Joshua P. Swanson — Page 2

arXiv2018

Cyclic sieving, necklaces, and branching rules related to Thrall's problem

We show that the cyclic sieving phenomenon of Reiner--Stanton--White together with necklace generating functions arising from work of Klyachko offer a remarkably unified, direct, and largely bijective approach to a series of results due to Kraskiewicz--Weyman, Stembridge, and Schocker related to the so-called higher Lie modules and branching rules for inclusions $ C_a \wr S_b \hookrightarrow S_{ab} $. Extending the approach gives monomial expansions for certain graded Frobenius series arising from a generalization of Thrall's problem.

math.CO↗

arXiv2017

On the Existence of Tableaux with Given Modular Major Index

We provide simple necessary and sufficient conditions for the existence of a standard Young tableau of a given shape and major index $r$ mod $n$, for all $r$. Our result generalizes the $r=1$ case due essentially to (1974) and proves a recent conjecture due to Sundaram (2016) for the $r=0$ case. A byproduct of the proof is an asymptotic equidistribution result for "almost all" shapes. The proof uses a representation-theoretic formula involving Ramanujan sums and normalized symmetric group character estimates. Further estimates involving "opposite" hook lengths are given which are well-adapted to classifying which partitions $λ\vdash n$ have $f^λ\leq n^d$ for fixed $d$. We also give a new proof of a generalization of the hook length formula due to Fomin-Lulov (1995) for symmetric group characters at rectangles. We conclude with some remarks on unimodality of symmetric group characters.

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