On permanental identities of symmetric and skew-symmetric matrices in characteristic \lowercase{$p$}

http://dx.doi.org/10.4153/CMB-1998-018-9
Canad. Math. Bull. 41(1998), 118-124

  Published:1998-03-01
 Printed: Mar 1998

Let $M_n(F)$ be the algebra of $n \times n$ matrices over a field $F$ of characteristic $p>2$ and let $\ast$ be an involution on $M_n(F)$. If $s_1, \ldots, s_r$ are symmetric variables we determine the smallest $r$ such that the polynomial $$ P_{r}(s_1, \ldots, s_{r}) = \sum_{\sigma \in {\cal S}_r}s_{\sigma(1)}\cdots s_{\sigma(r)} $$ is a $\ast$-polynomial identity of $M_n(F)$ under either the symplectic or the transpose involution. We also prove an analogous result for the polynomial $$ C_r(k_1, \ldots, k_r, k'_1, \ldots, k'_r) = \sum_ {\sigma, \tau \in {\cal S}_r}k_{\sigma(1)}k'_{\tau(1)}\cdots k_{\sigma(r)}k'_{\tau(r)} $$ where $k_1, \ldots, k_r, k'_1, \ldots, k'_r$ are skew variables under the transpose involution.