A KLR grading of the Brauer algebras Ge Li geli@maths.usyd.edu.au - PowerPoint PPT Presentation

A KLR grading of the Brauer algebras Ge Li geli@maths.usyd.edu.au September 9, 2014 University of Sydney School of Mathematics and Statistics Ge Li geli@maths.usyd.edu.au A KLR grading of the Brauer algebras

Introduction Recently, Brundan and Kleshchev showed that cyclotomic Hecke algebras of type G ( ℓ, 1 , n ) are isomorphic to the cyclotomic Khovanov-Lauda-Rouquier algebras R Λ n introduced by Khovanov and Lauda, and Rouquier, where a connection between the representation theory of Hecke algebras and Lusztig’s canonical bases was established. In this way, cyclotomic Hecke algebras inherit a Z -grading. Ge Li geli@maths.usyd.edu.au A KLR grading of the Brauer algebras

Introduction Recently, Brundan and Kleshchev showed that cyclotomic Hecke algebras of type G ( ℓ, 1 , n ) are isomorphic to the cyclotomic Khovanov-Lauda-Rouquier algebras R Λ n introduced by Khovanov and Lauda, and Rouquier, where a connection between the representation theory of Hecke algebras and Lusztig’s canonical bases was established. In this way, cyclotomic Hecke algebras inherit a Z -grading. Hu and Mathas proved that R Λ n is graded cellular over a field, or an integral domain with certain properties, by constructing a graded cellular basis { ψ st | s , t ∈ Std( λ ) , λ ⊢ n } . Ge Li geli@maths.usyd.edu.au A KLR grading of the Brauer algebras

Introduction Recently, Brundan and Kleshchev showed that cyclotomic Hecke algebras of type G ( ℓ, 1 , n ) are isomorphic to the cyclotomic Khovanov-Lauda-Rouquier algebras R Λ n introduced by Khovanov and Lauda, and Rouquier, where a connection between the representation theory of Hecke algebras and Lusztig’s canonical bases was established. In this way, cyclotomic Hecke algebras inherit a Z -grading. Hu and Mathas proved that R Λ n is graded cellular over a field, or an integral domain with certain properties, by constructing a graded cellular basis { ψ st | s , t ∈ Std( λ ) , λ ⊢ n } . As a speical case of cyclotomic Hecke algebras, the symmetric group algebras R S n inherit the above properties. Ge Li geli@maths.usyd.edu.au A KLR grading of the Brauer algebras

Introduction Recently, Brundan and Kleshchev showed that cyclotomic Hecke algebras of type G ( ℓ, 1 , n ) are isomorphic to the cyclotomic Khovanov-Lauda-Rouquier algebras R Λ n introduced by Khovanov and Lauda, and Rouquier, where a connection between the representation theory of Hecke algebras and Lusztig’s canonical bases was established. In this way, cyclotomic Hecke algebras inherit a Z -grading. Hu and Mathas proved that R Λ n is graded cellular over a field, or an integral domain with certain properties, by constructing a graded cellular basis { ψ st | s , t ∈ Std( λ ) , λ ⊢ n } . As a speical case of cyclotomic Hecke algebras, the symmetric group algebras R S n inherit the above properties. The goal of this talk is to study the Z -grading of the Brauer algebra B n ( δ ) over a field R of characteristic p = 0, and as a byproduct, show the Brauer algebras B n ( δ ) are graded cellular algebras. Ge Li geli@maths.usyd.edu.au A KLR grading of the Brauer algebras

The Brauer algebras Ge Li geli@maths.usyd.edu.au A KLR grading of the Brauer algebras

The Brauer algebras Let R be a commutative ring with identity 1 and δ ∈ R . Ge Li geli@maths.usyd.edu.au A KLR grading of the Brauer algebras

