Kac-Moody Groups, their Flag Varieties and Representation Theory

This paper explores the possible use of Schubert cells and Schubert varieties in finite geometry, particularly in regard to the question of whether these objects might be a source of understanding of ovoids or provide new examples. The main result provides a characterization of those Schubert cells for finite Chevalley groups which have the first property (thinness) of ovoids. More importantly, perhaps this short paper can help to bridge the modern language barrier between finite geometry and representation theory. For this purpose, this paper includes very brief surveys of the powerful lattice theory point of view from finite geometry and the powerful method of indexing points of flag varieties by Chevalley generators from representation theory.

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Dissimilarity Maps on Trees and the Representation Theory of $GL_n(\mathbb{C})$

The Electronic Journal of Combinatorics ◽

10.37236/2715 ◽

2012 ◽

Vol 19 (3) ◽

Author(s):

Christopher Manon

Keyword(s):

Representation Theory ◽

Phylogenetic Trees ◽

Type A ◽

Flag Variety ◽

Flag Varieties ◽

Tropical Varieties

We revisit representation theory in type $A,$ used previously to establish that the dissimilarity vectors of phylogenetic trees are points on the tropical Grassmannian variety. We use a different version of this construction to show that the space of phylogenetic trees $K_n$ maps to the tropical varieties of every flag variety of $GL_n(\mathbb{C}).$ Using this map, we find a tropical function on the space of phylogenetic trees for each semistandard tableaux, and we show that the functions satisfy the tropicalized equations which cut out $GL_n(\mathbb{C})$ flag varieties.

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Shadows in the Wild - Folded Galleries and Their Applications

Jahresbericht der Deutschen Mathematiker-Vereinigung ◽

10.1365/s13291-021-00244-2 ◽

2021 ◽

Author(s):

Petra Schwer

Keyword(s):

Representation Theory ◽

Coxeter Groups ◽

Arithmetic Geometry ◽

Flag Varieties ◽

Reflection Groups ◽

Wide Audience ◽

Combinatorial Objects ◽

Affine Grassmannians ◽

In The Wild ◽

Affine Flag

AbstractThis survey is about combinatorial objects related to reflection groups and their applications in representation theory and arithmetic geometry. Coxeter groups and folded galleries in Coxeter complexes are introduced in detail and illustrated by examples. Further it is explained how they relate to retractions in Bruhat-Tits buildings and to the geometry of affine flag varieties and affine Grassmannians. The goal is to make these topics accessible to a wide audience.

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