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A finite Coxeter group W has a natural metric d and if ${ℳ}$ is a subset of W, then for each $u\in W$, there is $q\in {ℳ}$ such that $d(u,q)=d(u,{ℳ})$. Such q is not unique in general but if ${ℳ}$ is a Coxeter matroid, then it is unique, and we define a retraction ${{ℛ}}_{{ℳ}}^m:W\to {ℳ}\subset W$ so that ${{ℛ}}_{{ℳ}}^m(u)=q$.

The T-fixed point set $Y^T$ of a T-orbit closure Y in a flag variety G/B is a Coxeter matroid, where G is a semi-simple algebraic group, B is a Borel subgroup, and T is a maximal torus of G contained in B. We define a retraction ${{ℛ}}_Y^g:W\to Y^T\subset W$ geometrically, where W is the Weyl group of $G$, and show that ${{ℛ}}_Y^g={{ℛ}}_{Y^T}^m$. We introduce another retraction ${{ℛ}}_{{ℳ}}^a:W\to {ℳ}\subset W$ algebraically for an arbitrary subset ${ℳ}$ of W when W is a Weyl group of classical Lie type, and show that ${{ℛ}}_{{ℳ}}^a={{ℛ}}_{{ℳ}}^m$ when ${ℳ}$ is a Coxeter matroid.

A homology class $d\in {\mathrm{\text{H}}}_2(X,\mathbb{Z})$ of a complex flag variety $X=G∕P$ is called a line degree if the moduli space ${\overline{\mathcal{ℳ}}}_{0,0}(X,d)$ of 0-pointed stable maps to X of degree d is also a flag variety $G∕P^{\prime }$. We prove a quantum equals classical formula stating that any n-pointed (equivariant, $\mathrm{\text{K}}$-theoretic, genus zero) Gromov–Witten invariant of line degree on X is equal to a classical intersection number computed on the flag variety $G∕P^{\prime }$. We also prove an n-pointed analogue of the Peterson comparison formula stating that these invariants coincide with Gromov–Witten invariants of the variety of complete flags $G∕B$. Our formulas make it straightforward to compute the big quantum $\mathrm{\text{K}}$-theory ring ${\mathrm{\text{QK}}}^{\mathrm{\text{big}}}(X)$ modulo the ideal $〈Q^d〉$ generated by degrees d larger than line degrees.

We study algebraic, combinatorial and geometric aspects of weighted Poincaré–Birkhoff–Witt (PBW)-type degenerations of (partial) flag varieties in type $A$. These degenerations are labeled by degree functions lying in an explicitly defined polyhedral cone, which can be identified with a maximal cone in the tropical flag variety. Varying the degree function in the cone, we recover, for example, the classical flag variety, its abelian PBW degeneration, some of its linear degenerations and a particular toric degeneration.

Characteristic classes of Schubert varieties can be used to study the geometry and the combinatorics of homogeneous spaces. We prove a relation between elliptic classes of Schubert varieties on a generalized full flag variety and those on its Langlands dual. This new symmetry is motivated by 3D mirror symmetry, and it is only revealed if Schubert calculus is elevated from cohomology or K theory to the elliptic level.