Work

The Combinatorics Behind the Leading Kazhdan-Lusztig Coefficients of Braid Matroids

Year: 2024

Type: article

Abstract: Ferroni and Larson gave a combinatorial interpretation of the braid Kazhdan-Lusztig polynomials in terms of series-parallel matroids. As a consequence, they confirmed an explicit formula for the leading Kazhdan-Lusztig coefficients of braid matroids with odd rank, as conjectured by Elias, Proudfoot, and Wakefield. Based on Ferroni and Larson’s work, we further explore the combinatorics behind the leading Kazhdan-Lusztig coefficients of braid matroids. The main results of this paper include an explicit formula for the leading Kazhdan-Lusztig coefficients of braid matroids with even rank, a simple expression for the number of simple series-parallel matroids of rank $k + 1$ on $2k$ elements, and explicit formulas for the leading coefficients of inverse Kazhdan-Lusztig polynomials of braid matroids. The binomial identity for the Abel polynomials plays an important role in the proofs of these formulas.

Source: The Electronic Journal of Combinatorics

DOI: 10.37236/12778

Authors: Alice L.L. Gao, Nicholas James Proudfoot, Arthur L.B. Yang, Zhong-Xue Zhang

Institutions: Northwestern Polytechnical University, University of Oregon, Nankai University

Language: English

Topic: Advanced Combinatorial Mathematics

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: National Science Foundation, National Natural Science Foundation of China, Natural Science Foundation of Shaanxi Province, Fundamental Research Funds for the Central Universities