# ALGEBRAIC COMBINATORICS

Skew Howe duality and random rectangular Young tableaux
Algebraic Combinatorics, Volume 1 (2018) no. 1, pp. 81-94.

We consider the decomposition into irreducible components of the external power ${\bigwedge }^{p}\left({ℂ}^{m}\otimes {ℂ}^{n}\right)$ regarded as a ${GL}_{m}×{GL}_{n}$-module. Skew Howe duality implies that the Young diagrams from each pair $\left(\lambda ,\mu \right)$ which contributes to this decomposition turn out to be conjugate to each other, i.e. $\mu ={\lambda }^{\prime }$. We show that the Young diagram $\lambda$ which corresponds to a randomly selected irreducible component $\left(\lambda ,{\lambda }^{\prime }\right)$ has the same distribution as the Young diagram which consists of the boxes with entries $\le p$ of a random Young tableau of rectangular shape with $m$ rows and $n$ columns. This observation allows treatment of the asymptotic version of this decomposition in the limit as $m,n,p\to \infty$ tend to infinity.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.8
Classification: 22E46,  20C30,  60C05
Keywords: Skew Howe duality, random Young diagrams, representations of general linear groups ${GL}_{m}$, representations of finite symmetric groups
@article{ALCO_2018__1_1_81_0,
author = {Panova, Greta and \'Sniady, Piotr},
title = {Skew {Howe} duality and random rectangular {Young} tableaux},
journal = {Algebraic Combinatorics},
pages = {81--94},
publisher = {MathOA foundation},
volume = {1},
number = {1},
year = {2018},
doi = {10.5802/alco.8},
zbl = {06882335},
mrnumber = {3857160},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.8/}
}
Panova, Greta; Śniady, Piotr. Skew Howe duality and random rectangular Young tableaux. Algebraic Combinatorics, Volume 1 (2018) no. 1, pp. 81-94. doi : 10.5802/alco.8. https://alco.centre-mersenne.org/articles/10.5802/alco.8/

[1] Biane, Philippe Quantum random walk on the dual of $\mathrm{SU}\left(n\right)$, Probab. Theory Related Fields, Volume 89 (1991) no. 1, pp. 117-129 | Article | MR 1109477 | Zbl 0746.46058

[2] Biane, Philippe Representations of unitary groups and free convolution, Publ. Res. Inst. Math. Sci., Volume 31 (1995) no. 1, pp. 63-79 | Article | MR 1317523 | Zbl 0856.22017

[3] Biane, Philippe Representations of symmetric groups and free probability, Adv. Math., Volume 138 (1998) no. 1, pp. 126-181 | Article | MR 1644993 | Zbl 0927.20008

[4] Bufetov, Alexey; Gorin, Vadim Representations of classical Lie groups and quantized free convolution, Geom. Funct. Anal., Volume 25 (2015) no. 3, pp. 763-814 | Article | MR 3361772 | Zbl 1326.22012

[5] Collins, Benoît; Novak, Jonathan; Śniady, Piotr Semiclassical asymptotics of ${GL}_{N}\left(ℂ\right)$ tensor products and quantum random matrices (2016) (Preprint arXiv:1611.01892) | Zbl 06904449

[6] Féray, Valentin Stanley’s formula for characters of the symmetric group, Ann. Comb., Volume 13 (2010) no. 4, pp. 453-461 | Article | MR 2581097 | Zbl 1234.20014

[7] Féray, Valentin; Śniady, Piotr Asymptotics of characters of symmetric groups related to Stanley character formula, Ann. of Math. (2), Volume 173 (2011) no. 2, pp. 887-906 | Article | MR 2776364 | Zbl 1229.05276

[8] Howe, Roger Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures (1992) (Tel Aviv) (Israel Math. Conf. Proc.), Volume 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 1-182 | Article | MR 1321638 | Zbl 0844.20027

[9] Landsberg, Joseph M. Distribution of Young diagrams, 2012 (Question asked on MathOverflow, https://mathoverflow.net/questions/88435/distribution-of-young-diagrams)

[10] Landsberg, Joseph M.; Ottaviani, Giorgio New lower bounds for the border rank of matrix multiplication, Theory Comput., Volume 11 (2015), pp. 285-298 | Article | MR 3376667 | Zbl 1336.68102

[11] Mkrtchyan, Sevak On a question of J. M. Landsberg (2017) (In preparation)

[12] Pittel, Boris; Romik, Dan Limit shapes for random square Young tableaux, Adv. in Appl. Math., Volume 38 (2007) no. 2, pp. 164-209 | Article | MR 2290809 | Zbl 1122.60009

[13] Śniady, Piotr Gaussian fluctuations of characters of symmetric groups and of Young diagrams, Probab. Theory Related Fields, Volume 136 (2006) no. 2, pp. 263-297 | Article | MR 2240789 | Zbl 1104.46035

[14] Stanley, Richard P. Irreducible Symmetric Group Characters of Rectangular Shape (2001) (Preprint arXiv:math/0109093) | Zbl 1068.20017