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 permutations can be generated by set-valued Young tableaux. In this paper, we introduce the structure of set-valued Rothe tableaux of permutations. Given the Rothe diagram $D(w)$ of a permutation $w$, a set-valued Rothe tableau of shape $D(w)$ is a filling of finite nonempty subsets of positive integers into the squares of $D(w)$ such that the rows are weakly decreasing and the columns are strictly increasing. We show that the double Grothendieck polynomials of 1432-avoiding permutations can be generated by set-valued Rothe tableaux. When restricted to 321-avoiding permutations, our formula specializes to the tableau formula for double Grothendieck polynomials due to Matsumura. Employing the properties of tableau complexes given by Knutson, Miller and Yong, we obtain two alternative tableau formulas for the double Grothendieck polynomials of 1432-avoiding permutations.
Edelman and Greene constructed a bijection between the set of standard Young tableaux and the set of balanced Young tableaux of the same shape. Fomin, Greene, Reiner and Shimozono introduced the notion of balanced Rothe tableaux of a permutation $w$, and established a bijection between the set of balanced Rothe tableaux of $w$ and the set of reduced words of $w$. In this paper, we introduce the notion of standard Rothe tableaux of $w$, which are tableaux obtained by labelling the cells of the Rothe diagram of $w$ such that each row and each column is increasing. We show that the number of stan…
We present a new family of hook-length formulas for the number of standard increasing tableaux which arise in the study of factorial Grothendieck polynomials. In the case of straight shapes our formulas generalize the classical hook-length formula and Stanley's formula. For skew shapes, our formulas generalize the Naruse hook-length formula and its $q$-analogues, which were studied in previous papers of the series.
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 Schützenberger gave a combinatorial formula for key polynomials using right keys; Buch gave a set-valued tableau formula for Grassmannian stable Grothendieck polynomials. We establish a novel combinatorial rule for Lascoux polynomials involving right keys and set-valued tableaux. Our rule recovers the tableaux formulas of key polynomials and Grassmannian stable Grothendieck…
We prove K-theoretic and shifted K-theoretic analogues of the bijection of Stanton and White between domino tableaux and pairs of semistandard tableaux. As a result, we obtain product formulas for pairs of stable Grothendieck polynomials and pairs of K-theoretic Q-Schur functions.
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 polynomials and k-K-Schur functions -- Schubert representatives for the K-theory of affine Grassmannians and their dual in the nil Hecke ring. We prove a number of combinatorial properties including Pieri rules.
We show that the flagged Grothendieck polynomials defined as the generating functions of flagged set-valued tableaux of Knutson-Miller-Yong can be expressed by a Jacobi-Trudi type determinant formula generalizing the work of Hudson-Matsumura. We also introduce the flagged skew Grothendieck polynomials in these two expressions and show that they coincide.
We give a $U_q(\mathfrak{sl}_n)$-crystal structure on multiset-valued tableaux, hook-valued tableaux, and valued-set tableaux, whose generating functions are the weak symmetric, canonical, and dual weak symmetric Grothendieck functions, respectively. We show the result is isomorphic to a (generally infinite) direct sum of highest weight crystals, and for multiset-valued tableaux and valued-set tableaux, we provide an explicit bijection. As a consequence, these generating functions are Schur positive; in particular, the canonical Grothendieck functions, which was not previously known. We also g…
We show that several families of polynomials defined via fillings of diagrams satisfy linear recurrences under a natural operation on the shape of the diagram. We focus on key polynomials, (also known as Demazure characters), and Demazure atoms. The same technique can be applied to Hall-Littlewood polynomials and dual Grothendieck polynomials. The motivation behind this is that such recurrences are strongly connected with other nice properties, such as interpretations in terms of lattice points in polytopes and divided difference operators.
