View source for Hecke algebra of a finite group

The '''Hecke algebra of a finite group''' is the [[Algebra over a field|algebra]] spanned by the [[Double coset|double coset]]s ''HgH'' of a [[Subgroup|subgroup]] ''H'' of a [[Finite group|finite group]] ''G''. It is a special case of a [[Hecke algebra of a locally compact group]].

== Definition ==
Let ''F'' be a [[Field (mathematics)|field]] of [[Characteristic (algebra)|characteristic]] zero, ''G'' a finite [[Group (mathematics)|group]] and ''H'' a subgroup of ''G''. Let <math>F[G]</math> denote the
[[Group ring|group algebra]] of ''G'': the space of ''F''-valued functions on ''G'' with the multiplication given by convolution. We write <math>F[G/H]</math> for the space of ''F''-valued functions on <math>G/H</math>. 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:
:<math>F[G/H] = F[G]^H.</math>
Similarly, there is the identification
:<math>R := \operatorname{End}_G(F[G/H]) = F[G]^{H \times H}</math>
given by sending a ''G''-linear map ''f'' to the value of ''f'' evaluated at the characteristic function of ''H''. For each double coset <math>HgH</math>, let <math>T_g</math> denote the characteristic function of it. Then those <math>T_g</math>'s form a [[Basis (linear algebra)|basis]] of ''R''.

== Application in representation theory ==
Let <math>\varphi : G \rightarrow GL(V)</math> be any finite-dimensional [[Complex number|complex]] representation of a finite group ''G'', the Hecke algebra <math>H = \operatorname{End}_G(V)</math> is the algebra of ''G''-[[Representation theory#Equivariant maps and isomorphisms|equivariant]] [[Endomorphism|endomorphisms]] of ''V''. For each [[Irreducible representation|irreducible representation]] <math>W</math> of ''G'', the [[Group action|action]] of ''H'' on ''V'' preserves <math>\tilde{W}</math> – the [[Isotypic component|isotypic component]] of <math>W</math> – and commutes with <math>W</math> as a ''G'' action.

== See also ==
*[[Gelfand pair]]

== References ==
*[[Biography:Claudio Procesi|Claudio Procesi]] (2007) ''Lie Groups: an approach through invariants and representations'', Springer, {{isbn|9780387260402}}.
*Mark Reeder (2011) Notes on representations of finite groups, [https://www.yumpu.com/en/document/read/34401000/representation-theory notes].

[[Category:Algebras]]
[[Category:Representation theory of Lie groups]]


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