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Register a new userCapelli elements of the group algebra
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Authors
Naoya Yamaguchi
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No Arabic abstract
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.
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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 $mathfrak g_flat(-1)cong J$. Set $V:=mathfrak g_flat(-1)^*$ and $mathfrak g:=mathfrak g_flat(0)$. In most cases, the space $mathcal P(V)$ of superpolynomials on $V$ is a completely reducible and multiplicity-free representation of $mathfrak g$, with a decomposition $mathcal P(V):=bigoplus_{lambdainOmega}V_lambda$, where $left(V_lambdaright)_{lambdainOmega}$ is a family of irreducible $mathfrak g$-modules parametrized by a set of partitions $Omega$. In these cases, one can define a natural basis $left(D_lambdaright)_{lambdainOmega}$ of Capelli operators for the algebra $mathcal{PD}(V)^{mathfrak g}$. In this paper we complete the solution to the Capelli eigenvalue problem, which is to determine the scalar $c_mu(lambda)$ by which $D_mu$ acts on $V_lambda$. We associate a restricted root system $mathit{Sigma}$ to the symmetric pair $(mathfrak g,mathfrak k)$ that corresponds to $J$, which is either a deformed root system of type $mathsf{A}(m,n)$ or a root system of type $mathsf{Q}(n)$. We prove a necessary and sufficient condition on the structure of $mathit{Sigma}$ for $mathcal{P}(V)$ to be completely reducible and multiplicity-free. When $mathit{Sigma}$ satisfies the latter condition we obtain an explicit formula for the eigenvalue $c_mu(lambda)$, in terms of Sergeev-Veselovs shifted super Jack polynomials when $mathit{Sigma}$ is of type $mathsf{A}(m,n)$, and Okounkov-Ivanovs factorial Schur $Q$-polynomials when $mathit{Sigma}$ is of type $mathsf{Q}(n)$.
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) ideal defining $hbox{rm Sing},g$ and let $2r$ be the dimension of a $G$-orbit of a regular element in $g$. Then ${cal L}^k = 0$ for any $k<r$. On the other hand, there exists a remarkable $G$-module $Ms {cal L}^r$ which already defines $hbox{rm Sing},g$. The main results of this paper are a determination of the structure of $M$.
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 partitions $lambda$ of length at most $r$, whose eigenvalues are given by specialization of Knop--Sahi interpolation polynomials. In this paper we consider a double fibration $Y longleftarrow X longrightarrow Z$ where $Y$ is the Grassmanian of $r$-dimensional subspaces of $mathbb F^n $ with $n geq 2r$. Using this we construct a family of invariant differential operators $D_{lambda,s}$ on $Y$ that we refer to as quadratic Capelli operators. Our main result shows that the eigenvalues of the $D_{lambda,s}$ are given by specializations of Okounkov interpolation polynomials.
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