Representation Theory for Oriented Matroids and Combinatorial Intersection Homology for Gorenstein* Meet-Semilattices
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Representation Theory for Oriented Matroids and Combinatorial Intersection Homology for Gorenstein* Meet-Semilattices

  • Author(s): Kowalenko, Ethan
  • Advisor(s): Mautner, Carl
  • et al.
Creative Commons 'BY' version 4.0 license
Abstract

This dissertation consist of two distinct parts, coming from different parts of algebraic combinatorics.

In the first part, we discuss representation theory for oriented matroids, a joint project with Carl Mautner. In particular, we consider oriented matroid programs, which are a combinatorial abstraction of linear programming problems. To such a program $\PB$, together with a \emph{parameter space} $U$, we associate a finite-dimensional quadratic algebra $A(\PB,U)$. These algebras generalize those of Braden-Licata-Proudfoot-Webster in the realizable case. When the program is Euclidean, we prove that our algebras are quasihereditary and Koszul. We also have partial related results in the non-Euclidean case.

In the second part, we develop intersection homology for certain posets.Combinatorial intersection homology for fans (IHF) is a sheaf-based theory modeled on the equivariant intersection cohomology for toric varieties, which was used to prove that the coefficients of toric $h$-polynomials of polytopes are nonnegative. More generally, Stanley conjectures that the toric $h$-polynomials of Cohen-Macaulay lower Eulerian meet-semilattices also have nonnegative coefficients. This paper builds an analogue of IHF for this more general class of posets, and we make a hard Lefschetz conjecture which would solve Stanley's conjecture.

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