The Brauer algebras Let R be a commutative ring with identity 1 and δ ∈ R . The Brauer algebras B n ( δ ) is a unital associative R -algebra with generators { s 1 , s 2 , . . . , s n − 1 } ∪ { e 1 , e 2 , . . . , e n − 1 } Ge Li geli@maths.usyd.edu.au A KLR grading of the Brauer algebras

The Brauer algebras Let R be a commutative ring with identity 1 and δ ∈ R . The Brauer algebras B n ( δ ) is a unital associative R -algebra with generators { s 1 , s 2 , . . . , s n − 1 } ∪ { e 1 , e 2 , . . . , e n − 1 } associated with relations (Inverses) s 2 k = 1. 1 (Essential idempotent relation) e 2 k = δ e k . 2 (Braid relations) s k s k +1 s k = s k +1 s k s k +1 and s k s r = s r s k if | k − r | > 1. 3 (Commutation relations) s k e l = e l s k and e k e r = e r e k if | k − r | > 1. 4 (Tangle relations) e k e k +1 e k = e k , e k +1 e k e k +1 = e k +1 , s k e k +1 e k = s k +1 e k 5 and e k e k +1 s k = e k s k +1 . (Untwisting relations) s k e k = e k s k = e k . 6 Ge Li geli@maths.usyd.edu.au A KLR grading of the Brauer algebras

The Brauer algebras The Brauer algebra B n ( δ ) has R -basis consisting of Brauer diagrams D , which consist of two rows of n dots, labelled by { 1 , 2 , . . . , n } , with each dot joined to one other dot. See the following diagram as an example: 1 2 3 4 5 6 D = . 1 2 3 4 5 6 Ge Li geli@maths.usyd.edu.au A KLR grading of the Brauer algebras

The Brauer algebras Two diagrams D 1 and D 2 can be composed to get D 1 ◦ D 2 by placing D 1 above D 2 and joining corresponding points and deleting all the interior loops. The multiplication of B n ( δ ) is defined by D 1 · D 2 = δ n ( D 1 , D 2 ) D 1 ◦ D 2 , where n ( D 1 , D 2 ) is the number of deleted loops. For example: 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 = δ 1 · × = 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 Ge Li geli@maths.usyd.edu.au A KLR grading of the Brauer algebras

The cyclotomic KLR algebras Ge Li geli@maths.usyd.edu.au A KLR grading of the Brauer algebras

The cyclotomic KLR algebras Suppose R is a field of characteristic p = 0 and fix δ ∈ R . Ge Li geli@maths.usyd.edu.au A KLR grading of the Brauer algebras

The cyclotomic KLR algebras Suppose R is a field of characteristic p = 0 and fix δ ∈ R . Let P = Z + δ − 1 and Γ δ be the oriented quiver with vertex set P and directed 2 edges i → i + 1, for i ∈ P . Thus, Γ δ is the quiver of type A ∞ . Ge Li geli@maths.usyd.edu.au A KLR grading of the Brauer algebras

The cyclotomic KLR algebras Suppose R is a field of characteristic p = 0 and fix δ ∈ R . Let P = Z + δ − 1 and Γ δ be the oriented quiver with vertex set P and directed 2 edges i → i + 1, for i ∈ P . Thus, Γ δ is the quiver of type A ∞ . Fix a weight Λ = Λ k for some k ∈ P . The cyclotomic KLR algebras , R Λ n of type Γ δ is the unital associative R -algebra with generators { e ( i ) | i ∈ P n } ∪ { y k | 1 ≤ k ≤ n } ∪ { ψ k | 1 ≤ k ≤ n − 1 } , and relations: � δ i 1 , k e ( i ) = 0 , e ( i ) e ( j ) = δ ij e ( i ) , i ∈ P n e ( i ) = 1 , y 1 y r e ( i ) = e ( i ) y r , ψ r e ( i ) = e ( s r · i ) ψ r , y r y s = y s y r , ψ r y s = y s ψ r , if s � = r , r + 1 , ψ r ψ s = ψ s ψ r , if | r − s | > 1 , � ( y r ψ r + 1) e ( i ) , if i r = i r +1 , ψ r y r +1 e ( i ) = y r ψ r e ( i ) , if i r � = i r +1 � ( ψ r y r + 1) e ( i ) , if i r = i r +1 , y r +1 ψ r e ( i ) = if i r � = i r +1 ψ r y r e ( i ) , Ge Li geli@maths.usyd.edu.au A KLR grading of the Brauer algebras

