Combinatorics in finite classical groups
Final Report Abstract
The results of this project contribute to extremal and algebraic combinatorics. The guiding theme was to transfer combinatorial questions and known results from set theory to the setting of finite-dimensional vector spaces over finite fields. This naturally leads to the combinatorics of the finite general linear group GL(n,q) over a finite field of order q, acting on a finite vector space. In the combinatorics of a finite set, the symmetric group, a naturally attached association scheme and the representation theory of the group play an important role. In the context of a finite linear group, again its association scheme and its character theory turned out to be useful tools in this project. And indeed, the character theory of the finite general linear group is related to that of the symmetric group, but much more involved. A first topic studied in the project was devoted to intersection theorems for GL(n,q) in the spirit of the classical Erdös-Ko-Rado Theorem (1961). This theorem answered the question how large a family of pairwise intersecting subsets of size k within a given set of n elements can be, and how the intersecting families of maximal size look like. In the setting of GL(n,q), the notion of intersection means coincidence on subsets or subspaces of the underlying finite vector space. Tight upper bounds for different variants of intersecting sets were found, and a characterization of the intersecting sets of maximal size could be given. The obtained results are essentially analogous to results of Ellis, Friedgut and Pilpel (2011) for the symmetric group acting by permutations on a finite set. A second major part of the project was devoted to questions concerning the transitivity of subsets of GL(n,q). Here the classical origin is work by Livingstone and Wagner (1965), who studied transitivity questions for subgroups of the symmetric group. There are interesting generalizations by Martin and Sagan (2006), dealing with subsets of the symmetric group instead of subgroups, and considering their action on set partitions. In analogy, the present project considers the action of subsets of GL(n,q) on flag-like structures built from subspaces of the full underlying vector space. Based on the theory of association schemes, it is shown that such transitive sets are so-called Delsarte T-designs in the association scheme of GL(n,q). This generalizes a group-theoretical result of Perin (1972) on subgroups of GL(n,q) acting transitively on subspaces over finite fields.
Publications
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Intersection theorems for finite general linear groups. Mathematical Proceedings of the Cambridge Philosophical Society, 175(1), 129-160.
ERNST, ALENA & SCHMIDT, KAI–UWE
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Subsets of finite general linear groups. Dissertation, Paderborn University, December 2024. 146 pages.
A. Ernst
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Transitivity in finite general linear groups. Mathematische Zeitschrift, 307(3).
Ernst, Alena & Schmidt, Kai-Uwe
