aa r X i v : . [ m a t h . QA ] N ov A NEW REALISATION OF THE i -QUANTUM GROUP U ( n ) JIE DU AND YADI WU
Abstract.
We follow the approach developed in [
BLM90 ] and modified in [
DF10 ] to in-vestigate a new realisation for the i -quantum groups U ( n ), building on the multiplicationformulas discovered in [ BKLW18 , Lem. 3.2]. This allows us to present U ( n ) via a basis andmultiplication formulas by generators. We also establish a surjective algebra homomorphismfrom a Lusztig type form of U ( n ) to integral q -Schur algebras of type B . Thus, base changesallow us to relate representations of the i -quantum hyperalgebras of U ( n ) to representationsof finite orthogonal groups of odd degree in non-defining characteristics. This generalises partof Dipper–James’ type A theory to the type B case. Contents
1. Introduction 12. The i -quantum group U ( n ): a first realisation 33. The q -Schur algebra of type B
54. Some multiplication formulas 85. The subalgebra A ( n ) 136. Lifting Bao–Wang’s Schur duality to the integral level 167. A new realisation of U ( n ) 20References 251. Introduction
Arising from representations of finite groups of Lie type, Iwahori–Hecke algebras play animportant role in the study of unipotent principal blocks. In late 1980s, Dipper–James [
DJ89 ]introduced q -Schur algebras to study the representations of finite general linear groups innon-defining characteristics. In an entirely different context, these algebras appeared earlierin the study of the Schur–Weyl duality for a quantum linear group or quantum gl n . Thus, q -Schur algebras naturally link representations of quantum gl n with those of finite generallinear groups. It is natural to expect that such a connection extends to finite orthogonaland symplectic groups. However, example calculations on characters showed that this is notthe case. In fact, from the invariant theory point of view, it is the Brauer algebra (or theBMW-algebra in the quantum case), instead of a group (or Hecke) algebra, that is involvedin the Schur–Weyl duality. Date : November 6, 2020.2010
Mathematics Subject Classification.
Key words and phrases. quantum group, q -Schur algebra, Hecke algebra, Schur–Weyl duality, quantum hy-peralgebra, finite orthogonal group.The paper was written while the second author was visiting the University of New South Wales as a practicumstudent for a year. She would like to thank UNSW for the hospitality and the China Scholarship Council forthe financial support. Recently, in their study of canonical bases for quantum symmetric pairs, Bao and Wang[
BW18 ] introduced certain co-ideal subalgebras U and U ı of quantum linear groups, whosecorresponding quantum symmetric pairs in [ Le03 ] are of type AIII. We follow [
CLW ] to callthem i -quantum groups. Bao–Wang further proved that these i -quantum groups pair withthe Hecke algebras of type B in a Schur–Weyl duality, where two types of Hecke endomor-phism algebras or q -Schur algebras play the bridging role. More interestingly, as revealedin [ BKLW18 ], these q -Schur algebras arise naturally from finite orthogonal groups of odd de-gree or finite symplectic groups.With an entirely different motivation, the structure and representations of general Heckeendomorphism algebras were investigated by B. Parshall, L. Scott, and the first author overtwenty years ago. They made a stratification conjecture about their structure. A slightlymodified version of this conjecture has recently been proved in [ DPS ] and applications torepresentations of finite groups of Lie type have been obtained. In order to extend theseapplications to i -quantum groups, we need to lift Bao-Wang’s duality to the integral level,and hence, to the roots-of-unity level. This is the aim of the current paper.Building on the work of Bao et al [ BKLW18 ], we will define an algebra homomorphism φ from the i -quantum groups U ( n ) into the direct product of the corresponding q -Schur algebras S ( n, r ) Q ( v ) of type B . We mainly focus on the determination of the homomorphic image of φ in terms of a BLM type basis { A ( j ) } A, j (cf. [ BLM90 ]). This then allows us to investigatethe Lusztig type forms and their associated hyperalgebras for the i -quantum groups througha certain monomial basis. We thus establish an algebra epimorphism from the integral i -quantum group to the integral q -Schur algebras of type B . Note that, with this epimorphism,the representation category of i - q -Schur algebras becomes a full subcategory of that of the i -quantum group (or i -quantum hyperalgebra). We further prove that the map φ is injective.This gives a new realisation of the i -quantum group U ( n ) in terms of the basis { A ( j ) } A, j together with explicit multiplication formulas by generators.Just like the original realisation by Beilinson–Lusztig–MacPherson for quantum gl n , thiswork laid down a foundation for a further study of the above mentioned i -quantum hyperal-gebra. On the other hand, it should be plausible that a similar realisation can be obtained for i -quantum groups U ı ( n ) although the triangular relation between two bases is a bit subtle inthis case. We will get these done in a forthcoming paper.We organise the paper as follows. We first review in § i -quantum groups U ( n ) and their realisation as coideal sublagebra of the quantum linear group U ( gl n +1 ).In particular, the natural representation Ω of U ( gl n +1 ) and its tensor product Ω ⊗ r restrictto become U ( n )-modules. In §
3, we introduce q -Schur algebra of type B through finiteorthogonal groups O r +1 as well as (Iwahori–)Hecke algebras of type B r . We will also re-view the Schur–Weyl duality discovered by Bao–Wang [ BW18 ]. Section 4 is quite parallelto [
BLM90 , § § BKLW18 , Lem. 3.2], we derive certain multiplicationformulas in S ( n, r ) Q ( v ) . The structure constants in these formulas are independent of r . Thisallows to extend these formulas to get similar formulas in the direct product S ( n ) Q ( v ) of S ( n, r ) Q ( v ) (Theorem 4.4). Using a triangular relation in [ BKLW18 , Thm. 3.10], we prove in § A ( n ) spanned by all A ( j ) is a subalgebra of S ( n ) Q ( v ) (Theorem 5.2).The above mentioned homomorphism φ has the image A ( n ) (Theorem 6.1). Using φ , wethen lift Bao–Wang’s epimorphism to the integral level (Theorem 6.5). Finally, in the lastsection, we prove that φ is injective and thus, we establish the new realisation (Theorem 7.1). Some notations.
For a positive integer a , let[1 , a ] = { , , . . . , a } , [1 , a ) = { , , . . . , a − } . NEW REALISATION OF THE i -QUANTUM GROUP U ( n ) 3 Let Z = Z [ v, v − ] be the integral Laurent polynomial ring. For n >
0, we set[[ n ]] = v n − v − , [ n ] = v n − v − n v − v − and [ n ]! = [1][2] . . . [ n ] . Set [[0]] = [0] = 0 and [0] ! = 1. We also define, for s, t ∈ Z with t > (cid:20) st (cid:21) = t Y i =1 v s − i +1 − v − ( s − i +1) v i − v − i , (cid:20) K ; st (cid:21) = t Y i =1 Kv s − i +1 − K − v − ( s − i +1) v i − v − i , where K is an element in a Q ( v )-algebra.2. The i -quantum group U ( n ) : a first realisation In the study of quantum symmetric pairs, Bao and Wang introduced the following quantumalgebra U ( n ), extracted from a quantum symmetric pair of type AIII in [ Le03 , § BKLW18 , § n indicates the rank of the i -quantum group. Definition 2.1.
The algebra U ( n ) is defined to be the associative algebra over Q ( v ) generatedby e i , f i , d a , d − a , i ∈ [1 , n ], a ∈ [1 , n + 1] subject to the following relations: for i, j ∈ [1 , n ], a, b ∈ [1 , n + 1],(iQG1) d a d − a = d − a d a = 1 , d a d b = d b d a ;(iQG2) d a e j d − a = v δ a,j − δ a,j +1 e j , d a f j d − a = v − δ a,j + δ a,j +1 f j , if a ≤ n ; d n +1 e j d − n +1 = v − δ n,j e j , d n +1 f j d − n +1 = v δ n,j f j ;(iQG3) e i f j − f j e i = δ i,j d i d − i +1 − d − i d i +1 v − v − , if i, j = n ;(iQG4) e i e j = e j e i , f i f j = f j f i , if | i − j | > e i e j + e j e i = [2] e i e j e i , f i f j + f j f i = [2] f i f j f i , if | i − j | = 1;(iQG6) f n e n + e n f n = [2] (cid:0) f n e n f n − ( vd n d − n +1 + v − d − n d n +1 ) f n (cid:1) ,e n f n + f n e n = [2] (cid:0) e n f n e n − e n ( vd n d − n +1 + v − d − n d n +1 ) (cid:1) . Note that the subalgebra generated by e i , f i , d a , d − a , i ∈ [1 , n ), a ∈ [1 , n ] is isomorphic tothe quantum linear group U ( gl n ) in the following sense. Definition 2.2.
