Similarity of matrices over local rings of length two

Abstract

Let $R$ be a local principal ideal ring of length two, for example, the ring $R=\mathbb{Z}/p^2\mathbb{Z}$ with $p$ prime. In this paper we develop a theory of normal forms for similarity classes in the matrix rings $M_n(R)$ by interpreting them in terms of extensions of $R[t]$-modules. Using this theory, we describe the similarity classes in $M_n(R)$ for $n\le 4$, along with their centralizers. Among these, we characterize those classes which are similar to their transposes. Non-self-transpose classes are shown to exist for all $n>3$. When $R$ has finite residue field of order $q$, we enumerate the similarity classes and the cardinalities of their centralizers as polynomials in $q$. Surprisingly, the polynomials representing the number of similarity classes in $M_n(R)$ turn out to have non-negative integer coefficients.

46 pages

References in corpus (4)

• A q-rious positivity
• On the motivic Donaldson-Thomas invariants of quivers with potentials
• Orbits of pairs in abelian groups
• Representations of automorphism groups of finite O-modules of rank two

Cited by in corpus (5)

• Similarity classes of integral $p$-adic matrices and representation zeta functions of groups of type $A_2$
• Simultaneous Similarity Classes of Commuting matrices over a finite field
• Effective and Infinite-Rank Superrigidity in the Context of Representation Stability
• z-classes in groups: a survey
• Equivalence classes of nodes in trees and rational generating functions