Highest weight modules and polarized embeddings of shadow spaces

Let Gamma be the K-shadow space of a spherical building Delta. An embedding V of Gamma is called polarized if it affords all "singular" hyperplanes of Gamma. Suppose that Delta is associated to a Chevalley group G. Then Gamma can be embedded into what we call the Weyl module for G of highest weight lambda_K. It is proved that this module is polarized and that the associated minimal polarized embedding is precisely the irreducible G-module of highest weight lambda_K. In addition a number of general results on polarized embeddings of shadow spaces are proved. The last few sections are devoted to the study of specific shadow spaces, notably minuscule weight geometries, polar grassmannians, and projective flag-grassmannians. The paper is in part expository in nature so as to make this material accessible to a wide audience.Comment: Improvement in exposition of Sections 1-3 and . Notation improved. References added. Main results unchange

The generating rank of the unitary and symplectic Grassmannians

We prove that the Grassmannian of totally isotropic $k$-spaces of the polar space associated to the unitary group $\mathsf{SU}_{2n}(\mathbb{F})$ ($n\in \mathbb{N}$) has generating rank ${2n\choose k}$ when $\mathbb{F}\ne \mathbb{F}_4$. We also reprove the main result of Blok [Blok2007], namely that the Grassmannian of totally isotropic $k$-spaces associated to the symplectic group $\mathsf{Sp}_{2n}(\mathbb{F})$ has generating rank ${2n\choose k}-{2n\choose k-2}$, when $\rm{Char}(\mathbb{F})\ne 2$

The hyperplanes of the U (4)(3) near hexagon

Combining theoretical arguments with calculations in the computer algebra package GAP, we are able to show that there are 27 isomorphism classes of hyperplanes in the near hexagon for the group U (4)(3). We give an explicit construction of a representative of each class and we list several combinatorial properties of such a representative

On extensions of hyperplanes of dual polar spaces

AbstractLet Δ be a thick dual polar space and F a convex subspace of diameter at least 2 of Δ. Every hyperplane G of the subgeometry F˜ of Δ induced on F will give rise to a hyperplane H of Δ, the so-called extension of G. We show that F and G are in some sense uniquely determined by H. We also consider the following problem: if e is a full projective embedding of Δ and if eF is the full embedding of F˜ induced by e, does the fact that G arises from the embedding eF imply that H arises from the embedding e? We will study this problem in the cases that e is an absolutely universal embedding, a minimal full polarized embedding or a Grassmann embedding of a symplectic dual polar space. Our study will allow us to prove that if e is absolutely universal, then also eF is absolutely universal

The pseudo-hyperplanes and homogeneous pseudo-embeddings of the generalized quadrangles of order (3, t)

In the paper "as reported by De Bruyn (Adv Geom, to appear)", we introduced the notions of pseudo-hyperplane and pseudo-embedding of a point-line geometry and proved that every generalized quadrangle of order (s, t), 2 a parts per thousand currency sign s < a, has faithful pseudo-embeddings. The present paper focuses on generalized quadrangles of order (3, t). Using the computer algebra system GAP and invoking some theoretical relationships between pseudo-hyperplanes and pseudo-embeddings obtained in "De Bruyn (Adv Geom, to appear)", we are able to give a complete classification of all pseudo-hyperplanes of . We hereby find several new examples of tight sets of generalized quadrangles, as well as a complete classification of all 2-ovoids of . We use the classification of the pseudo-hyperplanes of to obtain a list of all homogeneous pseudo-embeddings of

Notes on topological insulators

This paper is a survey of the $\mathbb{Z}_2$-valued invariant of topological insulators used in condensed matter physics. The $\mathbb{Z}$-valued topological invariant, which was originally called the TKNN invariant in physics, has now been fully understood as the first Chern number. The $\mathbb{Z}_2$ invariant is more mysterious, we will explain its equivalent descriptions from different points of view and provide the relations between them. These invariants provide the classification of topological insulators with different symmetries in which K-theory plays an important role. Moreover, we establish that both invariants are realizations of index theorems which can also be understood in terms of condensed matter physics.Comment: 62 pages, 3 figure