Related papers: Quantum Grothendieck Polynomials
The purpose of this paper is to prove a Pieri-type multiplication formula for quantum Grothendieck polynomials, which was conjectured by Lenart-Maeno. This formula would enable us to compute explicitly the quantum product of two arbitrary…
We study the algebraic aspects of (small) quantum equivariant $K$-theory of flag manifold. Lascoux-Sch\"utzenberger's type formula for quantum double and quantum double dual Grothendieck polynomials and the quantum Cauchy identity for…
In our previous paper, we gave a presentation of the torus-equivariant quantum $K$-theory ring $QK_{H}(Fl_{n+1})$ of the (full) flag manifold $Fl_{n+1}$ of type $A_{n}$ as a quotient of a polynomial ring by an explicit ideal. In this paper,…
We prove a `Whitney' presentation, and a `Coulomb branch' presentation, for the torus equivariant quantum K theory of the Grassmann manifold $\mathrm{Gr}(k;n)$, inspired from physics, and stated in an earlier paper. The first presentation…
Symmetric Grothendieck polynomials are analogues of Schur polynomials in the K-theory of Grassmannians. We build dual families of symmetric Grothendieck polynomials using Schur operators. With this approach we prove skew Cauchy identity and…
We study the algebraic aspects of equivariant quantum cohomology algebra of the flag manifold. We introduce and study the quantum double Schubert polynomials, which are the Lascoux-Schutzenberger type representatives of the equivariant…
Grothendieck polynomials, introduced by Lascoux and Sch\"utzenberger, are certain $K$-theory representatives for Schubert varieties. Symplectic Grothendieck polynomials, described more recently by Wyser and Yong, represent the $K$-theory…
We investigate the longstanding problem of finding a combinatorial rule for the Schubert structure constants in the $K$-theory of flag varieties (in type $A$). The Grothendieck polynomials of A. Lascoux-M.-P. Sch\"{u}tzenberger (1982) serve…
We show that (equivariant) K-theoretic 3-point Gromov-Witten invariants of genus zero on a Grassmann variety are equal to triple intersections computed in the ordinary (equivariant) K-theory of a two-step flag manifold, thus generalizing an…
Fulton's universal Schubert polynomials give cohomology formulas for a class of degeneracy loci, which generalize Schubert varieties. The K-theoretic quiver formula of Buch expresses the structure sheaves of these loci as integral linear…
We study the back stable $K$-theory Schubert calculus of the infinite flag variety. We define back stable (double) Grothendieck polynomials and double $K$-Stanley functions and establish coproduct expansion formulae. Applying work of…
Grothendieck polynomials are important objects in the study of the $K$-theory of flag varieties. Their many remarkable properties have been studied in the context of algebraic geometry and tableaux combinatorics. We explore a new tool,…
Kirillov and Naruse have constructed double Grothendieck polynomials to represent the equivariant K-theory classes of Schubert varieties in the complete flag manifolds of types B, C, and D. We derive a recursive formula for these…
Let $G$ be a split semisimple linear algebraic group over a field and let $X$ be a generic twisted flag variety of $G$. Extending the Hilbert basis techniques to Laurent polynomials over integers we give an explicit presentation of the…
We formulate a nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of [Fomin-Kirillov '94] in the basis of stable Grothendieck polynomials for partitions. This gives a common generalization, as well as…
A GL$(n)$ quantum integrable system generalizing the asymmetric five vertex spin chain is shown to encode the ring relations of the equivariant quantum cohomology and equivariant quantum K-theory ring of flag varieties. We also show that…
We provide a direct proof of Seidel representation in the quantum K-theory QK(Gr(k, n)) by studying projected Gromov-Witten varieties concretely. As applications, we give an alternative proof of the K-theoretic quantum Pieri rule by Buch…
We prove a formula for the structure sheaf of a quiver variety in the Grothendieck ring of its embedding variety. This formula generalizes and gives new expressions for Grothendieck polynomials. We furthermore conjecture that the…
We use Bott-Samelson resolutions of Schubert varieties in Grassmannians along with equiariant localization techniques to show that the factorial Schur functions and the factorial Grothendieck polynomials represent Schubert classes in…
In this paper, we provide an explicit description of the Schubert classes in the equivariant $K$-theory of weighted Grassmann orbifolds. We introduce the `twisted factorial Grothendieck polynomials', a family of symmetric polynomials by…