In this paper we study the interplay between the operation of circuit-hyperplane relaxation and the Kazhdan–Lusztig theory of matroids. We obtain a family of polynomials, not depending on the matroids but only on their ranks, that relate the Kazhdan–Lusztig, the inverse Kazhdan–Lusztig and the -polynomial of each matroid with those of its relaxations. As an application of our main theorem, we prove that all matroids having a free basis are non-degenerate. Additionally, we obtain bounds and explicit formulas for all the coefficients of the Kazhdan–Lusztig, inverse Kazhdan–Lusztig and -polynomial of all sparse paving matroids.
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Keywords: Kazhdan–Lusztig polynomials of matroids, Circuit-hyperplane relaxations, Geometric lattices, Real-rooted polynomials
Ferroni, Luis 1; Vecchi, Lorenzo 2
CC-BY 4.0
@article{ALCO_2022__5_4_745_0,
author = {Ferroni, Luis and Vecchi, Lorenzo},
title = {Matroid relaxations and {Kazhdan{\textendash}Lusztig} non-degeneracy},
journal = {Algebraic Combinatorics},
pages = {745--769},
year = {2022},
publisher = {The Combinatorics Consortium},
volume = {5},
number = {4},
doi = {10.5802/alco.244},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.244/}
}
TY - JOUR AU - Ferroni, Luis AU - Vecchi, Lorenzo TI - Matroid relaxations and Kazhdan–Lusztig non-degeneracy JO - Algebraic Combinatorics PY - 2022 SP - 745 EP - 769 VL - 5 IS - 4 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.244/ DO - 10.5802/alco.244 LA - en ID - ALCO_2022__5_4_745_0 ER -
%0 Journal Article %A Ferroni, Luis %A Vecchi, Lorenzo %T Matroid relaxations and Kazhdan–Lusztig non-degeneracy %J Algebraic Combinatorics %D 2022 %P 745-769 %V 5 %N 4 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.244/ %R 10.5802/alco.244 %G en %F ALCO_2022__5_4_745_0
Ferroni, Luis; Vecchi, Lorenzo. Matroid relaxations and Kazhdan–Lusztig non-degeneracy. Algebraic Combinatorics, Volume 5 (2022) no. 4, pp. 745-769. doi: 10.5802/alco.244
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