# ALGEBRAIC COMBINATORICS

Matroid relaxations and Kazhdan–Lusztig non-degeneracy
Algebraic Combinatorics, Volume 5 (2022) no. 4, pp. 745-769.

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 $Z$-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 $Z$-polynomial of all sparse paving matroids.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.244
Classification: 05B35,  05E10,  52B40,  11B83
Keywords: Kazhdan–Lusztig polynomials of matroids, Circuit-hyperplane relaxations, Geometric lattices, Real-rooted polynomials
Ferroni, Luis 1; Vecchi, Lorenzo 2

1 KTH Royal Institute of Technology Department of Mathematics Stockholm Sweden
2 Università di Bologna Dipartimento di Matematica Bologna Italy
Copyrights: The authors retain unrestricted copyrights and publishing rights
@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},
publisher = {The Combinatorics Consortium},
volume = {5},
number = {4},
year = {2022},
doi = {10.5802/alco.244},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.244/}
}
TY  - JOUR
TI  - Matroid relaxations and Kazhdan–Lusztig non-degeneracy
JO  - Algebraic Combinatorics
PY  - 2022
DA  - 2022///
SP  - 745
EP  - 769
VL  - 5
IS  - 4
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.244/
UR  - https://doi.org/10.5802/alco.244
DO  - 10.5802/alco.244
LA  - en
ID  - ALCO_2022__5_4_745_0
ER  - 
%0 Journal Article
%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://doi.org/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. https://alco.centre-mersenne.org/articles/10.5802/alco.244/

[1] Ardila, Federico; Fink, Alex; Rincón, Felipe Valuations for matroid polytope subdivisions, Canad. J. Math., Volume 62 (2010) no. 6, pp. 1228-1245 | DOI | MR | Zbl

[2] Ardila, Federico; Sanchez, Mario Valuations and the Hopf Monoid of Generalized Permutahedra, Int. Math. Res. Not. IMRN (2022), Paper no. rnab355, 76 pages | DOI

[3] Bansal, Nikhil; Pendavingh, Rudi A.; van der Pol, Jorn G. On the number of matroids, Combinatorica, Volume 35 (2015) no. 3, pp. 253-277 | DOI | MR | Zbl

[4] Braden, Tom; Huh, June; Matherne, Jacob P.; Proudfoot, Nicholas; Wang, Botong Singular Hodge theory for combinatorial geometries (2020) | arXiv

[5] Brenti, Francesco Twisted incidence algebras and Kazhdan-Lusztig-Stanley functions, Adv. Math., Volume 148 (1999) no. 1, pp. 44-74 | DOI | MR | Zbl

[6] Derksen, Harm; Fink, Alex Valuative invariants for polymatroids, Adv. Math., Volume 225 (2010) no. 4, pp. 1840-1892 | DOI | MR

[7] Elias, Ben; Proudfoot, Nicholas; Wakefield, Max The Kazhdan-Lusztig polynomial of a matroid, Adv. Math., Volume 299 (2016), pp. 36-70 | DOI | MR | Zbl

[8] Elias, Ben; Williamson, Geordie The Hodge theory of Soergel bimodules, Ann. of Math. (2), Volume 180 (2014) no. 3, pp. 1089-1136 | DOI | MR | Zbl

[9] Ferroni, Luis On the Ehrhart Polynomial of Minimal Matroids, Discrete Comput. Geom. (2021), 19 pages | DOI | MR | Zbl

[10] Ferroni, Luis Matroids are not Ehrhart positive, Adv. Math., Volume 402 (2022), Paper no. 108337, 27 pages | DOI | MR | Zbl

[11] Gao, Alice L. L.; Lu, Linyuan; Xie, Matthew H. Y.; Yang, Arthur L. B.; Zhang, Philip B. The Kazhdan-Lusztig polynomials of uniform matroids, Adv. in Appl. Math., Volume 122 (2021), Paper no. 102117, 24 pages | DOI | MR | Zbl

