Some complete intersection symplectic quotients in positive characteristic: invariants of a vector and a covector
Résumé
Given a linear action of a group $G$ on a $K$-vector space $V$, we consider the invariant ring $K[V \oplus V^*]^G$, where $V^*$ is the dual space. We are particularly interested in the case where $V =\gfq^n$ and $G$ is the group $U_n$ of all upper unipotent matrices or the group $B_n$ of all upper triangular matrices in $\GL_n(\gfq)$. In fact, we determine $\gfq[V \oplus V^*]^G$ for $G = U_n$ and $G =B_n$. The result is a complete intersection for all values of~$n$ and~$q$. We present explicit lists of generating invariants and their relations. This makes an addition to the rather short list of ``doubly parametrized'' series of group actions whose invariant rings are known to have a uniform description.
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