[Paper Review] Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures

Katie Gedeon, Nicholas Proudfoot+1 more|arXiv (Cornell University)|Nov 22, 2016

Advanced Combinatorial MathematicsMathematics15 references19 citations

TL;DR

This paper surveys recent results and conjectures on Kazhdan-Lusztig polynomials of matroids, proposing that these polynomials are always real-rooted and log-concave, with roots interlacing under contraction. It introduces equivariant versions with symmetric function coefficients and establishes functional equations for generating functions, particularly for uniform, thagomizer, and braid matroids, while highlighting open problems on non-negativity and structure of coefficients.

ABSTRACT

We report on various results, conjectures, and open problems related to Kazhdan-Lusztig polynomials of matroids. We focus on conjectures about the roots of these polynomials, all of which appear here for the first time.

Motivation & Objective

• To summarize known results and conjectures on Kazhdan-Lusztig polynomials of matroids, particularly focusing on structural properties like real-rootedness and log-concavity.
• To investigate the non-negativity of coefficients in these polynomials, especially in the context of realizable matroids and open conjectures for general matroids.
• To explore equivariant Kazhdan-Lusztig polynomials for symmetric matroids, where coefficients are replaced by symmetric functions, and to analyze their generating functions.
• To establish functional equations for generating functions of Kazhdan-Lusztig polynomials in key families, such as uniform, thagomizer, and braid matroids.
• To identify and formalize new conjectures, including root interlacing under contraction and coefficient structures in braid matroids.

Proposed method

• The paper uses a recursive definition of Kazhdan-Lusztig polynomials based on the lattice of flats and Möbius function, following the axiomatic characterization from [EPW16].
• It applies geometric and cohomological interpretations: for realizable matroids, the polynomial equals the intersection cohomology Poincaré polynomial of the reciprocal plane, implying non-negative coefficients.
• For families like uniform, thagomizer, and braid matroids, the authors use symmetric group actions to define equivariant polynomials with coefficients in symmetric functions.
• Functional equations for generating functions are derived using plethysm and power series manipulation, such as $\mathcal{T}(t^{-1},tu)=(t-1)us(u)v(t,u)+\frac{s(tu)^2}{s(u)^2}\mathcal{T}(t,u)$.
• The paper employs SAGE for extensive computer calculations to verify conjectures on root interlacing and coefficient structures.
• It leverages the theory of $\operatorname{FS^{op}}$-modules to prove that generating functions of coefficients are rational with poles at rational reciprocals, leading to exponential generating function decompositions.

Experimental results

Research questions

• RQ1 Are the coefficients of Kazhdan-Lusztig polynomials of arbitrary matroids non-negative, as conjectured in Conjecture 2.2?
• RQ2 Do the roots of Kazhdan-Lusztig polynomials of matroids always interlace when one matroid is obtained by contracting an edge in another, as suggested in Conjecture 3.4?
• RQ3 Can the generating functions of coefficients in Kazhdan-Lusztig polynomials of braid matroids be expressed as rational functions with poles at reciprocal integers, as shown in Proposition 5.12?
• RQ4 Do the equivariant Kazhdan-Lusztig polynomials of braid matroids satisfy a functional equation involving plethysm and symmetric functions, as conjectured in [GPY]?
• RQ5 Is the leading coefficient of $P_{B_{2k}}(t)$ equal to $(2k-3)!!(2k-1)^{k-2}$, as proposed in Conjecture 5.11?

Key findings

• The Kazhdan-Lusztig polynomial of a realizable matroid has non-negative coefficients, as it equals the intersection cohomology Poincaré polynomial of the reciprocal plane, as proven in Theorem 2.3.
• For uniform matroids, the equivariant Kazhdan-Lusztig polynomial coefficients are Schur-positive symmetric functions, and their generating function satisfies a functional equation involving plethysm.
• The generating function $\mathcal{T}(t,u)$ for the $S_n$-equivariant polynomials of thagomizer matroids satisfies $\mathcal{T}(t^{-1},tu)=(t-1)us(u)v(t,u)+\frac{s(tu)^2}{s(u)^2}\mathcal{T}(t,u)$.
• The exponential generating function $Q(t,z)$ for braid matroid polynomials satisfies $Q(t^{-1},tz)=t\,Q(t,K(t,z))$, where $K(t,z)=t^{-1}((-1+(1+z)^t)$.
• The $i$-th coefficient of the Kazhdan-Lusztig polynomial of the braid matroid has an exponential generating function $G_i(z)$ that is a linear combination of $e^{jz}$ with polynomial coefficients, implying rational generating functions with poles at $\{j^{-1} \mid 1 \leq j \leq 2i\}$.
• Explicit rational generating functions for the first two coefficients are $H_1(z) = \frac{z^4}{(1-z)^3(1-2z)}$ and $H_2(z) = \frac{15z^6 - 50z^7 + 40z^8 + 4z^9}{(1-z)^5(1-2z)^3(1-4z)}$, confirming the pole structure.