Journal article
What makes a D-0 graph Schur positive?
Journal of algebraic combinatorics, v 44(3), pp 677-727
01 Nov 2016
Abstract
We define a D-0 graph to be a graph whose vertex set is a subset of permutations of n, with edges of the form . . . bac . . . <-> . . . bca . . . or . . . acb . . . <-> . . . cab . . . (Knuth transformations), or . . . bac . . . <->. . . acb . . . or . . . bca . . . <-> . . . cab . . . (rotation transformations), such that whenever the Knuth and rotation transformations at positions i - 1, i, i + 1 are available at a vertex, exactly one of these is an edge. The generating function of such a graph is the sum of the quasisymmetric functions associated to the descent sets of its vertices. Assaf studied D-0 graphs in (Dual equivalence and Schur positivity, http://www-bcf.usc.edu/similar to shassaf/degs.pdf, 2014) and showed that they provide a rich source of examples of the D graphs of (Dual equivalence graphs and a combinatorial proof of LLT and Macdonald positivity, http://www-bcf.usc.edu/similar to shassaf/positivity.pdf, 2014). A key construction of Assaf expresses the coefficient of q(t) in an LLT polynomial as the generating function of a certain D-0 graph. LLT polynomials are known to be Schur positive by work of Grojnowski-Haiman, and experimentation shows that many D-0 graphs have Schur positive generating functions, which suggests a vast generalization of LLT positivity in this setting. As part of a series of papers, we study D-0 graphs using the Fomin-Greene theory of noncommutative Schur functions. We construct a D-0 graph whose generating function is not Schur positive by solving a linear program related to a certain noncommutative Schur function. We go on to construct a D graph on the same vertex set as this D-0 graph.
Metrics
Details
- Title
- What makes a D-0 graph Schur positive?
- Creators
- Jonah Blasiak - Drexel University
- Publication Details
- Journal of algebraic combinatorics, v 44(3), pp 677-727
- Publisher
- Springer Nature
- Number of pages
- 51
- Grant note
- DMS-14071174 / NSF; National Science Foundation (NSF)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000387223000007
- Scopus ID
- 2-s2.0-84969242684
- Other Identifier
- 991019169524504721
InCites Highlights
Data related to this publication, from InCites Benchmarking & Analytics tool:
- Web of Science research areas
- Mathematics