Direct sum of linear representations

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.

Effect on character
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
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.