aa r X i v : . [ m a t h . R T ] F e b THE BRUNDAN-KLESHCHEV ISOMORPHISM REVISITED
FAN KONG AND ZHI-WEI LI*
Abstract.
We give a short and unified proof of the Brundan-Kleshchev isomor-phism between blocks of cyclotomic Hecke algebras and cyclotomic Khovanov-Lauda-Rouquier algebras of type A. Introduction
In 2008, Brundan and Kleshchev [1] gave an explicit isomorphism between blocksof cyclotomic affine Hecke algebras of symmetric groups, or the corresponding de-generated cyclotomic affine Hecke algebras, and cyclotomic KLR algebras of typeA. On the other hand, in 1989, Lusztig showed that there is a natural isomorphismbetween affine Hecke algebras and their graded versions [5]. Motivated by Lusztig’swork, we introduced the semi-rationalization extensions of affine Hecke algebras in[4], where an isomorphism between direct sums of blocks of cyclotomic affine Heckealgebra of all types and their cyclotomic semi-rationalization algebras was given.The aim of this paper is to give a shorter proof of Brundan and Kleshchev iso-morphism using the machinery of our general result.
Main result (Theorem 5.6)
The cyclotomic KLR algebra of type A and blocks of thecyclotomic ( degenerate ) affine Hecke algebra of a symmetric group can be realized asthe same quotient of an algebra. Date : February 15, 2021.*Corresponding author. FAN KONG AND ZHI-WEI LI
It can be illustrated by the following diagram of algebras.˜ LL ˜ L (Λ) R (Λ) H (Λ) e ( C ) H q (Λ) e ( C ) ∼ = ∼ = ∼ =The algebra L , called the Lusztig algebra, is some extension of a degenerate affineHecke algebra or a non-degenerate affine Hecke algebra. This algebra contains allcorresponding rational functions, which allow us to construct the KLR-basis in aunified way either in the degenerate case or in the non-degenerate case. The algebra˜ L , called the semi-rationalization algebra, can be viewed as a subalgebra of L onlycontaining part of the rational functions. The semi-rationalization algebra ˜ L hasthree bases: the first one is the KLR basis by which we construct an isomorphismbetween the cyclotomic semi-rationalization algebra ˜ L (Λ) and the cyclotomic KLRalgebra R (Λ); the second one is the degenerate Bernstein-Zelevinski basis by whichwe give an isomorphism between ˜ L (Λ) and the cyclotomic degenerate affine Heckealgebra H (Λ); the third one is the non-degenerate Bernstein-Zelevinski basis bywhich we give an isomorphism between ˜ L (Λ) and the cyclotomic non-degenerateaffine Hecke algebra H q (Λ). 2. Preliminaries
The Demazure operator.
Let S n be the symmetric group with basic trans-positions σ , · · · , σ n − . Let k be a field. Then S n acts on the left on the polynomialring k [ X ] and on the Laurent polynomial ring k [ X ± ] in X := X , · · · , X n by per-muting variables. Using the S n -action above, the Demazure operators are defined as ∂ r on k [ x ] for all 1 ≤ r < n as ∂ r ( f ) = σ r ( f ) − fX r − X r +1 . It is well-known that the Demazure operators satisfy the Leibniz rule ∂ r ( f g ) = ∂ r ( f ) g + σ r ( f ) ∂ r ( g ) , for all f, g ∈ k [ X ] and for 1 ≤ r < n , and the relations σ r ( ∂ r ( f )) = ∂ r ( f ) , ∂ r ( σ r ( f )) = − ∂ r ( f ) . HE BRUNDAN-KLESHCHEV ISOMORPHISM REVISITED 3
Let k ( X ) be the corresponding rational function field, then the S n -action on k [ X ]above can be extended to an action w : fg w ( f ) w ( g ) of S n on k ( X ) (by the fieldautomorphism). This means that the action of the Demazure operators on k [ X ]also extends to operators on k ( X ).Let F be the quotient field of the subalgebra Z := k [ X ] S n = { f ∈ k [ X ] | w ( f ) = f for every w ∈ S n } of S n -invariants. By [5, 3.12 (a)], there is a natural k -algebra isomorphism k [ X ] ⊗ Z F → k ( X ) , f ⊗ g f g. (2.1)2.2. Index sets.
Let I = Z /e Z = { , · · · , e − } ( e = 0 or 2 ≤ e ∈ Z ) be the vertexset of the quiver of type A ∞ if e = 0 or A (1) e − if e ≥ A ∞ : · · · − − · · · A (1) e − : 0 1 2 0 1 30 12 34 0 12 · · · For 1 ≤ r < n , we define the map σ r ( i ) s = i σ r ( s ) . Then S n acts on the set of n -tuples i = ( i , · · · , i n ) ∈ I n by the place permutation : w ( i ) s = i w − ( s ) .Throughout this paper, We fix an S n -orbit C of I n .3. Affine Hecke algebras and their rationalizations
The degenerate affine Hecke algebra.
