Hecke algebra of a finite group
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 [math]\displaystyle{ F[G] }[/math] denote the group algebra of G: the space of F-valued functions on G with the multiplication given by convolution. We write [math]\displaystyle{ F[G/H] }[/math] for the space of F-valued functions on [math]\displaystyle{ 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]\displaystyle{ F[G/H] = F[G]^H. }[/math]
Similarly, there is the identification
- [math]\displaystyle{ 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]\displaystyle{ HgH }[/math], let [math]\displaystyle{ T_g }[/math] denote the characteristic function of it. Then those [math]\displaystyle{ T_g }[/math]'s form a basis of R.
Application in representation theory
Let [math]\displaystyle{ \varphi : G \rightarrow GL(V) }[/math] be any finite-dimensional complex representation of a finite group G, the Hecke algebra [math]\displaystyle{ H = \operatorname{End}_G(V) }[/math] is the algebra of G-equivariant endomorphisms of V. For each irreducible representation [math]\displaystyle{ W }[/math] of G, the action of H on V preserves [math]\displaystyle{ \tilde{W} }[/math] – the isotypic component of [math]\displaystyle{ W }[/math] – and commutes with [math]\displaystyle{ W }[/math] 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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