorial g p -functions. A t the end of this section, w e state conjectures on the factorial g p -functions. Conjecture 3.22. L et Φ [ β ]( k ) i := k − 1 Y ℓ =1 (1 + β b ℓ ) ∞ X j =0 h j ( b 1 . . . , b k )  Φ [ β ] i + j − 1 2 ( − β 2 ) i + j  . | λ ⟩ g p,b := Φ [ β ]( λ 1 ) λ 1 e − θ Φ [ β ]( λ 2 ) λ 2 e − θ · · · Φ [ β ]( λ r ) λ r ( ϕ 0 + 1) | v ac ⟩ . Then we have g p λ ( y | b ) = ⟨ v ac | e H [ β ] ( y ) | λ ⟩ ( g p,b ) . In order to prov e this conjecture it suffices to prov e Conjecture 3.23. F or strict p artitions λ, µ , we have ( GQ,b ) ⟨ µ | λ ⟩ p,b = δ µ,λ , ( GQ,b ) ⟨ µ | := ( | µ ⟩ ( GQ,b ) ) ∗ . This leads to a formula of g p λ ( x | b ) as a sum of Pfaffians similar to Corollary 3.6. 4. Coefficients in terms of double Grothendieck pol ynomials In this section, we give form ulas expressing the general expansion co efficien ts in terms of Grothendieck p olynomials of type A. W e also discuss coincidences among these expansion coe fficien ts (Corollary 4.11). 4.1. Double Grothendiec k p olynomials. F or each p erm utation w ∈ S ∞ = S ∞ n =1 S n , there is a double Gr othendie ck p olynomial G w ( b ; c ) due to Lascoux and Sc h ¨ utzen b erger [11] defined as follows. Define the operators π i for i ≥ 1 on Z [ β ][ b ; c ] := Z [ β ][ b 1 , b 2 , . . . ][ c 1 , c 2 , . . . ] by π i = ∂ i ◦ (1 + β b i +1 ) , ∂ i := 1 − s ( b ) i b i − b i +1 , where s i exc hanges b i and b i +1 . Let w ( n ) 0 denote the longest element of S n . There is a unique family { G w ( b ; c ) | w ∈ S ∞ } of p olynomials such that for n ≥ 1 G w ( n ) 0 ( b ; c ) = Y i + j ≤ n b i ⊖ c j . and (4.1) π i G w ( b ; c ) = ( G ws i ( b ; c ) if ℓ ( w s i ) < w , − β G w ( b ; c ) if ℓ ( w s i ) > w . W e call the polynomial G w ( b ; c ) by double Grothendieck p olynomial. R emark 4.1 . In the existing literature G w ( n ) 0 ( b ; c ) is defined as the product of b i ⊕ c j instead of b i ⊖ c j . In this setting, our definition coincide with G w ( b, ⊖ c ). W e use this sign con v en tion to b e consistent with the factorial Grothendiec k p olynomial defined in (2.9). Our conv en tion is consisten t with [12]. W e define G w ( b ) := G w ( b ; 0) for w ∈ S ∞ . Then G w ( b ) are p olynomials in Z [ β ][ b 1 , b 2 , . . . ] known as Grothendieck p olynomial. NEUTRAL-FERMION CONSTRUCTIONS OF F ACTORIAL gp -AND g q -FUNCTIONS 23 Prop osition 4.2. [12, Prop osition 5.10] G w ( b ; c ) = G w − 1 ( ⊖ c ; ⊖ b ) . In p articular G w (0; c ) = G w − 1 ( ⊖ c ; 0) = G w − 1 ( ⊖ c ) . R emark 4.3 . F or i ≥ 1 and f ∈ Z [ β ][ b ], we hav e π i f = 1 c ( α i ) ( s i f − (1 + β c ( α i )) f ) , where c ( α i ) = b i +1 ⊖ b i . The right hand side has the same form of the op erator T i − β , where T i is defined in (2.22). Th us T i = π i + β . W e use this fact to prov e Theorem 4.10. F or a p ositiv e integer r , a p ermutation w ∈ S ∞ is called r - Gr assmannian if it satisfies w (1) < · · · < w ( r ) , w ( r + 1) < w ( r + 