SchurElements( H )
returns the list of constants arising from the Schur relations for the irreducible characters of the Iwahori-Hecke algebra H.
The Schur element corresponding to an irreducible character chi is
also equal to P/D_{chi} where P is the Poincare polynomial and
D_{chi} is the generic degree of chi. Note, however, that this only
works if D_{chi} neq 0. (We can have D_{chi}=0 if the parameters
of H are suitably chosen roots of unity, for example.) The ordering of
the degrees corresponds to the ordering of the characters as returned by
the function CharTable. Note that the Schur element corresponding to
the ind-character is the Poincare polynomial P.
gap> u := X( Rationals );; u.name := "u";;
gap> v := X( LaurentPolynomialRing( Rationals ) );; v.name := "v";;
gap> schur := SchurElements( Hecke( CoxeterGroup( "G", 2 ),
> [ u ^ 2, v ^ 2 ], [ u, v ] ) );
[ (u^6 + u^4)*v^6 + (u^6 + 2*u^4 + u^2)*v^4 + (u^4 + 2*u^2 + 1)*v^
2 + (u^2 + 1), (1 + u^(-2)) + (1 + 2*u^(-2) + u^(-4))*v^(
-2) + (u^(-2) + 2*u^(-4) + u^(-6))*v^(-4) + (u^(-4) + u^(-6))*v^(
-6), (u^(-4) + u^(-6))*v^6 + (u^(-2) + 2*u^(-4) + u^(-6))*v^4 + (
1 + 2*u^(-2) + u^(-4))*v^2 + (1 + u^(-2)),
(u^2 + 1) + (u^4 + 2*u^2 + 1)*v^(-2) + (u^6 + 2*u^4 + u^2)*v^(
-4) + (u^6 + u^4)*v^(-6), (2*u^0)*v^2 + (2*u - 2*u^(-1))*v + (2*u^
2 - 2 + 2*u^(-2)) + (-2*u + 2*u^(-1))*v^(-1) + (2*u^0)*v^(-2),
(2*u^0)*v^2 + (-2*u + 2*u^(-1))*v + (2*u^2 - 2 + 2*u^(-2)) + (2*u -
2*u^(-1))*v^(-1) + (2*u^0)*v^(-2) ]
The Poincaraccent19 e polynomial is just the Schur element corresponding to the trivial (or index) representation:
gap> schur[1];
(u^6 + u^4)*v^6 + (u^6 + 2*u^4 + u^2)*v^4 + (u^4 + 2*u^2 + 1)*v^
2 + (u^2 + 1)
(But note that the trivial character is not always the first character!) For further information about generic degrees and connections with the representation theory of finite groups of Lie type, see BC72 and Car85.
This function requires the package "chevie" (see RequirePackage).
GAP 3.4.4