Free relations for matrix invariants in the modular case

Abstract

A classical linear group G < GL(n) acts on d-tuples of n × 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 ch : S → J between the graded ring S = ⊗ ∞ d=0 S d , where S 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.

Matrix identities with forms

“…Necessary definitions are given in Section 5. To prove Theorem 5.5, in Section 6 we obtained an essentially smaller than in [9] generating set for K ′ n (see Theorem 6.3 and Remark 6.4). We also showed that K ′ n and T ′ n are finitely based if and only if p = 0 (see Lemma 6.16).…”

“…The next lemma describes the large free algebra σ Y as a quotient of the absolutely free algebra σ Y . Its proof follows immediately from the proof of Lemma 3.1 from [9]. Lemma 6.2.…”

“…In particular, K ′ n , T ′ n , T ′ n are finitely based in characteristic zero case. In the case of arbitrary characteristic an infinite generating set for the T-ideal K ′ n was described in [9], [10] (see Theorem 6.1).…”

“…We start this section with the known description of the ideal of relations K ′ n . We completed the proof of the following theorem in [9], using results from [10] and the approach described in [19].…”

Minimal generating and separating sets for O(3)-invariants of several matrices

“…The ideal of relations between the generators of R GL(n) was described in [17,16,19]. In case p = 0 relations between generators of R O(n) were computed in [16] and in case p = 2 relations between generators of matrix O(n)-invariants were obtained in [12] and [13].…”