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If $\chi^{\lambda}$ and $\chi^{\mu}$ are the characters of two irreducible representations $V^{\lambda}$ and $V^{\mu}$ of a finite group $G$, is there a simple way of proving that : $$ \chi^{\lambda} *\chi^{\mu} = \delta_{\lambda, \mu} \frac{|G|}{\dim V^{\lambda}} \,\chi^{\lambda}$$ where $\chi^{\lambda} *\chi^{\mu}(\sigma)=\sum_{g\in G}\chi^{\lambda}(\sigma g^{-1})\chi^{\mu}(g)$ is the product of convoluition and $\delta_{\lambda, \mu}$ equals 1 if $\lambda=\mu$ and 0 if $\lambda\neq\mu$.

I'm actually interested in representations of the symetric group $\mathcal{S}_n$, and this relation was instrumental in the definition of an isomorphism between the center of $\mathbb{C}[\mathcal{S}_n]$ and the complex functions on the conjugacy classes of $\mathcal{S}_n$.

I found a demonstration on this page http://drexel28.wordpress.com/2011/03/02/representation-theory-using-orthogonality-relations-to-compute-convolutions-of-characters-and-matrix-entry-functions/:

\begin{aligned}\left(\chi^{(\alpha)}\ast\chi^{(\beta)}\right)(x) &= \sum_{g\in G}\chi^{(\alpha)}\left(xg^{-1}\right)\chi^{(\beta)}(g)\\ &=\sum_{g\in G} \sum_{p=1}^{d_\alpha}\sum_{q=1}^{d_\beta}D^{(\alpha)}_{p,p}\left(xg^{-1}\right)D^{(\beta)}_{q,q}(g)\\ &=\sum_{p,s=1}^{d_\alpha}\sum_{q=1}^{d_\beta}D^{(\alpha)}_{p,s}(x)\sum_{g\in G}\overline{D^{(\alpha)}_{p,s}(g)}D^{(\beta)}_{q,q}(g)\\ &= \sum_{p,s=1}^{d_\alpha}\sum_{q=1}^{d_\beta}D^{(\alpha)}_{p,s}(x)\frac{|G|}{d_\alpha}\delta_{\alpha,\beta}\delta_{p,q}\delta_{q,s}\\ &=\frac{|G|}{d_\alpha}\delta_{\alpha,\beta}\sum_{p,s=1}^{d_\alpha}D^{(\alpha)}_{p,s}(x)\delta_{p,q}\delta_{p,s}^2\\ &= \frac{|G|}{d_\alpha}\delta_{\alpha,\beta}\sum_{p=1}^{d_\alpha}D^{(\alpha)}_{p,p}(x)\\ &= \frac{|G|}{d_\alpha}\delta_{\alpha,\beta}\chi^{(\alpha)}(x)\end{aligned}

but it is not clear for me why $\sum_{g\in G}\overline{D^{(\alpha)}_{p,s}(g)}D^{(\beta)}_{q,q}(g)$ is equal to $\frac{|G|}{d_\alpha}\delta_{\alpha,\beta}\delta_{p,q}\delta_{q,s} $. It seems to me that it is a not-so-trivial consequence of Schur's lemma. So my question is : Is there a simpler way of proving this relation about convolution of characters ?

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Do you need the full strength of the formula, or are you just in need of the orthogonality relations enjoyed by characters? Either way, we have already developed a purely algebraic approach to this relation, through the use of structure theorems and double centralizer theorems for non-commutative algebras. But, after that development, the proof is quite simple. In fact, that approach –  awllower Jun 1 '12 at 7:33
even said more about characters: they can express the primitive central idempotent elements of the module A in terms of characters. And this of course leads to general relations between representation matrices, for those elements are orthonormal. –  awllower Jun 1 '12 at 7:37

2 Answers 2

up vote 1 down vote accepted

Not only do characters enjoy orthogonality relations, the entries of any two matrix representations also exhibit orthogonality. Indeed, the relations for characters follow as corollary to the latter.

