The Coxeter Flag Variety
Nantel Bergeron, Lucas Gagnon, Hunter Spink, Vasu Tewari

TL;DR
This paper introduces the $c$-Coxeter flag variety, exploring its geometric structure, cohomology, and connections to noncrossing partitions, providing explicit descriptions and paving methods.
Contribution
It defines the $c$-Coxeter flag variety, constructs an affine paving, and describes its cohomology in terms of $c$-clusters and noncrossing partitions, linking geometry and combinatorics.
Findings
The $c$-Coxeter flag variety is the common vanishing locus of certain generalized Plücker coordinates.
An explicit affine paving of $ ext{CFl}_c$ is constructed.
The cohomology ring is described as permuted quasisymmetric coinvariants in type A.
Abstract
For a Coxeter element in a Weyl group , we define the -Coxeter flag variety as the union of left-translated Richardson varieties . This is a complex of toric varieties whose geometry is governed by the lattice of -noncrossing partitions. We show that is the common vanishing locus of the generalized Pl\"ucker coordinates indexed by . We also construct an explicit affine paving of and identify the -weights of each cell in terms of -clusters. This paving gives a GKM description of and in terms of the induced Cayley subgraph on , and we show these rings are naturally isomorphic for different choices of . In type…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Algebraic structures and combinatorial models · Advanced Algebra and Geometry
