I think I've found a (rather alarming) bug regarding Iwahori-Hecke algebras:
sage: L.<q>=LaurentPolynomialRing(ZZ)
sage: H=IwahoriHeckeAlgebra('A1',q^2)
sage: T=H.T(); Cp=H.Cp()
sage: T(q*Cp[1])
T[1] + 1
sage: M=MatrixSpace(H,1,1)
sage: M(q)*M(Cp[1])
[q*T[1]]
sage: q*T[1]==T[1]+1
False
Briefly, sage does not appear to multiply in the Iwahori-Hecke algebra
correctly when doing matrix multiplication! This affects larger matrices
too. The bug goes away if everything is in the T() basis or in the Cp()
basis.
~Amit
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