Following [3], the degenerate affineHecke algebra H of S n is defined to be the associated unital k -algebra with genera-tors { X , · · · , X n , T , · · · , T n − } subject to the following relations for all admissibleindices: X r X s = X s X r ; (3.1) T r X s = σ r ( X s ) T r + ∂ r ( X s ); (3.2) T r = 1; (3.3) T r T s = T s T r if | r − s | >
2; (3.4) T r T r +1 T r = T r +1 T r T r +1 ; (3.5) FAN KONG AND ZHI-WEI LI
For w = σ r σ r · · · σ r m ∈ S n a reduced expression we put T w := T r T r · · · T r m .Then T w is a well-defined element in H and the algebra H is a free k [ X ]-modulewith the basis { T w | w ∈ S n } .3.2. The non-degenerate affine Hecke algebra.
Assume 1 = q ∈ k × . We definethe affine Hecke algebra H q to be the associated unital k -algebra with generators { X ± , · · · , X ± n , T , · · · , T n − } subject to the following relations for all admissibleindices: X ± r X ± s = X ± s X ± r , X r X − r = X − r X r = 1; (3.6) T r X s = σ r ( X s ) T r + ( q − X r +1 ∂ r ( X s ); (3.7) T r = ( q − T r + q ; (3.8) T r T s = T s T r if | r − s | >
2; (3.9) T r T r +1 T r = T r +1 T r T r +1 ; (3.10)Similar to the degenerate case, H q is a free k [ X ± ]-module with the basis { T w | w ∈ S n } .3.3. The rationalization.
By [5, Proposition 3.11], the center of H is the commu-tative algebra Z of S n -invariants. Thus H can be seen as a Z -subalgebra (identifiedwith the subspace H ⊗
1) of the F -algebra H F := H ⊗ Z F .Similarly, the non-degenerate affine Hecke algebra H q also can be seen as a Z -subalgebra (identified with the subspace H q ⊗
1) of the F -algebra H q,F := H q ⊗ Z F . For any 1 ≤ r < n and f ∈ k ( X ), as a consequence of [5, 3.12 (d)], we have T r f = σ r ( f ) T r + ∂ r ( f ) if f ∈ H F ,σ r ( f ) T r + ( q − X r +1 ∂ r ( f ) if f ∈ H q,F . (3.11)In light of the Bernstein and Zelevinski basis of H and H q , there are decomposi-tions, see [5, 3.12(c)]: H F (resp. , H q,F ) = ⊕ w ∈ S n T w k ( x ) = ⊕ w ∈ S n k ( x ) T w . (3.12)3.4. Intertwining elements.
For 1 ≤ r < n , we define the intertwining element κ r as follows: κ r := T r + X r − X r +1 in H F ,T r + ( q − X r +1 X r − X r +1 in H q,F . HE BRUNDAN-KLESHCHEV ISOMORPHISM REVISITED 5
The elements κ r have the following properties [5, Proposition 5.2]: κ r f = σ r ( f ) κ r ∀ f ∈ k ( X ); (3.13) κ r = − X r − X r +1 ) in H F ,q − ( q − X r X r +1 ( X r − X r +1 ) in H q,F ; (3.14) κ r κ s = κ s κ r if s = r, r + 1; (3.15) κ r κ r +1 κ r = κ r +1 κ r κ r +1 . (3.16)If w = σ r σ r · · · σ r m is a reduced expression in S n , then we can define κ w := κ r κ r · · · κ r m in H F (resp., H q,F ). It is also a well-defined element by the braid rela-tions (3.15) and (3.16). By the decompositions (3.12), we know that { κ w | w ∈ S n } is a basis of H F (resp., H q,F ) as free k ( X )-module.4. The Lusztig extensions of affine Hecke algebras
From now on, we fix I = Z /e Z , where e = char k for the degenerate affine Heckealgebras, and e is the smallest positive integer such that 1 + q + · · · + q e − = 0 andsetting e = 0 if no such integer exists for the non-degenerate affine Hecke algebras.4.1. The Lusztig extensions.
Let E be the unital k -algebra with basis { ǫ ( i ) | i ∈C} . Multiplication is given by ǫ ( i ) ǫ ( j ) = δ ij ǫ ( i ) . (4.1)The Lusztig extension of H with respect to E is the k -algebra L which is equal as k -space to the tensor product L := H F ⊗ k E = ⊕ w ∈ S n , i ∈C κ w k ( X ) ǫ ( i )of the rationalization algebra H F and the semi-simple algebra E . Multiplication isdefined so that H F (identified with the subspace H F ⊗
1) and E (identified with thesubspace 1 ⊗ E ) are subalgebras of L , and in addition X r ǫ ( i ) = ǫ ( i ) X r , (4.2) κ r ǫ ( i ) = ǫ ( σ r ( i )) κ r , (4.3)for all 1 ≤ r < n and i ∈ C . FAN KONG AND ZHI-WEI LI
The
Lusztig extension L q of H q with respect to E is defined similarly. We havethe following relations T r ǫ ( i ) = ǫ ( σ r ( i )) T r + X r − X r +1 ( ǫ ( σ r ( i )) − ǫ ( i )) in L ,ǫ ( σ r ( i )) T r + ( q − X r +1 X r − X r +1 ( ǫ ( σ r ( i )) − ǫ ( i )) in L q . (4.4)4.2. The Brundan-Kleshchev auxiliary elements.
