Direct sum of linear representations
This article gives a basic definition in the following area: linear representation theory
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Definition in terms of linear representation as a homomorphism
Suppose and are two linear representations of a group over a field . In other words, are vector spaces over and and are homomorphisms. Then, the direct sum is given as follows:
- The vector space is
- Let be the natural map obtained by coordinate-wise action. Then, the homomorphism from to is given by . In other words, the map is:
In matrix terms, it and are finite-dimensional and are given a basis, we can think of as matrix-valued. Then the map is given as:
An analogous definition holds for infinite direct sums. Here, is replaced by a map from the direct product of the s to .
Definition in terms of linear representation as a group action
Suppose has two linear representations over a field : one on a vector space and the other on the vector space . Then, the direct sum representation on is given by making act coordinate-wise on the direct sum, i.e.:
An analogous coordinate-wise definition holds for infinite direct sums.
Definition in terms of linear representation as a module over the group ring
Suppose has two linear representations over a field : modules and over the group ring . The direct sum of these representations is the direct sum of and as -modules.
Similarly, we can take an infinite direct sum of linear representations as the infinite direct sum of the corresponding modules.
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:
The trace of is the sum of the traces of and .
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 is the product of the determinants of and .
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.