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A second proof of the Shareshian--Wachs conjecture, by way of a new Hopf algebra
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TLDR
In this paper, the authors give a second proof of the Shareshian-Wachs conjecture, based on recursively decomposing Hessenberg varieties, using a new Hopf algebra as the organizing principle for this recursion.Citations
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Chromatic quasisymmetric functions
TL;DR: In this paper, a quasisymmetric refinement of Stanley's chromatic symmetric function is introduced, and a conjectural refinement of the power sum basis expansion is shown in special cases.
Journal ArticleDOI
Unit interval orders and the dot action on the cohomology of regular semisimple Hessenberg varieties
Patrick Brosnan,Timothy Y. Chow +1 more
TL;DR: In this paper, it was shown that the local invariant cycle map is an isomorphism if and only if the special fiber has palindromic cohomology, which is independent of the Hessenberg variety context.
Journal ArticleDOI
The Cohomology Rings of Regular Nilpotent Hessenberg Varieties in Lie Type A
TL;DR: In this paper, it was shown that the Hessenberg function can be expressed by generators and relations of the cohomology ring of the regular semisimple Hessenberg variety.
Journal ArticleDOI
The cohomology of abelian Hessenberg varieties and the Stanley–Stembridge conjecture
Megumi Harada,Martha Precup +1 more
TL;DR: In this article, the authors define a subclass of Hessenberg varieties called abelian Hessenberg, inspired by the theory of abelians in a Lie algebra developed by Kostant and Peterson, and give an inductive formula for the $S_n$-representation on the cohomology of a regular semisimple Hessenberg variety.
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LLT polynomials, chromatic quasisymmetric functions and graphs with cycles
Per Alexandersson,Greta Panova +1 more
TL;DR: A Dyck path model for unit-interval graphs is used to study the chromatic quasisymmetric functions introduced by Shareshian and Wachs, as well as unicellular LLT polynomials, revealing some parallel structure and phenomena regarding their e -positivity.
References
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Book
Symmetric functions and Hall polynomials
TL;DR: In this paper, the characters of GLn over a finite field and the Hecke ring of GLs over finite fields have been investigated and shown to be symmetric functions with two parameters.
Journal ArticleDOI
Equivariant cohomology, Koszul duality, and the localization theorem
TL;DR: In this paper, the authors considered the action of a compact Lie group K on a space X and gave a description of equivariant homology and intersection homology in terms of Equivariant geometric cycles.
Journal ArticleDOI
A Symmetric Function Generalization of the Chromatic Polynomial of a Graph
TL;DR: In this paper, the authors consider the expansion of a finite graph G with d vertices in terms of various symmetric function bases, such as partitions of the vertices into stable subsets, the Mobius function of the lattice of contractions of G, and the structure of the acyclic orientations of G.
Book ChapterDOI
Chromatic quasisymmetric functions and Hessenberg varieties
TL;DR: In this paper, the authors discuss three distinct topics of independent interest; one in enumerative combinatorics, one in symmetric function theory, and one in algebraic geometry, and explore some remarkable connections between these topics.
Journal ArticleDOI
Unit interval orders and the dot action on the cohomology of regular semisimple Hessenberg varieties
Patrick Brosnan,Timothy Y. Chow +1 more
TL;DR: In this paper, it was shown that the local invariant cycle map is an isomorphism if and only if the special fiber has palindromic cohomology, which is independent of the Hessenberg variety context.