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

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This article gives specific information, namely, linear representation theory, about a family of groups, namely: unitriangular matrix group of degree three.
View linear representation theory of group families | View other specific information about unitriangular matrix group of degree three

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