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Inspired by the Capelli identities for group determinants obtained by T^oru Umeda, we give a basis of the center of the group algebra of any finite group by using Capelli identities for irreducible representations. The Capelli identities for irreducible representations are modifications of the Capelli identity. These identities lead to Capelli elements of the group algebra. These elements construct a basis of the center of the group algebra.
We give a further extension and generalization of Dedekinds theorem over those presented by Yamaguchi. In addition, we give two corollaries on irreducible representations of finite groups and a conjugation of the group algebra of the groups which have an index-two abelian subgroups.
For a finite dimensional unital complex simple Jordan superalgebra $J$, the Tits-Kantor-Koecher construction yields a 3-graded Lie superalgebra $mathfrak g_flatcong mathfrak g_flat(-1)oplusmathfrak g_flat(0)oplusmathfrak g_flat(1)$, such that $mathfr
Let $G$ be a complex simple Lie group and let $g = hbox{rm Lie},G$. Let $S(g)$ be the $G$-module of polynomial functions on $g$ and let $hbox{rm Sing},g$ be the closed algebraic cone of singular elements in $g$. Let ${cal L}s S(g)$ be the (graded) id
Given a complex reflection group W we compute the support of the spherical irreducible module of the rational Cherednik algebra of W in terms of the simultaneous eigenfunction of the Dunkl operators and Schur elements for finite Hecke algebras.
Let $Z$ be the symmetric cone of $r times r$ positive definite Hermitian matrices over a real division algebra $mathbb F$. Then $Z$ admits a natural family of invariant differential operators -- the Capelli operators $C_lambda$ -- indexed by partitio