Matrix identities with forms

TL;DR

This paper characterizes matrix identities with forms over fields of arbitrary characteristic, showing that a finite set of partial linearizations suffices to generate the T-ideal of identities, extending previous results and applying to invariants of matrix groups.

Contribution

It proves that the T-ideal of identities with forms for matrix algebras is finitely based using partial linearizations within a specific index range, generalizing Zubkov's earlier work.

Findings

T-ideal T(n) is finitely based.

Partial linearizations of _t for n<t;2n suffice.

Results apply to invariants of GL(n) and O(n) groups.

Abstract

Consider the algebra M(n,F) of n x n matrices over an infinite field F of arbitrary characteristic. An identity for M(n,F) with forms is such a polynomial in n x n generic matrices and in \sigma_k(x), 0<k\leq n, coefficients in the characteristic polynomial of monomials in generic matrices, that is equal to zero matrix. This notion is a characteristic free analogue of identities for M(n,F) with trace. In 1996 Zubkov established an infinite generating set for the T-ideal T(n) of identities for M(n,F) with forms. Namely, for t>n he introduced partial linearizations of \sigma_t and proved that they together with the well-known free relations and the Cayley--Hamilton polynomial generate T(n) as a T-ideal. We show that it is enough to take partial linearizations of \sigma_t for n<t\leq 2n. In particular, the T-ideal T(n) is finitely based. Working over a field of characteristic different from two, we obtain a similar result for the ideal of identities with forms for the F-algebra generated by n x n generic and transpose generic matrices. These results imply that ideals of identities for the algebras of matrix GL(n)- and O(n)-invariants are generated by the well-known free relations together with partial linearizations of \sigma_t for n<t\leq 2n and partial linearizations of \sigma_{t,r} for n<t+2r\leq 2n, respectively.