Asymptotics of characters of symmetric groups related to Stanley character formula

Abstract

We prove an upper bound for characters of the symmetric groups. Namely, we show that there exists a constant a>0 with a property that for every Young diagram λwith n boxes, r(λ) rows and c(λ) columns |Tr ρ^λ(π) / Tr ρ^λ(e)| < [a max(r(λ)/n, c(λ)/n,|π|/n) ]^{|π|}, where |π| is the minimal number of factors needed to write π\in S_n as a product of transpositions. We also give uniform estimates for the error term in the Vershik-Kerov's and Biane's character formulas and give a new formula for free cumulants of the transition measure.

arXiv commentVersion 4: Change of title, shortened to 20 pages. Version 3: 24 pages, the title and the list of authors were changed. Version 2: 14 pages, the title, abstract and the main result were changed. Version 1: 10 pages (mistake in Lemma 7- which is false)

References in corpus (4)

• On the impossibility of a quantum sieve algorithm for graph isomorphism: unconditional results
• Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations
• Proof of Stanley's conjecture about irreducible character values of the symmetric group
• Upper bound on the characters of the symmetric groups for balanced Young diagrams and a generalized Frobenius formula

Cited by in corpus (10)

• Zonal polynomials via Stanley's coordinates and free cumulants
• The quantum detection of projectors in finite-dimensional algebras and holography
• Asymptotics of Jack characters
• Distribution of Frobenius elements in families of Galois extensions
• Shifted symmetric functions and multirectangular coordinates of Young diagrams
• Asymptotics of characters of symmetric groups: structure of Kerov character polynomials
• A Spin Analogue of Kerov Polynomials
• Commutators of random matrices from the unitary and orthogonal groups
• Random matrices associated to Young diagrams
• $\lambda$-shaped random matrices, $\lambda$-plane trees, and $\lambda$-Dyck paths