Images of graded polynomials on matrix algebras

TL;DR

This paper investigates the possible images of graded polynomials on matrix algebras with a specific grading, confirming conjectures and classifying images for certain degrees and matrix sizes.

Contribution

It explicitly characterizes the linear spans of images of multilinear graded polynomials on matrix algebras and confirms a conjecture for prime-sized matrices of degree 2.

Findings

Confirmed the graded L'vov-Kaplansky conjecture for degree 2 polynomials over matrices of prime size.

Determined all possible images of semi-homogeneous graded polynomials on 2x2 matrices.

Classified the linear spans of images of multilinear graded polynomials over the rationals.

Abstract

The aim of this paper is to start the study of images of graded polynomials on full matrix algebras. We work with the matrix algebra $M_n(K)$ over a field $K$ endowed with its canonical $\mathbb{Z}_n$-grading (Vasilovsky's grading). We explicitly determine the possibilities for the linear span of the image of a multilinear graded polynomial over the field $\mathbb Q$ of rational numbers and state an analogue of the L'vov-Kaplansky conjecture about images of multilinear graded polynomials on $n\times n$ matrices, where $n$ is a prime number. We confirm such conjecture for polynomials of degree 2 over $M_n(K)$ when $K$ is a quadratically closed field of characteristic zero or greater than $n$ and for polynomials of arbitrary degree over matrices of order 2. We also determine all the possible images of semi-homogeneous graded polynomials evaluated on $M_2(K)$.