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The Kazhdan-Lusztig polynomial of a matroid. (English) Zbl 1341.05250
Summary: 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.

MSC:
05E10 Combinatorial aspects of representation theory
05B35 Combinatorial aspects of matroids and geometric lattices
52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
14M15 Grassmannians, Schubert varieties, flag manifolds
20C30 Representations of finite symmetric groups
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