Gaussian fluctuations of characters of symmetric groups and of Young diagrams

By Piotr Śniady. 2005

76 citations recorded by the source corpus.

We study asymptotics of reducible representations of the symmetric groups Sq 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, the left-regular representation (Plancherel measure), irreducible representations and 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.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c8f2c7f2159fccdd588fc1faa2bdef4233638e45

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In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Exponential Approximation by Stein's Method and Spectral Graph Theory (2006, background): “…explain, limit theorems for graph spectra (especially Cayley graphs and finite symmetric spaces) have been studied by many authors and from various perspectives; some references are Hora (1998b,a); Kerov (1993); Fulman (2005, 2006b,a, 2008); Śniady (2006); Shao and Su (2006); Terras (1999, 1996).”