Matching fields and lattice points of simplices

We show that the Chow covectors of a linkage matching field define a bijection between certain degree vectors and lattice points, and we demonstrate how one can recover the linkage matching field from this bijection. This resolves two open questions from Sturmfels and Zelevinsky (1993) [26] on linkage matching fields. For this, we give an explicit construction that associates a bipartite incidence graph of an ordered partition of a common set to each lattice point in a dilated simplex. Given a triangulation of a product of two simplices encoded by a set of spanning trees on a bipartite node set, we similarly prove that the bijection from left to right degree vectors of the trees is enough to recover the triangulation. As additional results, we show a cryptomorphic description of linkage matching fields and characterise the flip graph of a linkage matching field in terms of its prodsimplicial flag complex. Finally, we relate our findings to transversal matroids through the tropical Stiefel map

ON THE SIGNED MATCHINGS OF GRAPHS

For a graph $G$ and any $v\in V(G)$, $E_{G}(v)$ is the set of all edges incident with $v$. A function $f:E(G)\rightarrow \{-1,1\}$ is called a signed matching  of $G$ if  $\sum_{e\in E(v)}f(e) \leq 1$ for every ${v\in V(G)}$. For a signed matching $x$, set $x(E(G))=\sum_{e\in E(G))}x(e)$. The signed  matching number of $G$, denoted by $\beta_1'(G)$, is the maximum $x(E(G))$ where the maximum is taken over all signed matching over $G$. In this paper we obtain the signed matching number of some families of graphs and study the signed matching number of subdivision and edge deletion of edges of graph

Combinatorics in Schubert varieties and Specht modules

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, June 2011."June 2011." Cataloged from PDF version of thesis.Includes bibliographical references (p. 57-59).This thesis consists of two parts. Both parts are devoted to finding links between geometric/algebraic objects and combinatorial objects. In the first part of the thesis, we link Schubert varieties in the full flag variety with hyperplane arrangements. Schubert varieties are parameterized by elements of the Weyl group. For each element of the Weyl group, we construct certain hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial if and only if the Schubert variety is rationally smooth. For classical types the arrangements are (signed) graphical arrangements coning from (signed) graphs. Using this description, we also find an explicit combinatorial formula for the Poincaré polynomial in type A. The second part is about Specht modules of general diagram. For each diagram, we define a new class of polytopes and conjecture that the normalized volume of the polytope coincides with the dimension of the corresponding Specht module in many cases. We give evidences to this conjecture including the proofs for skew partition shapes and forests, as well as the normalized volume of the polytope for the toric staircase diagrams. We also define new class of toric tableaux of certain shapes, and conjecture the generating function of the tableaux is the Frobenius character of the corresponding Specht module. For a toric ribbon diagram, this is consistent with the previous conjecture. We also show that our conjecture is intimately related to Postnikov's conjecture on toric Specht modules and McNamara's conjecture of cylindric Schur positivity.by Hwanchul Yoo.Ph.D

Applications of tropical combinatorics and monomial modules

PhDWe study three aspects of tropical combinatorics and monomial modules. In the fi rst, we consider the tropical geometry specifi cally arising from convergent Puiseux series in multiple indeterminates. One application is a new view on stable intersections of tropical hypersurfaces. Another one is the study of families of ordinary convex polytopes depending on more than one parameter through tropical geometry. In the second, we consider matching fi elds and their connections to combinatorial geometry. We show that the Chow covectors of a linkage matching fi eld defi ne a bijection of lattice points, resolving two open questions from Sturmfels & Zelevinsky. We use a similar method to prove that, given a triangulation of a product of two simplices encoded by a set of bipartite trees, the bijection from left to right degree vectors of the trees is enough to recover the triangulation. As additional results, we show a cryptomorphic description of linkage matching fi elds and characterise the flip graph of a linkage matching fi eld in terms of its prodsimplicial flag complex. In the third, we study commutative algebra arising from generalised Frobenius numbers. We defi ne generalised lattice modules, a class of monomial modules whose Castelnuovo{ Mumford regularity captures the k-th Frobenius number. We study the fi ltration of generalised lattice modules providing an explicit characterisation of their minimal generators, and show that there are only finitely many isomorphism classes of generalised lattice modules. As a consequence of our commutative algebraic approach, we prove structural results on the sequence of generalised Frobenius numbers and also construct an algorithm to compute them.This work was supported by the EPSRC (1673882). Furthermore I am grateful to Queen Mary University of London and the Eileen Coyler Priz