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Transactions of the American Mathematical Society

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A class of Schur algebras

Author: M. Brender
Journal: Trans. Amer. Math. Soc. 248 (1979), 435-444
MSC: Primary 20C05
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 $ K\, \triangleleft \,H\, \subset \,G$. The constructed Schur algebra $ {\text{C}}\left[ G \right]_K^H$ 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 $ {\text{C}}\left[ G \right]_K^H$ 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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Keywords: Schur algebra, S-ring, semisimple algebra, finite group, spherical function, character, double coset algebra, centralizer ring
Article copyright: © Copyright 1979 American Mathematical Society

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