On even entries in the character table of the symmetric group

Peluse, Sarah

doi:10.48550/arxiv.2007.06652

Cited by 4 publications

(3 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let C n = [χ λ (μ)] λ, μ be the usual character table (see, for example, [6,13,14]) for the symmetric group S n , where the indices λ and μ both vary over the p(n) many integer partitions of n. Confirming conjectures of Miller [9], Peluse and Soundararajan [11,12] recently proved that if is prime, then almost all of the p(n) 2 entries in C n , as n → +∞, are multiples of . We note that Miller conjectured that the same conclusion holds for arbitrary prime powers, a claim which remains open.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 68%

Zeros in the character tables of symmetric groups with an -core index

Eleanor¹,

Ono²

2022

Can. Math. Bull.

View full text Add to dashboard Cite

Let Cn = [χ λ (µ)] λ,µ be the character table for Sn, where the indices λ and µ run over the p(n) many integer partitions of n. In this note we study Z ℓ (n), the number of zero entries χ λ (µ) in Cn, where λ is an ℓ-core partition of n. For every prime ℓ ≥ 5, we prove an asymptotic formula of the form. For primes ℓ and n > ℓ 6 /24, we show that χ λ (µ) = 0 whenever λ and µ are both ℓ-cores. Furthermore, if Z * ℓ (n) is the number of zero entries indexed by two ℓ-cores, then for ℓ ≥ 5 we obtain the asymptotic2020 Mathematics Subject Classification. 20C30, 11P82, 05A17. Key words and phrases. Primary: character tables, hook lengths, partitions, symmetric groups. E.M. acknowledges the support of a UVA Dean's Doctoral Fellowship. K.O. thanks the Thomas Jefferson Fund and the NSF (DMS-2002265 and DMS-2055118) for their support.1 This is a ``preproof'' accepted article for Canadian Mathematical Bulletin This version may be subject to change during the production process.

show abstract

Section: Introduction and Statement Of Resultsmentioning

confidence: 68%

Zeros in the character tables of symmetric groups with an -core index

Eleanor¹,

Ono²

2022

Can. Math. Bull.

View full text Add to dashboard Cite

show abstract

“…As discussed e.g. in [PPV16, §8] and [Mil19,Pel20], other values of the character table are of interest as well, notably the uniqueness and parity of the characters. The corresponding complexity problems χ λ (µ) = 1 and χ λ (µ) = 0 mod 2 are also very interesting and worth studying.…”

Section: Other Valuesmentioning

confidence: 99%

Positivity of the Symmetric Group Characters Is as Hard as the Polynomial Time Hierarchy

Ikenmeyer,

Pak,

Panova

2023

International Mathematics Research Notices

View full text Add to dashboard Cite

We prove that deciding the vanishing of the character of the symmetric group is $\textsf{C}_= \textsf{P}$-complete. We use this hardness result to prove that the absolute value and also the square of the character are not contained in $\textsf{#P}$, unless the polynomial hierarchy collapses to the second level. This rules out the existence of any (unsigned) combinatorial description for the square of the characters. As a byproduct of our proof, we conclude that deciding positivity of the character is $\textsf{PP}$-complete under many-one reductions, and hence $\textsf{PH}$-hard under Turing reductions.

show abstract

“…Based on this first observation, Miller [11,13] conjectured that as n goes to infinity, almost every entry of the character table of the symmetric group S n is even. Following partial progress due to McKay [10], Gluck [5], and Morotti [14], the first author proved this conjecture in [15]. Based on the second observation, Miller [11,13] also conjectured, more generally, that for any fixed prime p, almost every entry of the character table of S n is a multiple of p as n goes to infinity.…”

Section: Introductionmentioning

confidence: 98%