×

The equivariant inverse Kazhdan-Lusztig polynomials of uniform matroids. (English) Zbl 1503.05018

Summary: Motivated by the concepts of the inverse Kazhdan-Lusztig polynomial and the equivariant Kazhdan-Lusztig polynomial, N. Proudfoot [Algebr. Comb. 2, No. 4, 613–619 (2019; Zbl 1417.05024)] defined the equivariant inverse Kazhdan-Lusztig polynomial for a matroid. In this paper, we show that the equivariant inverse Kazhdan-Lusztig polynomial of a matroid is very useful for determining its equivariant Kazhdan-Lusztig polynomials, and we determine the equivariant inverse Kazhdan-Lusztig polynomials for Boolean matroids and uniform matroids. As an application, we give a new proof of Gedeon, Proudfoot, and Young’s formula [K. Gedeon et al., J. Comb. Theory, Ser. A 150, 267–294 (2017; Zbl 1362.05131)] for the equivariant Kazhdan-Lusztig polynomials of uniform matroids. Inspired by Lee, Nasr, and Radcliffe’s combinatorial interpretation for the ordinary Kazhdan-Lusztig polynomials of uniform matroids [K. Lee et al., Electron. J. Comb. 28, No. 4, Research Paper P4.44, 29 p. (2021; Zbl 1486.05037)], we further present a new formula for the corresponding equivariant Kazhdan-Lusztig polynomials.

MSC:

