The Coxeter Flag Variety

TL;DR

The Coxeter flag variety $ ext{CFl}_c$ is constructed as a toric complex inside $G/B$, assembled from translated Coxeter Richardson varieties and governed by the lattice of $c$-noncrossing partitions $ ext{NC}(W,c)$. The authors develop Plücker-vanishing descriptions, an affine paving, and a GKM framework that compute the (equivariant) cohomology of $ ext{CFl}_c$ in a way that is independent of the Coxeter element $c$ up to conjugacy; in type A this recovers the quasisymmetric flag variety and yields a presentation in terms of permuted quasisymmetric coinvariants. A moment-polytope perspective identifies the constituents of $ ext{CFl}_c$ as $(W,c)$-polypositroids and relates the topology to the combinatorics of $ ext{NC}(W,c)$ and $c$-clusters. The paper also develops duality bases for the cohomology via localization and provides a detailed type A treatment connecting to Billey-style Schubert calculus in the Coxeter setting. Overall, the work unifies Coxeter combinatorics, toric geometry, and Schubert calculus to describe a rich family of flag-variety-like objects with stable cohomology across Coxeter elements.

Abstract

For a Coxeter element $c$ in a Weyl group $W$, we define the $c$-Coxeter flag variety $\operatorname{CFl}_c\subset G/B$ as the union of left-translated Richardson varieties $w^{-1}X^{wc}_w$. This is a complex of toric varieties whose geometry is governed by the lattice $\operatorname{NC}(W,c)$ of $c$-noncrossing partitions. We show that $\operatorname{CFl}_c$ is the common vanishing locus of the generalized Plücker coordinates indexed by $W\setminus\operatorname{NC}(W,c)$. We also construct an explicit affine paving of $\operatorname{CFl}_c$ and identify the $T$-weights of each cell in terms of $c$-clusters. This paving gives a GKM description of $H^\bullet(\operatorname{CFl}_c)$ and $H^\bullet_{T_{ad}}(\operatorname{CFl}_c)$ in terms of the induced Cayley subgraph on $\operatorname{NC}(W,c)$, and we show these rings are naturally isomorphic for different choices of $c$. In type $\mathrm{A}$, this recovers the quasisymmetric flag variety for a special $c$, and for general $c$ we show the cohomology ring has a presentation as permuted quasisymmetric coinvariants.

Figures (8)

• Figure 1: $\operatorname{HHMP}$ decomposition of the type $\mathrm{A}_3$ permutahedron corresponding to $c=s_1s_3s_2$ (left) and $c=s_3s_2s_1$ (right). The corresponding Coxeter flag varieties are each unions of five $3$-dimensional toric varieties, whose moment polytopes are translates of the depicted polytopes (the two blue polytopes on the right are identified under this translation). See \ref{['fig:two-models-panels']} for how the polytopes on the left assemble to a polytopal complex whose face lattice is $\operatorname{CFl}_{s_1s_3s_2}$.
• Figure 2: The blue, red, and yellow trapezoids $P_{[\operatorname{id}, c]}$, $P_{[s_1, s_{0} s_{1} s_{0}]}$, and $P_{[c, w_{\circ}]}$ comprising the HHMP subdivision of the type $\mathrm{B}_{2}$-permutahedron for $c = s_{0}s_{1}$ (left) and the intersecting polytopes $P_{[\operatorname{id}, c]}$, $s_{1}P_{[s_1, s_{0} s_{1} s_{0}]}$, and $c^{-1} P_{[c, w_{\circ}]}$ (right), where we only glue the bold edges. The elements $s_{1} s_{0}$ and $w_{\circ}$ outside of $\operatorname{NC}(\mathrm{B}_{2}, c)$ correspond to the vertices $s_{1}s_{0} \cdot (\lambda_{1}, \lambda_{2}) = (\lambda_2, -\lambda_1)$ and $w_{\circ} \cdot (\lambda_{1}, \lambda_{2}) = (-\lambda_{1}, -\lambda_{2})$
• Figure 3: Cumulative overlay of the moment polytopes for $c=s_1s_3s_2$; The top row is an exploded view of the "HHMP" decomposition and the bottom is the polytopal complex. The moment polytopes $P_{[w,wc]}$ from left to right come from $w$ equal to $2314$, $2413$, $1423$, $1234$, and $1324$. The first four have the same face lattice as a cube, and the last has the face lattice of the $10$ vertex tetragonal trapezohedron. The red and blue dots indicate respectively the images of $w$ and $w\cdot c$ either before or after translation by $w^{-1}$.
• Figure 4: The Hasse diagram of $\operatorname{NC}(S_4,s_1s_3s_2)$ with the five decreasing chains in the $c$-reflection order $(1\, 2) \prec (3 \, 4)\prec (1\, 4) \prec (2\, 4) \prec (1\, 3) \prec (2\, 3)$ bolded.
• Figure 5: The $\mathrm{B}_{2}$ root system with its positive $c$-cluster fan highlighted, for $c=s_0s_1$. Note that the rays $\alpha_0+\alpha_1$ and $2\alpha_0+\alpha_1$ do not correspond to positive $c$-clusters.

Theorems & Definitions (140)

• Theorem A: \ref{['thm:everying_about_cfl']}
• Theorem B: \ref{['thm:everying_about_cfl']}
• Theorem C: \ref{['sec:Bruhatmax']}
• Theorem D: \ref{['cor:coho_iso']}
• Theorem 1.4: BGNST2
• Theorem E: \ref{['cor:permutedqsym']}
• Remark 2.1
• Theorem 2.2: \ref{['thm:polyposSect8']}
• Definition 3.1
• Definition 3.3: Dy93