A Combinatorial Formula for Kazhdan-Lusztig Polynomials of $ρ$-Removed Uniform Matroids

Lee, Kyungyong; D., Nasr, George; Jamie, Radcliffe,

doi:10.48550/arxiv.1911.04373

Cited by 2 publications

(4 citation statements)

References 12 publications

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“…When CH is a disjoint family, this formula has a manifestly positive interpretation, as stated in the introduction of this paper. More details can be found in [13]. Otherwise, for more general cases of sparse paving matroids, we are not yet able to give a manifestly non-negative expression.…”

Section: Non-negativity For Sparse Paving Matroidsmentioning

confidence: 90%

“…We now point out that remarkably, this formula no longer depends on the structure of CH, only the size. Hence, the proof proceeds as in the case of Theorem 3 in [13].…”

Section: The Kazhdan-lusztig Polynomials For Sparse Paving Matroidsmentioning

confidence: 98%

“…When CH is a disjoint collection, we have already shown in [13,Proposition 2] that the formula in Theorem 1.1 has a manifestly positive interpretation. Consider the subset of Skyt(m + 1, i, d − 2i + 1) satisfying at least one of the following three conditions.…”

Section: Introductionmentioning

confidence: 99%

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A Combinatorial Formula for Kazhdan-Lusztig Polynomials of Sparse Paving Matroids

Lee¹,

D.²,

Jamie³

2020

Preprint

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We prove the positivity of Kazhdan-Lusztig polynomials for sparse paving matroids, which are known to be logarithmically almost all matroids, but are conjectured to be almost all matroids. The positivity follows from a remarkably simple combinatorial formula we discovered for these polynomials using skew young tableaux. This supports the conjecture that Kazhdan-Lusztig polynomials for all matroids have non-negative coeffiecients. In special cases, such as uniform matroids, our formula has a nice combinatorial interpretation.2010 Mathematics Subject Classification. 05B35.

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Section: Non-negativity For Sparse Paving Matroidsmentioning

confidence: 90%

“…We now point out that remarkably, this formula no longer depends on the structure of CH, only the size. Hence, the proof proceeds as in the case of Theorem 3 in [13].…”

Section: The Kazhdan-lusztig Polynomials For Sparse Paving Matroidsmentioning

confidence: 98%

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A Combinatorial Formula for Kazhdan-Lusztig Polynomials of Sparse Paving Matroids

Lee¹,

D.²,

Jamie³

2020

Preprint

Self Cite

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“…Lu, Xie and Yang [16] used the method of generating functions to obtain explicit formulas for the Kazhdan-Lusztig polynomials of fan matroids, wheel matroids and whirl matroids. Lee, Nasr and Radcliffe studied the Kazhdan-Lusztig polynomials of ρ-removed uniform matroids in [15] and sparse paving matroids in [14]. By using the concept of Zpolynomials, Proudfoot, Xu and Young [18] obtained a faster algorithm for computing the Kazhdan-Lusztig polynomials for braid matroids.…”

Section: ]mentioning

confidence: 99%

The inverse Kazhdan-Lusztig polynomial of a matroid

Gao,

Xie

2020

Preprint

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In analogy with the classical Kazhdan-Lusztig polynomials for Coxeter groups, Elias, Proudfoot and Wakefield introduced the concept of Kazhdan-Lusztig polynomials for matroids. It is known that both the classical Kazhdan-Lusztig polynomials and the matroid Kazhdan-Lusztig polynomials can be considered as special cases of the Kazhdan-Lusztig-Stanley polynomials for locally finite posets. 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. We also compute these polynomials for boolean matroids and uniform matroids. As an unexpected application of the inverse Kazhdan-Lusztig polynomials, we obtain a new formula to compute the Kazhdan-Lusztig polynomials for uniform matroids. Similar to the Kazhdan-Lusztig polynomial of a matroid, we conjecture that the coefficients of its inverse Kazhdan-Lusztig polynomial are nonnegative and log-concave.

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A Combinatorial Formula for Kazhdan-Lusztig Polynomials of $ρ$-Removed Uniform Matroids

Cited by 2 publications

References 12 publications

A Combinatorial Formula for Kazhdan-Lusztig Polynomials of Sparse Paving Matroids

A Combinatorial Formula for Kazhdan-Lusztig Polynomials of Sparse Paving Matroids

The inverse Kazhdan-Lusztig polynomial of a matroid

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