Abstract Nonsense

Crushing one theorem at a time

The Irreps of the Product of Finitely Many Finite Groups (Pt. II)

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

Corollary: Let G_1,\cdots,G_n be finitely many finite groups then,

\text{ }

\displaystyle \widehat{\prod_{j=1}^{n}G_j}=\left\{\rho^{(\alpha_1)}\boxtimes\cdots\boxtimes\rho^{(\alpha_n)}:\alpha_k\in\widehat{G_k},\;k\in[n]\right\}

\text{ }

\text{ }

From the above it’s clear that we may index the elements of \displaystyle \widehat{\prod_{j=1}^{n}G_j} as \alpha_1\boxtimes\cdots\boxtimes\alpha_n and so we have the characters of \displaystyle \widehat{\prod_{j=1}^{n}G_j} may be written as \chi^{(\alpha_1\boxtimes\cdots\boxtimes\alpha_n)} and we have that

\displaystyle \chi^{(\alpha_1\boxtimes\cdots\boxtimes\alpha_n)}(g_1,\cdots,g_n)=\chi^{(\alpha_1)}(g_1)\cdots\chi^{(\alpha_n)}(g_n)

and that

\displaystyle \left\langle \chi^{(\alpha_1\boxtimes\cdots\boxtimes\alpha_n)},\chi^{(\beta_1\boxtimes\cdots\boxtimes\beta_n)}\right\rangle=\prod_{j=1}^{n}\left\langle \chi^{(\alpha_j)},\chi^{(\beta_j)}\right\rangle



The last thing we note which can often come in hand for practical computation (since, for instance it can be done on math world) is the following:

Theorem: Let G be a finite group with irreducible characters \chi^{\text{triv}},\chi^{(\alpha_1)},\cdots,\chi^{(\alpha_n)} and conjugacy classes \mathcal{C}_1,\cdots,\mathcal{C}_n. Moreover, let H a finite group with irreducible characters \chi^{\text{triv}},\chi^{(\beta_1)},\cdots,\chi^{(\beta_1)},\cdots,\chi^{(\beta_m)} and conjugacy classes \mathcal{D}_1,\cdots,\mathcal{D}_m. If M_G is character table of G thought of as a matrix with (i,j)^{\text{th}} entry \chi^{(\alpha_i)}\left(\mathcal{C}_j\right) and M_H is the character table of H thought of as the matrix with (i,j)^{\text{th}} entry \chi^{(\beta_i)}\left(\mathcal{D}_j\right). Then, if M_{G\times H} is the character table of G\times H where the rows are arranged left to right \mathcal{C}_1\times\mathcal{D}_1,\cdots,\mathcal{C}_1\times\mathcal{D}_m,\cdots,\mathcal{C}_n\times\mathcal{D}_1,\cdots,\mathcal{C}_n\times\mathcal{D}_m and the columns arranged up to down \chi^{\text{triv}\boxtimes(\beta_1)},\cdots,\chi^{\text{triv}\boxtimes(\beta_m)},\cdots,\chi^{(\alpha_n)\boxtimes(\beta_1)},\cdots,\chi^{(\alpha_n)\boxtimes(\beta_m)}. Then, M_{G\times H}=M_G\otimes M_H where \otimes is the Kronecker product.

\text{ }

\text{ }

Proof: This follows immediately from the definition \blacksquare


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


April 11, 2011 - Posted by | Algebra, Representation Theory | , , , , ,


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