Recall that [1, (3.12), (4.21)],for each 1 ≤ r ≤ n , Brundan and Kleshchev introduced the element y r := P i ∈C ( X r − i r ) ǫ ( i ) in L , P i ∈C (1 − q − i r X r ) ǫ ( i ) in L q . (4.5)Then y r ǫ ( i ) = ǫ ( i ) y r by 4.2, and y r is a unit in L with y − r = P i ∈C ( X r − i r ) − ǫ ( i ) in L , P i ∈C (1 − q − i r X r ) − ǫ ( i ) in L q . Let k [ y ] be the polynomial ring with y := y , y , · · · , y n and k ( y ) be the rationalfunction field. Lemma 4.6. k ( X ) ⊗ k E = k ( y ) ⊗ k E in L or L q as subalgebras.Proof. For the degenerate case, if g ( y ) is a polynomial in k [ y ], then g ( y ) = X i ∈C g ( y ) ǫ ( i ) = X i ∈C g ( X − i , · · · , X n − i n ) ǫ ( i )in L ( H , F, C ). Therefore, if g ( y ) = 0, then g ( X − i , · · · , X n − i n ) = 0 in k [ X ] for all i ∈ C . Thus g ( y ) − = P i ∈C g ( X − i , · · · , X n − i n ) − ǫ ( i ) exists in L . So k ( y ) ⊗ k E is a subalgebra of L .Combining (4.5) with X r = P i ∈C ( y r + i r ) ǫ ( i ), we know that k ( X ) ⊗ k E = k ( y ) ⊗ k E in L .Imitate the proof above, we can prove the non-degenerate case. (cid:3) For each 1 ≤ r < n , Brundan and Kleshchev defined the element Q r ( i ) = y r +1 − y r if i r = i r +1 , i r − i r +1 = 1 , e = 2 , y r +1 − y r (1+ y r +1 − y r ) if i r − i r +1 = − , e = 2 , y r +1 − y r if i r − i r +1 = 1 , e = 2 , − y r − y r +1 + i r − i r +1 if i r − i r +1 = 0 , ± HE BRUNDAN-KLESHCHEV ISOMORPHISM REVISITED 7 in L , and Q r ( i ) = − q + qy r +1 − y r if i r = i r +1 ,q i r if i r − i r +1 = 1 , e = 2 , − q + q y r +1 − y r q ir (1 − q + qy r +1 − y r ) if i r − i r +1 = − , e = 2 , − q + qy r +1 − y r if i r − i r +1 = 1 , e = 2 , q ir − ir +1 (1 − y r ) − q (1 − y r +1 ) q ir − ir +1 (1 − y r ) − (1 − y r +1 ) if i r − i r +1 = 0 , ± , in L q .Notice that the symmetric group S n acts on the left on k [ y ] by permuting variables: σ r ( y s ) = P i ∈C ( σ r ( X s ) − i s ) ǫ ( σ r ( i )) in L P i ∈C (1 − q − i r σ r ( X s )) ǫ ( σ r ( i )) in L q = y σ r ( s ) , where 1 ≤ r < n and 1 ≤ s ≤ n .The following Lemma is proved in [1, (3.27)-(3.29), (4.33)-(4.35)]. Lemma 4.7.
Let ≤ r < n and i ∈ C . Then σ r ( Q r ( σ r ( i ))) Q r ( i ) = − ( y r +1 − y r ) if i r − i r +1 = 0; y r − y r +1 (1+ y r − y r +1 ) if i r − i r +1 = 1 , e = 2; y r +1 − y r (1+ y r +1 − y r ) if i r − i r +1 = − , e = 2; y r +1 − y r ) if i r − i r +1 = 1 , e = 2;1 − y r − y r +1 + i r − i r +1 ) if i r − i r +1 = 0 , ± . in L , and in L q , σ r ( Q r ( σ r ( i ))) Q r ( i )= (1 − q + qy r +1 − y r )(1 − q + qy r − y r +1 ) if i r = i r +1 ; q (1 − q + q y r − y r +1 )(1 − q + qy r − y r +1 ) if i r − i r +1 = 1 , e = 2; q (1 − q + q y r +1 − y r )(1 − q + qy r +1 − y r ) if i r − i r +1 = − , e = 2; − q + qy r − y r +1 )(1 − q + qy r +1 − y r ) if i r − i r +1 = 1 , e = 2; q − (1 − q ) q ir + ir +1 (1 − y r +1 )(1 − y r )[ q ir +1 (1 − y r +1 ) − q ir (1 − y r )] if i r = 0 , ± . Surprisingly, either in L or in L q , we can set the same named elements θ r := κ r X i ∈C Q − r ( i ) ǫ ( i ) , which share the same relations. FAN KONG AND ZHI-WEI LI
Lemma 4.8.
For each ≤ r < n and i ∈ C , we have θ r ǫ ( i ) = ǫ ( σ r ( i )) θ r , (4.9) f θ r = θ r σ r ( f ) ∀ f ∈ k ( y ) , (4.10) θ r θ s = θ s θ r if s = r, r + 1 , (4.11) θ r θ r +1 θ r = θ r +1 θ r θ r +1 , (4.12) θ r ǫ ( i ) = − y r +1 − y r ) ǫ ( i ) if i r = i r +1 , ( y r − y r +1 ) ǫ ( i ) if i r − i r +1 = 1 , e = 2 , ( y r +1 − y r ) ǫ ( i ) if i r − i r +1 = − , e = 2 , ( y r − y r +1 )( y r +1 − y r ) ǫ ( i ) if i r − i r +1 = 1 , e = 2 ,ǫ ( i ) if i r − i r +1 = 0 , ± . (4.13) Proof.
