Free relations for matrix invariants in modular case

TL;DR

This paper proves that the ideal of free relations among matrix invariants for classical groups like O(n) and Sp(n) is zero, extending known results and completing the description of relations for these invariants.

Contribution

It establishes the absence of free relations for matrix invariants of classical groups in the modular case, generalizing previous characteristic isomorphism results.

Findings

The ideal of free relations for O(n)- and Sp(n)-invariants is zero.

Complete description of relations among generators for these invariants.

Provides an independent proof for the case of GL(n)-invariants.

Abstract

A classical linear group $G<GL(n)$ acts on $d$-tuples of $n\times n$ matrices by simultaneous conjugation. Working over an infinite field of characteristic different from two we establish that the ideal of free relations, i.e. relations valid for matrices of any order, between generators for matrix O(n)- and $\Sp(n)$-invariants is zero. We also prove similar result for invariants of mixed representations of quivers. These results can be considered as a generalization of the characteristic isomorphism ${\rm ch}:\Sym\to J$ between the graded ring $\Sym=\otimes_{d=0}^{\infty} \Sym_d$, where $\Sym_d$ is the character group of the symmetric group $S_d$, and the inverse limit $J$ with respect to $n$ of rings of symmetric polynomials in $n$ variables. As a consequence, we complete the description of relations between generators for O(n)-invariants as well as the description of relations for invariants of mixed representations of quivers. We also obtain an independent proof of the result that the ideal of free relations for $GL(n)$-invariants is zero, which was proved by Donkin in [Math. Proc. Cambridge Philos. Soc. 113 (1993), 23--43].