The quantum linear group is a Hopf algebra U ( gl N ) over Q ( v ) with generators E a , F a , K ± j , a ∈ [1 , N ) , j ∈ [1 , N ] , and relations:(QG1) K i K j = K j K i , K j K − j = K − j K j = 1;(QG2) K j E b = v δ j,b − δ j,b +1 E b K j , K j F b = v − δ j,b + δ j,b +1 F b K j ;(QG3) [ E a , F b ] = δ a,b e K a − e K − a v − v − , where e K a = K a K − a +1 ;(QG4) E a E b = E b E a , F a F b = F b F a , if | a − b | > a, b ∈ [1 , N ) with | a − b | = 1 ,E a E b − ( v + v − ) E a E b E a + E b E a = 0 ,F a F b − ( v + v − ) F a F b F a + F b F a = 0 . Its comultiplication and counit are defined, respectively, by∆ : U ( gl N ) −→ U ( gl N ) ⊗ U ( gl N ) ,E a ⊗ E a + E a ⊗ e K a ,F a F a ⊗ e K − a ⊗ F a ,K j K j ⊗ K j , and ǫ : U ( gl N ) −→ Q ( v ) ,E a ,F a ,K j . JIE DU AND YADI WU
The first realisation of U ( n ) is to embed it into the quantum group U ( gl n +1 ); see [ BKLW18 ,Prop. 4.5]. See also [
BW18 , Prop. 6.2] for the first version.
Lemma 2.3.
There is an injective Q ( v ) -algebra homomorphism ι : U ( n ) → U ( gl n +1 ) de-fined, for i ∈ [1 , n ] , by d i K − i K − n +2 − i , d n +1 v − K − n +1 ,e i F i + e K − i E n +1 − i , f i E i e K n +1 − i + F n +1 − i . Moreover, relative to the coalgebra structure, ι ( U ( n )) is a coideal of U ( gl n +1 ) . Let Ω = Ω n +1 be the natural representation of U ( gl n +1 ) with a Q ( v )-basis { ω , . . . , ω n +1 } via the following actions: E h ω i = δ i,h +1 ω h , F h ω i = δ i,h ω h +1 , K j ω i = v δi,j ω i . (2.3.1)Then Ω ⊗ r becomes a U ( gl n +1 )-module via the actions: E h .ω i = ∆ ( r − ( E h ) ω i , F h .ω i = ∆ ( r − ( F h ) ω i , K j .ω i = ∆ ( r − ( K j ) ω i , where ω i = ω i ⊗ · · · ⊗ ω i r for i = ( i , . . . , i r ), and∆ ( r − ( K j ) = K j ⊗ · · · ⊗ K j | {z } r , ∆ ( r − ( E h ) = r X j =1 ⊗ · · · ⊗ ⊗ E h ⊗ e K h ⊗ · · · ⊗ e K h | {z } j − , ∆ ( r − ( F h ) = r X j =1 e K − h ⊗ · · · ⊗ e K − h | {z } j − ⊗ F h ⊗ ⊗ · · · ⊗ . (2.3.2)Thus, we obtain a Q ( v )-algebra homomorphism ρ r : U ( gl n +1 ) → End(Ω ⊗ r ) . (2.3.3)It is well-known that the image im( ρ r ) is the commutant subalgebra relative to a right actionof the Hecke algebra of type A (see Theorem 3.5 below). This is called a q -Schur algebra (oftype A ). We will soon see in next section that when we restrict ρ r to the subalgebra im( ι ): ρ r := ρ r ◦ ι : U ( n ) ι −→ U ( gl n +1 ) ρ r −→ End(Ω ⊗ r ) , the image im( ρ r ) is the commutant subalgebra relative to a right action of the Hecke algebraof type B . This is called the q -Schur algebra of type B ; see [ BW18 , Thm. 6.27].
Lemma 2.4.
The algebra U ( n ) has an involutive automorphism ω defined by ω ( e i ) = f i , ω ( f i ) = e i , ω ( d i ) = d − i (1 ≤ i ≤ n ) , ω ( d n +1 ) = v − d − n +1 . Proof.
This can be directly checked by the relations above or modified from [
BW18 , Lem.6.1](1); compare [
DDPW08 , Lem 6.5(1)]. (cid:3) We corrected a typo in the correspondence d n +1 vK − n +1 there. NEW REALISATION OF THE i -QUANTUM GROUP U ( n ) 5 The q -Schur algebra of type B For any field k , let GL n ( k ) be the genernal linear group over k and consider the groupisomorphism ϑ : GL n ( k ) −→ GL n ( k ) , x J − ( x t ) − J, where J has entries J i,j = 1 whenever i + j = n + 1 and 0 otherwise. The orthogonal groupO n ( k ) := { x ∈ GL n ( k ) | J = x t J x } (where char( k ) = 2)is the fixed-point group of ϑ . Let G ( q ) := O r +1 ( k ) for k = F q , the finite field of q elements.Let Λ( n + 1 , r ) = { λ = ( λ , . . . , λ n , λ n +1 ) ∈ N n +1 | λ + · · · + λ n +1 = r } . Define the bijection e : Λ( n + 1 , r ) −→ e Λ( n + 1 , r ) ⊆ Λ(2 n + 1 , r + 1) ,λ e λ := ( λ , . . . , λ n , λ n +1 + 1 , λ n , . . . , λ ) , (3.0.1)where e Λ( n + 1 , r ) is the image of Λ( n + 1 , r ) of the map. For λ ∈ Λ( n + 1 , r ), let P e λ ( q )be the standard parabolic subalgebra of GL r +1 ( F q ) associated with e λ , consisting of upperquasi-triangular matrices with blocks of sizes e λ i on the diagonal. Let P λ ( q ) = P e λ ( q ) ∩ G ( q ) . Then G ( q ) acts on the set P λ ( q ) \ G ( q ) of left cosets P λ ( q ) g in G ( q ). For any commutativering R , this action induces a permutation representation over R which is isomorphic to theinduced representations Ind G ( q ) P λ ( q ) R of the trivial representation 1 R to G ( q ) and define E q,R ( n, r ) = End RG ( q ) (cid:18) M λ ∈ Λ( n +1 ,r ) Ind G ( q ) P λ ( q ) R (cid:19) op . (3.0.2)This is called the q -Schur algebra of type B (compare the type A case in [ DJ89 ]).This algebra has the following interpretation of Hecke endomorphism algebra. Let H ( B r )be the Hecke algebra over Z = Z [ v, v − ] associated with the Coxeter system ( W, S ) of type B r , where S = { s , . . . , s r − , s r } has the Dynkin diagram: r − r Then it is generated by T i = T s i for 1 ≤ i ≤ r subject to the relations: T i = ( v − T i + v , ∀ i ; T i T j = T j T i , | i − j | > ,T j T j +1 T j = T j +1 T j T j +1 , ≤ j < r − T r − T r T r − T r = T r T r − T r T r − . It has a basis { T w } w ∈ W . The subalgebra generated by T , . . . , T r − is the Hecke algebra H ( S r )associated with the symmetric group S r .For λ ∈ Λ( n + 1 , r ), let W λ be the parabolic subgroup of W generated by S \{ s λ + ··· + λ i | i ∈ [1 , n ] } , and let x λ = P w ∈ W λ T w . The Hecke endomorphism Z -algebra: S ( n, r ) = End H ( B r ) (cid:0) T ( n, r ) (cid:1) , where T ( n, r ) = M λ ∈ Λ( n +1 ,r ) x λ H ( B r ) (3.0.3)is called the (generic) q -Schur algebra of type B . JIE DU AND YADI WU
In order to label the standard basis of S ( n, r ) by matrices, we consider the graph automor-phism of the symmetric group S r +1 = S [1 , r +1] : σ : S r +1 → S r +1 , ( i, j ) (2 r + 2 − i, r + 2 − j ) for all i, j ∈ [1 , r + 1] . If θ ∈ S r +1 denotes the permutation sending i to 2 r + 2 − i , then, for any π ∈ S r +1 , σ ( π ) = θ ◦ π ◦ θ and σ ( i, j ) = ( θ ( i ) , θ ( j )). Further, we may identify W as the fixed-pointsubgroup S σ r +1 of σ with s i = ( i, i + 1)(2 r + 2 − i, r + 1 − i ) (1 ≤ i < r ) , s r = ( r, r + 1)( r + 1 , r + 2)( r, r + 1) . For the parabolic subgroup W λ , λ ∈ Λ( n + 1 , r ), if e λ = ( λ , . . . , λ n , λ n +1 + 1 , λ n , . . . , λ ) ∈ Λ(2 n + 1 , r + 1) . as in (3.0.1), then σ stabilises the Young subgroup S e λ and W λ = S σ e λ is the fixed-pointsubgroup. Lemma 3.1.
