
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…
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Taxonomy
TopicsArchitecture and Computational Design