The assertions follow straightforwardly from (3.13)-(3.16), (4.9), Lemma 4.7and (3.14). (cid:3)
The KLR basis of the Lusztig extensions.
For each 1 ≤ r < n , set ψ r = X i ∈C [ θ r − δ ir +1 ir y r − y r +1 ] ǫ ( i ) . (4.14) Proposition 4.15.
The Lusztig extension L or L q is generated by { y , · · · , y n , ψ , · · · , ψ n − , f − , ǫ ( i ) | = f ∈ k [ y ] , i ∈ C} subject to the following relations. f f − = f − f = 1 , ∀ = f ∈ k [ y ]; (4.16) y r ǫ ( i ) = ǫ ( i ) y r , y r y s = y s y r ; (4.17) ψ r ǫ ( i ) = ǫ ( σ r ( i )) ψ r ; (4.18) ψ r y s ǫ ( i ) = [ σ r ( y s ) ψ r + δ i r i r +1 ∂ r ( y s )] ǫ ( i ); (4.19) ψ r ǫ ( i ) = if i r − i r +1 = 0 , ( y r − y r +1 ) ǫ ( i ) if i r − i r +1 = 1 , e = 2 , ( y r +1 − y r ) ǫ ( i ) if i r − i r +1 = − , e = 2 , − ( y r − y r +1 ) ǫ ( i ) if i r − i r +1 = 1 , e = 2 ,ǫ ( i ) if i r = 0 , ± . (4.20) ψ r ψ s = ψ s ψ r , if s = r, r + 1; (4.21) HE BRUNDAN-KLESHCHEV ISOMORPHISM REVISITED 9 ( ψ r ψ r +1 ψ r − ψ r +1 ψ r ψ r +1 ) ǫ ( i )= − ǫ ( i ) if i r +2 = i r , i r − i r +1 = 1 , e = 2 ,ǫ ( i ) if i r +2 = i r , i r − i r +1 = − , e = 2 , − ( y r + y r +2 − y r +1 ) ǫ ( i ) if i r +2 = i r , i r − i r +1 = 1 , e = 2 , else . (4.22) Proof.
The relations (4.16) and (4.17) can be proved by straightforward calculationsusing Lemma 4.6.As a result of (4.9) and (4.14), we see that ψ r ǫ ( i ) = ( θ r − δ irir +1 y r − y r +1 ) ǫ ( i ) = ǫ ( σ r ( i ))( θ r − δ irir +1 y r − y r +1 ) = ǫ ( σ r ( i )) ψ r , where the third identity holds since σ r ( i ) = i whenever i r = i r +1 .By (4.14) and (4.10), we obtain ψ r y s ǫ ( i ) = ( θ r − δ irir +1 y r − y r +1 ) y s ǫ ( i )= ( σ r ( y s ) θ r − δ irir +1 y s y r − y r +1 ) ǫ ( i )= ( σ r ( y s ) ψ r + δ irir +1 σ r ( y s ) y r − y r +1 − δ irir +1 y s y r − y r +1 ) ǫ ( i )= ( σ r ( y s ) ψ r + δ i r i r +1 ∂ r ( y s )) ǫ ( i ) . If s = r, r + 1, using (4.18), (4.14), (4.10) and (4.11), then we have ψ r ψ s ǫ ( i ) = ψ r ǫ ( σ s ( i )) ψ s ǫ ( i )= ( θ r − δ irir +1 y r − y r +1 )( θ s − δ isis +1 y s − y s +1 ) e ( i )= ( θ r θ s − δ irir +1 y r − y r +1 θ s − θ r δ isis +1 y s − y s +1 + δ irir +1 δ isis +1 ( y r − y r +1 )( y s − y s +1 ) ) ǫ ( i )= ( θ s θ r − θ s δ irir +1 y r − y r +1 − δ isis +1 y s − y s +1 θ r + δ irir +1 δ isis +1 ( y r − y r +1 )( y s − y s +1 ) ) ǫ ( i )= ( θ s − δ isis +1 y s − y s +1 )( θ r − δ irir +1 y r − y r +1 ) ǫ ( i )= ψ s ψ r ǫ ( i )So ψ r ψ s = ψ s ψ r .As a result of (4.14), (4.18) and Lemma 4.13, we get ψ r ǫ ( i ) = ψ r ǫ ( σ r ( i )) ψ r ǫ ( i )= ( θ r − δ ir +1 ir y r − y r +1 )( θ r − δ irir +1 y r − y r +1 ) ǫ ( i ) FAN KONG AND ZHI-WEI LI = ( θ r + δ irir +1 ( y r − y r +1 ) ) ǫ ( i )= i r − i r +1 = 0 , ( y r − y r +1 ) ǫ ( i ) if i r − i r +1 = 1 , e = 2 , ( y r +1 − y r ) ǫ ( i ) if i r − i r +1 = − , e = 2 , ( y r − y r +1 )( y r +1 − y r ) ǫ ( i ) if i r − i r +1 = 1 , e = 2 ,ǫ ( i ) if i r = 0 , ± . In the light of (4.18), (4.14) and (4.10), there holds that ψ r ψ r +1 ψ r ǫ ( i )= ( θ r − δ ir +1 ir +2 y r − y r +1 )( θ r +1 − δ irir +2 y r +1 − y r +2 )( θ r − δ irir +1 y r − y r +1 ) ǫ ( i )= [ θ r θ r +1 θ r − δ irir +1 y r +1 − y r +2 θ r θ r +1 − δ ir +1 ir +2 y r − y r +1 θ r +1 θ r + δ irir +1 δ ir +1 ir +2 ( y r − y r +2 )( y r +1 − y r +2 ) θ r + δ irir +1 δ ir +1 ir +2 ( y r − y r +1 )( y r − y r +2 ) θ r +1 − δ irir +2 y r − y r +2 θ r − δ irir +1 δ irir +2 δ ir +1 ir +2 ( y r − y r +1 ) ( y r +1 − y r +2 ) ] ǫ ( i )Similarly, we have ψ s ψ r ψ s ǫ ( i )= ( θ r +1 − δ irir +1 y r +1 − y r +2 )( θ r − δ irir +2 y r − y r +1 )( θ r +1 − δ ir +1 ir +2 y r +1 − y r +2 ) ǫ ( i )= [ θ r +1 θ r θ r +1 − δ irir +1 y r +1 − y r +2 θ r θ r +1 − δ ir +1 ir +2 y r − y r +1 θ r +1 θ r + δ irir +1 δ ir +1 ir +2 ( y r − y r +2 )( y r +1 − y r +2 ) θ r + δ irir +1 δ ir +1 ir +2 ( y r − y r +1 )( y r − y r +2 ) θ r +1 − δ irir +2 y r − y r +2 θ r +1 − δ irir +1 δ irir +2 δ ir +1 ir +2 ( y r − y r +1 )( y r +1 − y r +2 ) ] ǫ ( i )By (4.12) and 4.13, we arrive at[ ψ r ψ r +1 ψ r − ψ r +1 ψ r ψ r +1 ] ǫ ( i )= δ i r i r +2 [ y r − y r +2 ( θ r +1 − θ r ) + δ irir +1 ( y r + y r +2 − y r +1 )( y r − y r +1 ) ( y r +1 − y r +2 ) ] ǫ ( i )= − ǫ ( i ) if i r +2 = i r , i r − i r +1 = 1 , e = 2 ,ǫ ( i ) if i r +2 = i r , i r − i r +1 = − , e = 2 , − ( y r + y r +2 − y r +1 ) ǫ ( i ) if i r +2 = i r , i r − i r +1 = 1 , e = 2 , . To finish the proof of the theorem, we need to prove the relations (4.16)-(4.22)generate all relations. In fact, for each w ∈ S n , we fix a reduced decomposition HE BRUNDAN-KLESHCHEV ISOMORPHISM REVISITED 11 w = σ r σ r · · · σ r m and define the element ψ w := ψ r ψ r · · · ψ r m ∈ L . Note that ψ w in general does depend on the choice of reduced decomposition of w [2, Proposition 2.5]. Since ψ r = θ r − X i ∈C δ ir +1 ir y r − y r +1 ǫ ( i ) = κ r X i ∈C Q − r ( i ) ǫ ( i ) − X i ∈C δ ir +1 ir y r − y r +1 ǫ ( i )and { κ w | w ∈ S n } is a k ( x )-basis of H F , we can show that { ψ w | w ∈ S n } isa k ( y ) ⊗ k E -basis of L by Lemma 4.6 since every element in L can be written as P w ∈ S n , i ∈C ψ w f w, i ( y ) ǫ ( i ) with f w, i ( y ) in k ( y ) by the relations (4.16)-(4.22). Thereforethe generating set { y , · · · , y n , ψ , · · · , ψ n − , f − , ǫ ( i ) | = f ∈ k [ y ] , i ∈ C} subject to relations (4.16)-(4.22) is complete.The same proof above is valid for L q , and then we are done. (cid:3) According to the Proposition above, we will not distinguish between Lusztig ex-tensions in the sequel.4.4.
The cyclotomic KLR algebras.
Recall [1, Subsection 2.2], the
KLR algebraof type A is defined to be the k -algebra R generated by { y , · · · , y n , ψ , · · · , ψ n , ǫ ( i ) | i ∈ C , } subject to the relations (4.16)-(4.22) of Proposition 4.15. Thus R can be viewed asa subalgebra of L .From now on, we fix an index Λ = (Λ i ) i ∈ I ∈ N I (we follow the convention that N = { , , , · · · } ) with P i ∈ I Λ i < ∞ . We call the quotient R (Λ) := R / h y Λ i ǫ ( i ) | i ∈ Ci a cyclotomic KLR algebra of type A .Recall that, Brund and Kleshchev proved the following result in [1, Lemma 2.1]. Lemma 4.23.
The elements y r are nilpotent in R (Λ) . FAN KONG AND ZHI-WEI LI
The semi-rationalizations.
