Maximal Newton points and the quantum Bruhat graph

We discuss a surprising relationship between the partially ordered set of Newton points associated to an affine Schubert cell and the quantum cohomology of the complex flag variety. The main theorem provides a combinatorial formula for the unique maximum element in this poset in terms of paths in the quantum Bruhat graph, whose vertices are indexed by elements in the finite Weyl group. Key to establishing this connection is the fact that paths in the quantum Bruhat graph encode saturated chains in the strong Bruhat order on the affine Weyl group. This correspondence is also fundamental in the work of Lam and Shimozono establishing Peterson's isomorphism between the quantum cohomology of the finite flag variety and the homology of the affine Grassmannian. One important geometric application of the present work is an inequality which provides a necessary condition for non-emptiness of certain affine Deligne-Lusztig varieties in the affine flag variety.Comment: 39 pages, 4 figures best viewed in color; final version to appear in Michigan Math.

Generic Newton points and the Newton poset in Iwahori double cosets

We consider the Newton stratification on Iwahori double cosets in the loop group of a reductive group. We describe a group-theoretic condition on the generic Newton point, called cordiality, under which the Newton poset (i.e. the index set for non-empty Newton strata) is saturated and Grothendieck's conjecture on closures of the Newton strata holds. Finally, we give several large classes of Iwahori double cosets for which this condition is satisfied by studying certain paths in the associated quantum Bruhat graph.Comment: 17 pages, 1 figure; expanded introduction, generalized main theorem, changed section numbers; final version to appear in Forum of Mathematics, Sigm

Equivariant Quantum Cohomology of the Grassmannian via the Rim Hook Rule

A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the product of two classes in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton provided a way to compute quantum products of Schubert classes in the Grassmannian of k-planes in complex n-space by doing classical multiplication and then applying a combinatorial rim hook rule which yields the quantum parameter. In this paper, we provide a generalization of this rim hook rule to the setting in which there is also an action of the complex torus. Combining this result with Knutson and Tao's puzzle rule then gives an effective algorithm for computing all equivariant quantum Littlewood-Richardson coefficients. Interestingly, this rule requires a specialization of torus weights modulo n, suggesting a direct connection to the Peterson isomorphism relating quantum and affine Schubert calculus.Comment: 24 pages and 4 figures; typos corrected; final version to appear in Algebraic Combinatoric

Enumerations relating braid and commutation classes

We obtain an upper and lower bound for the number of reduced words for a permutation in terms of the number of braid classes and the number of commutation classes of the permutation. We classify the permutations that achieve each of these bounds, and enumerate both cases.Comment: 19 page

Alcove Walks and GKM Theory for Affine Flags

We develop the GKM theory for the torus-equivariant cohomology of the affine flag variety using the combinatorics of alcove walks. Dual to the usual GKM setup, which depicts the orbits of the small torus action on a graph, alcove walks take place in tessellations of Euclidean space. Walks in affine rank two occur on triangulations of the plane, providing a more direct connection to splines used for approximating surfaces. Alcove walks in GKM theory also need not be minimal length, and can instead be randomly generated, giving rise to more flexible implementation. This work reinterprets and recovers classical results in GKM theory on the affine flag variety, generalizing them to both non-minimal and folded alcove walks, all motivated by applications to splines.Comment: 30 pages, 8 figures best viewed in color; final version to appear in Springer INdAM Series, "Approximation Theory and Numerical Analysis Meet Algebra, Geometry, Topology

Crystal Chute Moves on Pipe Dreams

Schubert polynomials represent a basis for the cohomology of the complete flag variety and thus play a central role in geometry and combinatorics. In this context, Schubert polynomials are generating functions over various combinatorial objects, such as rc-graphs or reduced pipe dreams. By restricting Bergeron and Billey's chute moves on rc-graphs, we define a Demazure crystal structure on the monomials of a Schubert polynomial. As a consequence, we provide a method for decomposing Schubert polynomials as sums of key polynomials, complementing related work of Assaf and Schilling via reduced factorizations with cutoff, as well as Lenart's coplactic operators on biwords.Comment: 29 pages, 15 figures best viewed in colo

An affine approach to Peterson comparison

The Peterson comparison formula proved by Woodward relates the three-pointed Gromov-Witten invariants for the quantum cohomology of partial flag varieties to those for the complete flag. Another such comparison can be obtained by composing a combinatorial version of the Peterson isomorphism with a result of Lapointe and Morse relating quantum Littlewood-Richardson coefficients for the Grassmannian to k-Schur analogs in the homology of the affine Grassmannian obtained by adding rim hooks. We show that these comparisons on quantum cohomology are equivalent, up to Postnikov's strange duality isomorphism.Comment: 28 page

Affine Deligne-Lusztig varieties and folded galleries governed by chimneys

We characterize the nonemptiness and dimension problems for an affine Deligne-Lusztig variety $X_x(b)$ in the affine flag variety in terms of galleries that are positively folded with respect to a chimney. If the parabolic subgroup associated to the Newton point of b has rank 1, we then prove nonemptiness for a certain class of Iwahori-Weyl group elements x by explicitly constructing such galleries.Comment: 53 pages, 14 figures best viewed in colo

An equivariant quantum Pieri rule for the Grassmannian on cylindric shapes

The quantum cohomology ring of the Grassmannian is determined by the quantum Pieri rule for multiplying by Schubert classes indexed by row or column-shaped partitions. We provide a direct equivariant generalization of Postnikov's quantum Pieri rule for the Grassmannian in terms of cylindric shapes, complementing related work of Gorbounov and Korff in quantum integrable systems. The equivariant terms in our Graham-positive rule simply encode the positions of all possible addable boxes within one cylindric skew diagram. As such, unlike the earlier equivariant quantum Pieri rule of Huang and Li and known equivariant quantum Littlewood-Richardson rules, our formula does not require any calculations in a different Grassmannian or two-step flag variety.Comment: 27 pages, 9 figures best viewed in color; updated discussion of several reference