First, a quick corollary of Schur's. Let $A$ be an irreducible matrix representation of $G$ over $\mathbb{C}$, and suppose $T$ is a matrix that commutes with $A(g)$ for all $g\in G$. Then the same is true for $T-cI$ for any $c\in\Bbb C$; choosing an eigenvalue $c$ of $T$ means that $T-cI$ is noninvertible, so it must be the zero matrix by Schur's, and hence $T$ is a scalar multiple of the identity matrix. (For arbitrary fields $k$ this can probably be adapted by tensoring $V$ with the algebraic closure $k^\mathrm{alg}$ and then restricting back down to $k$, unsure.)

Let $k$ be a field, $A:G\to M_{r\times r}(k)$ and $B:G\to M_{s\times s}(k)$ two irreducible matrix representations of $G$, and then finally $X$ an $r\times s$ matrix of unknowns. Consider the matrix

$$Y=\sum_{g\in G} A(g)XB(g^{-1}). \tag{1}$$

With $h\in G$ arbitrary, we have

$$\begin{array}{c l} A(h)Y & =\sum_{g\in G} A(hg)XB(g^{-1}) \\ & = \sum_{g\in G} A(g)XB\big(\underbrace{(h^{-1}g)^{-1}}_{g^{-1}h}\big) \\ & = \sum_{g\in G}A(g)XB(g^{-1})B(h) \\ & = YB(h). \end{array} \tag{2}$$

If $Y$ is invertible then $A\cong B$ are equivalent representations. Otherwise, if $A\not\cong B$ are inequivalent, we find $Y$ is not invertible and so by Schur's lemma is the zero matrix. Assume the latter case. Then the $(i,j)$ entry of the equation $Y=0$ is

$$\sum_{g\in G}\sum_{k,l} a_{ik}(g)x_{kl}b_{lj}(g^{-1})=\sum_{k,l}x_{kl}\left(\sum_{g\in G}a_{ik}(g)b_{lj}(g^{-1})\right)=0. \tag{3}$$

The $x_{lk}$'s are arbitrary unknowns however, yet the equality above holds regardless, therefore the coefficient of each is zero. In other words, $\langle a_{ik},b_{lj}\rangle_G=0$ when $A,B$ are inequivalent. (This is the inner product defined on class functions of $G$, i.e. functions that are constant on conjugacy classes.)

Otherwise we might as well say $A=B$ (if they're equivalent it's not going to make any difference to the character's values anyway), and we consider

$$\operatorname{tr}Y=\operatorname{tr} \frac{1}{|G|}\sum_{g\in G} A(g)XA(g^{-1})=\frac{1}{|G|}\sum_{g\in G}\operatorname{tr} X=\operatorname{tr}X. \tag{4}$$

Let $k$ be algebraically closed. Since $Y$ is a scalar multiple of the identity, we have $y_{ii}=\operatorname{tr}X/r$. Hence we write $Y=\frac{\operatorname{tr}X}{r}I$ as

$$\frac{1}{|G|}\sum_{k,l}x_{kl}\left(\sum_{g\in G}a_{ik}(g)a_{lj}(g^{-1})\right)=\begin{cases} \frac{x_{11}+\cdots+x_{rr}}{r} & i=j \\ 0 & \text{otherwise}.\end{cases} \tag{5}$$

Equating coefficients above and then putting this together with $A,B$ inequivalent, we have derived

$$\langle a_{ik},b_{lj}\rangle_G=\frac{\delta_{AB}\delta_{ij}\delta_{kl}}{r}. \tag{6}$$

Note that in $\Bbb C$, $B(g^{-1})=\overline{B(g)^T}$ because $B$ is unitary, which switches the indices in $b_{\circ\circ}$ above too.

This is still a brutish method of index juggling, and I'm not familiar with an easier way in these matters. (Of course I wouldn't though; I just came across the above a few hours ago.) Perhaps someone else has enlightenment on that count.

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Thank you very much for the clarification of the orthogonality of matrix coefficients as a consequence of Schur's lemma. Thus the proof is a little bit longer than what i expected, but we cannot do better using coefficients. –  saposcat Jun 1 '12 at 13:40

Actually Awllower replied to my question.

The theorem 2.13 page 19 in "Character Theory of finite groups" by I. Martin Isaacs, is exactly the relation I wanted, and the author prove it in a pure algebraic way, though not as elementary as I expected.

The relation on the convolution product of irreducible characters appears here to be a consequence of a decomposition of the primitive central idempotent elements (noted $e_i$ in Isaac's book) of $\mathbb{C}[G]$ in terms of irreducible characters, as awllower said.

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