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The Kazhdan-Lusztig polynomials of uniform matroids [article]

Alice L.L. Gao and Linyuan Lu and Matthew H.Y. Xie and Arthur L.B. Yang and Philip B. Zhang
2018 arXiv   pre-print
The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias, Proudfoot, and Wakefield [ Adv. Math. 2016]. Let U_m,d denote the uniform matroid of rank d on a set of m+d elements.  ...  A, 2017] pointed out that they can derive an explicit formula of the Kazhdan-Lusztig polynomials of U_m,d using equivariant Kazhdan-Lusztig polynomials.  ...  The second author is supported in part by the National Science Foundation of USA grant DMS-1600811.  ... 
arXiv:1806.10852v1 fatcat:a4pxijn4s5favnl4nd45iily64

The equivariant inverse Kazhdan-Lusztig polynomials of uniform matroids [article]

Alice L.L. Gao, Matthew H.Y. Xie, Arthur L.B. Yang
2021 arXiv   pre-print
Motivated by the concepts of the inverse Kazhdan-Lusztig polynomial and the equivariant Kazhdan-Lusztig polynomial, Proudfoot defined the equivariant inverse Kazhdan-Lusztig polynomial for a matroid.  ...  As an application, we give a new proof of Gedeon, Proudfoot and Young's formula for the equivariant Kazhdan-Lusztig polynomials of uniform matroids.  ...  The equivariant Kazhdan-Lusztig polynomials Note that the proof of Theorem 3.2 only relies on the evaluation of the equivariant characteristic polynomials for uniform matroids and the inverse Kazhdan-Lusztig  ... 
arXiv:2105.08546v1 fatcat:vzuwoosjvfevfly7cysp24vzcq

The inverse Kazhdan-Lusztig polynomial of a matroid [article]

Alice L.L. Gao, Matthew H.Y. Xie
2020 arXiv   pre-print
As an unexpected application of the inverse Kazhdan-Lusztig polynomials, we obtain a new formula to compute the Kazhdan-Lusztig polynomials for uniform matroids.  ...  In the framework of Kazhdan-Lusztig-Stanley polynomials, we study the inverse of Kazhdan-Lusztig-Stanley functions and define the inverse Kazhdan-Lusztig polynomials for matroids.  ...  The authors are very grateful to Arthur L.B. Yang for his helpful comments on the improvement of this paper. The first author is supported by the National Science Foundation of China (No.11801447).  ... 
arXiv:2007.15349v1 fatcat:kd7aajjegveq5ccv4eizhrcknq

Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures [article]

Katie Gedeon, Nicholas Proudfoot, Benjamin Young
2017 arXiv   pre-print
We report on various results, conjectures, and open problems related to Kazhdan-Lusztig polynomials of matroids.  ...  We focus on conjectures about the roots of these polynomials, all of which appear here for the first time.  ...  C m,d,i be the coefficient of t i in the S m+d -equivariant Kazhdan-Lusztig polynomial of the uniform matroid U m,d .  ... 
arXiv:1611.07474v2 fatcat:g5dxk2go7jcz5l7l25wautrnfe

Kazhdan-Lusztig polynomials of thagomizer matroids [article]

Katie R. Gedeon
2017 arXiv   pre-print
We also give a conjecture for the S_n-equivariant Kazhdan-Lusztig polynomial of a thagomizer matroid.  ...  We introduce thagomizer matroids and compute the Kazhdan-Lusztig polynomial of a rank n+1 thagomizer matroid by showing that the coefficient of t^k is equal to the number of Dyck paths of semilength n  ...  Kazhdan-Lusztig polynomials of a matroid.  ... 
arXiv:1610.05349v2 fatcat:c2kimkpenjb6zn53qkcyq3bpwi

Equivariant Kazhdan-Lusztig polynomials of q-niform matroids [article]

Nicholas Proudfoot
2018 arXiv   pre-print
We show that the equivariant Kazhdan-Lusztig polynomial of a q-niform matroid is the unipotent q-analogue of the equivariant Kazhdan-Lusztig polynomial of the corresponding uniform matroid, thus providing  ...  evidence for the positivity conjecture for equivariant Kazhdan-Lusztig polynomials.  ...  Introduction For any matroid M , the Kazhdan-Lusztig polynomial P M (t) ∈ Z[t] was introduced in [EPW16] .  ... 
arXiv:1808.07855v2 fatcat:brt6flnrpradbbnel2imxutdou

Equivariant Kazhdan–Lusztig polynomials of $q$-niform matroids

Nicholas Proudfoot
2019 Algebraic Combinatorics  
We show that the equivariant Kazhdan-Lusztig polynomial of a q-niform matroid is the unipotent q-analogue of the equivariant Kazhdan-Lusztig polynomial of the corresponding uniform matroid, thus providing  ...  evidence for the positivity conjecture for equivariant Kazhdan-Lusztig polynomials.  ...  The author is indebted to June Huh for help with formulating the main result and to Olivier Dudas for help with proving it.  ... 
doi:10.5802/alco.59 fatcat:magyfv5xnjgh3okc7sohhc7o2u

The Kazhdan-Lusztig polynomial of a matroid [article]