For λ, µ ∈ Λ( n + 1 , r ) and e d ∈ S r +1 , suppose the double coset S e λ e d S e µ definesa (2 n + 1) × (2 n + 1) matrix A = ( a i,j ) over N whose entries sum to r + 1 . Then S e λ e d S e µ isstabilised by σ if and only if a i,j = a N +1 − i,N +1 − j for all i, j ∈ [1 , N ] ( N = 2 n + 1 ).Proof. Let R e λi = { e λ + · · · e λ i − + 1 , . . . , e λ + · · · e λ i − + e λ i } . Then a i,j = | R e λi ∩ e dR e µj | and θ ( R e λi ) = R e λN +1 − i . Note that σ stabilises S e λ and S e µ . Thus, σ stabilises S e λ e d S e µ (i.e., S e λ e d S e µ = S e λ σ ( e d ) S e µ ) if and only if | R e λN +1 − i ∩ σ ( e d ) R e µN +1 − j | = | R e λN +1 − i ∩ e dR e µN +1 − j | = a N +1 − i,N +1 − j for all i, j . (See, e.g., [ DDPW08 , Lem. 4.14].) However, | R e λN +1 − i ∩ σ ( e d ) R e µN +1 − j | = | θ ( R e λi ∩ e dR e µj ) | = | R e λi ∩ e dR e µj | = a i,j . (cid:3) For λ, µ ∈ Λ( n + 1 , r ), let D λ,µ be the minimal length representatives of all double cosets W λ wW µ .For N = 2 n + 1, letΞ n +1 = (cid:8) A = ( a i,j ) ∈ Mat N ( N ) | a i,j = a N +1 − i,N +1 − j , ∀ i, j ∈ [1 , N ] } , Ξ n +1 = (cid:8) A − diag( a , , a , , . . . , a N,N ) | A = ( a i,j ) ∈ Ξ n +1 } , Ξ n +1 , r +1 = { A = ( a i,j ) ∈ Ξ n +1 | | A | := X i,j a i,j = 2 r + 1 } . (3.1.1) Corollary 3.2.
There is a bijection m : { ( λ, d, µ ) | λ, µ ∈ Λ( n + 1 , r ) , d ∈ D λ,µ } −→ Ξ n +1 , r +1 . Proof.
The matrix m ( λ, d, µ ) is the matrix associated with the double coset S e λ e d S e µ . Theassertion follows from the lemma above. (cid:3) Recall the basis { e A | A ∈ Ξ n +1 , r +1 } for E q,R ( n, r ) introduced in [ BKLW18 , § Theorem 3.3.
By specialising v to q = q R ∈ R , there is an algebra isomorphism ξ : E q,R ( n, r ) −→ S ( n, r ) R := S ( n, r ) ⊗ Z R, e A ξ dλ,µ , where m ( λ, d, µ ) = A and ξ dλ,µ is the H ( B r ) -module homomorphism defined by ξ dλ,µ ( x ν ) = δ µ,ν X w ∈ W λ dW µ T w . We will identify the two basis elements e A = ξ dλ,µ in the sequel. NEW REALISATION OF THE i -QUANTUM GROUP U ( n ) 7 Remark 3.4.
The proof is omitted as it is standard, extending Iwahori’s original isomorphismEnd RG ( q ) (ind G ( q ) P (1 r ) ( q ) R ) ∼ = H ( B r ) R . See [ DDPW08 , Thm. 13.15]. It should be noted that amore general result for all finite types can be found in [ LW , Thm. 4.2].We record the fact that, if D = diag( e λ ) ∈ Ξ n +1 , r +1 for some λ ∈ Λ( n + 1 , r ) is diagonal,then e D e A = δ e λ, ro( A ) e A , e A e D = δ co( A ) , e λ e A (3.4.1)for all A = ( a i,j ) ∈ Ξ n +1 , r +1 , wherero( A ) := (cid:0) X j a ,j , X j a ,j , . . . , X j a n +1 ,j (cid:1) co( A ) := (cid:0) X i a i, , X i a i, , . . . , X i a i, n +1 (cid:1) . We end this section with Bao-Wang’s Schur duality. We need to interpret the H ( B r )-module T ( n, r ) in terms of the tensor space Ω ⊗ r ; see [ DPS ]. Let I (2 n + 1 , r ) = { i = ( i , . . . , i r ) | i j ∈ [1 , n + 1] , ∀ j } . (3.4.2)Then ω i := ω i ⊗ · · · ⊗ ω i r , i ∈ I (2 n + 1 , r ), form a basis for Ω ⊗ r .Recall the action of H ( S r ) on Ω ⊗ r defined in [ DDPW08 , (14.6.4)]: for 1 ≤ j < r , ω i T j = vω i s j , if i j < i j +1 ; v ω i , if i j = i i +1 ;( v − ω i + vω i s j , if i j > i j +1 . (3.4.3)We extend the action to H ( B r ) by setting ω i T r = vω i s r , if i j < r + 1; v ω i , if i j = r + 1;( v − ω i + vω i s r , if i j > r + 1 . (3.4.4)Here s j = ( j, j + 1) and i s j is the place permutation:( i , . . . , i r − , i r ) s j = ( ( i , . . . , i j − , i j +1 , i j , i j +2 , . . . , i r ) , if j < r ;( i , . . . , i j − , i j , i j +1 , . . . , i r − , i r +2 ) , if j = r, where i r +2 = N + 1 − i r − . As usual, we will call a surjective homomorphism an epimorphism . Let H ( S r ) = H ( S r ) Q ( v ) ,etc. Theorem 3.5. (1) The U ( gl n +1 ) action on Ω ⊗ r commutes with the action of H ( S r ) and thebimodule structure induces epimorphisms (cf. (2.3.3) ) ρ r : U ( gl n +1 ) −→ End H ( S r ) (Ω ⊗ r ) , ρ ∨ r : H ( S r ) −→ End U ( gl n +1 ) (Ω ⊗ r ) . (2) [ DPS , Lem. 5.3.8]
There is an H ( B r ) -module isomorphism T ( n, r ) Q ( v ) ∼ = Ω ⊗ r . Hence, End H ( B r ) (Ω ⊗ r ) ∼ = S ( n, r ) Q ( v ) . (3) [ BW18 , Thm.6.27]
The actions of U ( n ) and H ( B r ) commute and satisfy the doublecentraliser property as stated in (1). In particular, the algebra homomorphism ρ r restricts toan algebra epimorphism ρ r = ρ r ◦ ι : U ( n ) −→ End H ( B r ) (Ω ⊗ r ) . (3.5.1) JIE DU AND YADI WU
Remarks 3.6. (1) The Q ( v )-algebra S (2 n + 1 , r ) Q ( v ) := End H ( S r ) Q ( v ) (Ω ⊗ r ) is known as the q -Schur algebra ( q = v ). By identifying S ( n, r ) Q ( v ) with End H ( B r ) Q ( v ) (Ω ⊗ r ) under the isomor-phism in (2), we may regard S ( n, r ) Q ( v ) as a subalgebra of S (2 n + 1 , r ) Q ( v ) . Clearly, theserelations hold at the integral level.(2) The epimorphisms ρ r induce an algebra homomorphism ρ : U ( n ) −→ Y r ≥ End H ( B r ) Q ( v ) (Ω ⊗ r ) , u ( ρ r ( u )) γ ≥ . One of the aims of this paper is to determine a basis for the image im( ρ ) and to show that ρ induces an isomorphism U ( n ) ∼ = im( ρ ).(3) Lai, Nakano and Xiang considered the representation theory of S ( n, r ) k over a field k .In particular they realized the aforementioned algebra as the dual of the r th homogeneouscomponent of the quotient of the coordinate algebra of the quantum matrix space by a rightideal that is also a coideal. This shows there is a natural polynomial representation theory(see [ LNX , Section 2.4]).Moreover, under a certain invertibility condition (i.e., the “semisimple bottom” conditionin the sense of [
DR00 ]), the structure and representations of these algebras including quasi-hereditariness and cellularity were investigated [
LNX , Sections 5-6]. In turn, they obtaineda concrete realization for the category O of rational Cherednik algebras of type B togetherwith the Knizhnik–Zamolodchikov functor in terms of the module category of S ( n, r ) k andits corresponding Schur functor (see [ LNX , Section 8]).4.