The cyclotomic KLR algebras  0 , if i r = i r +1 ,     e ( i ) , if i r � = i r +1 ± 1 , ψ 2 r e ( i ) =  ( y r +1 − y r ) e ( i ) , if i r +1 = i r + 1 ,    ( y r − y r +1 ) e ( i ) , if i r +1 = i r − 1 ,  if i r +2 = i r = i r +1 − 1 ,  ( ψ r +1 ψ r ψ r +1 + 1) e ( i ) ,  ψ r ψ r +1 ψ r e ( i ) = ( ψ r +1 ψ r ψ r +1 − 1) e ( i ) , if i r +2 = i r = i r +1 + 1 ,   ψ r +1 ψ r ψ r +1 e ( i ) , otherwise. for i , j ∈ P n and all admissible r and s . Moreover, R Λ n is naturally Z -graded with degree function determined by  − 2 , if i k = i k +1 ,   deg e ( i ) = 0 , deg y r = 2 and deg ψ k e ( i ) = if i k � = i k +1 ± 1, 0 ,   if i k = i k +1 ± 1. 1 , for 1 ≤ r ≤ n , 1 ≤ k ≤ n and i ∈ P n . Ge Li geli@maths.usyd.edu.au A KLR grading of the Brauer algebras

The cyclotomic KLR algebras We have an diagrammatic representation of R Λ n . Ge Li geli@maths.usyd.edu.au A KLR grading of the Brauer algebras

The cyclotomic KLR algebras We have an diagrammatic representation of R Λ n .To do this, we associate to each generator of R Λ n an P -labelled decorated planar diagram on 2 n dots in the following way: i 1 i 2 in e ( i ) = , i 1 i 2 in ir − 1 ir +1 i 1 ir in e ( i ) y r = , i 1 ir − 1 ir ir +1 in ik − 1 ik ik +1 ik +2 i 1 in e ( i ) ψ k = , i 1 ik − 1 ik +1 ik ik +2 in for i = ( i 1 , . . . , i n ) ∈ P n , 1 ≤ r ≤ n and 1 ≤ k ≤ n − 1. The labels connected by a string have to be the same. Ge Li geli@maths.usyd.edu.au A KLR grading of the Brauer algebras

The cyclotomic KLR algebras We have an diagrammatic representation of R Λ n .To do this, we associate to each generator of R Λ n an P -labelled decorated planar diagram on 2 n dots in the following way: i 1 i 2 in e ( i ) = , i 1 i 2 in ir − 1 ir +1 i 1 ir in e ( i ) y r = , i 1 ir − 1 ir ir +1 in ik − 1 ik ik +1 ik +2 i 1 in e ( i ) ψ k = , i 1 ik − 1 ik +1 ik ik +2 in for i = ( i 1 , . . . , i n ) ∈ P n , 1 ≤ r ≤ n and 1 ≤ k ≤ n − 1. The labels connected by a string have to be the same. Diagrams are considered up to isotopy, and multiplication of diagrams is given by concatenation, subject to the relations of R Λ n listed before. Ge Li geli@maths.usyd.edu.au A KLR grading of the Brauer algebras

The cyclotomic KLR algebras Theorem (Brundan-Kleshchev) The symmetric group algebras R S n are isomorphic to the cyclotomic KLR algebras R Λ n . Ge Li geli@maths.usyd.edu.au A KLR grading of the Brauer algebras