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Trace identities and central polynomials in the matrix superalgebras $M_{n,k}$
Yu. P. Razmyslov
Abstract:
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.
UDC:
512
MSC: 16A38 Received: 15.02.1984
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