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.
GAP 3.4.4