Abstract
In previous work of this author it was conjectured that the sum of power sums $p_\lambda,$ for partitions $\lambda$ ranging over an interval $[(1^n), \mu]$ in reverse lexicographic order, is Schur-positive. Here we investigate this conjecture and establish its truth in the following special cases: for $\mu\in [(n-4,1^4), (n)]$ or $\mu\in [(1^n), (3,1^{n-3})], $ or $\mu=(3, 2^k, 1^r)$ when $k\geq 1$ and $0\leq r\leq 2.$ Many new Schur positivity questions are presented.
Highlights
Introduction and PreliminariesIn this paper we consider Schur positivity questions related to the reverse lexicographic order on integer partitions
Μ of the same integer n, we say a partition λ is preceded by a partition μ in reverse lexicographic order if λ1 > μ1 or there is an index j 2 such that λi = μi for i < j and λj > μj
In general the character values form a sequence of 1’s and (−1)’s, with the partitions written in reverse lexicographic order; it is not obvious why the resulting partial sums should be nonnegative
Summary
In this paper we consider Schur positivity questions related to the reverse lexicographic order on integer partitions. In general the character values form a sequence of 1’s and (−1)’s, with the partitions written in reverse lexicographic order; it is not obvious why the resulting partial sums should be nonnegative. For n = 8 : 7, −5, 3, −1, −1, 4, −2, 0, 1, 1, −3, 1, 1, 0, −1, 2, 0, −1, −1, −1, 0, 1 with partial sums: 7, 2, 5, 4, 3, 7, 5, 5, 6, 7, 4, 5, 6, 6, 5, 7, 7, 6, 5, 4, 4, 5 These examples highlight the fact that there are many ways of reordering the conjugacy classes so that the resulting partial (row) sums in the character table may be negative, and the corresponding sum of power sums will fail to be Schur-positive.
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