Central polynomials of minimal degree for matrices

TL;DR

The paper tackles Formanek's conjecture on the minimal degree of central polynomials for $M_n(K)$ in characteristic $0$, focusing on the two-variable case for $n=4$. It develops and deploys a suite of representation-theoretic methods—$S_m$- and $GL_d(K)$-theoretic analyses, multihomogeneous considerations, and PI-theory—to reduce the search to a small set of linear systems. Through explicit weight-vector constructions and targeted evaluations, it establishes that $M_4(K)$ has no central polynomials in two variables of degree $\le 12$, complementing known degree-13 examples and reinforcing the Formanek bound in this case. The results also yield a byproduct: no two-variable polynomial identities in degree $\le 12$, illustrating the power of the combined representation-theoretic/PI approach for computational PI-theory problems.

Abstract

Formanek made the conjecture that the minimal degree of the central polynomials for the $n\times n$ matrix algebra over a field of characteristic 0 is $(n^2+3n-2)/2$ and this is true for $n\leq 3$. For $n=4$ there are examples of central polynomials of degree $13=(4^2+3\cdot 4-2)/2$ and we do not know whether there are central polynomials of lower degree. In this paper we discuss methods for searching for central polynomials of low degree and prove that the algebra of $4\times 4$ matrices does not have central polynomials in two variables of degree $\leq 12$. As a byproduct of our computations we obtain that this algebra does not have also polynomial identities in two variables of degree $\leq 12$.

Theorems & Definitions (22)

• Conjecture 1.3
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Lemma 3.4
• proof
• Corollary 3.5
• proof
• Lemma 3.6
• proof