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The Kazhdan-Lusztig polynomials of uniform matroids
[article]

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. ...

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

2021
*
arXiv
*
pre-print

Motivated by

arXiv:2105.08546v1
fatcat:vzuwoosjvfevfly7cysp24vzcq
*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*...##
###
The inverse Kazhdan-Lusztig polynomial of a matroid
[article]

2020
*
arXiv
*
pre-print

As an unexpected application

arXiv:2007.15349v1
fatcat:kd7aajjegveq5ccv4eizhrcknq
*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). ...##
###
Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures
[article]

2017
*
arXiv
*
pre-print

We report on various results, conjectures, and open problems related to

arXiv:1611.07474v2
fatcat:g5dxk2go7jcz5l7l25wautrnfe
*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 . ...##
###
Kazhdan-Lusztig polynomials of thagomizer matroids
[article]

2017
*
arXiv
*
pre-print

We also give a conjecture for

arXiv:1610.05349v2
fatcat:c2kimkpenjb6zn53qkcyq3bpwi
*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*. ...##
###
Equivariant Kazhdan-Lusztig polynomials of q-niform matroids
[article]

2018
*
arXiv
*
pre-print

We show that

arXiv:1808.07855v2
fatcat:brt6flnrpradbbnel2imxutdou
*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] . ...##
###
Equivariant Kazhdan–Lusztig polynomials of $q$-niform matroids

2019
*
Algebraic Combinatorics
*

We show that

doi:10.5802/alco.59
fatcat:magyfv5xnjgh3okc7sohhc7o2u
*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. ...##
###
The Kazhdan-Lusztig polynomial of a matroid
[article]

2016
*
arXiv
*
pre-print

We associate to every

arXiv:1412.7408v3
fatcat:tqkshgrgbzapvjfdwdh5ecyxye
*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. ...##
###
The Kazhdan–Lusztig polynomial of a matroid

2016
*
Advances in Mathematics
*

We associate to every

doi:10.1016/j.aim.2016.05.005
fatcat:pdsync7ncjfd5ofv3lmtesdlhi
*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, ...##
###
On matroid modularity and the coefficients of the inverse Kazhdan-Lusztig polynomial of a matroid
[article]

2021
*
arXiv
*
pre-print

Following

arXiv:2103.08580v4
fatcat:jtfv3swpfbgrnec4c6wed5hq4y
*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. ...##
###
Matroid relaxations and Kazhdan-Lusztig non-degeneracy
[article]

2022
*
arXiv
*
pre-print

We obtain a family

arXiv:2104.14531v3
fatcat:7n6aknskwnbsfdglgwzwi2np74
*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. ...##
###
The Z-polynomial of a matroid
[article]

2017
*
arXiv
*
pre-print

We introduce

arXiv:1706.05575v1
fatcat:x6cfevxfn5evnjexnohbttxtte
*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 ...##
###
Kazhdan-Lusztig polynomials of matroids under deletion
[article]

2020
*
arXiv
*
pre-print

We present a formula which relates

arXiv:1909.09888v2
fatcat:if3zpvfcbvhknhd6y3kk3thjj4
*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*. ...##
###
Kazhdan-Lusztig Polynomials Under Deletion

2020
*
Electronic Journal of Combinatorics
*

We present a formula which relates

doi:10.37236/9026
fatcat:mbrpr2h24ba3rp6eyp4jxuipgq
*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. ...##
###
Equivariant incidence algebras and equivariant Kazhdan–Lusztig–Stanley theory

2021
*
Algebraic Combinatorics
*

This gives a new way

doi:10.5802/alco.174
fatcat:ffzycfki4je5jllcuchcdlefqa
*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. ...
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