Upper bound on the characters of the symmetric groups for balanced Young diagrams and a generalized Frobenius formula

Abstract

We study asymptotics of an irreducible representation of the symmetric group S_n corresponding to a balanced Young diagram λ(a Young diagram with at most C\sqrt{n} rows and columns for some fixed constant C) in the limit as n tends to infinity. We show that there exists a constant D (which depends only on C) with a property that |χ^λ(π)| = | Tr ρ^λ(π)/Tr ρ^λ(e) | < [ D max(1,|π|^2/n) / \sqrt{n}} ]^{|π|}, where |π| denotes the length of a permutation (the minimal number of factors necessary to write πas a product of transpositions). Our main tool is an analogue of Frobenius character formula which holds true not only for cycles but for arbitrary permutations.

26 pages

References in corpus (7)

• Gaussian fluctuations of characters of symmetric groups and of Young diagrams
• On the impossibility of a quantum sieve algorithm for graph isomorphism: unconditional results
• Irreducible Symmetric Group Characters of Rectangular Shape
• The Algebra of Conjugacy Classes in Symmetric Groups and Partial Permutations
• Asymptotics of characters of symmetric groups, genus expansion and free probability
• On the formula of Goulden and Rattan for Kerov polynomials
• On the Impossibility of a Quantum Sieve Algorithm for Graph Isomorphism

Cited by in corpus (10)

• 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
• Combinatorial interpretation and positivity of Kerov's character polynomials
• Asymptotics of characters of symmetric groups related to Stanley character formula
• Zonal polynomials via Stanley's coordinates and free cumulants
• Refined estimates for some basic random walks on the symmetric and alternating groups
• Stanley character polynomials
• Linear versus spin: representation theory of the symmetric groups
• Normalized characters of symmetric groups and Boolean cumulants via Khovanov's Heisenberg category
• Planar algebras for the Young graph and the Khovanov Heisenberg category