2) < · · · . Note that such a permutation w is determined by the v alues w (1) , . . . , w ( r ) . Let λ = ( λ 1 , λ 2 , . . . , λ r ) b e a partition of length at most r. Then there exists a unique r -Grassmannian p erm utation w ( r ) λ suc h that (4.2) w ( r ) λ ( i ) − i = λ r − i +1 (1 ≤ i ≤ r ) . Let λ = ( λ 1 , . . . , λ r ) b e a partition of length at most r . W e define ρ ( r ) λ i := s r − i + λ i · · · s r − i +2 s r − i +1 if λ i ≥ 1 . If λ i = 0 we define ρ ( r ) λ i := e then (4.3) w ( r ) λ = ρ ( r ) λ r · · · ρ ( r ) λ 1 . Example 4.4 . Let λ = (3 , 1 , 0) b e a partition of length 3. Consider the following tableau is for r = 3. s 3 s 4 s 5 · · · s 2 s 3 s 4 · · · s 1 s 2 s 3 · · · The 3-Grassmannian p erm utation corresp onding to λ is given by w (3) λ = ρ (3) 0 ρ (3) 1 ρ (3) 3 = ( s 2 )( s 5 s 4 s 3 ) . The following fact is w ell-kno wn (cf. [18], [5]). Prop osition 4.5. F or any p artition λ of length at most r , we have (4.4) G w ( r ) λ ( b ; c ) = G λ ( b 1 , . . . , b r | c ) = det([ b i | c ] λ j + r − j (1 + β b i ) j − 1 ) Q 1 ≤ i δ r ) β G w µ w − 1 λ s i ( b ; c ) , wher e i ∈ { 1 , 2 , . . . } is the c ontent of the rightmost b ox in the b ottom r ow in the shifte d skew diagr am λ \ δ r . Pr o of. W e address the pro of for the co efficien t C g p λµ ( b, c ). The argument pro ceeds b y induction on the length of w λ w − 1 δ r . F or λ = δ r using Lemma 4.7 (4.10) g p δ r ( y | b ) = X µ ∈ SP ( r ) G w µ w − 1 δ r ( b ; c ) g p µ ( y | c ) . Consider the strict partition ξ r of length r b y ξ r := ( r + 1 , r − 1 , r − 2 , . . . , 2 , 1). Note that ξ r is the unique strict partition of size | δ r | + 1 containing δ r . Thus w ξ r = s r w δ r > w δ r . F or any ξ r ⊆ µ we hav e w µ w − 1 δ r s r = w µ w − 1 ξ r < w µ w − 1 δ r . Apply T r on b oth sides of (4.10) and using Theorem 2.6 we ha v e g p ξ r ( y | b ) = X µ ∈ SP ( r ) T r  G w µ w − 1 δ r ( b ; c )  g p µ ( y | c ) . If µ = δ r then T r ( G w µ w − 1 δ r ( b ; c )) = T r ( G e ( b ; c )) = 0. F or µ ⊃ ξ r , using (4.1) we ha ve T r ( G w µ w − 1 δ r ( b ; c )) = ( π r + β )( G w µ w − 1 δ r ( b ; c )) = G w µ w − 1 ξ r ( b ; c ) + β G w µ w − 1 δ r ( b ; c ) . Th us g p ξ r ( y | b ) = X µ : ξ r ⊂ µ ; ℓ ( µ )= r  G w µ w − 1 ξ r ( b ; c ) + β G w µ w − 1 δ r ( b ; c )  g p µ ( y | c ) . Th us (4.9) holds for λ = ξ r . Let s i b e the righ tmost b o x in the b ottom row in λ \ δ r and η b e obtained from λ by remo ving b ox s i . By the induction hypothesis (4.11) g p η ( y | b ) = X µ : η ⊂ µ ; ℓ ( µ )= ℓ ( η )  G w µ w − 1 η ( b ; c ) + β G w µ w − 1 η s j ( b ; c )  g p µ ( y | c ) , for some j. One sees j  = i . Note that | λ | = 1 + | η | and w λ = s i w η > w η . 