The Representations of Automorphism Groups of $\mathfrak{o}$-modules of type $(\ell,1^n)$

TL;DR

This paper develops a method to compute the representation zeta polynomial of automorphism groups of certain modules over valuation rings, revealing that their representation dimensions are polynomially determined by the residue field size.

Contribution

It introduces an inductive procedure to explicitly determine the representation zeta polynomial for automorphism groups of modules of type $( extstyle{ extstyleigoplus_{i=1}^n} ext{o}_ ext{1})$ over valuation rings.

Findings

Representation zeta polynomial can be computed inductively.

Representation dimensions are given by finitely many polynomials evaluated at $q=| ext{o}_1|$.

Explicit formulas for automorphism groups of modules of type $( extstyle{ extstyleigoplus_{i=1}^n} ext{o}_ ext{1})$.

Abstract

Let $\mathfrak{o}$ be the valuation ring of a non-Archimedean local field with finite residue field. We give a procedure to find the representation zeta polynomial of $\mathrm{Aut}_\mathfrak{o}(\mathfrak{o}_\ell\oplus\mathfrak{o}_1^{\oplus n})$ by induction on $n$. In particular, we show that the dimensions of the representations are given by evaluating finitely many polynomials at $q=|\mathfrak{o}_1|$.