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=Hnq of type An-1 is the associative algebra with 1 given by generators T1,…,Tn-1 and relations TiTj=TjTi,if  i-j>1,TiTi+1Ti=Ti+1TiTi+1,1≤i≤n-2,Ti2=1,1≤i≤n-1. If q=1 then Hn=ℂSn, the group algebra of the symmetric group.

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

dimHn≤n!

		Proof.
	We will show two things. Every element of Hn can be written as a linear combination of elements of the form w1Tn-1w2, where w1,w2∈Hn-1. Every element of Hn can be written as a linear combination of elements of the form aTn-1…Tl, where 1≤l≤n and a∈Hn-1. Every element of Hn can be written as a linear combination of elements of the form w1Tn-1w2Tn-2w3Tn-1…wl-1Tn-1wl,where  wj∈Hn-1. Then, by induction, w2 is a linear combination of elements of the form aTk-2b where a,b∈Hn-2. So Tn-1w2Tn-1=Tn-1aTn-2bTn-1=aTn-1Tn-2Tn-1b=aTn-2Tn-1Tn-2b, since all elements of Hn-2 commute with Tn-1. So w1Tn-1w2Tn-2w3Tn-1…wlTn-1wl is a linear combination of elements w1'Tn-1w2'Tn-2w4Tn-1w5…wl-1Tn-1wl, where w1'=w1aTn-1 and w2'=Tn-2bw3 are in Hn-1. In this way the number of Tn-1 factors has benn reduced by one. Thus, by induction, h is a linear combination of elements h'∈Hn-1 and elements h1Tn-1h2 with h1,h2∈Hn-1. By (a), any element h∈Hn-1 can be written as a linear combination of elements of h1Tn-1h2 with h1,h2∈Hk-2. By induction, h2 can be written as a linear combination of elements of the form aTn-2Tn-3…Tl=h1aTn-1Tn-2…Tl,where  h1a∈Hn-1. So every element h∈Hn-1 is a linear combination of elements h'∈Hn-1 and elements of the form h1Tn-1…Tl,with  h1∈Hn-1. So dimHn<dimHn-1.n. Thus, by induction, dimHn≤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. dimHλ=# of standard tableaux of shape λ.
3. The irreducible Hn-module Hλ is given by the vector space Hλ=ℂ-spanvT|T  is a standard tableau of shape  λ with basis vT and with Hn action given by TivTq-q-11-q2cTi-cTi+1+q-1+q-q-11-q2cTi-cTi+1vsiT, where
   cTi is the content of the box containing i in T,
   siT is the same as T except that i and i+1 are switched,
   vsiT=0 if siT 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 Xεk,1≤k≤n, in the Iwahori-Hecke algebra Hn of type A by Xε1=1,  and  Xεk=Tk-1Tk-2…T2T12T2…Tk-2Tk-1,2≤k≤n. The action of Xεk on the Hn-module Hλ is given by XεkvT=q2cTkvT,for all standard tableaux  T, where cTk is the content of the box containing k in T.

		Proof.
	The proof is by induction on k using the relation Xεk=Tk-1Xεk-1Tk-1. Clearly, Xε1 vT=vT=q2cT1vT, since cT1 for all standard tableaux T. By the induction assumption and the definition of the action of Tk-1 on Hλ, XεkvT=Tk-1Xεk-1Tk-1vT=Tk-1q2cTk-1q-q-11-q2cTk-cTk-1vT+q2cTkq-1+q-q-11-q2cTk-cTk-1vsk-1T=q2cTkTk-1Tk-1-q-q-1vT=q2cTkvT.□

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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