# 81 Iwahori-Hecke algebras

In this chapter we describe functions for dealing with Iwahori-Hecke algebras associated to finite Coxeter groups.

Let W be a finite Coxeter group, with generators S={s_1,ldots,s_n}. As before, let m(i,j) denote the order of the product s_is_j. Let R be a commutative ring with~1 and q_1,ldots,q_n be elements in R such that q_i=q_j whenever m(i,j) is odd. Thus, we have q_i=q_j whenever s_i and s_j are conjugate in W. The corresponding Iwahori-Hecke algebra with parameters {q_i} is a deformation of the group algebra of W over R where the multiplication of basis elements involves the parameters q_i and where the deformation is trivial if all q_i are equal to 1.

More precisely, the Iwahori-Hecke algebra H=H(W,R, {q_i}) is the associative R-algebra with 1 = T_1 generated by elements T_{s_1},ldots,T_{s_n} subject to the following relations. [ beginarrayccll (T_s_i-q_i)(T_s_i+1) & = & 0 & mbox for all; i
T_s_iT_s_jT_s_i cdots & = & T_s_jT_s_iT_s_j cdots & mbox for i neq j and with m(i,j) factors on each side. endarray ] Since the generators T_{s_i} satisfy the braid relations, the algebra H is in fact a quotient of the group algebra of the braid group associated with W. It follows that, if w=s_{i_1} cdots s_{i_m}= s_{j_1} cdots s_{j_m} are two reduced expressions of w in W as products of fundamental reflections then the corresponding products of the generators T_{s_i} respectively T_{s_j} will give the same element of H, which we may therefore denote by T_w. Then the elements {T_w mid w in W} actually form a free R-basis of H. The multiplication of two arbitrary basis elements T_v,T_w (for v,w in W) is then performed as follows. Choose a reduced expression for v, say v=s_{i_1} cdots s_{i_k}. Then T_v is the product of the corresponding generators T_{s_i} hence we are reduced to the case where v=s_i for some i. In this case, we have [ T_s_iT_w = left{ beginarraycl T_s_iw & mbox if l(s_iw)=l(w)+1
q_iT_s_iw+(q_i-1)T_w & mbox if l(s_iw)=l(w)-1. endarrayright. ] There is a universal choice for R and {q_i}: Let u_1,ldots,u_n be indeterminates over QQ such that u_i=u_j whenever m(i,j) is odd, and let A_0={ZZ}[u_1,ldots,u_n] be the corresponding polynomial ring. Then H_0:=H(W,A_0,{u_i}) is called the generic Iwahori-Hecke algebra associated with W. If R and {q_i} are given as above then H(W,R,{q_i}) can be obtained by specialization from H_0: There is a unique ring homomorphism f:A_0 rightarrow R such that f(u_i)=q_i for all i. Then we can view R as an A_0-module via f and we can identify H(W,R,{q_i})=R otimes _{A_0} H_0.

If all u_i are equal we call the corresponding algebra the one-parameter Iwahori-Hecke algebra associated with W. Certain invariants associated with the irreducible characters of this algebra play a special role in the representation theory of the underlying finite Coxeter groups, namely the a- and A-invariants which were already used in chapter Character tables for Coxeter groups (see LowestPowerGenericDegrees, JInductionTable).

For basic properties of Iwahori-Hecke algebras and their relevance to the representation theory of finite groups of Lie type, we refer to CR87, Sections~67 and 68.

In the following example, we compute the multiplication table for the 0-Iwahori--Hecke algebra associated with the Coxeter group of type A_2.

```    gap> W := CoxeterGroup( "A", 2 );
CoxeterGroup("A", 2)```

Algebra with all parameters equal to 0:

```    gap> H := Hecke( W, 0 );
Hecke(CoxeterGroup("A", 2),[ 0, 0 ],[  ])```

Create the T-basis:

```    gap> T := Basis( H, "T" );
function ( arg ) ... end
gap> el := CoxeterWords( W );
[ [  ], [ 2 ], [ 1 ], [ 2, 1 ], [ 1, 2 ], [ 1, 2, 1 ] ]```

Multiply any two T-basis elements:

```    gap> mat := []; for i in [1..6] do mat[i]:=[]; for j in [1..6] do
> Add( mat[i], T( el[i]) * T( el[j] ) ); od; od;
gap> PrintArray(mat);
[ [     T(),       T(2),      T(1),    T(2,1),    T(1,2),  T(1,2,1) ],
[    T(2),      -T(2),    T(2,1),   -T(2,1),  T(1,2,1), -T(1,2,1) ],
[    T(1),     T(1,2),     -T(1),  T(1,2,1),   -T(1,2), -T(1,2,1) ],
[  T(2,1),   T(1,2,1),   -T(2,1), -T(1,2,1), -T(1,2,1),  T(1,2,1) ],
[  T(1,2),    -T(1,2),  T(1,2,1), -T(1,2,1), -T(1,2,1),  T(1,2,1) ],
[ T(1,2,1), -T(1,2,1), -T(1,2,1),  T(1,2,1),  T(1,2,1), -T(1,2,1) ] ]```
Thus, we can not only work with generic algebras where the parameters are indeterminates. In the following chapter we will see that this also works on the level of characters and representations.

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GAP 3.4.4
April 1997