Ben Elias, Nicholas Proudfoot, Max Wakefield
2016 arXiv   pre-print
We associate to every matroid M a polynomial with integer coefficients, which we call the Kazhdan-Lusztig polynomial of M, in analogy with Kazhdan-Lusztig polynomials in representation theory.  ...  We also introduce a q-deformation of the Mobius algebra of M, and use our polynomials to define a special basis for this deformation, analogous to the canonical basis of the Hecke algebra.  ...  In an appendix, written jointly with Ben Young, we give tables of Kazhdan-Lusztig polynomials of uniform matroids and braid matroids of low rank.  ... 
arXiv:1412.7408v3 fatcat:tqkshgrgbzapvjfdwdh5ecyxye

The Kazhdan–Lusztig polynomial of a matroid

Ben Elias, Nicholas Proudfoot, Max Wakefield
2016 Advances in Mathematics  
We associate to every matroid M a polynomial with integer coefficients, which we call the Kazhdan-Lusztig polynomial of M , in analogy with Kazhdan-Lusztig polynomials in representation theory.  ...  Despite these parallels, the behavior of the polynomials for matroids differs drastically from the behavior of ordinary Kazhdan-Lusztig polynomials for Coxeter groups. In particular,  ...  Acknowledgments: The authors would like to thank June Huh, Joseph Kung, Emmanuel Letellier,  ... 
doi:10.1016/j.aim.2016.05.005 fatcat:pdsync7ncjfd5ofv3lmtesdlhi

On matroid modularity and the coefficients of the inverse Kazhdan-Lusztig polynomial of a matroid [article]

Lorenzo Vecchi
2021 arXiv   pre-print
Following the work of Gao and Xie in [2], we state some properties of the inverse Kazhdan-Lusztig polynomial of a matroid.  ...  We link the degeneracy of a matroid to the inverse Kazhdan-Lusztig polynomial and we show that the Conjecture holds for modular matroids, by proving that degenerate modular matroids are not regular.  ...  In 2020, Gao and Xie introduced in [2] the analogue of the inverse Kazhdan-Lusztig polynomial for a matroid, employing the general framework of Kazhdan-Lusztig-Stanley theory.  ... 
arXiv:2103.08580v4 fatcat:jtfv3swpfbgrnec4c6wed5hq4y

Matroid relaxations and Kazhdan-Lusztig non-degeneracy [article]

Luis Ferroni, Lorenzo Vecchi
2022 arXiv   pre-print
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  ...  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.  ...  many important aspects of the exposition.  ... 
arXiv:2104.14531v3 fatcat:7n6aknskwnbsfdglgwzwi2np74

The Z-polynomial of a matroid [article]

Nicholas Proudfoot, Ben Young, Yuan Xu
2017 arXiv   pre-print
We introduce the Z-polynomial of a matroid, which we define in terms of the Kazhdan-Lusztig polynomial.  ...  We then exploit a symmetry of the Z-polynomial to derive a new recursion for Kazhdan-Lusztig coefficients.  ...  Acknowledgments: The authors are grateful to Sara Billey for originally suggesting the study of the Z-polynomial, and to Katie Gedeon and Max Wakefield for discussions regarding the relationship between  ... 
arXiv:1706.05575v1 fatcat:x6cfevxfn5evnjexnohbttxtte

Kazhdan-Lusztig polynomials of matroids under deletion [article]

Tom Braden, Artem Vysogorets
2020 arXiv   pre-print
We present a formula which relates the Kazhdan-Lusztig polynomial of a matroid M, as defined by Elias, Proudfoot and Wakefield, to the KazhdanLusztig polynomials of the matroid obtained by deleting an  ...  We give a number of applications of our formula to KazhdanLusztig polynomials of graphic matroids, including a simple formula for the KazhdanLusztig polynomial of a parallel connection graph.  ...  The papers [PWY16, GPY17, GLX + ] actually compute a richer invariant, the equivariant Kazhdan-Lusztig polynomial, for uniform matroids.  ... 
arXiv:1909.09888v2 fatcat:if3zpvfcbvhknhd6y3kk3thjj4

Kazhdan-Lusztig Polynomials Under Deletion

Tom Braden, Artem Vysogorets
2020 Electronic Journal of Combinatorics  
We present a formula which relates the KazhdanLusztig polynomial of a matroid $M$, as defined by Elias, Proudfoot and Wakefield, to the KazhdanLusztig polynomials of the matroid obtained by deleting  ...  We give a number of applications of our formula to KazhdanLusztig polynomials of graphic matroids, including a simple formula for the KazhdanLusztig polynomial of a parallel connection graph.  ...  Acknowledgements The authors thank Jacob Matherne and Nicholas Proudfoot for helpful suggestions on a draft of this paper, and the anonymous referee for numerous corrections and improvements.  ... 
doi:10.37236/9026 fatcat:mbrpr2h24ba3rp6eyp4jxuipgq

Equivariant incidence algebras and equivariant Kazhdan–Lusztig–Stanley theory

Nicholas Proudfoot
2021 Algebraic Combinatorics  
This gives a new way of thinking about the equivariant Kazhdan-Lusztig polynomial and equivariant Z-polynomial of a matroid.  ...  We establish a formalism for working with incidence algebras of posets with symmetries, and we develop equivariant Kazhdan-Lusztig-Stanley theory within this formalism.  ...  We thank Tom Braden for his feedback on a preliminary draft of this work. The author is supported by NSF grant DMS-1954050.  ... 
doi:10.5802/alco.174 fatcat:ffzycfki4je5jllcuchcdlefqa
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