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 $R=\Z/p^2\Z$ 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 $n\leq 4$, 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.Comment: 46 page

Simultaneous Similarity Classes of Commuting matrices over a finite field

“…The results of Singla [9], Jambor and Plesken [6] show that c n,2 (q) is the number of similarity classes of matrices in M n (R/P 2 ). Comparing the results in this paper with those of Avni, Onn, Prasad and Vaserstein [1], and Prasad, Singla and Spallone [8], we find that the number of similarity classes in M 3 (R/P k ) is equal to c 3,k (q) for all k. The calculations of this paper and the results of the papers cited above lead us to conjecture the following:…”

Similarity classes of integral p-adic matrices and representation zeta functions of groups of type A2

“…for ℓ = 1, a classification is achieved, for instance, by the Frobenius normal form. For the case ℓ = 2 see, for example, [31,49]. In Theorem 2.11 we give a complete and irredundant list of representatives of the similarity classes in gl 3 (o ℓ ), for any ℓ ∈ N, building on and refining results from [7].…”

Effective and Infinite-Rank Superrigidity in the Context of Representation Stability

“…What about for non-algebraic representations? Well it turns out that for n ≥ 3 not every matrix GL n (Z) is conjugate to its transpose, and in fact there are matrices over Z/p 2 Z which are not conjugate to their transposes (see [16]). In particular this implies there exists representations of GL n (Z) for which the dual and the inverse-transpose twist representations are non-isomorphic, as their characters will be different on these non self-transpose conjugacy classes.…”

“…Theorem 3. 16 Let V be a finitely generated pointwise finite dimensional V IC(Z)module over C, V admits a finite filtration…”