The Irreps of the Product of Finitely Many Finite Groups
Point of post: In this post we shall discuss how one can find the set of all irreps, up to equivalence, of a group of the form given the irreps of for .
We’ve developed quite an extensive theory regarding how to find the irreps of a finite group and how to relate those irreps to one another, as well as their characters.The question remains though if there is a natural way that the irreps of relate naturally to constructions based on . In particular, in this post we are interested in determining the relationships between the irreps (in particular the characters) of the product of two groups and given the knowledge of the characters of and . So for example, it’s easy (as we’ve shown) to construct the character table for and it’s equally easy to construct the character table for . A next logical step would be to combine them and find the character table for . It would be nice if one would have to not go tromping through all the extra work to create this (larger) character table having gone through the (admittedly small amount of work) to construct the ones for and . In this post we shall show that our greatest wish is true–we can’t just easily get some characters of from those of and but we can easily get all of them.
Finding The Irreducible Characters of the Finite Product of Finitely Many Groups
Let and be groups, and and be representations on and respectively. We can define a new representation, called the tensor product of and , on by
by where is the tensor product of vector spaces and is the tensor product of linear transformations (compare with the tensor product of representations given one group). This is indeed a representation since the tensor product of two transformations is unitary and it’s a homomorphism since
The interesting difference this type of tensor representation and the one we previously discussed is that the tensor product of two irreps of and is an irrep on . The fascinating thing is that the set of all such tensor product of representations constitutes all of the irreps. Indeed
Theorem: Let and be finite groups and for each and each choose a representative and . Then, for every such one has that is an irreducible representation for and if and only if . Moreover
Proof: Let and be arbitrary. To prove that is an irrep for we note that evidently and so
and thus by our alternate characterization of irreducibility we may conclude that is an irrep of .
To prove our second we actually prove the stronger claim that
To prove the last claim we note that we’ve defined a map by which by the second part of our theorem is an injection. But, by a previous theorem we have that and thus by a simple set theoretic fact we may conclude that is also a surjection from where the conclusion follows.
1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Math. Soc., 1996. Print.