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Let $G$ be a finite symmetric group, and let $\widehat G$ denote the set of equivalence classes of irreducible unitary representations of $G$.

For any non-trivial $\rho\in \widehat G$, we know that $\sum_{g\in G} \rho(g)=0$.

Question

Let $\rho_1, \rho_2,\dots ,\rho_k\in\widehat{G}$, and let $\rho=\bigotimes_{i=1}^k \rho_i$. Denote $P=\sum_{g\in G}\rho(g)$.

1.Special case when $k=2$. If $k=2$, what's the possible value of $\text{rank}\ (P)$? Is it only $\{0,1\}$? And if so what's the iff condition $\rho_1, \rho_2$ satisfying such that $\text{rank}\ (P)=1$?

2.General case. What's the possible value of $\text{rank}\ (P)$? And what's the iff condition $\rho_1, \rho_2\dots ,\rho_k$ satisfying such that $\text{rank}\ (P)$ to be the max possible value?

Any answers or references are welcome!

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  • $\begingroup$ You should link to your previous related question: mathoverflow.net/questions/503413. $\endgroup$ Commented Nov 8 at 15:33
  • $\begingroup$ @SamHopkins Does the first special case can be solved by Schur Lemma? $\endgroup$ Commented Nov 8 at 16:04
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    $\begingroup$ $P/|G|$ is the projection to the trivial isotypic component of $\rho$, so the rank is the multiplicity of the trivial representation in $\rho$. When $k=2$ the multiplicity of the trivial representation in $\rho_1\otimes\rho_2$ is the dimension of $Hom_G(\rho_1^\vee,\rho_2)$, which by Schur's lemma is 1 if $\rho_1^\vee\simeq\rho_2$ and 0 otherwise. $\endgroup$ Commented Nov 8 at 16:21

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The operator $P$ is up to a scalar multiple the orthogonal projector to the isotypical component of the trivial representation. The rank of this project is then the dimension of this isotypical component $\dim \operatorname{Hom}_G(1, \rho_1\otimes \dotsb \rho_k) = \dim \operatorname{Hom}_{S_n}(\rho_k^\vee, \rho_1\otimes \dotsb \rho_{k-1})$.

As noted by Kenta Suzuki in comments, for $k=2$ you get rank either 0 or 1 based on the Schur lemma.

For $k=3$, as explained in an answer to your previous question, you get the Kronecker coefficient $g(\rho_1, \rho_2, \rho_3^\vee).$ The maximum value (given fixed $n$) is attained for so called Vershik–Kerov–Logan–Shepp shapes. See On the largest Kronecker and Littlewood–Richardson coefficients by Igor Pak, Greta Panova and Damir Yeliussizov.

For $k=4$ one can start by decomposing the tensor product $$ \begin{align} \operatorname{Hom}_{S_n}(\rho_4^\vee, \rho_1\otimes \rho_{2}\otimes \rho_3) &= \operatorname{Hom}_{S_n}\left(\rho_4^\vee, \left( \bigoplus_\nu g(\rho_1, \rho_2, \nu)\nu\right)\otimes \rho_3\right)\\ &= \operatorname{Hom}_{S_n}\left(\rho_4^\vee, \left(\bigoplus_{\nu,\mu} g(\rho_1, \rho_2, \nu)g(\nu, \rho_3, \mu) \mu\right)\right)\\ \end{align} $$ and then using the result for $k=3$ to obtain a lower bound.

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  • $\begingroup$ I only had a brief look at the paper by Pak et al., but it doesn't seem to me that they discuss the maximum for fixed $n$. Their results are about the limit when $n\rightarrow\infty$ and $\rho_j$ grow in some "nice" way. $\endgroup$ Commented Nov 10 at 5:00

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