The Kazhdan–Lusztig polynomial of a matroid

Ben Elias, Nicholas Proudfoot, Max Wakefield
2016 Advances in Mathematics  
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
more » ... asis 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. which one takes the stalk is determined by x. This proves that P x,y (t) has non-negative coefficients when W is a finite Weyl group. The non-negativity of the coefficients of P x,y (t) for arbitrary Coxeter groups was conjectured in [KL79], but was only recently proved by Williamson and the first author [EW14, 1.2(1)]. • Algebra: The polynomials P x,y (t) are the entries of the matrix relating the Kazhdan-Lusztig basis (or canonical basis) to the standard basis of the Hecke algebra of W , a q-deformation of the group algebra C[W ]. When W is a finite Weyl group, Kazhdan and Lusztig showed that the structure coefficients for multiplication in the Kazhdan-Lusztig basis are polynomials with non-negative coefficients. For general Coxeter groups, this is proved in [EW14, 1.2(2)]. In our analogy, the Coxeter group W is replaced by a matroid M , and the elements x, y ∈ W are replaced by flats F and G of M . We only define a single polynomial P M (t) for each matroid, but one may associate to a pair F ≤ G the polynomial P M F G (t), where M F G is the matroid whose lattice of flats is isomorphic to the interval 3 [F, G]. The role of the R-polynomial is played by the characteristic polynomial of the matroid. The analogue of being a finite Weyl group is being a representable matroid; that is, the matroid M A associated to a collection A of vectors in a vector space. The analogue of a Schubert variety is the reciprocal plane X A , also known as the spectrum of the Orlik-Terao algebra of A. The analogue of the group algebra C[W ] is the Möbius algebra E(M ); we introduce a q-defomation E q (M ) of this algebra which plays the role of the Hecke algebra. All of these analogies may be summarized as follows: • Combinatorics: We give a recursive definition of the polynomial P M (t) in terms of the characteristic polynomial of a matroid (Theorem 2.2), and we conjecture that the coefficients are non-negative (Conjecture 2.3). • Geometry: If M is representable over a finite field, we show that P M (t) is equal to the ℓ-adić etale intersection cohomology Poincaré polynomial of the reciprocal plane 4 (Theorem 3.10). Any matroid that is representable over some field is representable over a finite field, thus we obtain a proof of Conjecture 2.3 for all representable matroids (Corollary 3.11). • Algebra: We use the polynomials P M F G (t) to define the Kazhdan-Lusztig basis of the qdeformed Möbius algebra E q (M ). We conjecture that the structure constants for multiplication in this basis are polynomials in q with non-negative coefficients (Conjecture 4.2), and we verify this conjecture in a number of cases. Remark 1.1. Despite these parallels, the behavior of the polynomials for matroids differs drastically from the behavior of ordinary Kazhdan-Lusztig polynomials for Coxeter groups. In particular,
doi:10.1016/j.aim.2016.05.005 fatcat:pdsync7ncjfd5ofv3lmtesdlhi