Tilted Richardson Varieties

Abstract

The flag variety $\mathrm{Fl}n$ and its subvarieties, such as Schubert and Richardson varieties, are central objects in algebraic geometry and algebraic combinatorics. In this thesis, we introduce and study tilted Richardson varieties $\mathcal{T}{u,v}$, a new family of subvarieties of $\mathrm{Fl}_n$ defined for all pairs of permutations $u$ and $v$. This family generalizes classical Richardson varieties when $u\leq v$ in the Bruhat order and provides a geometric framework for the quantum Bruhat graph. We establish their fundamental geometric properties, including irreducibility, explicit dimension formulas, and a well-defined stratification indexed by tilted Bruhat intervals, a generalization of classical Bruhat intervals introduced by Brenti, Fomin, and Postnikov. Moreover, we introduce a tilted generalization of the classical Deodhar decomposition, 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 their totally nonnegative parts and prove that they form CW-complexes. This generalizes previous work on the totally nonnegative flag variety and addresses Bj"orner's questions regarding geometric realizations of tilted Bruhat intervals. Finally, we establish explicit connections between tilted Richardson varieties and quantum Schubert calculus. We prove that $\mathcal{T}_{u,v}$ coincides with minimal-degree two-point curve neighborhoods, allowing us to compute their cohomology classes and derive new relationships among Gromov--Witten invariants of the flag variety.