26 KOUSHIK BRAHMA, T AKESHI IKEDA, SHINSUKE IW AO, YI Y ANG W e apply T i to the b oth hand sides of (4.11). F or the left hand side, T i ( g p η ( y | b )) = g p λ ( y | b ) using Theorem 4.10. F or the right hand side, the co efficient of g p η ( y | c ) v anishes b ecause T i G s j ( b ; c ) = 0 . Also, for an y µ such that λ ⊂ µ w e ha v e w µ w − 1 λ = w µ w − 1 η s i < w µ w − 1 η . Therefore using (4.1) T i  G w µ w − 1 η ( b ; c ) + β G w µ w − 1 η s j ( b ; c )  = T i  G w µ w − 1 η ( b ; c )  + T i  β G w µ w − 1 η s j ( b ; c )  =  G w µ w − 1 λ ( b ; c ) + β G w µ w − 1 λ s i ( b ; c )  +  ( − β ) β G w µ w − 1 η s j ( b ; c ) + β 2 G w µ w − 1 η s j ( b ; c )  =  G w µ w − 1 λ ( b ; c ) + β G w µ w − 1 λ s i ( b ; c )  . Th us applying T i on b oth sides of (4.11), and we ha ve our desired result. The argumen t for the co efficient C g q λµ ( b, c ) pro ceeds in the same manner using the Lemma 4.8. One needs to observe that for all i ≥ 1, the operators T i in types B ∞ and C ∞ coincide, and the pro of follows accordingly . □ Corollary 4.11. Consider λ, µ ∈ SP ( r ) such λ ⊂ µ . Then C g p λµ ( b, c ) = C g q λµ ( b, c ) = C GP µλ ( c, b ) = C GQ µλ ( c, b ) . R emark 4.12 . Let λ, µ ∈ SP ( r ) and δ r = ( r, r − 1 , . . . , 1). F or µ ⊂ λ , using Prop osition 4.2 we hav e C GP λµ ( b, c ) = C GQ λµ ( b, c ) = G w µ w − 1 λ ( ⊖ b ; ⊖ c ) + χ ( µ > δ r ) β G s i w µ w − 1 λ ( ⊖ b ; ⊖ c ) , where i ∈ { 1 , 2 , . . . } is the conten t of the rightmost b o x in the b ottom row in the shifted skew tableau µ \ δ r . Corollary 4.13. If η is a p artition of length less than or e qual to r . If G η ( x 1 , . . . , x r | b ) = X ν : ν ⊆ η C η ν ( b, c ) G ν ( x 1 , . . . , x r | c ) , then C η ν ( b, c ) =  G w ( r ) ν ( w ( r ) η ) − 1 ( ⊖ b ; ⊖ c ) + ξ ( ν > ∅ ) β G s i w ( r ) ν ( w ( r ) η ) − 1 ( ⊖ b ; ⊖ c )  , wher e i ∈ { 1 , 2 , . . . } is the c ontent of the rightmost b ox in the b ottom r ow in the table au ν . The expansion of double Grothendiec k p olynomials in terms of Grothendieck p olynomials is know due to [12, Prop osition 5.8]. Corollary 4.14. L et λ = ( λ 1 , . . . , λ r ) ∈ SP ( r ) and δ r = ( r , r − 1 , . . . , 1) . Then the c o efficients in (3.50) and (3.51) ar e given by the fol lowing: C g p λµ ( b ) = C g q λµ ( b ) = G w µ w − 1 λ ( b ) + χ ( λ > δ r ) β G w µ w − 1 λ s i ( b ) , wher e i ∈ { 1 , 2 , . . . } is the c ontent of the rightmost b ox in the b ottom r ow in the shifte d skew table au λ \ δ r . Mor e over, C GP λµ ( b ) = C GQ λµ ( b ) = G w µ w − 1 λ ( ⊖ b ) + χ ( µ > δ r ) β G s j w µ w − 1 λ ( ⊖ b ) , wher e j ∈ { 1 , 2 , . . . } is the c ontent of the rightmost b ox in the b ottom r ow in the shifte d skew table au µ \ δ r . NEUTRAL-FERMION CONSTRUCTIONS OF F ACTORIAL gp -AND g q -FUNCTIONS 27 Comparing the co efficients in the expansion form ulae in Lemma 4.7 and Theorem 3.15 we get a Jacobi-T rudi type formulae of the factorial Grothendieck p olynomial G η ( b 1 , . . . , b r | c ) defined in (2.9), for any partition η of length less than or equal to r . Corollary 4.15. L et η b e a p artition of length less than or e qual to r . 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