---
title: "Decomposing a Representation from Its Character"
description: "Build the character table of a finite group and use it to read off how any representation breaks into irreducibles."
canonical: https://lightmysky.com/learn/mathematics/decomposing-a-representation-from-its-character-mt_ccEcWW9xAa
source: https://lightmysky.com/learn/mathematics/decomposing-a-representation-from-its-character-mt_ccEcWW9xAa.md
retrieved: 2026-09-12
---

> **Agent view.** This is the Markdown twin of the page, for tools and assistants.
> When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md
> API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md
> Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt
> Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json
> Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`.

# Decomposing a Representation from Its Character

Build the character table of a finite group and use it to read off how any representation breaks into irreducibles.

Subject: Mathematics · Area: Abstract Algebra · Ages 23 to 24
Page: https://lightmysky.com/learn/mathematics/decomposing-a-representation-from-its-character-mt_ccEcWW9xAa

## Ready when they can

- Construct the character table of a small non-abelian group
- Decompose a permutation representation by taking inner products with the irreducible characters
- Read a normal subgroup off the table from the kernel of a character

## Lesson: Reading a group from its character table

A character table lists every irreducible character of a group. Each row is one irreducible character and each column is one conjugacy class, so each entry is that character's value there.

To break a representation into irreducibles, take an inner product with each row. Multiply the matching entries and add, the way a dot product multiplies matching components and adds. The result is a whole number: how many copies of that irreducible sit inside.

**Example.** Take the symmetries of a triangle and their shuffling of the three corners. Its character dotted with the all-ones row gives 1, so one copy of the trivial piece sits inside, and the remaining pieces are read off the same way.

**Tip.** To spot a normal subgroup, look at one row's kernel: the elements whose value equals the row's dimension. Those elements always form a normal subgroup, straight from the table.

**Recap.** Dot each row of the table into your character to count the pieces, and read normal subgroups from the kernels.

## Practice

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

## Needs first

- [Characters and the Orthogonality Relations](https://lightmysky.com/learn/mathematics/characters-and-the-orthogonality-relations-mt_RleR7YGXjX)
- [Normal Subgroups and Quotient Groups](https://lightmysky.com/learn/mathematics/normal-subgroups-and-quotient-groups-mt_RU7A-t7wQ5)