Affine Schubert Calculus

Abstract

The cohomology of the affine flag varietyFl G of a complex reductive group G is a comodule over the cohomology of the affine Grassmannian Gr G . We give positive formulae for the coproduct of an affine Schubert class in terms of affine Stanley classes and finite Schubert classes, in (torus-equivariant) cohomology and K-theory. As an application, we deduce monomial positivity for the affine Schubert polynomials of the second author. THOMAS LAM, SEUNGJIN LEE, AND MARK SHIMOZONOThe class ξ v G/B is considered an element of H * T (Fl G ) via pullback under evaluation at the identity (see (3.5)). The same formulae hold in non-equivariant cohomology.In the majority of this article (Sections 2-4) we will work in torus-equivariant K-theory K * T (Fl) of the affine flag variety. The coproduct formula for holds in (torus-equivariant) K-theory with Demazure product replacing length-additive products (see Theorem 4.7). Our proof relies heavily on the action of the affine nilHecke ring on K * T (Fl). Let us note that there are a number of different geometric approaches [KK, KS, LSSa] for constructing Schubert classes in K * T (Fl), see [LLMS, Section 3] for a comparison. However, our results holds at the level of Grothendieck groups and the precise geometric model (thick affine flag variety, thin affine flag variety, or based loop group) is not crucial.In Section 5, the proofs for cohomology are indicated. In Section 6, we give examples of our formula in classical type. In particular, we prove (Theorem 6.1) that the affine Schubert polynomials [Lee] are monomial positive, and we explain how the Billey-Haiman formula [BH] for type C or D Schubert polynomials (see also [IMN]) is a consequence of our coproduct formula.By taking an appropriate limit, the coproduct formula for backstable (double) Schubert polynomials [LLS] can be deduced from Theorem 1.1. Whereas the proofs in [LLS] are essentially combinatorial, the present work relies heavily on equivariant localization and the nilHecke algebra. Affine nilHecke ring and the equivariant K-theory of the affine flag varietyThe proofs of our results for a complex reductive group easily reduces to that of a semisimple simply-connected group. To stay close to our main references [KK, LSSa], we work with the latter. Henceforth, we fix a complex semsimple simply-connected group G.The results of this section are due to Kostant and Kumar [KK]. Our notation agrees with that of [LSSa].2.1. Small-torus affine K-nilHecke ring. Let T ⊂ G be the maximal torus with character group, or weight lattice P . We have P = i∈I Zω i where ω i denotes a fundamental weight and I denotes the finite Dynkin diagram of G. LetP = Zδ ⊕ i∈Î ZΛ i be the affine weight lattice with fundamental weights Λ i for i in the affine Dynkin node set I = I ∪ {0}, and let δ denote the null root. The natural projection cl :P → P has kernel Zδ ⊕ ZΛ 0 and satisfies cl(Λ i ) = ω i for i ∈ I. Let af : P →P be the section of cl given by afThe finite Weyl group W acts naturally on P and on R(T ), where R(T ) ∼ = Z[P ] = λ∈P Ze λ is t...

Zero-one Schubert polynomials

“…Knutson and Miller also showed them to be multidegrees of matrix Schubert varieties [7]. There are a number of combinatorial formulas for the Schubert polynomials [1,2,5,6,9,12,14,17], yet only recently has the structure of their supports been investigated: the support of a Schubert polynomial S w is the set of all integer points of a certain generalized permutahedron P(w) [4,15]. The question motivating this paper is to characterize when S w equals the integer point transform of P(w), in other words, when all the coefficients of S w are equal to 0 or 1.…”

A note on specializations of Grothendieck polynomials

“…The proof is by induction on ℓ(v). We first consider the case ℓ(v) = 0, that is, v is the identity permutation e. In this case, the argument is the same as the proof of formula (2.19) in [9] for double Schubert polynomials. If u = e, then it is trivial that G e (y e ; y) = 1.…”

Tropical critical points of the superpotential of a flag variety

“…This form was first introduced in [R] where it was defined as a natural generalisation of the unique torus-invariant volume form on a torus inside a toric variety. It is the meromorphic differential form on G/B with simple poles exactly along the divisor given by the union of all the Schubert divisors and all the opposite Schubert divisors, see [Lam,Section 2]. Similar volume forms also appear more recently in work on mirror symmetry and cluster varieties, see [GHK, BMRS].…”