Linear representation theory of unitriangular matrix group of degree five over a finite field

Contents

This article gives specific information, namely, linear representation theory, about a family of groups, namely: unitriangular matrix group of degree five. This article restricts attention to the case where the underlying ring is a finite field.
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This article describes the linear representation theory of the unitriangular matrix group of degree five over a finite field of size $q$, where $q$ is a prime power $p^r$ with underlying prime $p$. $p$ is the characteristic of the field.

Summary

Item Value
number of conjugacy classes (equals number of irreducible representations over a splitting field) $5q^4 - 5q^2 + 1$. See number of irreducible representations equals number of conjugacy classes, element structure of unitriangular matrix group of degree four over a finite field
degrees of irreducible representations 1 (occurs $q^4$ times) $q$ (occurs $2q^4 - q^3 - q^2 = q^2(q - 1)(2q + 1)$ times) $q^2$ (occurs $2q^4 - q^3 - 2q^2 + q = q(q - 1)(q + 1)(2q - 1)$ times) $q^3$ (occurs $2q^3 - 3q^2 + q = q(q-1)(2q - 1)$ times) $q^4$ (occurs $q^2 - 2q + 1 = (q - 1)^2$ times)
sum of squares of degrees of irreducible representations $q^{10}$ (equals order of the group)
see sum of squares of degrees of irreducible representations equals order of group
lcm of degrees of irreducible representations $q^4$