The Kazhdan–Lusztig polynomial of a matroid

Abstract

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.