Consider $\Lambda^p(C^n\otimes C^n)=\oplus_{\pi}S_{\pi}C^n\otimes S_{\pi'}C^n$ as a $GL_n\times GL_n$module. This space has dimension $\binom {n^2}p$. I would like any information on the shapes of pairs of Young diagrams $(\pi,\pi')$ that give the largest contribution to the dimension asymptotically. I am most interested in the case where $p$ is near $n^2/2$. Is there a slowly growing function $f(n)$ such that partitions with fewer than $f(n)$ steps contribute negligibly? If so, can the fastest growing such $f$ be determined?

$\begingroup$ What is denoted by $S_{\pi}C^n$? $\endgroup$ – Fedor Petrov Feb 28 '15 at 19:18

$\begingroup$ The Schur functor corresponding to $\pi$ applied to $\mathbb{C}^n$, I think. $\endgroup$ – alpoge Feb 28 '15 at 20:30
For more details on this answer, together with two proofs of the following theorem, see arXiv:1705.07604 preprint ``External powers of tensor products as representations of general linear groups'' by Greta Panova and myself.
The following result converts the original question into a problem about the representation theory of the symmetric groups for which more asymptotic tools are available.
Theorem. The random irreducible component of the external power \begin{equation} \label{eq:decomposition} \Lambda^p(\mathbb{C}^m\otimes \mathbb{C}^n)=\bigoplus_{\lambda}S^{\lambda}\mathbb{C}^m\otimes S^{\lambda'}\mathbb{C}^n \end{equation} regarded as a $\operatorname{GL}_m\times \operatorname{GL}_n$module corresponds to a pair of Young diagrams $(\lambda,\lambda')$, where $\lambda$ has the same distribution as the Young diagram which consists of the boxes with entries $\leq p$ of a uniformly random Young tableau with rectangular shape $n^m$ with $m$ rows and $n$ columns.
Alternatively: the random Young diagram $\lambda$ has the same distribution as a Young diagram which corresponds to a random irreducible component of the restriction $V^{n^m}\big\downarrow^{\mathfrak{S}_{mn}}_{\mathfrak{S}_{p}}$ of the irreducible representation $V^{n^m}$ of the symmetric group $\mathfrak{S}_{mn}$ which corresponds to the rectangular diagram $n^m$.
Above, when we speak about random irreducible component of a representation we refer to the following concept. For a representation $V$ of a group $G$ we consider its decomposition into irreducible components $$ V = \bigoplus_{\zeta \in \widehat{G} } m_\zeta V^\zeta, $$ where $m_\zeta\in\{0,1,\dots\}$ denotes the multiplicity of an irreducible representation $V^\zeta$ in $V$. This defines a probability measure $\mathbb{P}_V$ on the set $\widehat{G}$ of irreducible representations given by $$ \mathbb{P}_V(\zeta) := \frac{m_\zeta \operatorname{dim} V^{\zeta}}{\operatorname{dim} V}.$$
As I mentioned, the above theorem converts the original question into a problem about the representations of the symmetric groups for which several results are already available. In particular, the law of large numbers for the corresponding random Young diagrams has been proved in much wider generality by Biane ["Representations of symmetric groups and free probability", Adv. Math., 138(1):126181, 1998, Theorem 1.5.1] using the language of free cumulants of Young diagrams. The asymptotic Gaussianity of their fluctuations around the limit shape has been proved in [Piotr Śniady, "Gaussian fluctuations of characters of symmetric groups and of Young diagrams". Probab. Theory Related Fields 136 (2006), no. 2, 263–297, Example 7 combined with Theorem 8] using the same language.
In the specific case of the restriction $V^{n^m}\big\downarrow^{\mathfrak{S}_{mn}}_{\mathfrak{S}_{p}}$, the abovementioned generic tools can be applied in the scaling when $m,n,p\to\infty$ tend to infinity in such a way that the rectangle ratio $\frac{m}{n}$ converges to a strictly positive limit and the fraction $\frac{p}{mn}$ converges to some limit. [Boris Pittel and Dan Romik, "Limit shapes for random square Young tableaux", Adv. in Appl. Math. 38 (2007), no. 2, 164–209] have worked out this specific example and, among other results, found explicit asymptotic limit shapes of typical Young diagrams which contribute to such representations.
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