[论文解读] Inverse Kazhdan--Lusztig polynomials of fan matroids

Alice L. L. Gao, Yaxing Li+1 more|arXiv (Cornell University)|Mar 26, 2026

Advanced Combinatorial Mathematics被引用 0

一句话总结

该论文推导出生成函数和逆 Kazhdan–Lusztig 多项式 Q_{F_n}(t) 与逆 Z-多项式 Y_{F_n}(t) 的显式公式，并证明 Q_{F_n}(t) 的系数具有对数凹性且无内部零，同时给出多种证明和生成函数技术。

ABSTRACT

The inverse Kazhdan--Lusztig polynomial of a matroid was introduced by Gao and Xie, and the inverse $Z$-polynomial of a matroid was introduced by Ferroni, Matherne, Stevens, and Vecchi. In this paper, we study these two polynomials for fan matroids, a family of graphic matroids associated with fan graphs. We first derive the generating functions for the inverse Kazhdan--Lusztig polynomials of fan matroids using their recursive definition, and then deduce the explicit formulas of these polynomials therefrom. For the inverse $Z$-polynomials of fan matroids, we obtain their generating functions using a parallel generating function approach, and further derive their explicit expansions based on these generating functions. Additionally, we provide alternative proofs for the above generating functions using the deletion formulas for inverse Kazhdan--Lusztig and inverse $Z$-polynomials. As an application of the explicit formula for inverse Kazhdan--Lusztig polynomials, we prove that the coefficients of the inverse Kazhdan--Lusztig polynomial of the fan matroid form a log-concave sequence with no internal zeros.

研究动机与目标

• Motivate and study inverse Kazhdan–Lusztig polynomials Q_M(t) and inverse Z-polynomials Y_M(t) for fan matroids, a graphic matroid class.
• Derive generating functions for Q_{F_n}(t) and Y_{F_n}(t) via recursive and parallel generating-function methods.
• Obtain explicit formulas for Q_{F_n}(t) and Y_{F_n}(t) and prove log-concavity with no internal zeros for Q_{F_n}(t).
• Provide alternative deletion-based proofs of the generating functions for these invariants.

提出的方法

• Define generating functions Ψ(t,u)=sum_{n≥0} Q_{F_n}(t) u^n and derive a closed form Ψ(t,u)=1+(1-4u-√(1-4tu^2))/(2(-2+4u+tu)).
• Use a recursive definition for Q_{F_n}(t) and translate it into a functional equation for Ψ(t,u).
• Express flats of fan graphs via combinatorial decompositions and use a bijection with C_n′ to evaluate summands and weights.
• Decompose structures into odd/even parts and use composition formulas to obtain generating-function equations.
• Derive an explicit formula for Q_{F_n}(t) by matching generating-function recurrences with coefficient-wise recurrences.
• Similarly, derive Ψ_Y(t,u) for inverse Z-polynomials and obtain explicit expansions for Y_{F_n}(t).

实验结果

研究问题

• RQ1 What is the inverse Kazhdan–Lusztig polynomial Q_{F_n}(t) for the fan matroid F_n and its generating function Ψ(t,u) ?
• RQ2 What is the explicit closed form for Q_{F_n}(t) and how do its coefficients behave (e.g., log-concavity, no internal zeros) ?
• RQ3 What is the inverse Z-polynomial Y_{F_n}(t) for fan matroids and its generating function Ψ_Y(t,u) ?
• RQ4 Can we provide alternative proofs of the generating functions via deletion formulas?
• RQ5 Do the coefficients of Q_{F_n}(t) form a log-concave sequence with no internal zeros? (Yes, for fan matroids)

主要发现

• Q_{F_n}(t) has an explicit coefficient formula: Q_{F_n}(t)= sum_{k=0}^{⌊(n-1)/2⌋} ((n-2k)2^{n-2k-1}/n) binom(n,k) t^k.
• The coefficients of Q_{F_n}(t) form a log-concave sequence with no internal zeros.
• The inverse Z-polynomial Y_{F_n}(t) admits an explicit threefold summation formula: Y_{F_n}(t)= sum_{k=0}^{n} sum_{j=0}^{⌊n/2⌋} sum_{i=0}^{n-1} [(-2)^j 3^{n-1-i} /(2^{n+2})] binom(n-2j}{k-j} binom(n-1}{i} (binom((i-1)/2}{j}+3 binom((i+1)/2}{j}+4 binom(i/2}{j}) t^k.
• Generating function for inverse Kazhdan–Lusztig polynomials: Ψ(t,u)=1+(1-4u-√(1-4tu^2))/(2(-2+4u+tu)).
• Generating function for inverse Z-polynomials: Ψ_Y(t,u)=2(-1+u+tu)/(-3+4(1+t)u+√(1-4tu^2)).
• Alternative deletion-proof approaches are provided for the generating functions of both invariants.