Linear representation

This article describes a notion of representation, or a group action on a certain kind of object.
View a complete list of types of representations

This article gives a basic definition in the following area: linear representation theory
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Definition

Definition with symbols

Let $G$ be a group and $k$ be a field. A linear representation of $G$ over $k$ is a homomorphism $\rho :G\to GL(V)$ where $V$ is a vector space over $k$ and $GL(V)$ denotes the general linear group of $V$, viz the group of automorphisms of $V$ as a $k$-vector space.

Equivalently, a linear representation is a group action $G\times V\to V$ such that the permutation of $V$ induced by any element of $G$ is a linear map.

A representation is typically specified by giving the pair $(V,\rho )$ where $V$ is the vector space and $\rho$ is the homomorphism.

Constructs

Direct sum

Let $(V_1,{\rho }_1)$ and $(V_2,{\rho }_2)$ be two representations of a group $G$. Then the direct sum of these is defined as follows:

• The vector space for it is $V_1\oplus V_2$
• The action is: $g.(v_1\oplus v_2)=g.v_1\oplus g.v_2$.

In other words, it acts on each vector space separately.

Tensor product

Let $(V_1,{\rho }_1)$ and $(V_2,{\rho }_2)$ be two representations of a group $G$. The tensor product of these is defined as follows:

• The vector space for it is $V_1\otimes V_2$
• The action is such that $g.(v_1\otimes v_2)=g.v_1\otimes g.v_2$

Dual space

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Particular cases

Equivalence of representations

Two representations $(V_1,{\rho }_1)$ and $(V_2,{\rho }_2)$ of a group $G$ are said to be equivalent representations if there is an isomorphism $\sigma :V_1\to V_2$ such that ${\rho }_2(g)=\sigma \circ {\rho }_1(g)\circ {\sigma }^{-1}$.

When $V_1=V_2=V$, this is equivalent to demanding that there exists a $\sigma \in GL(V)$ such that ${\rho }_2(g)=\sigma \circ {\rho }_1(g)\circ {\sigma }^{-1}$, in other words, that ${\rho }_1$ and ${\rho }_2$ differ by an inner automorphism of $V$.

There are also further (weaker) notions of equivalence of representations. Further information: Conjugacy class-representation duality

Finite-dimensional representations

A representation is termed finite-dimensional if the vector space is finite-dimensional. When the vector space is finite-dimensional, we can choose a basis for the vector space and hence write the image of every element of $G$ as a matrix in that basis.

There are some properties of automorphisms of linear transformations over finite-dimensional vector spaces that are easily computed from the matrix, but are in fact independent of the choice of basis. The most important among these are the coefficients of the characteristic polynomial. In particular, there is a notion of trace (which is the sum of the diagonal entries) and the determinant.

Trace leads to the notion of character of a representation. Given a representation $(V,\rho )$ where $V$ is a finite-dimensional vector space, the character of $\rho$, denoted as ${\chi }_{\rho }$ is defined as the map from $G$ to $k$ that sends $g\in G$ to $Trace(\rho (g))$. The character is a special kind of class function (a class function being a function on the group that takes the same value within every conjugacy class).

Invariant subspaces and irreducible representations

Given a representation $(V,\rho )$ of a group $G$, an invariant subspace is a subspace $W\le V$ such that $\rho (g)$ takes $W$ to $W$ for every $g\in G$.

A representation is termed irreducible if it has no proper nontrivial invariant subspace (that is, the only invariant subspace is the zero subspace or the whole space).

A representation is termed completely reducible if it is expressible as a direct sum of irreducible representations.

Decomposition of representations

A direct sum decomposition of a linear representation is an expression of the linear representation as a direct sum of linear representations. A linear representation is said to be direct sum-indecomposable if for any direct sum decomposition of the representation, one of the summands is the zero-dimensional representation.

Note that any irreducible representation is direct sum-indecomposable, but the converse may not be true. In fact, the converse is true provided that every representation is completely reducible.