We give a combinatorial expansion of the stable Grothendieck polynomials of skew Young diagrams in terms of skew Schur functions, using a new row insertion algorithm for set-valued semistandard tableaux of skew shape. This expansion unifies some previous results: it generalizes a combinatorial formula obtained in earlier joint work with López Martín and Teixidor i Bigas concerning Brill-Noether curves, and it generalizes a 2000 formula of Lenart and a recent result of Reiner-Tenner-Yong to skew shapes. We also give an expansion in the other direction: expressing skew Schur functions in terms o…
Edelman and Greene constructed a correspondence between reduced words of the reverse permutation and standard Young tableaux. We prove that for any reduced word the shape of the region of the insertion tableau containing the smallest possible entries evolves exactly as the upper-left component of the permutation's (Rothe) diagram. Properties of the Edelman-Greene bijection restricted to 132-avoiding and 2143-avoiding permutations are presented. We also consider the Edelman-Greene bijection applied to non-reduced words.
The symmetric Grothendieck polynomials representing Schubert classes in the $K$-theory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type $A_n$ crystal structure on these tableaux. This crystal yields a new combinatorial formula for decomposing symmetric Grothendieck polynomials into Schur polynomials. For single-columns and single-rows, we give a new combinatorial interpretation of Lascoux polynomials (K-analogs of Demazure characters) by constructing a K-theoretic analog of crystals with an appropriate analog of a Demazure crystal. We relate o…
The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the K-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters, and we prove that this generalization still defines symmetric functions. For this fact, we give two self-contained proofs, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries 1…
Set-valued tableaux play an important role in combinatorial $K$-theory. Separately, semistandard skyline fillings are a combinatorial model for Demazure atoms and key polynomials. We unify these two concepts by defining a set-valued extension of semistandard skyline fillings and then give analogues of results of J. Haglund, K. Luoto, S. Mason, and S. van Willigenberg. Additionally, we give a bijection between set-valued semistandard Young tableaux and C. Lenart's Schur expansion of the Grothendieck polynomial $G_λ$, using the uncrowding operator of V. Reiner, B. Tenner, and A. Yong.
The problem of computing products of Schubert classes in the cohomology ring can be formulated as the problem of expanding skew Schur polynomials into the basis of ordinary Schur polynomials. In contrast, the problem of computing the structure constants of the Grothendieck ring of a Grassmannian variety with respect to its basis of Schubert structure sheaves is not equivalent to expanding skew stable Grothendieck polynomials into the basis of ordinary stable Grothendiecks. Instead, we show that the appropriate K-theoretic analogy is through the expansion of skew reverse plane partitions into t…
We show that the factorial flagged Grothendieck polynomials defined by flagged set-valued tableaux of Knutson-Miller-Yong can be expressed by a Jacobi-Trudi type determinant formula, generalizing the work of Hudson-Matsumura. In particular, we obtain an alternative proof of the tableau and determinant formulas of vexillary double Grothendieck polynomials, that have been obtained by Knutson-Miller-Yong and Hudson-Matsumura respectively. Moreover, we show that each factorial flagged Grothendieck polynomial can be obtained from a product of linear polynomials by applying K-theoretic divided diffe…
We investigate products J of ideals of "row initial" minors in the polynomial ring K[X] defined by a generic m-by-n matrix. Such ideals are shown to be generated by a certain set of standard bitableaux that we call superstandard. These bitableaux form a Gröbner basis of J, and J has a linear minimal free resolution. These results are used to derive a new generating set for the Grothendieck group of finitely generated (T_m x GL_n(K))-equivariant modules over K[X]. We employ the Knuth--Robinson--Schensted correspondence and a toric deformation of the multi-Rees algebra that parameterizes the ide…
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, similar to generating sequences, which we call the iterated residue technique. We prove new formulas on the calculus of iterated residues and use them to prove straightening laws and multiplication formulas for stable Grothendieck polynomials. As a further application of our method, we give new proofs that the $K$-Pieri rule and the expansions of Grothendieck po…
Let X,Y be finite sets and T a set of functions from X -> Y which we will call "tableaux". We define a simplicial complex whose facets, all of the same dimension, correspond to these tableaux. Such "tableau complexes" have many nice properties, and are frequently homeomorphic to balls, which we prove using vertex decompositions. In our motivating example, the facets are labeled by semistandard Young tableaux, and the more general interior faces are labeled by Buch's set-valued semistandard tableaux. One vertex decomposition of this "Young tableau complex" parallels Lascoux's transition formula…