Some multiplication formulas
We now derive some multiplication formulas and their associated stabilisation property inthe q -Schur algebra of type B . This work is built on the formulas in [ BKLW18 , Lem. 3.2].For i, j ∈ [1 , n + 1], let E i,j be the standard matrix units in Mat n +1 ( N ). Let E θi,j = E i,j + E n +2 − i, n +2 − j = E θ n +2 − i, n +2 − j . Note that E θn +1 ,n +1 = 2 E n +1 ,n +1 . Let ǫ θi,j = i = j = n + 1 , , and e θi = ro( E θi,i ) . Then ǫ θi,j is the ( i, j )-entry of E θi,j and e θi = e i + e n +2 − i , where e i = (0 , . . . , , ( i ) , , . . . ∈ Z n +1 form the standard basis for Z n +1 . Recall the dimension d ( A ) of the orbit O A and the dimension r ( A ) of the image of O A underthe first projection (see [ BKLW18 , (3.16)]). d ( A ) − r ( A ) = 12 (cid:0) X i ≥ k,j BKLW18 , Thm. 3.7] by taking R = 1. We can also derive them directly by [ BKLW18 , Lem. 3.2]. For notational clarity, we We thank Yiqiang Li for sending us this simplified version. NEW REALISATION OF THE i -QUANTUM GROUP U ( n ) 9 extend the usual Kronecker delta δ i,j to define δ ≤ i,j = ( i ≤ j, i > j. Let, for A ∈ Ξ n +1 , r +1 and h ∈ [1 , n ], β p ( A, h ) = X j ≥ p a h,j − X j>p a h +1 ,j + δ h,n δ ≤ p,n ,β ′ p ( A, h ) = X j ≤ p a h +1 ,j − X j
Suppose that h ∈ [1 , n ] , λ ∈ Λ( n + 1 , r − , and A ∈ Ξ n +1 , r +1 . The followingmultiplication formulas holds in S ( n, r ) : (a) [ E θh,h +1 + e λ ] · [ A ] = δ e θh +1 + e λ, ro( A ) X p ∈ [1 , n +1] ah +1 ,p ≥ ǫθh +1 ,p v β p ( A,h ) [[ a h,p + 1]][ A + E θh,p − E θh +1 ,p ] . (b) [ E θh +1 ,h + e λ ] · [ A ] = δ e θh + e λ, ro( A ) X p ∈ [1 , n +1] ah,p ≥ v β ′ p ( A,h ) [[ a h +1 ,p + 1]][ A − E θh,p + E θh +1 ,p ] . We now extend these formulas to a certain spanning set for S ( n, r ). Recall the notationΞ n +1 in (3.1.1).For A ∈ Ξ n +1 and j = ( j , j , . . . , j N ) ∈ Z n +1 , define A ( j , r ) = X λ ∈ Λ( n +1 ,r − | A | ) v e λ (cid:5) j [ A + e λ ] , if | A | ≤ r, , if | A | > r, (4.1.1)where e λ (cid:5) j = P n +1 i =1 e λ i j i with e λ defined as in (3.0.1). Note that ne λ | λ ∈ Λ( n + 1 , r − | A | ) o = { µ ∈ N n +1 | A + µ ∈ Ξ n +1 , r +1 } . In particular, if O denotes the zero matrix, e i ∈ Z n +1 as above, and = (0 , . . . , ∈ Z n +1 ,we have E θh,h +1 ( , r ) = P λ ∈ Λ( n +1 ,r − [ E θh,h +1 + e λ ] and O ( e i , r ) = O ( e n +2 − i , r ) = X λ ∈ Λ( n +1 ,r ) v λ i [ e λ ] , if 1 ≤ i ≤ n ; X λ ∈ Λ( n +1 ,r ) v λ n +1 +1 [ e λ ] , if i = n + 1 . (4.1.2)For 1 ≤ h ≤ n , we put α h = e h − e h +1 and α − h = − e h − e h +1 . The following multiplicationformulas are the type B counterpart of [ BLM90 , Lem. 5.3]. Theorem 4.2. Maintain the notations introduced above. For N = 2 n +1 , j = ( j , j , . . . , j N ) ∈ Z N , h ∈ [1 , n ] , and A = ( a i,j ) ∈ Ξ n +1 , the following multiplication formulas hold in S ( n, r ) for all r ≥ | A | : (1) O ( j , r ) A ( j ′ , r ) = v ro( A ) (cid:5) j A ( j + j ′ , r ) , A ( j ′ , r ) O ( j , r ) = v co( A ) (cid:5) j A ( j + j ′ , r ); (2) E θh,h +1 ( , r ) · A ( j , r ) = X ≤ p 1, then ε = 1 and v β h ( A + e µ,h )+ e µ (cid:5) j [[ a µh,h + 1]][ A + e µ + E θh,h − E θh +1 ,h ]= v β h ( A,h )+ e µ (cid:5) ( j + α h ) [[ µ h + 1]][ A − E θh +1 ,h + e µ + e θh ]= v β h ( A,h ) − v − v e µ (cid:5) ( j + α h ) (cid:0) − v − e µ h +1) (cid:1) [ A − E θh +1 ,h + e µ + e θh ]= v β h ( A,h ) − j h − j N +1 − h − − v − v e µ (cid:5) ( j + α h ) v j h + j N +1 − h +1 (cid:0) − v − e µ h +1) (cid:1) [ A − E θh +1 ,h + e µ + e θh ]= v β h ( A,h ) − j h − j N +1 − h − − v − (cid:0) v ( e µ + e θh ) (cid:5) ( j + e h − e h +1 ) − v ( e µ + e θh ) (cid:5) ( j − e h − e h +1 ) (cid:1) [ A − E θh +1 ,h + e µ + e θh ] . Since v e λ (cid:5) ( j + e h − e h +1 ) − v e λ (cid:5) ( j − e h − e h +1 ) = 0 whenever λ h = 0 (so e λ h = e λ N +1 − h = 0), it follows that X µ ∈ Λ( n +1 ,r − | A | ) (cid:0) v ( e µ + e θh ) (cid:5) ( j + e h − e h +1 ) − v ( e µ + e θh ) (cid:5) ( j − e h − e h +1 ) (cid:1) [ A − E θh +1 ,h + e µ + e θh ]= X λ ∈ Λ( n +1 ,r − | A | +1) (cid:0) v e λ (cid:5) ( j + α h ) − v e λ (cid:5) ( j + α − h ) (cid:1) [ A − E θh +1 ,h + e λ ]= ( A − E θh +1 ,h )( j + α h , r ) − ( A − E θh +1 ,h )( j + α − h , r ) . giving the second term in (2). Finally, for the third summation, we have X µ ∈ Λ( n +1 ,r − | A | e µh +1 ≥ ǫθh +1 ,h +1 v β h +1 ( A + e µ,h )+ e µ (cid:5) j [[ a h,h +1 + 1]][ A + e µ + E θh,h +1 − E θh +1 ,h +1 ]= v β h +1 ( A,h )+ j h +1 + j N − h [[ a h,h +1 + 1]] X µ e µh +1 ≥ ǫθh +1 ,h +1 v ( e µ − e h +1 − e N − h ) (cid:5) j [ A + E θh,h +1 + e µ − e θh +1 ]= v β h +1 ( A,h )+ j h +1 + j N − h [[ a h,h +1 + 1]]( A + E θh,h +1 )( j , r ) . Here, we have used an obvious bijection n µ ∈ Λ( n + 1 , r − | A | | µ h +1 ≥ o −→ Λ( n + 1 , r − | A | − µ , . . . , µ h , µ h +1 , µ h +2 , . . . , µ n +1 ) ( µ , . . . , µ h , µ h +1 − , µ h +2 , . . . , µ n +1 ) . This proves (2). The proof of (3) is similar. (cid:3) Remark 4.3. If one compares these multiplication formulas with those given in [ BLM90 ,Lem. 5.3] (or [ DDPW08 , Thm. 13.27]), they are very similar except the adjustments neededfor the h = n case. Of course, you may also see the difference arising from the symmetry ofthe matrices involved.Let S ( n ) Q ( v ) = Y r ≥ S ( n, r ) Q ( v ) . We will write the elements in S ( n ) Q ( v ) as formal infinite series. Define, for A ∈ Ξ n +1 and j ∈ Z n +1 , A ( j ) := X r ≥ A ( j , r ) ∈ S ( n ) Q ( v ) . For convenient use later, Theorem 4.2 is rewritten as follows. Theorem 4.4. For N = 2 n + 1 , j = ( j , j , . . . , j N ) ∈ Z N , h ∈ [1 , n ] , and A = ( a ij ) ∈ Ξ n +1 ,the following multiplication formulas hold in S ( n ) Q ( v ) : (1) O ( j ) A ( j ′ ) = v ro( A ) (cid:5) j A ( j + j ′ ) , A ( j ′ ) O ( j ) = v