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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Minimal identities of symmetric matrices

Authors: Wen Xin Ma and Michel L. Racine
Journal: Trans. Amer. Math. Soc. 320 (1990), 171-192
MSC: Primary 16A38; Secondary 17C05
MathSciNet review: 961598
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Abstract: Let $ {H_n}(F)$ denote the subspace of symmetric matrices of $ {M_n}(F)$, the full matrix algebra with coefficients in a field $ F$. The subspace $ {H_n}(F)\subset {M_n}(F)$ does not have any polynomial identity of degree less than $ 2n$. Let

$\displaystyle T_k^i({x_1}, \ldots ,{x_k}) = \sum\limits_{\begin{array}{*{20}{c}... ... {{{( - 1)}^\sigma }{x_{\sigma (1)}}} {x_{\sigma (2)}} \cdots {x_{\sigma (k)}},$

, and $ e(n) = n$ if $ n$ is even, $ n + 1$ if $ n$ is odd. For all $ n \geq 1,T_{2n}^i$ is an identity of $ {H_n}(F)$. If the characteristic of $ F$ does not divide $ e(n)!$ and if $ n \ne 3$, then any homogeneous polynomial identity of $ {H_n}(F)$ of degree $ 2n$ is a consequence of $ T_{2n}^i$. The case $ n = 3$ is also dealt with. The proofs are algebraic, but an equivalent formulation of the first result in graph-theoretical terms is given.

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