A class of Schur algebras

Author:
M. Brender

Journal:
Trans. Amer. Math. Soc. **248** (1979), 435-444

MSC:
Primary 20C05

DOI:
https://doi.org/10.1090/S0002-9947-1979-0522268-7

MathSciNet review:
522268

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Abstract: This paper delineates a class of Schur algebras over a finite group *G*, parametrized by two subgroups . The constructed Schur algebra is maximal for the two properties (a) centralizing the elements of *H*, and (b) containing the elements of *K* in the identity. Most commonly considered examples of Schur algebras fall into this class.

A complete set of characters of is given in terms of the spherical functions on the group *G* with respect to the subgroup *H*. Necessary and sufficient conditions are given for this Schur algebra to be commutative, in terms of a condition on restriction multiplicities of characters. This leads to a second-orthogonality-type relation among a subset of the spherical functions. Finally, as an application, a particular Schur algebra of this class is analyzed, and shown to be a direct sum of centralizer rings.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1979-0522268-7

Keywords:
Schur algebra,
*S*-ring,
semisimple algebra,
finite group,
spherical function,
character,
double coset algebra,
centralizer ring

Article copyright:
© Copyright 1979
American Mathematical Society