UC Riverside

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.