co( A ) (cid:5) j A ( j + j ′ );(2) E θh,h +1 ( ) · A ( j ) = X ≤ p This is clear since the coefficients in the multiplication formulas in Theorem 4.2 areindependent of r for all r ≥ | A | . (cid:3) Corollary 4.5. For h ∈ [1 , n ] , we have in S ( n ) Q ( v ) (1) E θh,h +1 ( ) m = [ m ] ! ( mE θh,h +1 )( );(2) E θh +1 ,h ( ) m = [ m ] ! ( mE θh +1 ,h )( ) . Proof. We only prove (1); the proof of (2) is similar. For A = mE θh,h +1 , the only non-zeroentry in row h + 1 is the diagonal entry and β h +1 ( A, h ) = X j ≥ h +1 a h,j − X j>h +1 a h +1 ,j + δ h,n δ ≤ h +1 ,n = m. Thus, by Theorem 4.4(2), E θh,h +1 ( ) = v [[1 + 1]](2 E θh,h +1 )( ) = [2] ! (2 E θh,h +1 )( ) . Now the general case follows from an induction. (cid:3) The subalgebra A ( n )We now prove that the subspace of S ( n ) Q ( v ) : A ( n ) = span { A ( j ) | A ∈ Ξ n +1 , j ∈ Z N } is indeed a subalgebra.As in [ BKLW18 ] or [ BLM90 , 5.3], we define a preorder (cid:22) on Ξ n +1 as follows. A (cid:22) B ⇐⇒ X r ≤ i ; s ≥ j a rs ≤ X r ≤ i ; s ≥ j b rs , for all 1 ≤ i < j ≤ n + 1 . Clearly, A (cid:22) B ⇐⇒ P r ≥ i ; s ≤ j a rs ≤ P r ≥ i ; s ≤ j b rs , for all i > j . We write A ≺ B if A (cid:22) B and B A .Let T n +1 = { ( i, h, j ) | ≤ j ≤ h < i ≤ n + 1 } . We order the set as in [ BKLW18 , Thm. 3.10]:( i, h, j ) ≤ ( i ′ , h ′ , j ′ ) ⇐⇒ i < i ′ or i = i ′ , j < j ′ or i = i ′ , j = j ′ , h > h ′ . (5.0.1)This order modifies the order ≤ i defined in [ DDPW08 , (13.7.1)]. For example, the first fewelements in ( T n +1 , ≤ ) are(2 , , , (3 , , , (3 , , , (3 , , , (4 , , , . . . (4 , , , (4 , , , (4 , , , . . . , ( N, N − , N − . (5.0.2) For A ∈ Ξ n +1 , let m A, := Y ( i,h,j ) ∈ ( T n +1 , ≤ ) ( a i,j E θh +1 ,h )( ) , (5.0.3)where the product is taken with respect to the order ≤ . Thus, by (5.0.2), the leading term ofthe product m A, is ( a , E θ , )( ) and the ending term is ( a N,N − E θN,N − )( ) ( N = 2 n + 1). Lemma 5.1. For each A ∈ Ξ n +1 , we have m A, = A ( ) + X B ∈ Ξ0diag2 n +1 , j ∈ Z NB ≺ A g A,B, j B ( j ) = A ( ) + ( lower terms ) . Proof. Repeatedly applying Theorem 4.4 yields m A, = X B ∈ Ξ n +1 j ∈ Z N g A,B, j B ( j )It suffices to prove that g A,A, = 1 and B ≺ A whenever g A,B, j = 0. Consider the r -thcomponent of m A, : π r ( m A, ) = ≤ Y ≤ j ≤ h
The vector space A ( n ) = span { A ( j ) | A ∈ Ξ n +1 , r +1 , j ∈ Z n +1 } of thealgebra S ( n ) Q ( v ) is a subalgebra which is generated by E θh,h +1 ( ) , E θh +1 ,h ( ) , O ( ± e i ) , for all ≤ h ≤ n and ≤ i ≤ n + 1 , and is presented by the multiplication formulas inTheorem 4.4.Proof. Let A ′ be the subalgebra generated by E θh,h +1 ( ) , E θh +1 ,h ( ) , O ( ± e i ) . By Theorem 4.4, we have A ′ ⊆ A ( n ). We now prove A ( n ) ⊆ A ′ . We prove all A ( j ) ∈ A ′ byinduction on k A k , where k A k = X ≤ i The Q ( v ) -algebra A ( n ) has bases B = { A ( j ) | A ∈ Ξ n +1 , j ∈ Z N } and M = { m A, j | A ∈ Ξ n +1 , j ∈ Z N } . Proof. The assertion for B is standard with an argument involving Vandermonde determinant.The assertion for M follows from the assertion for B and the triangular relation in Lemma5.1. (cid:3) Corollary 5.4. The canonical projection from S ( n ) Q ( v ) onto S ( n, r ) Q ( v ) restricts to a Q ( v ) -algebra epimorphism π r : A ( n ) → S ( n, r ) Q ( v ) . Proof. It suffice to prove that, for a fixed A ∈ Ξ n +1 , r +1 ,span { A ( j , r ) | j ∈ Z n +1 } = span { [ A + e λ ] | λ ∈ Λ( n + 1 , r − | A | ) } . This is clear from the definition of A ( j , r ) in (4.1.1). (cid:3) Lifting Bao–Wang’s Schur duality to the integral level In [ BW18 , Thm. 6.27], a ( U ( n ) , H ( B r ))-duality via the tensor space Ω ⊗ r is established;see Theorem 3.5(3). In this section, we will define an algebra epimorphism φ r : U ( n ) → S ( n, r ) Q ( v ) via the subalgebra A ( n ) and prove that φ r maps a Lusztig type form U ( n ) Z of U ( n ) onto the integral q -Schur algebra S ( n, r ). We will compare φ r with the epimorphism ρ r given in (3.5.1) in next section. Theorem 6.1. There is a Q ( v ) -algebra epimorphism φ : U ( n ) −→ A ( n ) such that e h E θh,h +1 ( ) , f h E θh +1 ,h ( ) , d ± h O ( ± e h ) , and d ± n +1 v ∓ O ( ± e n +1 ) for all ≤ h ≤ n .Proof. We must prove that the relations (iQG1)–(iQG6) in Definition 2.1 are all satisfied for e h = E θh,h +1 ( ) , f h = E θh +1 ,h ( ) , d h = O ( e h ) , d n +1 = v − O ( e n +1 ) . Since relations (iQG1)–(iQG5) are more or less the defining relations for quantum gl n , byRemark 4.3, the proof is almost the same as the proof of [ DDPW08 , Thm. 13.33]. We nowprove (iQG6). We only check the first relation here: f n e n + e n f n = [2] (cid:0) f n e n f n − ( vd n d − n +1 + v − d − n d n +1 ) f n (cid:1) . (6.1.1)First, compute f n e n = E θn +1 ,n ( ) E θn,n +1 ( ). By Theorem 4.4(3), we have f n e n = E θn +1 ,n ( ) E θn,n +1 ( ) = v β ′ n ( E θn,n +1 ,n ) ( E θn,n +1 + E θn +1 ,n )( )+ v β ′ n +1 ( E θn,n +1 ,n ) − v − (cid:16) v − ( E θn,n +1 − E θn,n +1 )( − α n ) − ( E θn,n +1 − E θn,n +1 )( α − n ) (cid:17) = ( E θn,n +1 + E θn +1 ,n )( ) + v − − v − O ( − α n ) − O ( α − n )1 − v − Further, we see that, E θn +1 ,n ( )( E θn,n +1 + E θn +1 ,n )( ) = v β ′ n ( E θn,n +1 + E θn +1 ,n ,n ) [[2]]( E θn,n +1 + 2 E θn +1 ,n )( )+ v β ′ n +1 ( E θn,n +1 + E θn +1 ,n ,n ) − v − (cid:16) v − E θn +1 ,n ( − α n ) − E θn +1 ,n ( α − n ) (cid:17) = v [[2]]( E θn,n +1 + 2 E θn +1 ,n )( ) + v − − v − E θn +1 ,n ( − α n ) − v − v − E θn +1 ,n ( α − n ) , (6.1.2)and E θn +1 ,n ( ) O ( − α n ) = v − E θn +1 ,n ( − α n ) , E θn +1 ,n ( ) O ( α − n ) = v − E θn +1 ,n ( α − n ) . Thus, we have f n e n = E θn +1 ,n ( ) E θn,n +1 ( )= v [[2]]( E θn,n +1 + 2 E θn +1 ,n )( ) + v − − v − E θn +1 ,n ( − α n ) − v − v − E θn +1 ,n ( α − n )+ v − − v − E