[Paper Review] Power sum expansion of chromatic quasisymmetric functions

Christos A. Athanasiadis|arXiv (Cornell University)|Sep 9, 2014

Advanced Combinatorial Mathematics8 references4 citations

TL;DR

This paper proves a conjecture by Shareshian and Wachs that the chromatic quasisymmetric function of the incomparability graph of a natural unit interval order expands explicitly in the power sum symmetric function basis. Using Roichman's formula for symmetric group characters and a bijection between linear extensions and acyclic orientations, the authors derive a combinatorial formula expressing the function in terms of power sum symmetric functions, providing evidence for positivity conjectures in chromatic symmetric function theory.

ABSTRACT

The chromatic quasisymmetric function of a graph was introduced by Shareshian and Wachs as a refinement of Stanley's chromatic symmetric function. An explicit combinatorial formula, conjectured by Shareshian and Wachs, expressing the chromatic quasisymmetric function of the incomparability graph of a natural unit interval order in terms of power sum symmetric functions, is proven. The proof uses a formula of Roichman for the irreducible characters of the symmetric group.

Motivation & Objective

• To prove a conjecture by Shareshian and Wachs on the power sum expansion of chromatic quasisymmetric functions for incomparability graphs of natural unit interval orders.
• To establish a combinatorial formula for the chromatic quasisymmetric function in the power sum basis, refining Stanley's theorem on symmetric functions.
• To provide evidence for the positivity conjecture of Shareshian and Wachs on the expansion in the elementary symmetric function basis.
• To connect the chromatic quasisymmetric function to symmetric group representation theory via Roichman's character formula.
• To reformulate the main result using acyclic orientations and descent statistics via a known bijection between linear extensions and acyclic orientations.

Proposed method

• The proof uses Roichman's formula for irreducible characters of the symmetric group to relate character values to power sum symmetric functions.
• The chromatic quasisymmetric function is first expressed in the fundamental quasisymmetric function basis via [4, Theorem 3.1].
• The expansion is then transformed into the power sum basis using the inverse of the transition matrix between fundamental and power sum symmetric functions.
• A key step involves using the involution ω on quasisymmetric functions to relate the chromatic quasisymmetric function to the power sum basis.
• The bijection between linear extensions of a poset and acyclic orientations of its incomparability graph is used to re-express the formula in terms of orientations.
• The formula is verified by showing that the number of ascents in an orientation matches the descent statistic in the corresponding linear extension.

Experimental results

Research questions

• RQ1 Does the chromatic quasisymmetric function of the incomparability graph of a natural unit interval order admit an explicit expansion in the power sum symmetric function basis?
• RQ2 Can the conjectured combinatorial formula for this expansion be rigorously proven using representation-theoretic tools?
• RQ3 How do the statistics of descents and ascents in linear extensions and acyclic orientations relate under the known bijection?
• RQ4 Does the power sum expansion provide evidence for the positivity of the chromatic quasisymmetric function in the elementary symmetric function basis?
• RQ5 Can the formula be reformulated in terms of acyclic orientations to yield a new combinatorial interpretation?

Key findings

• The chromatic quasisymmetric function $ X_G(x,t) $ for the incomparability graph of a natural unit interval order admits an explicit expansion in the power sum symmetric function basis.
• The expansion is given by $ X_G(x,t) = rac{1}{z_ u} ho ig( ext{coefficients in } ext{fundamental quasisymmetric basis} ig) $, where $ z_ u $ is the standard normalization factor.
• The coefficient of $ p_ u(x) $ in the expansion is $ rac{1}{z_ u} imes ext{sum over } u ext{-ordered partitions } u ext{ of } t^{ ext{asc}(o)} $, where $ o $ ranges over acyclic orientations satisfying specific sink and edge-direction conditions.
• The formula confirms the conjecture of Shareshian and Wachs on the power sum expansion, providing a combinatorial interpretation in terms of ordered partitions and acyclic orientations.
• The result implies that the chromatic quasisymmetric function is symmetric in $ x $, consistent with earlier results.
• An alternative formulation via acyclic orientations is given in Corollary 4.2, showing that $ ho X_G(x,t) = rac{1}{z_ u} imes ext{sum over } u ext{-ordered partitions } u ext{ of } t^{ ext{asc}(o)} $, where $ ext{asc}(o) $ counts directed edges from smaller to larger vertex in the total order.