Representation theory methods in extremal combinatorics
Date
2016
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Publisher
University of Delaware
Abstract
The research of this thesis lies in the area of extremal combinatorics. The word "extremal" comes from the kind of problems that are studied in this field. In fact, if a collection of finite objects (numbers, subsets, subspaces, graphs, etc.) satisfies some restrictions then the following questions are of interest from the perspective of extremal combinatorics: what is the maximum (minimum) size of those collections? what is the structure of the collections of maximum (minimum) size? ☐ For example, in extremal set theory one studies these questions for subsets of [n] = {1,2,…,n} subject to conditions such as the families of subsets are intersecting, anti-chain, included in another family of subsets, etc. This field has seen a tremendous growth in the past few decades. Remarkably, some of the results obtained in extremal set theory can be generalized when, instead of subsets, other objects are considered. The main results in this thesis are analogues of theorems in extremal set theory where, instead of subsets, objects like groups and subspaces are considered. First, we focus on generalizations of the Erdös-Ko-Rado theorem for permutation groups. In particular, for the group PGL(2, q) we prove that intersecting families of maximum size are stable. Moreover, for the group PSL(2,q) we prove that every intersecting family of maximum size is a coset of a point stabilizer. Secondly, we study rank resilience property of higher inclusion matrices of r-subsets vs. s-subsets. We prove that a q-analogue of this property holds, that is, the rank of the higher inclusion matrices of r-subspaces vs. s-subspaces is also resilient. Furthermore, we prove that this resilience property holds over any field in the set case and over any field of characteristic coprime to q in the vector space case. ☐ It is well known that, in general, these analogues of classical results are hard to prove. In fact, most of the proof ideas used to prove results in extremal set theory cannot be applied in a straightforward way. The main tools used here to prove our results come from representation theory. ☐ Representation theory is a branch of mathematics that studies algebraic structures by representing their elements as linear transformations of vector spaces. Indeed, one of the objectives of this thesis is to highlight how some tools provided by representation theory can be used to prove analogues of classical results in extremal combinatorics.