The standard polynomial as an identity on symplectic matrices.

Abstract

The symplectic involution s is defined on $2n \times 2n$ matrices by $$\pmatrix{A&B\cr C&D\cr}\sp{s} = \pmatrix{D\sp{t}&-B\sp{t}\cr -C\sp{t}&A\sp{t}\cr}$$ where A, B, C, and D are $n\times n$ matrices, and t is the standard transpose operation. This thesis investigates the value of the standard polynomial $S\sb{k} := \sum\sb{\sigma\in{\cal S}\sb{k}} (-1)\sp{\sigma} x\sb{\sigma(1)}\cdots x\sb{\sigma(k)}$ evaluated over the ring of matrices which are symmetric with respect to the symplectic involution, denoted $H\sb{n}(F, s).$ In this thesis the value of $S\sb{4n-3}$ for $n \ge 3$ is studied. In particular, it is shown that $S\sb{4n-3}$ is not an identity for n = 3 and n = 4. Moreover, a reduction is obtained in the general case. We choose a particular basis where each basis element can be represented on a graph of 2n points by a pair of labelled, directed edges. A pseudo-Eulerian path on this graph is defined as a path in which exactly one edge of each pair of edges is traversed exactly once. By counting the number of pseudo-Eulerian paths and assigning a value of $-$1 or +1 to each, the value of $S\sb{k}$ can be determined. (Abstract shortened by UMI.)