05B35 Combinatorial aspects of matroids and geometric lattices
05E05 Symmetric functions and generalizations
20C30 Representations of finite symmetric groups
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] T. Braden, J. Huh, J. P. Matherne, N. Proudfoot, and B. Wang, Singular Hodge Theory for Combinatorial Geometries, preprint, https://arxiv.org/abs/2010.06088, 2020.
[2] T. Braden and A. Vysogorets, Kazhdan-Lusztig polynomials of matroids under deletion, Electron. J. Combin., 27 (2020), P1.17. · Zbl 1431.05150
[3] F. Brenti, Twisted incidence algebras and Kazhdan-Lusztig-Stanley functions, Adv. Math., 148 (1999), pp. 44-74, https://doi.org/10.1006/aima.1999.1839. · Zbl 0954.16022
[4] F. Brenti, \(P\)-kernels, IC bases and Kazhdan-Lusztig polynomials, J. Algebra, 259 (2003), pp. 613-627, https://doi.org/10.1016/S0021-8693(02)00575-6. · Zbl 1022.20015
[5] B. Elias, N. Proudfoot, and M. Wakefield, The Kazhdan-Lusztig polynomial of a matroid, Adv. Math., 299 (2016), pp. 36-70, https://doi.org/10.1016/j.aim.2016.05.005. · Zbl 1341.05250
[6] C. J. Eschenbrenner and M. J. Falk, Orlik-Solomon algebras and Tutte polynomials, J. Algebraic Combin., 10 (1999), pp. 189-199, https://doi.org/10.1023/A:1018735815621. · Zbl 0955.52010
[7] A. L. L. Gao, L. Lu, M. H. Y. Xie, A. L. B. Yang, and P. B. Zhang, The Kazhdan-Lusztig polynomials of uniform matroids, Adv. in Appl. Math., 122 (2021), 102117, https://doi.org/10.1016/j.aam.2020.102117. · Zbl 1457.05019
[8] A. L. L. Gao and M. H. Y. Xie, The inverse Kazhdan-Lusztig polynomial of a matroid, J. Combin. Theory Ser. B, 151 (2021), pp. 375-392, https://doi.org/10.1016/j.jctb.2021.07.004. · Zbl 1473.05312
[9] K. Gedeon, Kazhdan-Lusztig polynomials of thagomizer matroids, Electron. J. Combin., 24 (2017), P3.12. · Zbl 1369.05029
[10] K. Gedeon, N. Proudfoot, and B. Young, The equivariant Kazhdan-Lusztig polynomial of a matroid, J. Combin. Theory Ser. A, 150 (2017), pp. 267-294, https://doi.org/10.1016/j.jcta.2017.03.007. · Zbl 1362.05131
[11] K. Gedeon, N. Proudfoot, and B. Young, Kazhdan-Lusztig polynomials of matroids: A survey of results and conjectures, Sém. Lothar. Combin., 78B (2017), 80. · Zbl 1384.05172
[12] J. Haglund, The \(q,t\)-Catalan Numbers and the Space of Diagonal Harmonics, University Lecture Ser. 41, American Mathematical Society, Providence, RI, 2008, https://doi.org/10.1007/s10711-008-9270-0. · Zbl 1142.05074
[13] T. Karn, G. Nasr, N. Proudfoot, and L. Vecchi, Equivariant Kazhdan-Lusztig theory of paving matroids, J. Algebraic Combin., to appear. · Zbl 1516.05022
[14] K. Lee, G. D. Nasr, and J. Radcliffe, A combinatorial formula for Kazhdan-Lusztig polynomials of \(\rho \)-removed uniform matroids, Electron. J. Combin., 27 (2020), P4.7, https://doi.org/10.37236/9435. · Zbl 1450.05008
[15] K. Lee, G. D. Nasr, and J. Radcliffe, A combinatorial formula for Kazhdan-Lusztig polynomials of sparse paving matroids, Electron. J. Combin., 28 (2021), P4.28. · Zbl 1486.05037
[16] N. A. Loehr and J. B. Remmel, A computational and combinatorial exposé of plethystic calculus, J. Algebraic Combin., 33 (2011), pp. 163-198, https://doi.org/10.1007/s10801-010-0238-4. · Zbl 1229.05275
[17] L. Lu, M. H. Y. Xie, and A. L. B. Yang, Kazhdan-Lusztig polynomials of fan matroids, wheel matroids, and whirl matroids, J. Combin. Theory Ser. A, 192 (2022), 105665, https://doi.org/10.1016/j.jcta.2022.105665. · Zbl 1496.05021
[18] I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Math. Monogr., The Clarendon Press, Oxford University Press, New York, 1995. · Zbl 0824.05059
[19] N. Proudfoot, The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials, EMS Surv. Math. Sci., 5 (2018), pp. 99-127, https://doi.org/10.4171/EMSS/28. · Zbl 1445.14038
[20] N. Proudfoot, Equivariant Kazhdan-Lusztig polynomials of \(q\)-niform matroids, Algebr. Comb., 2 (2019), pp. 613-619, https://doi.org/10.5802/alco.59. · Zbl 1417.05024
[21] N. Proudfoot, Equivariant incidence algebras and equivariant Kazhdan-Lusztig-Stanley theory, Algebr. Comb., 4 (2021), pp. 675-681, https://doi.org/10.5802/alco.174. · Zbl 1473.05321
[22] R. P. Stanley, Subdivisions and local \(h\)-vectors, J. Amer. Math. Soc., 5 (1992), pp. 805-851, https://doi.org/10.2307/2152711. · Zbl 0768.05100
[23] R. P. Stanley, Enumerative Combinatorics. Vol. 2, Cambridge Stud. Adv. Math. 62, Cambridge University Press, Cambridge, UK, 1999, https://doi.org/10.1017/CBO9780511609589. · Zbl 0928.05001
[24] M. L. Wachs, Poset topology: Tools and applications, in Geometric Combinatorics, IAS/Park City Math. Ser. 13, American Mathematical Society, Providence, RI, 2007, pp. 497-615, https://doi.org/10.1090/pcms/013/09. · Zbl 1135.06001
[25] M. Wakefield, A flag Whitney number formula for matroid Kazhdan-Lusztig polynomials, Electron. J. Combin., 25 (2018), P1.22. · Zbl 1380.05023
[26] M. H. Y. Xie and P. B. Zhang, Equivariant Kazhdan-Lusztig polynomials of thagomizer matroids, Proc. Amer. Math. Soc., 147 (2019), pp. 4687-4695, https://doi.org/10.1090/proc/14608. · Zbl 1421.05028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.