Trace identities and central polynomials in the matrix superalgebras~$M_{n,k}$

Yu. P. Razmyslov

Sbornik. Mathematics, Tome 56 (1987) no. 1, pp. 187-206

Résumé

A complete description is given of trace identities for matrix superalgebras $M_{n,k}=\biggl\{\begin{pmatrix} a_{11} a_{12} \\ a_{21} a_{22} \end{pmatrix}\biggr\}$, where $a_{11}$ and $a_{22}$ are square matrices of orders $n$ and $k$ respectively over the even elements of a Grassmann algebra $G$ with countably many generators, while $a_{12}$ and $a_{21}$ are $n\times k$ and $k\times n$ rectangular matrices respectively over the odd elements of $G$. A relation is found between multilinear trace identities of degree $ l$ in the algebra $M_{n,k}$ and irreducible representations of a symmetric group of order $(l+1)!\,$. It is proved that over a field of characteristic zero all trace identities of $M_{n,k}$ follow from identities of degree $nk+n+k$ that hold in that algebra. For every algebra $M_{n,k}$ over a field of arbitrary characteristic a central polynomial is given explicitly. Bibliography: 7 titles.