We define the semi-rationalization algebra as the k -algebra ˜ L generated by { y , · · · , y n , ψ , · · · , ψ n − , f − ( y ) , ǫ ( i ) | i ∈ C , f ( y ) ∈ k [ y ] with f (0) = 0 } subject to the relations (4.16)-(4.22) of Proposition 4.15.The following result shows that we can construct a complete generating set fromthe degenerate affine Hecke algebras or from the non-degenerate affine Hecke alge-bras. This is the key point for proving the BK isomorphism in the sequel. Theorem 4.24. (1)
Denote by ǫ ( i ) = ǫ ( i ) . Then the algebra ˜ L is generated by { X , · · · , X n , T , · · · , T n − , f − ( X ) ǫ ( i ) | i ∈ C , f ( X ) ∈ k [ X ] with f ( i ) = 0 } subject to relations (3.1)-(3.5), (4.1), (4.2), (4.4) and for f ∈ k [ X ] with f ( i ) = 0 ǫ ( j ) · f − ǫ ( i ) = δ ji ǫ ( i ) = f − ǫ ( i ) · ǫ ( j ) , f · f − ǫ ( i ) = ǫ ( i ) = f − ǫ ( i ) · f. (4.25)(2) Denote by ǫ ( i ) = ǫ ( i ) . Then the algebra ˜ L is generated by { X , · · · , X n , T , · · · , T n − , f − ( X ) ǫ ( i ) | i ∈ C , f ( X ) ∈ k [ X ] with f ( q i ) = 0 } subject to relations (3.6)-(3.10), (4.1), (4.2), ( ?? ) and for f ∈ k [ X ] with f ( q i ) = 0 ǫ ( j ) · f − ǫ ( i ) = δ ji ǫ ( i ) = ( f − ǫ ( i ) · ǫ ( j ) , f · f − ǫ ( i ) = ǫ ( i ) = f − ǫ ( i ) · f. (4.26) Proof. (1) There is an obvious homomorphism α : ˜ L → L by sending generators tothe same named generators. This homomorphism is injective since using relations(4.16)-(4.22), every element in ˜ L can be written as X w ∈ S n , i ∈C ψ w f w, i ( y ) g w, i ( y ) ǫ ( i )with f w, i ( y ) , g w, i ( y ) in k [ y ] and g w, i (0) = 0, and { ψ w | w ∈ S n } is a k ( y ) ⊗ k E -basis of L . Thus Im α = ⊕ w ∈ S n ψ w P ( y, E ), where P ( y, E ) is the commutative algebra { f g − | f, g ∈ k [ y ] , g (0) = 0 } ⊗ k E .Assume A is the k -algebra generated by { X , · · · , X n , T , · · · , T n − , f − ( X ) ǫ ( i ) | i ∈ C , f ( X ) ∈ k [ X ] with f ( i ) = 0 } subject to relations (3.1)-(3.5), (4.1), (4.2), (4.4) and (4.25). Then, using theserelations, every element in A can be written as X w ∈ S n , i ∈C T w f w, i ( X ) · g − w, i ( X ) ǫ ( i ) HE BRUNDAN-KLESHCHEV ISOMORPHISM REVISITED 13 with f w, i ( X ) , g w, i ( X ) in k [ X ] and g w, i ( i ) = 0. Thus there also has an injectivehomomorphism α ′ : A → L by sending generators to the same named generators.Therefore Im α ′ = ⊕ w ∈ S n T w P ( X, E ), where P ( X, E ) is the commutative algebra { f · g − ǫ ( i ) | i ∈ C , f, g ∈ k [ X ] , g ( i ) = 0 } .We claim that Im α = Im α ′ . In fact, by Lemma 4.6, P ( X, E ) = P ( y, E ) in L .Notice that ψ r = X i ∈C i r = i r +1 ( T r + y r − y r +1 + i r − i r +1 ) Q − r ( i ) ǫ ( i ) + X i ∈C i r = i r +1 ( T r + 1) Q − r ( i ) ǫ ( i )and Q r ( i ) , Q − r ( i ) ∈ P ( y, E ) in L , we get Im α = Im α ′ , and then A = ˜ L .(2) Imitate the proof of the statement (1). (cid:3) Lemma 4.27. ˜ L / h y Λ i ǫ ( i ) | i ∈ Ci = ˜ L / h Q i ∈ I ( X − i ) Λ i i for H , ˜ L / h Q i ∈ I ( X − q i ) Λ i i for H q . Proof.
For the degenerate case, by the relation of y and X , we have Y i ∈ I ( X − i ) Λ i = X j ∈C Y i ∈ I ( y + j − i ) Λ i ǫ ( j )= X j ∈C Y i ∈ I,i = j ( y + j − i ) Λ i y Λ j ǫ ( j )is in h y Λ j ǫ ( j ) | j ∈ Ci . By Theorem (4.24) (1), Q i ∈ I,i = j [( X − i ) Λ i ] − ǫ ( j ) is in ˜ L ,thus y Λ j ǫ ( j ) = ( X − j ) Λ j ǫ ( j )= Y i ∈ I ( X − i ) Λ i Y i ∈ I,i = j [( X − i ) Λ i ] − ǫ ( j )is in h Q i ∈ I ( X − i ) Λ i i . Hence h y Λ i ǫ ( i ) | i ∈ Ci = h Q i ∈ I ( X − i ) Λ i i in ˜ L . For the non-degenerate case, by the relation of y and X , we get that Y i ∈ I ( X − q i ) Λ i = X j ∈C Y i ∈ I ( q j − q i − q j y ) Λ i ǫ ( j )= X j ∈C Y i ∈ I,i = j ( − q j ) Λ j ( q j − q i − q j y ) Λ i y Λ j ǫ ( j )is in h y Λ j ǫ ( j ) | j ∈ Ci . By Theorem (4.24) (1), Q i ∈ I,i = j [( X − q i ) Λ i ] − ǫ ( j ) is in ˜ L ,thus y Λ j ǫ ( j ) = − q − j ( X − q j ) Λ j ǫ ( j ) FAN KONG AND ZHI-WEI LI = − q − j Y i ∈ I ( X − q i ) Λ i Y i ∈ I,i = j [( X − q i ) Λ i ] − ǫ ( j )is in h Q i ∈ I ( X − q i ) Λ i i . So h y Λ i ǫ ( i ) | i ∈ Ci = h Q i ∈ I ( X − q i ) Λ i i in ˜ L , and then weare done. (cid:3) Remark . Denote by ˜ L (Λ) := ˜ L / h y Λ i ǫ ( i ) | i ∈ Ci . Similar to the proof of Lemma 4.23, the elements y r are nilpotent in ˜ L (Λ). Thereforethe elements Q i ∈ I ( X r − i ) (resp., Q i ∈ I ( X r − q i )) are also nilpotent in ˜ L (Λ).The following result is important for us. Lemma 4.29.