θn +1 ,n ( − α n ) − v − − v − E θn +1 ,n ( α − n ) , NEW REALISATION OF THE i -QUANTUM GROUP U ( n ) 17 Second, compute e n f n = E θn,n +1 ( ) E θn +1 ,n ( ) . Since E θn +1 ,n ( ) = [2] ! (2 E θn +1 ,n )( ), byCorollary 4.5, it follows from Theorem 4.4(2) that e n f n = E θn,n +1 ( ) E θn +1 ,n ( ) = [2] ! E θn,n +1 ( )(2 E θn +1 ,n )( )= [2] ! (cid:16) v β n (2 E θn +1 ,n ,n ) − − v − (cid:2) (2 E θn +1 ,n − E θn +1 ,n )( α n ) − (2 E θn +1 ,n − E θn +1 ,n )( α − n ) (cid:3) + v β n +1 (2 E θn +1 ,n ,n ) (2 E θn +1 ,n + E θn,n +1 )( ) + v β n +2 (2 E θn +1 ,n ,n ) ( E θn +1 ,n + E θn,n +2 )( ) (cid:17) = [2] ! ( E θn +1 ,n + E θn,n +2 )( ) + [2] ! v − (2 E θn +1 ,n + E θn,n +1 )( )+ [2] ! v − − v − E θn +1 ,n ( α n ) − [2] ! v − − v − E θn +1 ,n ( α − n )Finally, compute the right hand side f n e n f n − ( vd n d − n +1 + v − d − n d n +1 ) f n . Similar to e n f n , e n f n = E θn,n +1 ( ) E θn +1 ,n ( )= E θn,n +2 ( ) + v − ( E θn +1 ,n + E θn,n +1 )( ) + v − − v − (cid:0) O ( α n ) − O ( α − n ) (cid:1) , Further, we have E θn +1 ,n ( ) E θn,n +2 ( ) = v β ′ n +2 ( E θn,n +2 ,n ) E θn +1 ,n +2 ( − α n ) + v β ′ n ( E θn,n +2 ,n ) ( E θn +1 ,n + E θn,n +2 )( )= E θn +1 ,n ( − α n ) + ( E θn +1 ,n + E θn,n +2 )( ) , This together with (6.1.2) and Theorem 4.4(1) gives f n e n f n = E θn +1 ,n ( ) (cid:16) E θn,n +2 ( ) + v − ( E θn +1 ,n + E θn,n +1 )( ) + v − − v − (cid:0) O ( α n ) − O ( α − n ) (cid:1)(cid:17) = E θn +1 ,n ( − α n ) + ( E θn +1 ,n + E θn,n +2 )( )+ [[2]]( E θn,n +1 + 2 E θn +1 ,n )( ) + 11 − v − (cid:0) v − E θn +1 ,n ( − α n ) − E θn +1 ,n ( α − n ) (cid:1) + 11 − v − (cid:0) E θn +1 ,n ( α n ) − v − E θn +1 ,n ( α − n ) (cid:1) . Again, by Theorem 4.4(1), vd n d − n +1 f n + v − d − n d n +1 f n = v O ( e n − e n +1 ) E θn +1 ,n ( ) + v − O ( − e n + e n +1 ) E θn +1 ,n ( )= E θn +1 ,n ( α n ) + E θn +1 ,n ( − α n ) . Now (6.1.1) follows from coefficient equating of both sides: Term Coefficient ( E θn,n +1 + 2 E θn +1 ,n )( ) v [[2]] + [2] ! v − = [[2]][2] . ( E θn +1 ,n + E θn,n +2 )( ) [2] ! = [2] E θn +1 ,n ( − α n ) v − − v − + v − − v − = [2](1 + v − − v − − E θn +1 ,n ( α n ) [2] ! v − − v − = [2]( − v − − E θn +1 ,n ( α − n ) − v − v − − v − − v − − [2] ! v − − v − = [2]( − − v − − v − − v − )which can be checked easily. (cid:3) Recall the canonical projection map π r : A ( n ) → S ( n, r ) Q ( v ) in Corollary 5.4. This togetherwith the above result gives the following. Corollary 6.2. There is an algebra epimorphism φ r = π r ◦ φ : U ( n ) −→ S ( n, r ) Q ( v ) . Remark 6.3. We remark that the algebra homomorphimsm φ r has been established in[ BKLW18 , Prop. 3.1] (compare [ BKLW18 , Lem. 3.12]). However, the proof there is geometricvia a geometric setting of S ( n, r ). See [ LL ] for an algebraic approach involving type B Heckealgebras of two parameters.We now use a Lusztig type form U ( n ) Z for U ( n ) and show that φ r restricts to a Z -algebraepimorphism.Define k i := O ( e i ) , if 1 ≤ i ≤ n,v − O ( e n +1 ) , if i = n + 1 . and define k i,r = π r ( k i ). Recall the notations h k i,r ;0 λ i i := Q λ i j =1 k i,r v − j +1 − k − i,r v j − v j − v − j at the end of § e λ in (3.0.1). Lemma 6.4. For any λ ∈ Λ( n + 1 , r ) , we have in S ( n, r ) n Y i =1 (cid:20) k i,r ; 0 λ i (cid:21) · (cid:20) k n +1 ,r ; 0 λ n +1 (cid:21) v = [ e λ ] . Proof. We only outline the proof; missing details can be found in [ DDPW08 , p.572]. For1 ≤ i ≤ n + 1 , ≤ j ≤ r , let D i ( j ) = X λ ∈ Λ( n +1 ,r ) λ i = j [ e λ ]Then, by (4.1.2), we have P rj =0 v j D i ( j ) = k i,r for 1 ≤ i ≤ n , and k n +1 ,r = v − X λ ∈ Λ( n +1 ,r ) v λ n +1 +1 [ e λ ] = r X j =0 v j D n +1 ( j ) . Thus, for 1 ≤ i ≤ n , (cid:20) k i,r ; 0 λ i (cid:21) = P j ≥ λ i (cid:20) jλ i (cid:21) D i ( j ), and (cid:20) k n +1 ,r ; 0 λ n +1 (cid:21) v = λ n +1 Y s =1 k n +1 ,r v − s +1) − k − n +1 ,r v s − v s − v − s = λ n +1 Y s =1 (cid:0) r X j =0 v j − s +1) − v − j − s +1) v s − v − s D n +1 ( j ) (cid:1) = r X j =0 (cid:0) λ n +1 Y s =1 v j − s +1) − v − j − s +1) v s − v − s (cid:1) D n +1 ( j ) = X j ≥ λ n +1 (cid:20) jλ n +1 (cid:21) v D n +1 ( j ) . Hence, n Y i =1 (cid:20) k i,r ; 0 λ i (cid:21) · (cid:20) k n +1 ,r ; 0 λ n +1 (cid:21) v = X j ≥ λ ,...,j n +1 ≥ λ n +1 (cid:18) n Y i =1 (cid:20) j i λ i (cid:21) · (cid:20) j n +1 λ n +1 (cid:21) v (cid:19) D ( j ) . . . D n +1 ( j n +1 ) = [ e λ ] , as desired. (cid:3) The notation in [ BKLW18 , Lem. 3.12] has been twisted by the involution ω in Lemma 2.4. NEW REALISATION OF THE i -QUANTUM GROUP U ( n ) 19 Let U ( n ) Z be the Z -subalgebra generated by divided powers e ( m ) i := e mi [ m ] ! , f ( m ) i := f mi [ m ] ! , d i , (cid:20) d i ; 0 t (cid:21) , d n +1 , (cid:20) d n +1 ; 0 t (cid:21) v for all m, t ∈ N and 1 ≤ i ≤ n .Recall the elements m A, defined in (5.0.3). This is a product of ( a i,j E θh +1 ,h )( ) for 1 ≤ j ≤ h < i ≤ n + 1 in the order defined in (5.0.1). We now define M A, by replacing the factor( a i,j E θh +1 ,h )( ) for h ≤ n by f ( a i,j ) h and replacing ( a i,j E θh +1 ,h )( ) for h > n by e ( a i,j )2 n +1 − h . Then M A, ∈ U ( n ) Z .For any A ∈ Ξ n +1 and λ ∈ N n +1 , define elements in A ( n ) Z and U ( n ) Z : m A,λ = (cid:18) n Y i =1 (cid:20) k i ; 0 λ i (cid:21) · (cid:20) k n +1 ; 0 λ n +1 (cid:21) v (cid:19) m A, , M A,λ = (cid:18) n Y i =1 (cid:20) d i ; 0 λ i (cid:21) · (cid:20) d n +1 ; 0 λ n +1 (cid:21) v (cid:19) M A, . (6.4.1) Theorem 6.5. The epimorphism φ r in Corollary 6.2 induces by restriction a Z -algebra epi-morphism φ r : U ( n ) Z −→ S ( n, r ) such that e ( m ) h ( mE θh,h +1 )( , r ) , f ( m ) h ( mE θh +1 ,h )( , r ) , d i k i,r . Proof. Let A ( n ) Z be the Z -subalgebra generated by( mE θi,i +1 )( ) , ( mE θi +1 ,i )( ) , k i , (cid:20) k i ; 0 t (cid:21) , k n +1 , (cid:20) k n +1 ; 0 t (cid:21) v for all m, t ∈ N and 1 ≤ i ≤ n . Since the epimorphism φ in Theorem 6.1 sends the generatorsfor U ( n ) Z onto the generators of A ( n ) Z , it follows that restricting to U ( n ) Z results in a Z -algebra epimorphism φ : U ( n ) Z → A ( n ) Z . On the other hand, the canonical projection π r : A ( n ) → S ( n, r ) Q ( v ) sends the generators of A ( n ) Z to elements in S ( n, r ). Thus, φ r = π r ◦ φ defines a Z -algebra homomorphism φ r : U ( n ) Z −→ S ( n, r ) . It remains to prove that φ r is surjective.For A ∈ Ξ n +1 , let A ′ is obtained from A by replacing the diagonal entries with zeros andlet m ( A ) := m A ′ , ro( A ) . Then, by definition, φ ( M ( A ) ) = m ( A ) . Now apply π r to m ( A ) . For e λ = ro( A ), by Lemma 6.4and [ BKLW18 , Thm. 3.10] (cf. the proof of Lemma 5.1), we have π r ( m ( A ) ) = π r (cid:18) n Y i =1 (cid:20) k i ; 0 λ i (cid:21) · (cid:20) k n +1 ; 0 λ n +1 (cid:21) v (cid:19) π r ( m A ′ , ) , = [ro( A )] Y ( j,h,i ) ∈ ( T n +1 , ≤ ) ( a i,j E θh +1 ,h )( , r )= [ A ] + (lower terms) . Hence, the set { π r ( m ( A ) ) | A ∈ Ξ n +1 , r +1 } spans S ( n, r ). This proves the surjectivity of φ r . (cid:3) Remarks 6.6. (1) Due to the integral nature, we may specialise Z to any commutative ring k to get a k -algebra epimorphism φ r,k : U ( n ) k → S ( n, r ) k . Thus, if k is a field, the representation category of S ( n, r ) k is a full subcategory of that of thehyperalgebra U ( n ) k of U ( n ). In this way we link representations of the i -quantum groups(or i -quantum hyperalgebras) U ( n ) k with those of the Hecke algebras of of type B .(2) Consider now representations of finite orthogonal groups G ( q ) = GL n ( F q ) in non-definingcharacteristics. Those involves in (3.0.2) are related to the unipotent principal block which canbe determined through the q -Schur algebra defined in (3.0.2). Theorem 6.5 extends furtherthis relation to i -quantum groups. It would be conceivable that much part of the classicalDipper–James theory can be generalised to finite orthogonal groups.(3) We remark that a similar relation in terms of a certain i -quantum coordinate algebra isdeveloped in [ LNX ]. See Remark 3.6(3).(4) The set (see (6.4.1)) (cid:8) d τ · · · d τ n +1 n +1 M A,λ | A ∈ Ξ n +1 , λ ∈ N n +1 , τ i ∈ { , } (cid:9) forms a Q ( v )-basis for U ( n ) (cf. [ DDPW08 , Lem. 6.47]. It is reasonable to believe that theset forms a Z -basis for the integral form U ( n ) Z .(5) The integral form of the modified i -quantum group ˙ U ( n ) and its i -canonical basis havealready been studied in [ BKLW18 ] and its appendix by Bao–Li–Wang.7. A new realisation of U ( n )The main purpose in this section is to prove that the Q ( v )-algebra homomorphism φ in Theorem 6.1 is in fact an isomorphism. Thus, we obtain a new realisation for the i -quantum group U ( n ) by explicitly presenting its regular representation in terms of a basisand multiplication formulas in Theorem 4.4, i.e., the matrix representations for generators.We explain the idea of the proof as follows. Let S (2 n + 1 , r ) = S (2 n + 1 , r ) Q ( v ) and S ( n, r ) = S ( n, r ) Q ( v ) be the q -Schur algebras over Q ( v ) of type A and B , respectively. ByTheorem 3.5(1), the algebra epimorphisms ρ r : U ( gl n +1 ) → S (2 n +1 , r ) induce a Q ( v )-algebramonomorphism (see, e.g. [ DG14 ]) ρ = Y r ≥ ρ r : U ( gl n +1 ) −→ S (2 n + 1) := Y r ≥ S (2 n + 1 , r ) . It is known from Theorem 3.5(2) that S ( n, r ) ⊂ S (2 n + 1 , r ). Thus, by the embedding ι in Lemma 2.3, both U ( n ) and S ( n ) are subalgebras of U ( gl n +1 ) (via ι ) and S (2 n + 1),respectively. The idea of proving that φ is injective is to show that φ is the restriction ρ of ρ to the image of ι . In other words, we need to prove that φ coincides with ρ via theisomorphism given in Theorem 3.5(2). Thus, we must prove that the action of U ( n ) on Ω ⊗ r via ι coincides with the action of U ( n ) on Ω ⊗ r via φ .We need some preparation. There are two cases to consider.If n ≥ r , then the basis element e ∅ := [diag( ∅ )] ∈ S ( n, r ) is an idempotent, where ∅ = (1 , . . . , | {z } r , , . . . , | {z } n − r , , . . . , | {z } n − r , , . . . , | {z } r ) ∈ N n +1 , and e ∅ S ( n, r ) e ∅ ∼ = H ( B r ), S ( n, r ) e ∅ ∼ = T ( n, r ) (see (3.0.3)). This gives an S ( n, r )- H ( B r )-bimodule structure on S ( n, r ) e ∅ . On the other hand, the tensor space Ω ⊗ r is an S ( n, r )- H ( B r )-bimodule via (3.4.3) and (3.4.4). Moreover, there is an S ( n, r )- H ( B r )-bimodule iso-morphism η r : Ω ⊗ r −→ S ( n, r ) e ∅ , ω i [ A i ] , NEW REALISATION OF THE i -QUANTUM GROUP U ( n ) 21 where i = ( i , i , . . . , i r ) ∈ I (2 n + 1 , r ) and A i = ( a k,l ) ∈ Ξ n +1 , r +1 defined for N = 2 n + 1 by a k,l = δ k,i l , if l ∈ [1 , r ]; δ k,n +1 if l = n + 1 ,a N +1 − k,N +1 − l if l ∈ [ N + 1 − r, N ]0 , for the remaining columns. (7.0.1)Note that co( A i ) = (1 r , n − r , , n − r , r ) = ∅ .We remark that the isomorphism η r is given in [ BKLW18 ] (compare [ BKLW18 , (2.9),(2.10)]with [ BKLW18 , (4.11),(4.12)], where ˜ e i is the [ A i ] here and v i is the ω i here.If n < r , then we may identify Ξ n +1 , r +1 as a subset of Ξ r +1 , r +1 via the following embed-ding: Ξ n +1 , r +1 −→ Ξ r +1 , r +1 , A = X | Y — · — Y ′ | X ′ A ◦ = X | Y · Y ′ | X ′ , (7.0.2)where X, X ′ , Y, Y ′ are n × n matrices, — · — and | · | represent the n + 1st row and column of A , and the zeros in A ◦ represent zero matrices of appropriate sizes. Thus, if n < r , we mayregard S ( n, r ) as a centraliser subalgebra of S ( r, r ) via the induces embedding [ A ] [ A ◦ ].The embedding [ A ] [ A ◦ ] induces an embedding e