Intersection cohomology without spaces
Tom Braden, Nicholas Proudfoot
TL;DR
This survey develops a unified framework showing that three independent instances of Kazhdan–Lusztig–type polynomials—from Coxeter groups, polytopes, and matroids—can be realized as Poincaré polynomials of purely algebraic objects: Soergel bimodules, intersection cohomology of fans, and intersection cohomology of matroids. By recasting computations in terms of equivariant intersection cohomology and sheaves on posets, the authors provide a space-free interpretation of these polynomials, while preserving the geometric underpinnings such as hard Lefschetz and Hodge–Riemann-type properties where available. They develop a robust, inductive local-to-global procedure using torus localization, boundary modules, and Mayer–Vietoris sequences to define and compute the relevant IH-theoretic objects even when the geometric spaces do not exist, and they relate these constructions to KLS-polynomials, g-polynomials, and Z-polynomials across the three settings. The work also extends to positive characteristic via parity sheaves, yielding p-KLS polynomials and insights into modular representation theory, including implications for tilting modules and canonical bases, with inequalities like monotonicity and top-heaviness that illuminate the combinatorial structure of the underlying posets and lattices. Overall, the paper connects combinatorics, geometry, and representation theory through a coherent algebraic narrative that generalizes classical IH phenomena to nonspace settings while preserving their deep geometric and algebraic consequences.
Abstract
We survey three settings in which dimensions of intersection cohomology groups of algebraic varieties provide deep combinatorial and representation-theoretic information, and computations of the groups themselves have been made using combinatorial sheaves on finite posets. These settings are (1) intersection cohomology of Schubert varieties, the associated Kazhdan-Lusztig polynomials and their realizations via moment graph sheaves and Soergel bimodules; (2) intersection cohomology of toric varieties, the associated g-polynomials of convex polytopes, and their realization via the theory of intersection cohomology of fans; and (3) intersection cohomology of arrangement Schubert varieties, the associated Kazhdan-Lusztig polynomials of matroids, and their realization via intersection cohomology of matroids. In all three settings these constructions are valid in more general situations where the variety does not exist, leading to "intersection cohomology without spaces." We give parallel presentations of these three stories, highlighting applications to KLS-polynomials.