Related papers: Factorial Flagged Grothendieck Polynomials
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 2007 Lam and Pylyavskyy found a combinatorial formula for the dual stable Grothendieck polynomials, which are the dual basis of the stable Grothendieck polynomials with respect to the Hall inner product. In 2016 Galashin, Grinberg, and…
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…
The generating functions of the major index and of the flag-major index, with each of the one-dimensional characters over the symmetric and hyperoctahedral group, respectively, have simple product formulas. In this paper, we give a…
In this article, we prove a tableau formula for the double Grothendieck polynomials associated to $321$-avoiding permutations. The proof is based on the compatibility of the formula with the $K$-theoretic divided difference operators.
We prove new determinantal identities for a family of flagged Schur polynomials. As a corollary of these identities we obtain determinantal expressions of Schubert polynomials for certain vexillary permutations.
We introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize Grothendieck polynomials and k-Schur functions. We prove that the polynomials drawn from these tableaux are the affine Grothendieck…
In this paper, we introduce a family of symmetric polynomials by specializing the factorial Schur polynomials. These polynomials represent the weighted Schubert classes of the cohomology of the weighted Grassmannian introduced by…
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 give degree formulas for Grothendieck polynomials indexed by vexillary permutations and $1432$-avoiding permutations via tableau combinatorics. These formulas generalize a formula for degrees of symmetric Grothendieck polynomials which…
Let $K$ be a field. Let $f\in K[[x_{1},...,x_{r}]]$ and $g\in K[[y_{1},...,y_{s}]]$ be nonzero elements. If $X$ (resp. $Y$) is a matrix factorization of $f$ (resp. $g$), Yoshino had constructed a tensor product (of matrix factorizations)…
We consider pairs of a set-valued column-strict tableau and a reverse plane partition of the same shape. We introduce algortithms for them, which implies a bijective proof for the finite sum Cauchy identity for Grothendieck polynomials and…
Lascoux polynomials generalize Grassmannian stable Grothendieck polynomials and may be viewed as K-theoretic analogs of key polynomials. The latter two polynomials have combinatorial formulas involving tableaux: Lascoux and…
This paper proves an identity between flagged Schur polynomials, giving a duality between row flags and column flags. This identity generalises both the binomial determinant duality theorem due to Gessel and Viennot and the symmetric…
We give some special values of Grothendieck polynomials and an explicit formula for the number of set-valued tableaux. For Young diagrams consisting of a single row or a single column, both the value and number are written by the Gauss'…
The notion of set-valued Young tableaux was introduced by Buch in his study of the Littlewood-Richardson rule for stable Grothendieck polynomials. Knutson, Miller and Yong showed that the double Grothendieck polynomials of 2143-avoiding…
We find new universal factorization identities for generalized Macdonald polynomials on the topological locus. We prove the identities (which include all previously known forumlas of this kind) using factorization identities for matrix…
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…
This paper is the sequel of the paper under the same title with part 1, where we introduced refined canonical stable Grothendieck polynomials and their duals with two families of infinite parameters. In this paper we give combinatorial…
We give new formulas for Grothendieck polynomials of two types. One type expresses any specialization of a Grothendieck polynomial in at least two sets of variables as a linear combination of products Grothendieck polynomials in each set of…