# Linear representation theory of unitriangular matrix group of degree three over a finite discrete valuation ring

## Contents

This article gives specific information, namely, linear representation theory, about a family of groups, namely: unitriangular matrix group of degree three.
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This article describes the linear representation theory of the unitriangular matrix group of degree three over a finite discrete valuation ring. It builds on the discussion at linear representation theory of unitriangular matrix group of degree three over a finite field.

We assume that the residue field has size $q$ and characteristic $p$, with $r = \log_pq$. We denote by $l$ the length of the discrete valuation ring. The size of the ring is thus $q^l = p^{rl}$ and the order of the field is $q^{3l} = p^{3rl}$.

## Summary

Item Value
degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$) 1 (occurs $q^{2l}$ times). For $1 \le i \le l$, $q^i$ occurs $q^{2l - i} - q^{2l - i - 1}$ times.
number of conjugacy classes equals number of irreducible representations over a splitting field $q^{2l} + q^{2l-1} - q^{l-1}$
sum of squares of degrees of irreducible representations $q^{3l}$
maximum degree of irreducible representation $q^l$