---
title: "Characters and the Orthogonality Relations"
description: "Reduce a representation to the trace of each group element, and prove the resulting functions are orthogonal for an inner product on class functions."
canonical: https://lightmysky.com/learn/mathematics/characters-and-the-orthogonality-relations-mt_RleR7YGXjX
source: https://lightmysky.com/learn/mathematics/characters-and-the-orthogonality-relations-mt_RleR7YGXjX.md
retrieved: 2026-09-12
---

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# Characters and the Orthogonality Relations

Reduce a representation to the trace of each group element, and prove the resulting functions are orthogonal for an inner product on class functions.

Subject: Mathematics · Area: Abstract Algebra · Ages 23 to 24
Page: https://lightmysky.com/learn/mathematics/characters-and-the-orthogonality-relations-mt_RleR7YGXjX

## Ready when they can

- Show a character is constant on conjugacy classes and independent of the chosen basis
- Prove the first orthogonality relation for irreducible characters
- Count the irreducible representations against the number of conjugacy classes

## Lesson: Traces that tell representations apart

A representation gives each group element a square matrix, so the group can act by stretching and turning space. The character of the representation at that element is the trace of its matrix. The trace is a single number: add the entries down the main diagonal.

A new basis replaces every matrix by a conjugate, which pictures the same stretch drawn on a new grid. The trace survives conjugation, so the character never sees which basis you chose. Conjugate group elements get conjugate matrices, so they share one character value: the character is constant on each conjugacy class.

**Example.** Take class functions with an inner product that averages over the group. Two irreducible characters have inner product 1 when they are the same and 0 when they differ. That is the first orthogonality relation: distinct irreducible characters sit at right angles to each other.

**Tip.** When you finish, count. The number of irreducible representations equals the number of conjugacy classes, so a missing character means the table is incomplete.

**Recap.** The trace of each matrix gives a class function, and distinct irreducible characters are orthogonal under the group average.

## Practice

18 questions on this page, each with its working shown.

## Needs first

- [Maschke's Theorem and Complete Reducibility](https://lightmysky.com/learn/mathematics/maschkes-theorem-and-complete-reducibility-mt_FVNCwUqeTx)
- [Group Actions, Orbits and the Class Equation](https://lightmysky.com/learn/mathematics/group-actions-orbits-and-the-class-equation-mt_urut0TFTOR)
- [Inner Products, Length and Orthogonality](https://lightmysky.com/learn/mathematics/inner-products-length-and-orthogonality-mt_X0HPRGto4W)

## Opens up

- [Decomposing a Representation from Its Character](https://lightmysky.com/learn/mathematics/decomposing-a-representation-from-its-character-mt_ccEcWW9xAa)