Tilted Richardson Varieties

TL;DR

Tilted Richardson varieties $ ext{T}_{u,v}$ extend classical Richardson geometry to the tilted Bruhat framework, tying flag-variety subvarieties to quantum Bruhat graphs and total positivity. The authors develop a complete combinatorial and geometric toolkit: tilted Bruhat orders and reduced words, four equivalent tilted Richardson definitions, a tilted Deodhar decomposition with tilted $R$-polynomials, and a nonnegative theory giving CW complexes of totally nonnegative parts. They connect these objects to minimal-degree two-point curve neighborhoods, deducing cohomology- and Gromov–Witten-theoretic consequences for the flag variety. Altogether, the work provides a unified structure bridging Schubert calculus, quantum cohomology, and total positivity in the tilted setting, with implications for positroid varieties and birational projection analyses.

Abstract

The study of the flag variety $\mathrm{Fl}_n$ and its subvarieties, including Schubert and Richardson varieties, plays a fundamental role in algebraic geometry and algebraic combinatorics. In this paper, we introduce and develop the theory of tilted Richardson varieties $\mathrm{T}_{u,v}$, a new family of subvarieties of the flag variety that provides a geometric framework for the quantum Bruhat graphs. These varieties are defined for all pairs of permutations $u$ and $v$, extending the classical Richardson varieties in the case where $u\leq v$ in the Bruhat order. We establish their fundamental geometric properties, proving irreducibility and providing explicit dimension formulas. Moreover, we show that they have a well-defined stratification indexed by tilted Bruhat intervals, a generalization of classical Bruhat intervals previously introduced by Brenti, Fomin, and Postnikov. Additionally, we introduce a tilted generalization of the classical Deodhar decomposition of Richardson varieties, which leads to a combinatorial formula for tilted Kazhdan--Lusztig R-polynomials, a notion that arises naturally in our framework. We further develop a theory of total positivity for tilted Richardson varieties. In particular, we define and study the totally nonnegative parts of tilted Richardson varieties, proving they form a CW complex. This generalizes earlier results on the totally nonnegative flag variety and answers Björner's questions regarding geometric realizations of tilted Bruhat intervals. Finally, we establish explicit connections between tilted Richardson varieties and quantum Schubert calculus. Specifically, we prove that $\mathrm{T}_{u,v}$ coincides with minimal-degree two-point curve neighborhoods. As a result, we compute their cohomology classes and derive new relationships among Gromov--Witten invariants of the flag variety.

Figures (15)

• Figure 1: Canonical representatives of Schubert cells for $w=2314$
• Figure 2: The quantum Bruhat graph $\Gamma_3$ (unlabeled edges have weight $1$)
• Figure 3: The tilted Bruhat interval $[231,123]$
• Figure 4: The lattice path $\Lambda(A,B)$ for $A=\{3,4,6,7\}$ and $B=\{1,2,3,5\}$
• Figure 6: The Hasse diagram of $\leq_{\mathbf{a}}$ on $S_4$ for $\mathbf{a}=(1,2,3,3)$

Theorems & Definitions (223)

• Definition 1.3
• Theorem 1.4
• Theorem 1.5: tilted Deodhar decomposition
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 2.1
• Proposition 2.2: Ehresmann Criterion ehresmann
• Lemma 2.3: bjorner1984orderings
• Example 2.4