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The Kazhdan-Lusztig polynomials of uniform matroids
[article]
2018
arXiv
pre-print
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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]
2021
arXiv
pre-print
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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]
2020
arXiv
pre-print
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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]
2017
arXiv
pre-print
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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]
2017
arXiv
pre-print
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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]
2018
arXiv
pre-print
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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
2019
Algebraic Combinatorics
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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]
2016
arXiv
pre-print
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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
2016
Advances in Mathematics
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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]
2021
arXiv
pre-print
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2021 pre-print arXiv:2103.08580v3 fatcat:ogajo3mltnetrguziko3rjxwgaOther Versions
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]
2022
arXiv
pre-print
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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]
2017
arXiv
pre-print
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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]
2020
arXiv
pre-print
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We present a formula which relates the Kazhdan-Lusztig polynomial of a matroid M, as defined by Elias, Proudfoot and Wakefield, to the Kazhdan–Lusztig polynomials of the matroid obtained by deleting an ...
We give a number of applications of our formula to Kazhdan–Lusztig polynomials of graphic matroids, including a simple formula for the Kazhdan–Lusztig 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
2020
Electronic Journal of Combinatorics
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We present a formula which relates the Kazhdan–Lusztig polynomial of a matroid $M$, as defined by Elias, Proudfoot and Wakefield, to the Kazhdan–Lusztig polynomials of the matroid obtained by deleting ...
We give a number of applications of our formula to Kazhdan–Lusztig polynomials of graphic matroids, including a simple formula for the Kazhdan–Lusztig 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
2021
Algebraic Combinatorics
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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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