We have k -algebra isomorphsim R (Λ) ∼ = ˜ L (Λ) .Proof. First note that if f ( y ) ∈ k [ y ] with f (0) = 0, the polynomial f ( y ) − f (0) isnilpotent in ˜ L (Λ) by the nilpotency of y r ’s. So there exists some g ( y ) ∈ k [ y ] and m ∈ N such that g ( y ) m = 0 and f − ( y ) = f (0) − P ml =0 g ( y ) l in R (Λ). Thus thehomomorphism R ֒ → ˜ L ։ ˜ L (Λ) is surjective and induces a surjective homomor-phism π : R (Λ) → ˜ L (Λ). Let ˜ F be the localization of the commutative ring k [ y ] S n of S n -invariants in k [ y ] with respect to { f ∈ k [ y ] S n | f (0) = 0 } . Similar to the proof of(2.1), we have ˜ L = R ⊗ k [ y ] Sn ˜ F by Theorem 4.24. Since the elements y r are nilpotentin R (Λ), similar to the proof above, we know that if f ( y ) ∈ K [ y ] S n with f (0) = 0,then it is a unit in R (Λ). Thus the homomorphism K [ y ] S n ֒ → R π ։ ˜ R (Λ) inducesa morphism π : ˜ F → R (Λ). Therefore we have an induced algebra homomorphism π ⊗ π : ˜ L → R (Λ). The homomorphism π ⊗ π induces an algebra homomorphism π ′ : ˜ L (Λ) → R (Λ). It is easy to check that π and π ′ are two-sided inverses. So R (Λ) ∼ = ˜ L (Λ). (cid:3) The Brundan-Kleshchev isomorphism
Cyclotomic degenerate affine Hecke algebras.
Recall that the cyclotomicdegenerate affine Hecke algebra is defined as H (Λ) := H / h Y i ∈ I ( X − i ) Λ i i . By [1, Subsection 3.1], there is a system { e ( i ) | i ∈ C} of mutually orthogonalidempotents in H (Λ) such that 1 = P i ∈ I n e ( i ) and e ( i ) H (Λ) = { h ∈ H (Λ) | ( X r − i r ) m h = 0 for all 1 ≤ r ≤ n and m ≫ } . HE BRUNDAN-KLESHCHEV ISOMORPHISM REVISITED 15
It is easy to check that X r e ( i ) = e ( i ) X r for all 1 ≤ r ≤ n and i ∈ C , and the element f ( X ) e ( i ) with f ( X ) ∈ k [ X ] is a unit in e ( i ) H (Λ) if and only if f ( i ) = 0. In thiscase, we write f ( X ) − e ( i ) for the inverse. Lemma 5.1.
For ≤ r < n and i ∈ I n , we have that T r e ( i ) = e ( i ) T r if i r +1 = i r ,e ( σ r ( i )) T r + ( X r − X r +1 ) − ( e ( σ r ( i )) − e ( i )) if i r +1 = i r . (5.2) Proof.
For any 1 ≤ s ≤ n , the element( σ r ( X s ) − σ ( i ) s ) e ( i ) = ( X s − i s ) e ( i ) if s = r, r + 1,( X r +1 − i r +1 ) e ( i ) if s = r, ( X r − i r ) e ( i ) if s = r + 1 . is nilpotent in H (Λ) e ( i ). Similarly, we can show that ∂ r (( X s − i s ) m ) e ( i ) = 0 by thebinomial theorem for an integer m ≫ i r = i r +1 . Therefore, if i r = i r +1 ,we get from (3.11) that( X s − i s ) m T r e ( i ) = T r ( σ r ( X s ) − i s ) m e ( i ) + ∂ r (( X s − i s ) m ) e ( i ) = 0whenever m ≫
0. Thus T r e ( i ) ∈ e ( i ) H (Λ) and then T r e ( i ) = e ( i ) T r e ( i ) = e ( i ) T r . If i r +1 = i r , as a result of (3.11), it holds that( X s − σ r ( i ) s ) m [ T r ( X r − X r +1 ) + 1] e ( i ) = [ T r ( X r − X r +1 ) + 1]( σ r ( X s ) − σ r ( i ) s ) m e ( i )= 0whenever m ≫
0. Therefore T r ( X r − X r +1 ) e ( i ) + e ( i ) = e ( σ r ( i ))[ T r ( X r − X r +1 ) + 1]Then right-multiplying by ( X r − X r +1 ) − e ( i ), we obtain T r e ( i ) = e ( σ r ( i )) T r e ( i ) − ( X r − X r +1 ) − e ( i ) . Similarly, we can prove that e ( σ r ( i )) T r = e ( σ r ( i )) T r e ( i ) − ( X r − X r +1 ) − e ( σ r ( i )) . Therefore T r e ( i ) = e ( σ r ( i )) T r + ( X r − X r +1 ) − ( e ( σ r ( i )) − e ( i )) . (cid:3) FAN KONG AND ZHI-WEI LI
Cyclotomic non-degenerate affine Hecke algebras.
