## Irreps of an Abelian Group

**Point of Post:** In this post we discuss the general methodology for finding all the irreps (up to equivalence) of a finite abelian group.

*Motivation*

Just as we were able to combine the structure theorem and our knowledge of the dual group of a cyclic group to gain information about the dual group of an abelian group we so shall use similar knowledge to gain the general knowledge about the irreps of a finite abelian group up to equivalence.

*Irreps of a Finite Abelian Group Up To Equivalence*

There is, in a real sense, no ‘theorem’ that can be associated to this post. Just a couple of observations. Namely, if we are given an abelian group we know from the structure theorem that . Once one finds such a deomposition everything becomes easy. Namely, we know from our theorems regarding the irreps of the products of groups that every irrep of , up to equivalence, looks like where is an irrep of . But, since is cyclic, we know that it is of the form for some . Realistically then, that’s all there is to it. But, since any good math post is incomplete with out one formal theorem we make one last observation in theorem form:

**Theorem: ***Let be an abelian group and be the character table for . Then, *

*where denotes similarity of matrices and each is an non-normalized DFT matrix, and the Kronecker product.*

**Proof: **As we noted before there exists such that

It follows then from our previous observation about the character table of the product of groups that up to a rearrangement of the columns will be equal to the Kronecker product of the character tables of . But, we know that the character table of each is (up to a permutation of the columns) a DFT matrix. The conclusion follows.

**References:**

1. Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Math. Soc., 1996. Print.

No comments yet.

## Leave a Reply