Positivity and Kleiman transversality in equivariant $K$-theory of homogeneous spaces

Abstract

Abstract. We prove the conjectures of and concerning the alternation of signs in the structure constants for torus-equivariant K-theory of generalized flag varieties G/P . These results are immediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with finitely many orbits. The computation of the coefficients in the expansion of the equivariant K-class of a subvariety in terms of Schubert classes is reduced to an Euler characteristic using the homological transversality theorem for nontransitive group actions due to S. Sierra. A vanishing theorem, when the subvariety has rational singularities, shows that the Euler characteristic is a sum of at most one term-the top one-with a well-defined sign. The vanishing is proved by suitably modifying a geometric argument due to M. Brion in ordinary K-theory that brings Kawamata-Viehweg vanishing to bear.

Equivariant Chow Classes of Matrix Orbit Closures

“…In this section we address positivity of the class K(k[X v ]) in the sense of Anderson, Griffeth and Miller [AGM11]. We assume that our coefficient field k is the field of complex numbers in this section.…”

Littlewood–Richardson coefficients for Grothendieck polynomials from integrability

“…It is worth noting that our two main theorems provide rules that are positive in the sense of [1]: our Theorem 2 displays positivity as in [1,Cor. 5.3] (except we use Schubert classes, not opposite ones, so the secondary alphabet is reversed), and our Theorem 2 ′ displays positivity as in [1,Cor. 5.2].…”

Euler characteristics of cominuscule quantum K‐theory

“…In general the structure constants are conjectured to satisfy Griffeth–Ram positivity , that is, up to a sign these constants are polynomials with non‐negative coefficients in the classes , where is any one‐dimensional representation of defined by a negative root (see, for example, ). This conjecture has been proved for the structure constants of the equivariant ‐theory ring by Anderson, Griffeth, and Miller , and for the equivariant quantum cohomology ring by Mihalcea .…”

“…Conjectures for the ring structure of QK T (X) have also been given by Lenart and Maeno [16] and Lenart and Postnikov [17] when X = G/B is defined by a Borel subgroup of G. In general the structure constants N w,d u,v are conjectured to satisfy Griffeth-Ram positivity [13], that is, up to a sign these constants are polynomials with non-negative coefficients in the classes [C −α ] − 1 ∈ Γ, where C −α is any one-dimensional representation of T defined by a negative root (see, for example, [3]). This conjecture has been proved for the structure constants N w,0 u,v of the equivariant K-theory ring K T (X) by Anderson, Griffeth, and Miller [1], and for the equivariant quantum cohomology ring QH T (X) by Mihalcea [18].…”