The Kazhdan-Lusztig polynomial of a matroid

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• 作者：Ben Elias^a ; ^{belias@uoregon.edu" class="auth_mail" title="E-mail the corresponding author} ; Nicholas Proudfoot^a ; ¹ ; ^{njp@uoregon.edu" class="auth_mail" title="E-mail the corresponding author} ; Max Wakefield^b ; ² ; ^{wakefiel@usna.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Matroid ; Orlik&ndash ; Terao algebra ; Intersection cohomology ; Kazhdan&ndash ; Lusztig theory
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：20 August 2016
• 年：2016
• 卷：299
• 期：Complete
• 页码：36-70
• 全文大小：558 K

文摘

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 conjecture that the coefficients are always non-negative, and we prove this conjecture for representable matroids by interpreting our polynomials as intersection cohomology Poincaré polynomials. We also introduce a q-deformation of the Möbius algebra of M, and use our polynomials to define a special basis for this deformation, analogous to the canonical basis of the Hecke algebra. We conjecture that the structure coefficients for multiplication in this special basis are non-negative, and we verify this conjecture in numerous examples.