The Hecke algebra of a finite group is the algebra spanned by the double cosets HgH of a subgroup H of a finite group G. It is a special case of a Hecke algebra of a locally compact group.
Let F be a field of characteristic zero, G a finite group and H a subgroup of G. Let
denote the group algebra of G: the space of F-valued functions on G with the multiplication given by convolution.
for the space of F-valued functions on
An (F-valued) function on G/H determines and is determined by a function on G that is invariant under the right action of H. That is, there is the natural identification: Similarly, there is the identification given by sending a G-linear map f to the value of f evaluated at the characteristic function of H. For each double coset
denote the characteristic function of it.
be any finite-dimensional complex representation of a finite group G, the Hecke algebra
is the algebra of G-equivariant endomorphisms of V. For each irreducible representation
– the isotypic component of