Direct sum of linear representations

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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.