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Covering \(\textsf{Irrep}(S_n)\) With tensor products and powers

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Abstract

We study when a tensor product of irreducible representations of the symmetric group \(S_n\) contains all irreducibles as subrepresentations—we say such a tensor product covers \(\textsf{Irrep}(S_n)\). Our results show that this behavior is typical. We first give a general sufficient criterion for tensor products to have this property, which holds asymptotically almost surely for constant-sized collections of (Plancherel or uniformly) random irreducibles. We also consider the minimal tensor power of a single fixed irreducible representation needed to cover \(\textsf{Irrep}(S_n)\). Here a simple lower bound comes from considering dimensions, and we show it is always tight up to a universal constant factor as was recently conjectured by Liebeck, Shalev, and Tiep.

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Notes

  1. We show the latter from the former in Appendix A, simplifying [16] which used two separate arguments.

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Acknowledgements

The author gratefully acknowledges support of NSF and Stanford Graduate Fellowships. I thank Daniel Bump, Pavel Etingof, Xiaoyue Gong, Sammy Luo, Alex Malcolm, Chris Ryba and the anonymous referee for helpful comments, corrections, discussions, and references. I thank Greta Panova for bringing [17] to my attention after the initial posting of this paper.

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Appendix A: Alternate proof of fourth power Saxl theorem

Appendix A: Alternate proof of fourth power Saxl theorem

Here we give an alternate proof of Theorem 2.10 that \(\varvec{\varrho }_r^{\otimes 4}\) covers \(\textsf{Irrep}(S_n)\) for r sufficiently large (which is Theorem 1.4 of [16]) based on another main result from [16]. The implication is immediate from a lemma on the representation theory of an arbitrary finite group G which we suspect to be known but have not been able to locate in the literature.

Definition A.1

Let G be a finite group and let \(M_G\) be the Plancherel probability measure on \(\textsf{Irrep}(G)\) which assigns an irreducible representation \(\varvec{\lambda }\) a probability \(M_G(\varvec{\lambda })=\frac{\dim (\varvec{\lambda })^2}{|G|}\). For an arbitrary finite-dimensional G-representation V, let \(M_G(V)\) denote the Plancherel measure of the set of distinct irreducible subrepresentations of V.

Theorem 1.6 of [16] states that \(\varvec{\varrho }_r^{\otimes 2}\) contains Plancherel-asymptotically-almost-all of \({\mathcal {Y}}_n\) for \(n=\left( {\begin{array}{c}r+1\\ 2\end{array}}\right) \), i.e. \(\lim _{r\rightarrow \infty } M_{S_n}(\varvec{\varrho }_r^{\otimes 2})=1\). Therefore Theorem 2.10 follows immediately from the lemma below. We note that proving Theorem 1.6 of [16] relies on the deep work of [4], so the proof of Theorem 2.10 given in [16] is more elementary than the present proof. Nonetheless we find the connection enlightening.

Lemma A.2

Suppose \(M_G(V)+M_G(W)>1\). Then \(V\otimes W\) covers \(\textsf{Irrep}(G)\).

Proof

The conclusion is equivalent to the statement that \(\langle \chi ^V\chi ^W,\chi ^{\varvec{\lambda }^*}\rangle >0\) for any irreducible representation \(\varvec{\lambda }\), where \((\cdot )^{*}\) denotes the dual representation. As \(\langle \chi ^V\chi ^W,\chi ^{\varvec{\lambda }^*}\rangle =\langle \chi ^W,\chi ^{V^*}\chi ^{\varvec{\lambda }^*}\rangle \), this is equivalent to showing the tensor product \(V^*\otimes \varvec{\lambda }^*\) shares some irreducible subrepresentation with W. We will prove that \(M_G(V\otimes \varvec{\lambda })\ge M_G(V)\). This implies \(M_G(V^*\otimes \varvec{\lambda }^*)+M_G(W)=M_G(V\otimes \varvec{\lambda })+M_G(W)>1\) so that they share a subrepresentation by the pigeonhole principle.

To see this, we work in the standard inner product space \(L^2(G)\) and recall that irreducible characters \(\chi ^{\varvec{\mu }}\) for \(\varvec{\mu }\in \textsf{Irrep}(G)\) are orthonormal. We identify representations with their characters. Consider for any representation U the best \(L^2\) approximation to the regular representation \({{\textbf {Reg}}}\) of G lying in the linear space

$$\begin{aligned} \left\{ \sum _{\varvec{\mu }_U\in U} a_{\varvec{\mu }_U}{\chi ^{\varvec{\mu }_U}}~:~a_{\varvec{\mu }_U}\in {\mathbb {R}}\right\} . \end{aligned}$$

From the point of view of irreducible representations it is clear that the best approximation \({\tilde{U}}\) is obtained by projection via \(a_{\varvec{\mu }_U}=\dim (\varvec{\mu }_U)\), and the \(L^2\) error of this approximation \({\tilde{U}}\) is therefore \(|{{\textbf {Reg}}}-\tilde{U}|=\sqrt{|G|\cdot (1-M_G(U))}\).

We form \({\tilde{V}}\) and multiply its character by \(\frac{\chi ^{\varvec{\lambda }}}{\dim \varvec{\lambda }}\), and by abuse of notation treat this as a tensor product of fractional representations. The key point is that \({\tilde{V}}\otimes \frac{\varvec{\lambda }}{\dim \varvec{\lambda }}\) has the same character value at the identity element of G, and a smaller character value (in absolute value) at all other elements. Since \({{\textbf {Reg}}}\) has character value 0 at all non-identity elements, computing the distances using the character basis implies

$$\begin{aligned} \left| {\tilde{V}}\otimes \frac{\varvec{\lambda }}{\dim \varvec{\lambda }}-{{\textbf {Reg}}}\right| \le |{\tilde{V}}-{{\textbf {Reg}}}|. \end{aligned}$$

Moreover \({\tilde{V}}\otimes \frac{\varvec{\lambda }}{\dim \varvec{\lambda }}\) is in the \({\mathbb {R}}\)-span of the irreducible subrepresentations of \(V\otimes \varvec{\lambda }\). Since the function \(\sqrt{1-M_G(U)}\) is decreasing in \(M_G(U)\), the fact that by using subrepresentations of \(V\otimes \varvec{\lambda }\) we weakly improved upon the best \(L^2\) approximation to \({{\textbf {Reg}}}\) using subrepresentations of V implies \(M_G(V\otimes \varvec{\lambda })\ge M_G(V)\) as desired. \(\square \)

Note that Lemma A.2 becomes completely false if Plancherel measure is replaced by uniform measure. For instance, the group of invertible affine transformations of \({\mathbb {F}}_p\) has \(p-1\) irreducible representations of dimension 1 and one of dimension \(p-1\). If VW each contain exactly the 1-dimensional irreducible representations then \(V\otimes W\) still consists of only 1-dimensional irreducibles, hence does not cover \(\textsf{Irrep}(G)\).

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Sellke, M. Covering \(\textsf{Irrep}(S_n)\) With tensor products and powers. Math. Ann. 388, 831–865 (2024). https://doi.org/10.1007/s00208-022-02532-3

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