of irreducible submodules. The number of these irreducible submodules for a given group is equal to the number of conjugacy classes in the group [11]. Since representations can be thought of as modules or homomorphisms, Maschke’s Theorem applies to representations thought of as homomorphisms. Modules can be broken down until we reach irreducible submodules; in other words, until we get to a direct sum of irreducible representations. Similarly, we can break a homomorphism down into a direct sum of multiple irreducible homomorphisms. Concretely, since our homomorphism sends the elements of the group to matrices, this looks like breaking down a matrix into smaller matrices along the diagonal. As an example, let us think about A5, the group of even permutations of a five element set. Suppose we wanted to represent A5 in 18 dimensions and hence each element of A5 maps to an 18 by 18 matrix. An example of what such a matrix would look like is , where the black squares represent elements in C. Given that A5 has irreducible representations of dimension 5, dimension 4, two different representations of dimension 3, and one of dimension 1, this representation can be broken down into irreducibles, perhaps a 5 by 5, two 4 by 4, a 3 by 3, and two 1 by 1 matrices (note that 5 + 4 + 4 + 3 + 1 + 1 = 18). Hence, our irreducible matrix would look like this 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . These two representations could be conjugate to each other, since the original matrix is conjugate to some block diagonal sum of irreducible representations. We could also create a different irreducible representation, as long as we use the dimensions of irreducible Channels • 2022 • Volume 6 • Number 2 Page 33
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