Equivalence of linear representations
Contents
Definition
In terms of a homomorphism of linear representations
An equivalence of linear representations between a linear representation and a linear representation
(where
is a group and
are vector spaces over a field
) is a homomorphism of linear representations from
to
having a two-sided inverse that is also a homomorphism of linear representations. In other words, it is a vector space isomorphism
such that:
For representations given as matrices
Given two linear representations and
of a group
over a field
, an equivalence of representations between
is given by a matrix
such that:
Equivalence of definitions
The second definition is a special case of the first if we view and
as the matrix for the isomorphism
.
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.