Skew Howe duality and random rectangular Young tableaux. (English) Zbl 1391.05268

Panova, Greta; Śniady, Piotr

Algebr. Comb. 1, No. 1, 81-94 (2018).

Summary: We consider the decomposition into irreducible components of the external power \(\bigwedge^p(\mathbb{C}^m\otimes \mathbb{C}^n)\) regarded as a \(\operatorname{GL}_m\times \operatorname{GL}_n\)-module. Skew Howe duality implies that the Young diagrams from each pair \((\lambda,\mu)\) which contributes to this decomposition turn out to be conjugate to each other, i.e. \(\mu =\lambda'\). We show that the Young diagram \(\lambda \) which corresponds to a randomly selected irreducible component \((\lambda ,\lambda')\) has the same distribution as the Young diagram which consists of the boxes with entries \(\leq p\) 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 \(m,n,p\rightarrow \infty \) tend to infinity.

MSC:

05E10	Combinatorial aspects of representation theory
22E46	Semisimple Lie groups and their representations
20C30	Representations of finite symmetric groups
60C05	Combinatorial probability

Keywords:

skew Howe duality; random Young diagrams; representations of general linear groups \(\operatorname{GL}_m\); representations of finite symmetric groups

References:

[1]	Biane, Philippe, Quantum random walk on the dual of \({\rm SU}(n)\), Probab. Theory Related Fields, 89, 1, 117-129, (1991) · Zbl 0746.46058
[2]	Biane, Philippe, Representations of unitary groups and free convolution, Publ. Res. Inst. Math. Sci., 31, 1, 63-79, (1995) · Zbl 0856.22017
[3]	Biane, Philippe, Representations of symmetric groups and free probability, Adv. Math., 138, 1, 126-181, (1998) · Zbl 0927.20008
[4]	Bufetov, Alexey; Gorin, Vadim, Representations of classical Lie groups and quantized free convolution, Geom. Funct. Anal., 25, 3, 763-814, (2015) · Zbl 1326.22012
[5]	Collins, Benoît; Novak, Jonathan; Śniady, Piotr, Semiclassical asymptotics of \(\operatorname{GL}_N(\mathbb{C})\) tensor products and quantum random matrices, (2016) · Zbl 1404.22032
[6]	Féray, Valentin, Stanley’s formula for characters of the symmetric group, Ann. Comb., 13, 4, 453-461, (2010) · Zbl 1234.20014
[7]	Féray, Valentin; Śniady, Piotr, Asymptotics of characters of symmetric groups related to Stanley character formula, Ann. of Math. (2), 173, 2, 887-906, (2011) · Zbl 1229.05276
[8]	Howe, Roger, The Schur lectures (1992) (Tel Aviv), 8, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, 1-182, (1995), Bar-Ilan Univ., Ramat Gan · Zbl 0844.20027
[9]	Landsberg, Joseph M., Distribution of Young diagrams, (2012)
[10]	Landsberg, Joseph M.; Ottaviani, Giorgio, New lower bounds for the border rank of matrix multiplication, Theory Comput., 11, 285-298, (2015) · Zbl 1336.68102
[11]	Mkrtchyan, Sevak, On a question of J. M. Landsberg, (2017)
[12]	Pittel, Boris; Romik, Dan, Limit shapes for random square Young tableaux, Adv. in Appl. Math., 38, 2, 164-209, (2007) · Zbl 1122.60009
[13]	Śniady, Piotr, Gaussian fluctuations of characters of symmetric groups and of Young diagrams, Probab. Theory Related Fields, 136, 2, 263-297, (2006) · Zbl 1104.46035
[14]	Stanley, Richard P., Irreducible Symmetric Group Characters of Rectangular Shape, (2001) · Zbl 1068.20017