---
tag: RK8RV2C998
kind: definition
name: Group representation
msc: [20C15, 20C05]
defines: WYGRXCG2RN
uses: [G5S1M9, 000AYPA9AJ, 0JYTG5K9Y4, 0FCHE902D7, 0NRYHJPQG2]
sources:
  - url: https://en.wikipedia.org/wiki/Group_representation
---

## Definition

Let $G$ be a [[group|G5S1M9]] and let $F$ be a field.
Let $V$ be a [[vector space|000AYPA9AJ]] over $F$, and let $GL(V)$ be the group of invertible [[linear maps|0NRYHJPQG2]] $V \to V$ under composition.
A *representation* of $G$ on $V$ is a [[group homomorphism|0JYTG5K9Y4]] $\rho \colon G \to GL(V)$.
That is, $\rho(gh) = \rho(g) \circ \rho(h)$ for all $g, h \in G$.
The dimension of $V$ is the degree of $\rho$.
Two representations $\rho \colon G \to GL(V)$ and $\sigma \colon G \to GL(W)$ are isomorphic if there is a linear isomorphism $T \colon V \to W$ with $T \circ \rho(g) = \sigma(g) \circ T$ for all $g \in G$.

## Remarks

Example: let $G = \Z / n\Z$ and $V = \C$.
The map $\rho(k) = e^{2\pi i k / n}$, viewed as multiplication on $\C$, is a representation of degree $1$, because $e^{2\pi i (k + l)/n} = e^{2\pi i k/n} e^{2\pi i l/n}$, and the value does not depend on the representative $k$ of its class.
Non-example: on $G = \Z / 2\Z$, the map sending $0$ to $\id$ and $1$ to $2 \cdot \id$ on $\C$ is not a representation, because $\rho(1 + 1) = \rho(0) = \id$ but $\rho(1)\rho(1) = 4 \cdot \id$.
When $V = F^n$, the group $GL(V)$ is the [[general linear group|0FCHE902D7]] $GL_n(F)$, and a representation is a matrix-valued homomorphism.
Equivalently, a representation is a linear action of $G$ on $V$, or a module over the [[group algebra|MNP2HBWB5Q]] $F[G]$.