Double coset algebras

Abstract

For a finite monoid M with unit e and an indexed family G={Gi:i∈I} of subgroups of the group G(M) of invertible elements in M, the complex vector space A(M,G) with basis the double cosets of the form GimGj, m∈M, has a natural multiplication yielding an associative C-algebra which we call a double coset algebra. We construct two Z-algebras with identity, LGS(M,G) and RGS(M,G), called the left and right generalized Schur algebras, which as Z-modules are free with basis the double cosets. When M is the symmetric group Sr and G is the family of Young subgroups indexed by compositions of r with at most n parts, A(M,G) is isomorphic to the usual Schur algebra S(n,r) and LGS(M,G) corresponds to its standard Z-form.The structure constants we derive for these algebras provide multiplication rules useful for further analysis of the generalized Schur algebras. Our main result is then the observation that LGS(M,G) and RGS(M,G) are always Z-forms for A(M,G), that is, A(M,G)≅C⊗ZLGS(M,G)≅C⊗ZRGS(M,G).The Iwahori–Hecke algebra H(M,G) corresponding to a finite monoid M and subgroup G is isomorphic to the double coset algebra A(M,G) where G={G} consists of a single set. So, as a special case of our result, we obtain Z-forms LGS(M,G) and RGS(M,G) for arbitrary Iwahori–Hecke algebras. The existence of a Z-form for any such H(M,G) appears to be a new result.