Abstract Nonsense

Using Orthogonality Relations to Compute Convolutions of Characters and Matrix Entry Functions

Point of post: In this post we will use our past results about the pairwise inner product of characters and matrix entry functions to compute their convolutions.

Motivation

In past posts we obainted certain relations between the pairwise inner product (as elements of the group algebra) of matrix entry functions and irreudcible characters. We shall use these relations to compute the convolution of matrix entry functions with eachother and likewise for characters.

March 2, 2011

Matrix Functions Form an (almost) Orthonormal Basis (Pt. II)

Point of post: This is a continuation of this post.

February 23, 2011

Matrix Entry Functions Form an (almost) Orthonormal Basis

Point of post: In this post we derive the result that the matrix entry functions form an orthonormal basis for the group algebra, thus deriving the fundamental result that $\displaystyle \sum_{\alpha\in\widehat{G}}d_\alpha^2=|G|$.

Motivation

In our last post we showed how fixing, for each $\alpha\in\widehat{G}$, some representative $\rho^{(\alpha)}:G\to\mathscr{V}^{(\alpha)}$ and some orthonormal ordered basis $\mathcal{B}^{(\alpha)}$ enabled us to form $d_\alpha^2$ ‘matrix entry functions’ which were elements of the group algebra $\mathcal{A}(G)$. We further derived some important properties about the span $\Lambda$ of these matrix entry functions (that it is closed under pointwise product and that it separates points). In this post we take this further and show, using a simple case of the Stone-Weierstrass theorem, that $\Lambda=\mathcal{A}(G)$. Moreover, we show that in the usual inner product on $\mathcal{A}(G)$ we have that the set of matrix entry functions is  almost (up to a scalar factor) orthonormal. Thus, it will follow that the set of matrix entry functions is an orthonormal basis for $\mathcal{A}(G)$ and thus we will derive the fundamental result that $\#\left(\widehat{G}\right)<\infty$ and much sharper that $\displaystyle \sum_{\alpha\in\widehat{G}}d_\alpha^2=|G|$.

February 23, 2011

Matrix Entry Functions (Pt. II)

Point of post: This post is a continuation of this one.

February 22, 2011

Matrix Entry Functions

Point of post: In this post we discuss the concept of how representations give rise naturally to a wide variety of elements of the group algebra. Namely, we discuss the matrix entries of the matrix realizations of an irrep.

Motivation

We’ve seen that every finite group $G$ gives rise naturally to a set of equivalence classes of irreps $\widehat{G}$. Suppose for a second though that for each equivalence class $\alpha\in\widehat{G}$ we picked some representative $\rho^{(\alpha)}$ and fixed a basis $\mathcal{B}^{(\alpha)}$ for $\rho^{(\alpha)}$‘s representation space $\mathscr{V}^{(\alpha)}$. Then, we canonically have defined a mapping $G\to\text{Mat}_{\deg \alpha}\left(\mathbb{C}\right)$ by $g\mapsto \left[\rho^{(\alpha)}\right]_{\mathscr{B}}$ where when it’s clear which $\rho^{(\alpha)}$ and $\mathcal{B}^{(\alpha)}$ we’re discussing we simply write $g\mapsto D^{(\alpha)}(g)$. From this we then have defined $\left(\deg\alpha\right)^2$ elements of the group algebra $\mathcal{A}(G)$. Namely, if we denote $D^{(\alpha)}_{i,j}(g)$ to be the $ij^{\text{th}}$ entry of $D^{(\alpha)}$ then the mapping $g\mapsto D_{i,j}^{(\alpha)}(g)$ is a mapping $G\to\mathbb{C}$ and thus an element of the group algebra as stated. In this post we discuss some of the important properties of these matrix entry functions as they shall prove absolutely crucial in all of the theory to come.

February 22, 2011