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Sira Busch

Publications and source records attributed to Sira Busch.

6 recordsLinked to original sources
arXiv2026

Lines and Opposition in Lie Incidence Geometries of Exceptional Type

We characterise sets of points of exceptional Lie incidence geometries, that is, the natural geometries arising from spherical buildings of exceptional types $\mathsf{F_4}$, $\mathsf{E_6}$, $\mathsf{E_7}$, $\mathsf{E_8}$ and $\mathsf{G_2}$, that form a line using the opposition relation. With that, we obtain a classification of so-called ``geometric lines'' in many of these geometries. Furthermore, our results lead to a characterisation of geometric lines in finite exceptional Lie incidence geometries as minimal blocking sets, that is, point sets of the size of a line admitting no object opposite to all of their members, in most cases, and we classify all exceptions. As a further consequence, we obtain a characterisation of automorphisms of exceptional spherical buildings as certain opposition preserving maps.

math.CO↗
arXiv2025

Groups of Projectivities in Spherical Buildings of Non-Simply Laced Type

In this article we describe the general and special projectivity groups for all irreducible residues of all thick, irreducible, spherical buildings of type $ \mathsf{B_{n}}$, $ \mathsf{C_{n}}$ and $\mathsf{F_4}$, and rank at least 3. This determines the exact structure and action of Levi subgroups of parabolic subgroups of groups of Lie type related to those buildings.

math.GR↗
arXiv2025

The Moufang Condition and Root Automorphisms for Spherical Buildings of Rank 3

We give direct, geometric constructions for nontrivial root elations for rank $2$ residues of higher rank buildings $\Delta$ of type $\mathsf{B_n}, \mathsf{C_n}$ and $\mathsf{H_m}$ for $n \in \mathbb{N}$ and $m \in \{3,4\}$. We show that we can extend these to the ambient building in the case that $\Delta$ has type $\mathsf{B_n}$ or $\mathsf{C_n}$. With that, we obtain a different proof for the fact that buildings of type $\mathsf{B_n}$ and $\mathsf{C_n}$ are Moufang. This geometric approach enables us to gain more insight into the root groups associated to these buildings and we obtain new results; Namely, that certain root elations generically fix more points than we previously knew and that every root elation in each point residual can be written as an even self-projectivity. Concerning $\mathsf{H_m}$, we will be able to see in a novel way why thick, spherical buildings of type $\mathsf{H_m}$ cannot exist. Altogther, this provides an alternative proof for the fact that all thick, irreducible, spherical buildings $\Delta$ of rank 3 have the Moufang property.

math.GR↗
arXiv2024

A Characterization of Lines in Finite Lie Incidence Geometries of Classical Type

We consider any classical Grassmannian geometry $\Gamma$; that is, any projective or polar Grassmann space. Suppose every line in $\Gamma$ contains $s+1$ points. Then we classify all sets of points in $\Gamma$ of cardinality $s+1$, with the property, that no object of opposite type in the corresponding building, is opposite every point of the set. It turns out that such sets are either lines, or hyperbolic lines in symplectic residues, or ovoids in large symplectic subquadrangles of rank 2 residues in characteristic 2. This is a far-reaching extension of a famous and fundamental result of Bose & Burton from the 1960s. We describe a new way to classify geometric lines in finite classical geometries and how our results correspond to blocking sets.

math.CO↗
arXiv2024

Groups of Projectivities and Levi Subgroups in Spherical Buildings of Simply Laced Type

We introduce the special and general projectivity groups attached to a simplex $F$ of a thick irreducible spherical building of simply laced type. If the residue of $F$ is irreducible, we determine the permutation group of both projectivity groups of $F$, acting on the residue of $F$ and show that the special projectivity group determines the precise action of the Levi subgroup of a parabolic subgroup on the corresponding residue. This reveals three special cases for the exceptional types $\mathsf{E_6,E_7,E_8}$. Furthermore, we establish a general diagrammatic rule to decide when exactly the special and general projectivity groups of $F$ coincide.

math.GR↗