Recall that the non-degenerate cyclotomic affine Hecke algebra H q (Λ) is H q (Λ) := H q / h Y i ∈ I ( X − q i ) Λ i i . Similar to [1, Subsection 4.1], there is a system { e ( i ) | i ∈ C} of mutually orthogonalidempotents in H q (Λ) such that 1 = P i ∈ I n e ( i ) and e ( i ) H q (Λ) = { h ∈ H q (Λ) | ( X r − q i r ) m h = 0 for all 1 ≤ r ≤ n and m ≫ } . Accordingly, it holds that X r e ( i ) = e ( i ) X r for all 1 ≤ r ≤ n and i ∈ C , and theelement f ( X ) e ( i ) with f ( X ) ∈ k [ X ] is a unit in e ( i ) H q (Λ) if and only if f ( q i ) = 0.We write f ( X ) − e ( i ) for the inverse in this case.Similar to the proof of Lemma 5.1, we can show that Lemma 5.3.
For ≤ r < n and i ∈ I n , we have T r e ( i ) = e ( i ) T r if i r = i r +1 ,e ( σ r ( i )) T r + (1 − q ) X r +1 X r +1 − X r ( e ( σ r ( i )) − e ( i )) if i r = i r +1 . (5.4)Let e ( C ) := X i ∈C e ( i ) . Then e ( C ) is a primitive central idempotent by (5.2) and (5.4).5.3. The BK isomorphism.
By Lemma 4.27, Theorem (4.24) and 5.1, (5.4)thereis a homomorphism ρ : ˜ L (Λ) → H (Λ) e ( C )(resp . H q (Λ) e ( C ))sending the generators X r , T r to the same named elements, and f − ( X ) ǫ ( i ) with f ( i ) = 0 (resp. f − ( X ) ǫ ( i ) with f ( q i ) = 0) to f − ( X ) e ( i ). We have the followingimportant result. Lemma 5.5. ρ is an algebra isomorphism.Proof. We only prove the claim for degenerate case since the proof of non-degeneratecase is similar. Apparently, ρ is surjective, so we only need to construct a left-inverseof ρ . By Lemma 4.27, there is a homomorphism τ : H (Λ) → ˜ L (Λ) HE BRUNDAN-KLESHCHEV ISOMORPHISM REVISITED 17 sending the generators X r , T r to the same named elements. Let i ∈ C and j ∈ I n . If i = j , then there is some 1 ≤ r ≤ n such that j r = i r . We claim that ǫ ( i ) τ ( e ( j )) = 0. In fact, by the construction of e ( j ), there is an integer m ≫ X r − j r ) m e ( j ) = 0. Thus we have( X r − j r ) m ǫ ( i ) τ ( e ( j )) = ǫ ( i ) τ (( X r − j r ) m e ( j )) = 0 . The assumption j r = i r implies that the element ( X r − j r ) − ǫ ( i ) ∈ ˜ L (Λ). Thus weget ǫ ( i ) τ ( e ( j )) = ( X r − j r ) − m ( X r − j r ) m ǫ ( i ) τ ( e ( j )) = ( X r − j r ) − m ǫ ( i )0 = 0 . Therefore, if j ∈ I n \ C we have τ ( e ( j )) = P i ∈C ǫ ( i ) τ ( e ( j )) = 0 . Therefore, if j ∈ C ,we obtain τ ( e ( j )) = X i ∈C ǫ ( i ) τ ( e ( j )) = ǫ ( j ) τ ( e ( j )) = ǫ ( j ) X i ∈ I n τ ( e ( i )) = ǫ ( j ) τ (1) = ǫ ( j ) . These show that τ | H (Λ) e ( C ) : H (Λ) e ( C ) → ˜ L (Λ) is an algebra homomorphism. Itis easy to check that τ ρ is the identity on each generator of ˜ L (Λ). Thus ρ is anisomorphism. (cid:3) By Lemma 4.29 and Lemma 5.5, we have arrive at our main result.
Theorem 5.6.
The cyclotomic KLR algebra of type A and blocks of the cyclotomic ( degenerate ) affine Hecke algebra of a symmetric group can be realized as the samequotient of an algebra. References [1] J. Brundan and A. Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Laudaalgebra. Invent. math. (3) (2009), 451-484.[2] J. Brundan, A. Kleshchev and Weiqiang Wang, The graded Specht modules. J. reine angew.Math. (2011), 61-87.[3] V. Drinfeld, Degenerate affine Hecke algebras and Yangians, Func. Anal. Appl. (1986),56-58.[4] Fan Kong and Zhi-Wei Li, The semi-rationalizations of affine Hecke algebras, preprint.[5] G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. (1989),599-635. Fan Kong, School of Mathematics and Statistics, Southwest University, Chongqing400715, P. R. China.
Email address : [email protected] FAN KONG AND ZHI-WEI LI
Zhi-Wei Li, School of Mathematics and Statistics, Jiangsu Normal University,Xuzhou 221116 Jiangsu, P. R. China.
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