Closure Operator And Lattice Property of Root Systems

The concept of root systems arises in the study of many mathematical areas. In particular it is a crucial tool for understanding Coxeter groups and various groupoids which demonstrate Coxeter like properties.

In this dissertation we study matroidal properties of various root systems. First we consider the signed groupoid set, a concept which was developed by Dyer and generalizes many notions including Coxeter groups, Coxeter groupoids and Coxeter-like groupoid studied by Brink and Howlett. We show that the finite, connected, simply connected, real, compressed, principal, complete and rootoidal signed groupoid sets correspond bijectively to the oriented simplicial geometries.

In the second part, we studied the closure operator and lattice property of the root system of an infinite Coxeter group. We establish a bijection between the infinitely long words of an affine Weyl group and certain biclosed sets of its positive system. Using this bijection, we show first that the biclosed sets in the standard positive system of rank 3 affine Weyl groups when ordered by inclusion form a complete algebraic ortholattice and secondly that the (generalized) braid graphs of those Coxeter groups are connected, which can be thought as an infinite version of Tit's solution to the word problem. Finally we investigate the relationship between infinitely long words and twisted weak orders and show that for affine Weyl groups, a biclosed set is an inversion set if and only if the associated twisted weak order on the group is a meet semilattice. These results are analogous to the fact that the weak order of a simplicial base chamber of a finite simple oriented matroid is a complete lattice (such order is defined on the family of biconvex sets of a given hemispace under inclusion).

Finally, we treat the root systems of an affine Weyl group as an infinite oriented matroid in the sense of Buchi and Fenton. We compute all of their hemispaces, covectors and the stabilizers of the covectors. Also we describe a complete rootoid structure from locally finite Coxeter groups which is analogous to the structure of a simplicial oriented geometry.