Doctor of Philosophy
Matroids are combinatorial structures that capture various notions of independence. Recently there has been great interest in studying various matroid invariants. In this thesis, we study two such invariants: Volume of matroid base polytopes and the Tutte polynomial. We gave an approach to computing volume of matroid base polytopes using cyclic flats and apply it to the case of sparse paving matroids. For the Tutte polynomial, we recover (some of) its coefficients as degrees of certain forms in the Chow ring of underlying matroid. Lastly, we study the stability of characters of the symmetric group via character polynomials. We show a combinatorial identity in the ring of class functions that implies stability results for certain class of Kronecker coefficients.
Summary for Lay Audience
This thesis is mainly about a mathematical object called matroid. A matroid can be understood as an abstraction of a matrix (a list of vectors, read point), but without caring specifically about the coordinates of the vectors forming the matrix. What we care about is the dependency relations of these vectors, for example, which among these vectors are collinear, which of them are coplanar et cetera. To a matroid, we can associate a polytope that records that matroid. This is called the base polytope of the underlying matroid. Polytopes are generalizations of polygons to higher dimensions. Like the notion of area is associated to a polygon, the notion of volume is well-defined for a polytope. The problem we tackled in the second chapter is to have a combinatorial approach to computing the volume of the base polytope of an arbitrary matroid. We outlined the previously known methods and contrast them with ours. In the third chapter, we study another invariant of a matroid. Like polytopes, we can also associate a fan to a matroid, called Bergman fan. This fan remembers some properties of the matroid. A famous invariant is the Tutte polynomial of the matroid. We showed that the Bergman fan when decorated with certain weights remembers the Tutte polynomial of the matroid. The fourth chapter is about permutations. We study special functions on the set of permutations on $n$ letters. These functions are of importance in representation theory of symmetric group. We discovered and proved a relation between alternating sums of two classes of such functions. We developed combinatorics of tilings of Young diagrams for its proof. The relation simplifies certain calculations and help us reprove previously known results.
Ashraf, Ahmed, "Of matroid polytopes, chow rings and character polynomials" (2019). Electronic Thesis and Dissertation Repository. 6492.
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