The Iwahori-Hecke algebra of type <math> <mi>A</mi> </math>

The Iwahori-Hecke algebra of type A

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 28 May 2010

The Iwahori-Hecke algebra of type A

Fix q∈ ℂ* . The Iwahori-Hecke algebra Hn = Hn q of type A n-1 is the associative algebra with 1 given by generators T1 ,…, T n-1 and relations Ti Tj = Tj Ti ,if   i-j >1, Ti T i+1 Ti = T i+1 Ti T i+1 ,1≤i≤n-2, T i2 =1,1≤i≤n-1. If q=1 then Hn =ℂ Sn , the group algebra of the symmetric group.

The Iwahori-Hecke algebra Hn of type A as basis Tw ,w∈ Sn .

dim Hn ≤n!

Proof.
We will show two things.
  1. Every element of Hn can be written as a linear combination of elements of the form w1 T n-1 w2 , where w1 , w2 ∈ H n-1 .
  2. Every element of Hn can be written as a linear combination of elements of the form a T n-1 … Tl , where 1≤l≤n and a∈ H n-1 .
  1. Every element of Hn can be written as a linear combination of elements of the form w1 T n-1 w2 T n-2 w3 T n-1 … w l-1 T n-1 wl ,where   wj ∈ H n-1 . Then, by induction, w2 is a linear combination of elements of the form a T k-2 b where a,b∈ H n-2 . So T n-1 w2 T n-1 = T n-1 a T n-2 b T n-1 =a T n-1 T n-2 T n-1 b=a T n-2 T n-1 T n-2 b, since all elements of H n-2 commute with T n-1 . So w1 T n-1 w2 T n-2 w3 T n-1 … wl T n-1 wl is a linear combination of elements w1' T n-1 w2' T n-2 w4 T n-1 w5 … w l-1 T n-1 wl , where w1' = w1 a T n-1 and w2' = T n-2 b w3 are in H n-1 . In this way the number of T n-1 factors has benn reduced by one. Thus, by induction, h is a linear combination of elements h' ∈ H n-1 and elements h1 T n-1 h2 with h1 , h2 ∈ H n-1 .
  2. By (a), any element h∈ H n-1 can be written as a linear combination of elements of h1 T n-1 h2 with h1 , h2 ∈ H k-2 . By induction, h2 can be written as a linear combination of elements of the form a T n-2 T n-3 … Tl = h1 a T n-1 T n-2 … Tl ,where   h1 a∈ H n-1 . So every element h∈ H n-1 is a linear combination of elements h' ∈ H n-1 and elements of the form h1 T n-1 … Tl ,with   h1 ∈ H n-1 . So dim Hn <dim H n-1 .n. Thus, by induction, dim Hn ≤n!.□

Assume that qk ≠1 for all k=2,…,n.

  1. The irreducible representations Hλ of the Iwahori-Hecke algebra Hn of type A are indexed by partitions λ with n boxes.
  2. dim Hλ = # of standard tableaux of shape λ.
  3. The irreducible Hn -module Hλ is given by the vector space Hλ =ℂ-span vT | T  is a standard tableau of shape  λ with basis vT and with Hn action given by Ti vT q- q -1 1- q 2 c T i -c T i+1 + q -1 + q- q -1 1- q 2 c T i -c T i+1 v si T , where
    c T i is the content of the box containing i in T,
    si T is the same as T except that i and i+1 are switched,
    v si T =0 if si T is not standard.

The proof of this theorem is exactly analogous to the proof of theorem ???? once one knows what the analogues of the Murphy elements are in this setting.

Define elements X εk ,1≤k≤n, in the Iwahori-Hecke algebra Hn of type A by X ε1 =1,  and   X εk = T k-1 T k-2 … T2 T12 T2 … T k-2 T k-1 ,2≤k≤n. The action of X εk on the Hn -module Hλ is given by X εk vT = q 2c T k vT ,for all standard tableaux  T, where c T k is the content of the box containing k in T.

Proof.
The proof is by induction on k using the relation X εk = T k-1 X ε k-1 T k-1 . Clearly, X ε1 vT = vT = q 2c T 1 vT , since c T 1 for all standard tableaux T. By the induction assumption and the definition of the action of T k-1 on Hλ , X εk vT = T k-1 X ε k-1 T k-1 vT = T k-1 q 2c T k-1 q- q -1 1- q 2 c T k -c T k-1 vT + q 2c T k q -1 + q- q -1 1- q 2 c T k -c T k-1 v s k-1 T = q 2c T k T k-1 T k-1 - q- q -1 vT = q 2c T k vT . □

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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