Linear representation

Definition
Let $$G$$ be a group and $$k$$ be a field. A linear representation or linear action of $$G$$ over $$k$$ can be defined in a number of ways, as outlined below. Each of these involves a vector space $$V$$ over $$k$$.

Definition as a homomorphism
A linear representation of $$G$$ over $$k$$ 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.

A linear representation is typically described by the pair $$(V,\rho)$$ where $$V$$ is the vector space and $$\rho$$ is the homomorphism. Sometimes, we simply talk of $$\rho$$ as the representation.

Definition as a group action
A linear representation of $$G$$ over $$k$$ is a group action $$G \times V \to V$$ (where $$V$$ is a vector space over $$k$$) such that the permutation of $$V$$ induced by any element of $$G$$ is a linear map. Explicitly, we want that:


 * Group action: $$e.v = v \ \forall \ v \in V; \qquad g.(h.v) = (gh).v \ \forall g,h \in G, v \in V$$
 * Commutes with scalar multiplication: $$g.(\alpha v) = \alpha (g.v) \ \forall \ g \in G, \alpha \in k, v \in V$$
 * Additivity: $$g.(v + w) = g.v + g.w \ \forall \ g \in G, v,w \in V$$

Definition as a module over a group ring
Let $$kG$$ denote the group ring of $$G$$ over the field $$k$$. A representation of $$G$$ over $$k$$ is then, a left $$kG$$-module $$V$$. In other words $$V$$ is a vector space over $$k$$, with a $$G$$-action such that the action gives rise to a $$kG$$-module structure.

In this language, we sometimes talk of $$V$$ as a $$G$$-module and hence simply describe the representation as being $$V$$, with the action implicit in the way $$V$$ is given a $$G$$-module structure.

Definition as a homomorphism from the group ring to endomorphisms of a vector space
Let $$kG$$ denote the group ring of $$G$$ over the field $$k$$. A representation of $$G$$ over $$k$$ is then, a vector space $$V$$ over $$k$$, and a homomorphism of $$k$$-algebras from $$kG$$ to the endomorphism algebra of $$V$$.

Equivalence in the language of group homomorphisms
As per the first definition, two representations $$(V_1,\rho_1)$$ and $$(V_2,\rho_2)$$ of a group $$G$$ are said to be equivalent linear 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 $$V$$.

Equivalence in the language of modules
Two linear representations of $$G$$ over $$k$$ are equivalent if the corresponding $$kG$$-modules are isomorphic as $$kG$$-modules: in other words, there is a vector space isomorphism between them that preserves the $$kG$$-action.

Equivalence in the language of algebra homomorphisms
Two linear representations of $$G$$ over $$k$$, say $$\alpha_1:kG \to \operatorname{End}(V_1)$$ and $$\alpha_2:kG \to operatorname{End}(V_2)$$, are equivalent, if there is an isomorphism $$\sigma:V_1 \to V_2$$ such that for any $$x \in kG$$, we have:

$$\alpha_2(x) = \sigma \circ \alpha_1(x) \circ \sigma^{-1}$$

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 $$g.(v_1 \otimes v_2) = g.v_1 \otimes g.v_2$$

Trivial representation
Every group has a trivial repreesntation on any vector space: the representation where every element of the group acts as the identity on every vector. We typically use the term trivial representation for the trivial one-dimensional representation: the trivial representation on a one-dimensional space.

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.