# Clans, sects, and symmetric spaces of Hermitian type

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## Description

This thesis examines the geometry and combinatorics of Borel subgroup orbits in classical symmetric spaces G/L where G is complex linear algebraic group and L is a Levi factor of a maximal parabolic subgroup P in G. In these cases, known as symmetric spaces of Hermitian type, we show that the canonical projection map $\pi : G/L \to G/P$ has the structure of an affine bundle. This fact yields a cell decomposition of G/L as well as isomorphisms of the cohomology and Chow rings of G/L and G/P, and motivates the study of the Borel orbits of G/L in relation to their images under the equivariant map $\pi$. For all of the cases of interest (symmetric spaces of types AIII, CI, DIII and BDI), G/P is a Grassmannian variety with Borel orbits called Schubert cells. Borel orbits of most of these symmetric spaces are parametrized by combinatorial objects called clans. This thesis provides enumerative formulae for the orbits in type CI, DIII and BDI , and gives bijections between sets of clans and other families of objects such as (fixed-point free) partial involutions, rook placements, and set partitions. Clans come with a poset structure given by the closure containment relation of the corresponding Borel orbits, and we supply rank polynomials for these posets in types CI and DIII. We give a combinatorial description of the closure order relations in types AIII, CI, and DIII which allows us to resolve part of a conjecture of Wyser on the restriction of this order from type AIII to other types. In the course of this description, we identify the preimages of Schubert cells under the map \pi as collections of clans called “sects.” Our combinatorial description of the sects identifies Borel orbits whose closures generate the Chow ring of G/L and reveals additional structure in the closure poset of clans. In particular, the preimage of the largest Schubert cell coincides variously with well-known posets of matrix Schubert varieties and congruence Borel orbit closures. Furthermore, we show that in type AIII the closure order restricted to a given sect can be described combinatorially in terms of “rank tableaux.”