Gaussian fluctuations of characters of symmetric groups and of Young diagrams

Abstract

We study asymptotics of reducible representations of the symmetric groups S_q for large q. We decompose such a representation as a sum of irreducible components (or, alternatively, Young diagrams) and we ask what is the character of a randomly chosen component (or, what is the shape of a randomly chosen Young diagram). Our main result is that for a large class of representations the fluctuations of characters (and fluctuations of the shape of the Young diagrams) are asymptotically Gaussian; in this way we generalize Kerov's central limit theorem. The considered class consists of representations for which the characters almost factorize and this class includes, for example, left-regular representation (Plancherel measure), tensor representations. This class is also closed under induction, restriction, outer product and tensor product of representations. Our main tool in the proof is the method of genus expansion, well known from the random matrix theory.

arXiv comment37 pages; version 3: conceptual change in the proofs

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• Jeu de taquin dynamics on infinite Young tableaux and second class particles
• Asymptotics of characters of symmetric groups, genus expansion and free probability
• Fourier transform on high-dimensional unitary groups with applications to random tilings
• Zonal polynomials via Stanley's coordinates and free cumulants
• Upper bound on the characters of the symmetric groups for balanced Young diagrams and a generalized Frobenius formula
• Gaussian fluctuations of Jack-deformed random Young diagrams
• Representations of Lie groups and random matrices
• Asymptotics of Jack characters
• Random strict partitions and random shifted tableaux
• Gaussian fluctuations of representations of wreath products
• Strong factorization property of Macdonald polynomials and higher-order Macdonald's positivity conjecture
• Stanley character polynomials
• Mod-phi convergence I: Normality zones and precise deviations
• Linear versus spin: representation theory of the symmetric groups
• Zero biasing and growth processes
• Macdonald cumulants, $G$-inversion polynomials and $G$-parking functions
• A central limit theorem for singular graphons
• Quantum Mechanics of Plancherel Growth
• Universality of global asymptotics of Jack-deformed random Young diagrams at varying temperatures
• Modulus of continuity of Kerov transition measure for continual Young diagrams
• Spectra of random linear combinations of matrices defined via representations and Coxeter generators of the symmetric group