The Kazhdan-Lusztig polynomial of a matroid

摘要

We associate to every matroid M a polynomial with integer coefficients, whichwe call the Kazhdan-Lusztig polynomial of M, in analogy with Kazhdan-Lusztigpolynomials in representation theory. We conjecture that the coefficients arealways non-negative, and we prove this conjecture for representable matroids byinterpreting our polynomials as intersection cohomology Poincare polynomials.We also introduce a q-deformation of the Mobius algebra of M, and use ourpolynomials to define a special basis for this deformation, analogous to thecanonical basis of the Hecke algebra. We conjecture that the structurecoefficients for multiplication in this special basis are non-negative, and weverify this conjecture in numerous examples.