### The Log-Concavity of Kazhdan-Lusztig Polynomials of Uniform Matroids

XIE Matthew H Y1, ZHANG Philip B2

1. 1. College of Science, Tianjin University of Technology, Tianjin 300384, China;
2. College of Mathematical Science, Tianjin Normal University, Tianjin 300387, China
• Received:2021-08-13 Revised:2021-11-24 Online:2023-01-25 Published:2023-02-09
• Supported by:
This paper was supported by the National Natural Science Foundation of China under Grant Nos. 11901431 and 12171362.

XIE Matthew H Y, ZHANG Philip B. The Log-Concavity of Kazhdan-Lusztig Polynomials of Uniform Matroids[J]. Journal of Systems Science and Complexity, 2023, 36(1): 117-128.

Elias, et al. (2016) conjectured that the Kazhdan-Lusztig polynomial of any matroid is logconcave. Inspired by a computer proof of Moll’s log-concavity conjecture given by Kauers and Paule, the authors use a computer algebra system to prove the conjecture for arbitrary uniform matroids.
 [1] Elias B, Proudfoot N, and Wakefield M, The Kazhdan-Lusztig polynomial of a matroid, Adv. Math., 2016, 299:36-70.[2] Braden T, Huh J, Matherne J P, et al., Singular Hodge theory for combinatorial geometries, arXiv:2010.06088.[3] Gedeon K, Proudfoot N, and Young B, Kazhdan-Lusztig polynomials of matroids:A survey of results and conjectures, Sém. Lothar. Combin., 2017, 78B:Art. 80.[4] Lu L, Xie M H Y, and Yang A L B, Kazhdan-Lusztig polynomials of fan matroids, wheel matroids, and whirl matroids, J. Comb. Theory, Ser. A, 2022, 192:105665.[5] Gao A L L and Xie M H Y, The inverse Kazhdan-Lusztig polynomial of a matroid, J. Combin. Theory Ser. B, 2021, 151:375-392.[6] Gao A L L, Lu L, Xie M H Y, et al., The Kazhdan-Lusztig polynomials of uniform matroids, Adv. in Appl. Math., 2021, 122:102117.[7] Gedeon K, Proudfoot N, and Young B, The equivariant Kazhdan-Lusztig polynomial of a matroid, J. Combin. Theory Ser. A, 2017, 150:267-294.[8] Lee K, Nasr G D, and Radcliffe J, A combinatorial formula for Kazhdan-Lusztig polynomials of ρ-removed uniform matroids, Electron. J. Combin., 2020, 27:P4.7.[9] Chen H Z Q, Yang A L B, and Zhang P B, The real-rootedness of generalized Narayana polynomials related to the Boros-Moll polynomials, Rocky Mountain J. Math., 2018, 48:107-119.[10] Chen X, Yang A L B, and Zhao J J Y, Recurrences for Callan's generalization of Narayana polynomials, Journal of Systems Science & Complexity, 2022, 35(4):1573-1585.[11] Kauers M and Paule P, A computer proof of Moll's log-concavity conjecture, Proc. Amer. Math. Soc., 2007, 135:3847-3856.[12] Koutschan C, Advanced applications of the holonomic systems approach, Doctoral Thesis, Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria, 2009.[13] Chen S and Kauers M, Some open problems related to creative telescoping, Journal of Systems Science & Complexity, 2017, 30(1):154-172.
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