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Lattice Path Matroids: Base Polytopes & Ehrhart Theory
Morales, Dania A
Morales, Dania A
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Abstract
We study matroid base polytopes of lattice path matroids, $\mathscr{P}(M(\nu))\subset \Rr^n$ with skew shape $\nu$. Our goal is to compute their corresponding Ehrhart polynomials. This story comes in three parts.\textit{Part 1:} We produce a minimal linear inequality description for the base polytope. Applying this description, we give the Ehrhart function for the base polytopes of lattice path matroids with straight shape $\nu$. That is, the shape $\nu$ corresponds to a partition. This class of lattice path matroids include uniform matroids and Schubert matroids.\textit{Part 2:} We consider a base polytope subdivision. Given any skew shape $\nu$, we decompose the base polytope of the lattice path matroid $\mathscr{P}(M(\nu))$ into base polytopes of lattice path matroids with ribbon shapes. We call the base polytope of a lattice path matroid a \emph{ribbon polytope}. We organize intersections of ribbon polytopes in a partially ordered set called the \emph{ribbon lattice}. Equipped with our subdivision and lattice structure, we express the Ehrhart polynomial for the base polytope of any lattice path matroid in terms of addition and multiplication of Ehrhart polynomials for ribbon polytopes.\textit{Part 3:} We focus on ribbon polytopes and we give a combinatorial formula for the Ehrhart $h^*$-polynomial. We achieve this by first observing that a ribbon polytope is Ehrhart equivalent to a corresponding \emph{order polytope}. We produce a unimodular triangulation for the order polytope that is shellable and we give a combinatorial formula for the restriction sets of our shellings. This gives the $h$-polynomial of the triangulation and thus the Ehrhart $h^*$-polynomial of the ribbon polytope.
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Date
2024-01-01
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University of Kansas
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- Embargoed until 2174-05-31
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Keywords
Mathematics, Base Polytopes, Ehrhart Polynomial, Lattice Path Matroids, Matroids, Subdivision