组合数学讨论班

报告题目: Geometric representations of projective spaces and generalized quadrangles

报 告 人: Koen Thas(Ghent University)

时      间: 2026年4月10日, 09:30

地      点:海纳苑2幢303

摘      要: 

(1) Title: Geometric representations of projective lines

Abstract: Given a division ring B and a division ring A such that the left dimension [A : B] = 2, one naturally has a projective line PG(1,B) at one’s disposal. If one considers a (left) vector space V of dimension n over A, a representation of PG(1,B) arises as a spread in PG(2n - 1,B). In this lecture, we study special “twisted” representations in the case n = 2 with remarkable geometric and automorphic properties. (This is joint work with Hendrik Van Maldeghem.)

(2)Title: Modular representation theory of skew translation generalized quadrangles

Abstract：Suppose Q is a skew translation generalized quadrangle (STGQ) with associated central symmetry group S. In a recent paper, we have shown that S cannot be a subgroup of (ℤ,+) (whereas in the finite case, one indeed has examples of STGQs for which the central symmetry group is cyclic).  More generally, call an STGQ modular if Aut(S) is a subgroup of the modular group GL(n,ℤ). In the current lecture, we present new nonexistence results for modular STGQs. 

(3)Title: Free automorphic actions on projective spaces

Abstract：The theory of Singer groups has been very fruitful in finite geometry over the years, but lacks the proper foundations in the infinite case. (Recall that a Singer group of a projective space is an automorphism group which acts sharply transitively on the point set.) The standard formalism to construct Singer groups for a projective space PG(n,K) (with K a field and infinite) consists of taking a field extension E of K degree n + 1, and interpreting E as an (n + 1)-dimensional vector space over K. The field E then naturally acts freely and transitively on the points of  PG(n,K) by multiplication. But what happens if there are no field extensions of degree n + 1 (and in particular if K is algebraically closed)? This is the question we want to explore in the present lecture.