Séminaire Lotharingien de Combinatoire, 82B.49 (2019), 12 pp.
Sheila Sundaram
On the Schur positivity of sums of power sums
Abstract.
Let T be a nonempty subset of positive integers and pn the nth power sum symmetric function. Consider the multiplicity-free sum of power sums FnT = Σλ partition of n pλ where the sum ranges over all partitions of n with parts in the set T. We define a new symmetric function fT and give two descriptions of the (possibly virtual) symmetric group representation associated to the series Πn in T (1-pn)-1 = Σn >= 0 FnT: one in terms of the Lie representation, and another as the symmetric or exterior power of (again possibly virtual) modules induced from centralisers of the symmetric group.
When T = {1}, the degree n term of fT reduces to the Frobenius characteristic of the Lie representation Lien. At the other extreme, when T is the set of all positive integers, it is the conjugacy action of Sn.
The function fT allows us to unify previous results on the Schur positivity of multiplicity-free sums of power sums, as well as investigate new ones. We also uncover some curious plethystic relationships between the conjugacy action and the Lie representation.
Finally we establish some special cases of an earlier conjecture of this author on the Schur positivity of sums of power sums in the intervals [(1n), μ] in reverse lexicographic order.
Received: November 15, 2018.
Accepted: February 17, 2019.
Final version: April 1, 2019.
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