Λ( n + 1 , r ) = { ro( A ) | A ∈ Ξ n +1 , r +1 }−→ e Λ( r + 1 , r ) , ro( A ) ro( A ) ◦ := ro( A ◦ ) . Let f = P λ ∈ Λ( n +1 ,r ) [ e λ ◦ ]. Then there is an algebra isomorphism f S ( r, r ) f ∼ = S ( n, r ). Thisinduces an S ( n, r )- H ( B r )-bimodule isomorphism η r : Ω ⊗ r −→ f S ( r, r ) e ∅ , ω i [ A i ] . Thus, as stated in Theorem 3.5(2), η r induces an Q ( v )-algebra isomorphism in both cases: e η r : End H ( B r ) (Ω ⊗ r ) −→ S ( n, r ) . Recall the automorphism ω in Lemma 2.4. Theorem 7.1. The Q ( v ) -algebra epimorphism φ r ◦ ω : U ( n ) −→ S ( n, r ) factors through theepimorphism ρ r = ρ r ◦ ι : U ( n ) −→ End H ( B r ) (Ω ⊗ r ) , that is, e η r ◦ ρ r = φ r ◦ ω . Hence, the Q ( v ) -algebra epimorphism φ = Π r ≥ φ r : U ( n ) −→ A ( n ) ⊂ Y r ≥ S ( n, r ) is an isomorphism.Proof. The commutative relations e η r ◦ ρ r = φ r ◦ ω for all r ≥ U ( n ) ρ φ ω e η Q r ≥ End H ( B r ) (Ω ⊗ r ) U ( n ) A ( n ) ⊂ Q r ≥ S ( n, r ) where e η = Q r ≥ e η r . Since ρ = Π r ≥ ρ r : U ( gl n +1 ) → Q r ≥ End H ( S r ) (Ω ⊗ r ) is a monomorphism,it follows that the ρ is injective. The commutative diagram shows that φ is injective. Hence, φ is an isomorphism.It remains to prove e η r ◦ ρ r = φ r ◦ ω . We simply compare the actions of ι ( d i ) , ι ( e h ) , ι ( f h ) on ω i = ω i ⊗ ω i ⊗ · · · ⊗ ω i r , respectively, with the actions on [ A i ] of π r φ ω ( d i ) = O ( − e i , r ) , π r φ ω ( e h ) = E θh +1 ,h ( , r ) , π r φ ω ( f h ) = E θh,h +1 ( , r ) , where A i is defined in (7.0.1). By the embedding in (7.0.2), it suffices to assume that n ≥ r .Let N = 2 n + 1 as usual.For 1 ≤ h ≤ n and i = ( i , . . . , i r ) ∈ I ( N, r ) as in (3.4.2), we have by (2.3.1), ι ( d h ) . ( ω i ⊗ ω i ⊗ · · · ⊗ ω i r ) = K − h K − N +1 − h .ω i ⊗ · · · ⊗ K − h K − N +1 − h .ω i r = v g h ω i ⊗ ω i ⊗ · · · ⊗ ω i r where g h = −|{ k | ≤ k ≤ r, i k = h }| − |{ k | ≤ k ≤ r, i k = N + 1 − h }| .On the other hand, O ( − e h , r ) . [ A i ] = X λ ∈ Λ( n +1 ,r ) v − λ h [ e λ ] · [ A i ] = v − ro( A i ) h [ro( A i )][ A i ] = v − ro( A i ) h [ A i ] , where, by (7.0.1) and noting h = n + 1,ro( A i ) h = |{ l | l ∈ [1 , r ] , a h,l = 1 = δ h,i l } ∪ { l | l ∈ [ N − r + 1 , N ] , a h,l = 1 }| = |{ l | l ∈ [1 , r ] , i l = h } ∪ { l | l ∈ [ N − r + 1 , N ] , i N +1 − l = N + 1 − h }| = |{ k | ≤ k ≤ r, i k = h }| + |{ k | ≤ k ≤ r, i k = N + 1 − h }| = − g h Now, for h = n + 1, ι ( d n +1 ) . ( ω i ⊗ ω i ⊗ · · · ⊗ ω i r ) = v − K − n +1 .ω i ⊗ · · · ⊗ K − n +1 .ω i r | {z } r = v g n +1 ω i ⊗ ω i ⊗ · · · ⊗ ω i r where g n +1 = − − |{ k | ≤ k ≤ r, i k = n + 1 }| . But O ( − e n +1 , r ) . [ A i ] = X λ ∈ Λ( n +1 ,r ) v − e λ n +1 [ e λ ] · [ A i ] = v − ro( A i ) n +1 [ro( A i )][ A i ] = v − ro( A i ) n +1 [ A i ] , where, by (7.0.1),ro( A i ) n +1 = 2 |{ k | ≤ k ≤ r, a n +1 ,k = δ n +1 ,i k = 1 }| + a n +1 ,n +1 = 2 |{ k | ≤ k ≤ r, i k = n + 1 }| + 1 = − g n +1 . This proves e η r ◦ ρ r ( d i ) = φ r ◦ ω ( d i ) for all i = 1 , , . . . , n + 1. NEW REALISATION OF THE i -QUANTUM GROUP U ( n ) 23 We now use the short notation ω i ω i · · · ω i r for a tensor product ω i ⊗ ω i ⊗ · · · ⊗ ω i r . ByLemma 2.3, (2.3.2), and noting e K i = K i K − i +1 , we have ι ( e h ) . ( ω i ω i . . . ω i r ) = ( F h + e K − h E N − h ) . ( ω i ω i . . . ω i r )= r X l =1 e K − h ω i · · · e K − h ω i l − · F h ω i l · ω i l +1 . . . ω i r + r X l =1 e K − h ω i · · · e K − h ω i l − e K − h · E N − h ω i l · e K − h e K N − h ω i l +1 . . . e K − h e K N − h ω i r = r X l =1 il = h v f ( l ) ω i . . . ω i l − ω h +1 ω i l +1 . . . ω i r + r X l =1 il = N − h +1 ( v f ( l ) ω i . . . ω i l − ) · ( v δ N − h,h +1 − δ N − h,h ω N − h ) · ( v f ( l ) ω i l +1 . . . ω i r ) , (7.1.1)where f ( l ) = |{ k | ≤ k < l, i k = h + 1 }| − |{ k | ≤ k < l, i k = h }| and f ( l ) = −|{ k | l < k ≤ r, i k = N − h + 1 }| + |{ k | l < k ≤ r, i k = N − h }| + |{ k | l < k ≤ r, i k = h + 1 }| − |{ k | l < k ≤ r, i k = h }| . Thus, since δ N − h,h +1 = δ h,n , δ N − h,h = 0, and the summands in the second sum survive onlywhen i l = N + 1 − h , we may assume i l = h, h + 1 and f ( l ) + f ( l ) + δ N − h,h +1 − δ N − h,h = −|{ k | l < k ≤ r, i k = N − h + 1 }| + |{ k | l < k ≤ r, i k = N − h }| + |{ k | ≤ k ≤ r, i k = h + 1 }| − |{ k | ≤ k ≤ r, i k = h }| + δ h,n . On the other hand, for the same i with A i = ( a k,l ) and e µ = ro( A i ) − ( e h + e N − h +1 ), since a h,l = 1 forces a h +1 ,l = 0 and a k,n +1 = 0 for all n + 1 = k ∈ [1 , N ] by (7.0.1), it follows that E θh +1 ,h ( , r ) . [ A i ] = X λ ∈ Λ( n +1 ,r − [ E θh +1 ,h + e λ ] · [ A i ] = [ E θh +1 ,h + e µ ][ A i ]= X l ∈ [1 ,N ] ,a h,l ≥ v β ′ l ( A i ,h ) [ A i − E θh,l + E θh +1 ,l ]= X l ∈ [1 ,r ] a h,l = δ h,il =1 v β ′ l ( A i ,h ) [ A i − E θh,l + E θh +1 ,l ]+ X l ∈ [ N − r +1 ,N ] a h,l = δ N +1 − h,iN +1 − l =1 v β ′ l ( A i ,h ) [ A i − E θh,l + E θh +1 ,l ] where β ′ l ( A i , h ) = P k ≤ l a h +1 ,k − P k The i -quantum group U ( n ) is a Q ( v ) -algebra with basis { A ( j ) | A ∈ Ξ n +1 , r +1 , j ∈ Z n +1 } , NEW REALISATION OF THE i -QUANTUM GROUP U ( n ) 25 which has generators E θh,h +1 ( ) , E θh +1 ,h ( ) , O ( ± e i ) , for all ≤ h ≤ n, ≤ i ≤ n + 1 , and relations (1) O ( ± e i ) A ( j ) = v ± ro( A ) i A ( j ± e i ) together with (2) and (3) in Theorem 4.4. In other words, the multiplication formulas (1)–(3) give rise to the matrix form of the regularrepresentation of U ( n ). Remark 7.3. We expect to investigate applications of this new realisation for U ( n ). Forexample, the existence of PBW type bases for U ( n ) Z seems not clear. It is very plausible toconstruct such a basis by using the devided powers of “root vectors” E θi,j ( ) for all 1 ≤ j
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