# Graded Polynomial Identities for Matrices with the Transpose Involution over an Infinite Field

## TL;DR

This paper characterizes the polynomial identities of matrix algebras with transpose involution under certain gradings over infinite fields, extending previous results beyond complex numbers.

## Contribution

It provides a basis for graded polynomial identities of matrices with transpose involution over infinite fields with arbitrary characteristic, generalizing prior complex-field results.

## Key findings

- Established a basis for graded polynomial identities
- Extended results to arbitrary characteristic fields
- Generalized previous complex-specific findings

## Abstract

Let $F$ be an infinite field, and let $M_{n}(F)$ be the algebra of $n\times n$ matrices over $F$. Suppose that this algebra is equipped with an elementary grading whose neutral component coincides with the main diagonal. In this paper, we find a basis for the graded polynomial identities of $M_{n}(F)$ with the transpose involution. Our results generalize for infinite fields of arbitrary characteristic previous results in the literature which were obtained for the field of complex numbers and for a particular class of elementary G-gradings.