Equivalence of linear representations

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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.