[12] Gao, Alice L. L.; Xie, Matthew H. Y. The inverse Kazhdan-Lusztig polynomial of a matroid, J. Combin. Theory Ser. B, Volume 151 (2021), pp. 375-392 | DOI | MR | Zbl

[13] Gedeon, Katie; Proudfoot, Nicholas; Young, Benjamin Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures, Sém. Lothar. Combin., Volume 78B (2017), p. Art. 80, 12 pages | MR | Zbl

[14] Gedeon, Katie R. Kazhdan-Lusztig polynomials of thagomizer matroids, Electron. J. Combin., Volume 24 (2017) no. 3, Paper no. 3.12, 10 pages | MR | Zbl

[15] Inc., OEIS Foundation The On-Line Encyclopedia of Integer Sequences, http://oeis.org, 2022

[16] Johnson, Selmer M. A new upper bound for error-correcting codes, IRE Trans., Volume IT-8 (1962), pp. 203-207 | DOI | MR | Zbl

[17] Lee, Kyungyong; Nasr, George D.; Radcliffe, Jamie A Combinatorial Formula for Kazhdan-Lusztig Polynomials of Sparse Paving Matroids, Electron. J. Combin., Volume 28 (2021) no. 4, Paper no. 4.44, 29 pages | DOI | MR | Zbl

[18] Lu, Linyuan; Xie, Matthew H. Y.; Yang, Arthur L. B. Kazhdan-Lusztig polynomials of fan matroids, wheel matroids and whirl matroids (2018) | arXiv

[19] Mayhew, Dillon; Newman, Mike; Welsh, Dominic; Whittle, Geoff On the asymptotic proportion of connected matroids, European J. Combin., Volume 32 (2011) no. 6, pp. 882-890 | DOI | MR | Zbl

[20] Merino, Criel; Noble, Steven D.; Ramírez-Ibáñez, Marcelino; Villarroel-Flores, Rafael On the structure of the $h$-vector of a paving matroid, European J. Combin., Volume 33 (2012) no. 8, pp. 1787-1799 | DOI | MR | Zbl

[21] Mills, Allan D. On matroids with many common bases, Discrete Math., Volume 203 (1999) no. 1-3, pp. 195-205 | DOI | MR | Zbl

[22] Nelson, Peter Almost all matroids are nonrepresentable, Bull. Lond. Math. Soc., Volume 50 (2018) no. 2, pp. 245-248 | DOI | MR | Zbl

[23] Oxley, James Matroid theory, Oxford Graduate Texts in Mathematics, 21, Oxford University Press, Oxford, 2011, xiv+684 pages | DOI | MR

[24] Polo, Patrick Construction of arbitrary Kazhdan-Lusztig polynomials in symmetric groups, Represent. Theory, Volume 3 (1999), pp. 90-104 | DOI | MR | Zbl

[25] Proudfoot, Nicholas; Wakefield, Max; Young, Ben Intersection cohomology of the symmetric reciprocal plane, J. Algebraic Combin., Volume 43 (2016) no. 1, pp. 129-138 | DOI | MR | Zbl

[26] Proudfoot, Nicholas; Xu, Yuan; Young, Ben The $Z$-polynomial of a matroid, Electron. J. Combin., Volume 25 (2018) no. 1, Paper no. 1.26, 21 pages | MR | Zbl

[27] Sage Developers, The SageMath, the Sage Mathematics Software System (Version 9.1) (2020) https://www.sagemath.org | DOI

[28] Stanley, Richard P. Subdivisions and local $h$-vectors, J. Amer. Math. Soc., Volume 5 (1992) no. 4, pp. 805-851 | DOI | MR | Zbl

[29] Truemper, Klaus Alpha-balanced graphs and matrices and $\mathrm{GF}\left(3\right)$-representability of matroids, J. Combin. Theory Ser. B, Volume 32 (1982) no. 2, pp. 112-139 | DOI | MR | Zbl

Cited by Sources: