Abstract Nonsense

Crushing one theorem at a time

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.

Induced Matrix Representation

Let G be a finite group, and (as usual) \widehat{G} the set of all equivalence classes if irreps on G. Then select once and for all a representative \rho^{(\alpha)}\in\alpha for each \alpha\in\widehat{G}. Then if \mathscr{V}^{(\alpha)} is the representation space for \rho^{(\alpha)} fix once and for all some ordered orthonormal basis \mathcal{B}^{(\alpha)}. We then define the map D^{(\alpha)}:G\to\text{Mat}_{d_\alpha}\left(\mathbb{C}\right) (where we’ve denoted the degree of \rho^{(\alpha)} by d_\alpha for notational convenience) by D^{(\alpha)}(g)=\left[\rho^{(\alpha)}(g)\right]_{\mathcal{B}^{(\alpha)}}. We call this the matrix realization of \alpha. We first claim that each D^{(\alpha)} is itself an irrep with representation space \mathbb{C}^{d_\alpha}. Indeed,

Theorem: Let D^{(\alpha)}:G\to\text{Mat}_{d_\alpha}\left(\mathbb{C}\right) be defined as above. Then D^{(\alpha)} is itself and irrep on G.

Proof: It’s evident that \text{im}\left(D^{(\alpha)}\right)\subseteq\mathcal{U}\left(\mathbb{C}^{d_\alpha}\right). Thus, to verify that D^{(\alpha)} is a representation of G it suffices to show that D^{(\alpha)} is a homomorphism. But, using the fact that the map [\cdot]_{\mathcal{B}^{(\alpha)}}:\text{End}\left(\mathscr{V}^{(\alpha)}\right)\to\text{Mat}_{d_\alpha}\left(\mathbb{C}\right) is an associative unital algebra isomorphism we know then that

\begin{aligned}D^{(\alpha)}\left(gg'\right) &= \left[\rho^{(\alpha)}(gg')\right]_{\mathcal{B}^{(\alpha)}}\\ &= \left[\rho^{(\alpha)}(g)\rho^{(\alpha)}(g')\right]_{\mathcal{B}^{(\alpha)}}\\ &= \left[\rho^{(\alpha)}(g)\right]_{\mathcal{B}^{(\alpha)}}\left[\rho^{(\alpha)}(g')\right]_{\mathcal{B}^{(\alpha)}}\\ &= D^{(\alpha)}(g)D^{(\alpha)}(g')\end{aligned}

 

from where the conclusion follows.  Now, to see that D^{(\alpha)} is irreducible let \phi_{\mathcal{B}^{(\alpha)}}:\mathscr{V}^{(\alpha)}\to\mathbb{C}^{d_\alpha} be the canonical identification (the isomorphism associating with each element of \mathscr{V}^{(\alpha)} the column vector of the coefficients of that element with respect to the ordered basis). Recall then that \phi_{\mathcal{B}^{(\alpha)}}^{-1}D^{(\alpha)}\phi_{\mathcal{B}^{(\alpha)}}=\rho^{(\alpha)} and let \mathscr{W}\leqslant \mathbb{C}^{d_\alpha} be D^{(\alpha)}-invariant. Note then that for any g\in G

\begin{aligned}\left(\rho^{(\alpha)}(g)\right)\left(\phi_{\mathcal{B}^{(\alpha)}}^{-1}\left(\mathscr{W}\right)\right)&= \phi_{\mathcal{B}^{(\alpha)}}^{-1}\phi_{\mathcal{B}^{(\alpha)}}\left(\rho^{(\alpha)}(g)\right)\phi_{\mathcal{B}^{(\alpha)}}^{-1}\left(\mathscr{W}\right)\\ &=\phi_{\mathcal{B}^{(\alpha)}}^{-1}\left(D^{(\alpha)}(g)\right)\left(\mathscr{W}\right)\\ &\subseteq \phi_{\mathcal{B}^{(\alpha)}}^{-1}\left(\mathscr{W}\right)\end{aligned}

 

and since g was arbitrary it follows that \phi_{\mathcal{B}^{(\alpha)}}^{-1}\left(\mathscr{W}\right) is \rho^{(\alpha)}-invariant and so \phi_{\mathcal{B}^{(\alpha)}}^{-1}\left(\mathscr{W}\right) is equal to either \{\bold{0}\} or \mathscr{V} which is equivalent to saying that \mathscr{W} is equal to either \{\bold{0}\} or \mathbb{C}^{d_\alpha}. Thus, since \mathscr{W} was arbitrary it follows that D^{(\alpha)} is irreducible. \blacksquare

References:

1.Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.

2. Serre, Jean Pierre. Linear Representations of Finite Groups. New York: Springer-Verlag, 1977. Print.

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February 22, 2011 - Posted by | Algebra, Representation Theory | , , ,

8 Comments »

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