A class of Schur algebras

Brender, M.

doi:10.1090/S0002-9947-1979-0522268-7

A class of Schur algebras
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by M. Brender PDF

Trans. Amer. Math. Soc. 248 (1979), 435-444 Request permission

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.

References

Charles W. Curtis and Timothy V. Fossum, On centralizer rings and characters of representations of finite groups, Math. Z. 107 (1968), 402–406. MR 237665, DOI 10.1007/BF01110070
John Karlof, The subclass algebra associated with a finite group and subgroup, Trans. Amer. Math. Soc. 207 (1975), 329–341. MR 367040, DOI 10.1090/S0002-9947-1975-0367040-2

The subclass algebra associated with a finite group and subgroup

Olaf Tamaschke, $S$-rings and the irreducible representations of finite groups, J. Algebra 1 (1964), 215–232. MR 167541, DOI 10.1016/0021-8693(64)90020-1
Olaf Tamaschke, On Schur-rings which define a proper character theory on finite groups, Math. Z. 117 (1970), 340–360. MR 280617, DOI 10.1007/BF01109852
Dennis Travis, Spherical functions on finite groups, J. Algebra 29 (1974), 65–76. MR 340399, DOI 10.1016/0021-8693(74)90111-2
Helmut Wielandt, Zur Theorie der einfach transitiven Permutationsgruppen. II, Math. Z. 52 (1949), 384–393 (German). MR 33817, DOI 10.1007/BF02230702
Helmut Wielandt, Finite permutation groups, Academic Press, New York-London, 1964. Translated from the German by R. Bercov. MR 0183775