Direct sum of linear representations

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

Definition in terms of linear representation as a homomorphism

Suppose $(V_{1},\rho _{1})$ and $(V_{2},\rho _{2})$ are two linear representations of a group $G$ over a field $k$. In other words, $V_{1},V_{2}$ are vector spaces over $k$ and $\rho _{1}:G\to Gl(V_{1})$ and $\rho _{2}:G\to GL(V_{2})$ are homomorphisms. Then, the direct sum is given as follows:

• The vector space is $V_{1}\oplus V_{2}$
• Let $\alpha :GL(V_{1})\times GL(V_{2})\to GL(V_{1}\oplus V_{2})$ be the natural map obtained by coordinate-wise action. Then, the homomorphism from $G$ to $GL(V_{1}\oplus V_{2})$ is given by $\alpha \circ (\rho _{1}\times \rho _{2})$. In other words, the map is:

$g\mapsto \alpha (\rho _{1}(g),\rho _{2}(g))$

In matrix terms, it $V_{1}$ and $V_{2}$ are finite-dimensional and are given a basis, we can think of $\rho _{1},\rho _{2}$ as matrix-valued. Then the map is given as:

$g\mapsto {\begin{pmatrix}\rho _{1}(g)&0\\0&\rho _{2}(g)\end{pmatrix}}$

An analogous definition holds for infinite direct sums. Here, $\alpha$ is replaced by a map from the direct product of the $GL(V_{i})$s to $GL(\bigoplus V_{i})$.

Definition in terms of linear representation as a group action

Suppose $G$ has two linear representations over a field $k$: one on a vector space $V_{1}$ and the other on the vector space $V_{2}$. Then, the direct sum representation on $V_{1}\oplus V_{2}$ is given by making $G$ act coordinate-wise on the direct sum, i.e.:

$g.(v_{1},v_{2})=(g.v_{1},g.v_{2})$

An analogous coordinate-wise definition holds for infinite direct sums.

Definition in terms of linear representation as a module over the group ring

Suppose $G$ has two linear representations over a field $k$: modules $V_{1}$ and $V_{2}$ over the group ring $kG$. The direct sum of these representations is the direct sum of $V_{1}$ and $V_{2}$ as $kG$-modules.

Similarly, we can take an infinite direct sum of linear representations as the infinite direct sum of the corresponding modules.

Facts

Effect on character

Further information: sum of characters equals character of direct sum

The character of the direct sum of two finite-dimensional linear representations is the sum of their characters. This is because under the map:

$\alpha :GL(V_{1})\times GL(V_{2})\to GL(V_{1}\oplus V_{2})$

The trace of $\alpha (a,b)$ is the sum of the traces of $a$ and $b$.

Effect on determinant

The determinantal representation of the direct sum of two finite-dimensional linear representations is the product of their determinantal representations. That's because, the determinant of $\alpha (a,b)$ is the product of the determinants of $a$ and $b$.

Indecomposable, irreducible, and completely reducible linear representations

Further information: indecomposable linear representation, irreducible linear representation, completely reducible linear representation

A linear representation that cannot be expressed as a direct sum of two linear representations on nonzero vector spaces, is termed indecomposable.

A stronger notion than indecomposability is the notion of an irreducible linear representation: a linear representation that has no proper nonzero invariant subspace. Maschke's lemma says that in the non-modular case for a finite group (i.e., the case where the characteristic of the field does not divide the order of the group), every indecomposable representation is irreducible.

A representation that can be expressed as a direct sum (possibly infinite) of irreducible linear representations is termed a completely reducible linear representation. It turns out that if every indecomposable linear representation is irreducible, then every representation is completely reducible.