Zeros in the character tables of symmetric groups with an -core index. Issue 2 (27th June 2023) - McSpirit, Eleanor; Ono, Ken
Abstract: Let C_n =[χ _λ (μ )]_λ, μ be the character table for S_n, where the indices λ and μ run over the p(n) many integer partitions of n. In this note, we study Z_(n), the number of zero entries χ _λ (μ ) in C_n, where λ is an -core partition of n. For every prime ≥ 5, we prove an asymptotic formula of the form align*Z_(n)α_· σ_(n+δ_)p(n)_ n^-52e^π2n/3, align* where σ _(n) is a twisted Legendre symbol divisor function, δ _:=(^2-1)/24, and 1/α _>0 is a normalization of the Dirichlet L -value L(( · ), -12). For primes and n>^6/24, we show that χ _λ (μ )=0 whenever λ and μ are both -cores. Furthermore, if Z^*_(n) is the number of zero entries indexed by two -cores, then, for ≥ 5, we obtain the asymptotic align*Z^*_(n)α_^2 · σ_( n+δ_)^2 _ n^-3. align*