Equivalence of linear representations

Definition

In terms of a homomorphism of linear representations

An equivalence of linear representations between a linear representation $\rho_1:G \to GL(V)$ and a linear representation $\rho_2:G \to GL(V_2)$ (where $G$ is a group and $V_1,V_2$ are vector spaces over a field $K$) is a homomorphism of linear representations from $\rho_1$ to $\rho_2$ having a two-sided inverse that is also a homomorphism of linear representations. In other words, it is a vector space isomorphism $f:V_1 \to V_2$ such that:

$f \circ \rho_1(g) \ =\rho_2(g) \circ f \ \forall \ g \in G$

For representations given as matrices

Given two linear representations $\rho_1:G \to GL(n,K)$ and $\rho_2:G \to GL(n,K)$ of a group $G$ over a field $K$, an equivalence of representations between $\rho_1, \rho_2$ is given by a matrix $A \in GL(n,K)$ such that:

$A\rho_1(g)A^{-1} = \rho_2(g) \ \forall \ g \in G$

Equivalence of definitions

The second definition is a special case of the first if we view $V_1 - V_2 = K^n$ and $A$ as the matrix for the isomorphism $K^n \to K^n$.

Notion of equivalent linear representations

Two linear representations are said to be equivalent if there exists an equivalence of linear representations.

Most notions related to linear representations are studied up to equivalence. When we say that two representations are distinct, or seek to count the number of representations of a certain type, we are doing this up to equivalence.

Facts

• Character determines representation in characteristic zero
• Equivalent linear representations of finite group over field are equivalent over subfield in characteristic zero: In particular, at least in characteristic zero, the notion of being equivalent does not depend on which field we are thinking of our matrices as living in.