Central polynomials and matrix invariants
Since the early 70's when Formanek and Razmyslov discovered central polynomials for matrices, many new central polynomials have been constructed. In the paper under review the authors consider multilinear polynomials \(F^{\lambda,\mu}\) of degree \(2n^2\), skew-symmetric in two disjoint sets of \(n^2\) variables and indexed by a pair of partitions \(\lambda,\mu\) of \(n^2\). These polynomials appeared first in \textit{A. Regev} [Commun. Algebra 8, 1417-1467 (1980; Zbl 0437.16012)] and are known to be either polynomial or central identities for the \(n\times n\) matrix algebra \(M_n(K)\) over a field \(K\) of characteristic 0. The purpose of the present paper is to describe conditions which guarantee that \(F^{\lambda,\mu}\) is a central polynomial. It turns out that this happens only in the case when \(\lambda\) and \(\mu\) are derived in a suitable way from the partition \(\delta=(1,3,\ldots,2n-3,2n-1)\). The authors give an explicit formula which involves characters of the symmetric group and allows to evaluate \(F^{\lambda,\mu}\) in \(M_n(K)\). As an application they give a new class of central polynomials for \(M_n(K)\). The proof of the main result is based on techniques from invariant theory of matrices and develops further ideas of \textit{E. Formanek} [J. Algebra 109, 93-114 (1987; Zbl 0625.16015)].
scientific article; zbMATH DE number 997620
scientific article