Geodesic
The combinatorics behind the leading Kazhdan-Lusztig coefficients of braid matroids
Alice L.L. Gao1 ; Nicholas James Proudfoot2 ; Arthur L.B. Yang3 ; Zhong-Xue Zhang3
1School of Mathematics and Statistics, Northwestern Polytechnical University
2University of Oregon
3Center for Combinatorics, LPMC, Nankai University
The electronic journal of combinatorics, Tome 31 (2024) no. 3
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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.
DOI : 10.37236/12778
Classification : 05B35, 05A15, 05A19, 52B40
Mots-clés : intersection cohomology, Kazhdan-Lusztig theory

Alice L.L. Gao  1   ; Nicholas James Proudfoot  2   ; Arthur L.B. Yang  3   ; Zhong-Xue Zhang  3

1 School of Mathematics and Statistics, Northwestern Polytechnical University
2 University of Oregon
3 Center for Combinatorics, LPMC, Nankai University 
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     author = {Alice L.L. Gao and Nicholas James Proudfoot and Arthur L.B. Yang and Zhong-Xue Zhang},
     title = {The combinatorics behind the leading {Kazhdan-Lusztig} coefficients of braid matroids},
     journal = {The electronic journal of combinatorics},
     year = {2024},
     volume = {31},
     number = {3},
     doi = {10.37236/12778},
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Alice L.L. Gao; Nicholas James Proudfoot; Arthur L.B. Yang; Zhong-Xue Zhang. The combinatorics behind the leading Kazhdan-Lusztig coefficients of braid matroids. The electronic journal of combinatorics, Tome 31 (2024) no. 3. doi: 10.37236/12778

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