Centralizers, Clifforders, Polynomial Equivalence and $ω$-equivalence of Matrices
Hechun Zhang, Chengyi Zhu
TL;DR
The paper develops a unified framework linking matrix centralizers, clifforders, and quasi-commutative relations via polynomial-type equivalences. It proves that two matrices have identical centralizers if and only if they are polynomial equivalent, and extends this insight to anti-commuting and $\omega$-centralizer contexts. Key contributions include a new proof of Potter's theorem, a complete characterization of clifforders for balanced (including nilpotent) matrices as equivalent to odd polynomial equivalence, and a nilpotent $\omega$-equivalence result yielding $q$-polynomial equivalence. The results offer a cohesive approach to understanding how polynomial transforms preserve structural relations in matrix algebras and point toward generalizations to Lie and non-associative settings.
Abstract
This article is devoted to the study of the centralizer and the clifforder of a matrix over a field $\mathbb{F}$ of characteristic zero, as well as the quasi-commutative relations between matrices over the complex field $\mathbb{C}$. We introduce several new concepts, including polynomial equivalence, odd polynomial equivalence, $q$-polynomial equivalence, the clifforder of a matrix, balanced matrices, and $ω$-equivalence. The clifforder of a matrix is defined via an anti-commuting relation. We present a new proof establishing that two matrices $A$ and $B$ share the same centralizer if and only if they are in polynomial equivalence. Moreover, we extend this to a broader generalization. For balanced matrices (including nilpotent matrices), we prove that their clifforders coincide if and only if they are odd polynomial equivalence. In addition, we investigate quasi-commutative relations defined using a primitive $q$-th root of unity $ω$, provide a new and elementary proof of a classical theorem of H. S. A. Potter, and further explore the properties and connections of $ω$-equivalence.