Minimal identities of symmetric matrices

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 \[ T_k^i({x_1}, \ldots ,{x_k}) = \sum \limits _{\begin {array}{*{20}{c}} {\alpha \in {\mathcal {S}_k}} \\ {1 \leq i \leq k,{\sigma ^{ - 1}}(i) \equiv 1,2\operatorname {mod} 4} \\ \end {array} } {{{( - 1)}^\sigma }{x_{\sigma (1)}}} {x_{\sigma (2)}} \cdots {x_{\sigma (k)}},\], 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.