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.

Definition

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. We write 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 , let denote the characteristic function of it. Then those 's form a basis of R.

Application in representation theory

Let 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 of G, the action of H on V preserves – the isotypic component of – and commutes with as a G action.

See also

References

  • Claudio Procesi (2007) Lie Groups: an approach through invariants and representations, Springer, ISBN 9780387260402.
  • Mark Reeder (2011) Notes on representations of finite groups, notes.
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