The right key of a semistandard Young tableau is a tool used to find Demazure characters for $sl_n(\mathbb{C})$. This thesis gives methods to obtain the right and left keys by inspection of the semistandard Young tableau. Given a partition $λ$ and a Weyl group element $w$, there is a semistandard Young tableau $Y_λ(w)$ of shape $λ$ that corresponds to $w$. The Demazure character for $λ$ and $w$ is known to be the sum of the weights of all tableaux whose right key is dominated by $Y_λ(w)$. The set of all such tableaux is denoted $\mathcal{D}_λ(w)$. Exploiting the method mentioned above for obtaining right keys, this thesis describes the entry at each location in any $T \in \mathcal{D}_λ(w)$. Lastly, we will consider $\mathcal{D}_λ(w)$ as an integral subset of Euclidean space. The final results present a condition that is both necessary and sufficient for this subset to be convex.
The Schur function indexed by a partition lambda with at most n parts is the sum of the weight monomials for the Young tableaux of shape lambda. Let pi be an n-permutation. We give two descriptions of the tableaux that contribute their monomials to the key polynomial indexed by pi and lambda. (These polynomials are the characters of the Demazure modules for GL(n).) The "atom" indexed by pi is the sum of weight monomials of the tableaux whose right keys are the "key" tableau for pi. Schur functions and key polynomials can be decomposed into sums of atoms. We also describe the tableaux that cont…
We compute, mimicking the Lascoux-Schützenberger type A combinatorial procedure, left and right keys for a Kashiwara-Nakashima tableau in type C. These symplectic keys have a similar role as the keys for semistandard Young tableaux. More precisely, our symplectic keys give a tableau criterion for the Bruhat order on the hyperoctahedral group and cosets, and describe Demazure atoms and characters in type C. The right and the left symplectic keys are related through the Lusztig involution. A type C Schützenberger evacuation is defined to realize that involution.
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 compute, mimicking the Lascoux-Schützenberger type A combinatorial procedure, left and right keys for a Kashiwara-Nakashima tableau in type C. These symplectic keys have a similar role as the keys for semistandard Young tableaux. More precisely, our symplectic keys give a tableau criterion for the Bruhat order on the hyperoctahedral group and cosets, and describe Demazure atoms and characters in type C. The right and the left symplectic keys are related through the Lusztig involution. A type C Schützenberger evacuation is defined to realize that involution.
This paper studies the properties of Demazure atoms and characters using linear operators and also tableaux-combinatorics. It proves the atom-positivity property of the product of a dominating monomial and an atom, which was an open problem. Furthermore, it provides a combinatorial proof to the key-positivity property of the product of a dominating monomial and a key using skyline fillings, an algebraic proof to the key-positivity property of the product of a Schur function and a key using linear operator and verifies the first open case for the conjecture of key-positivity of the product of t…
Answering a question of Kuniba, Misra, Okado, Takagi, and Uchiyama, it is shown that certain Demazure characters of affine type A, coincide with the graded characters of coordinate rings of closures of conjugacy classes of nilpotent matrices. This entails a translation of the affine type A crystal theory into the language of tableaux following Nakayashiki and Yamada, for the case of tensor products of the classical crystals indexed by rectangular partitions. In particular the explicit action of the zero-th crystal raising operator on the above crystals is given, and its direct connection with …
Lock polynomials and lock Kohnert tableaux are natural analogues to key polynomials and key Kohnert tableaux, respectively. In this paper, we compare lock polynomials to the much-studied key polynomials and show that the difference of a key polynomial and lock polynomial for the same composition is monomial positive. We also examine the conditions for which key and lock polynomials are symmetric or quasisymmetric. We accomplish these goals combinatorially using key Kohnert tableaux and lock Kohnert tableaux. In particular, for the difference of a key minus a lock, we focus on the behavior of c…
The Key map is an important tool in the determination of the Demazure crystals associated to Kac-Moody algebras. In finite type A, it can be computed in the tableau realization of crystals by a simple combinatorial procedure due to Lascoux and Sch{ü}tzenberger. We show that this procedure is a part of a more general construction holding in the Kac-Moody case that we illustrate in finite types and affine type A. In affine type A, we introduce higher level generalizations of core partitions which notably give interesting analogues of the Young lattice and are expected to parametrize distinguishe…
The Pieri rule is a nonnegative, multiplicity-free formula for the Schur function expansion of the product of an arbitrary Schur function with a single row Schur function. Key polynomials are characters of Demazure modules for the general linear group that generalize the Schur function basis of symmetric functions to a basis of the full polynomial ring. We prove a nonsymmetric generalization of the Pieri rule by giving a cancellation-free, multiplicity-free formula for the key polynomial expansion of the product of an arbitrary key polynomial with a single part key polynomial. Our proof is com…
Shuffles are n-multipermutations with suit multiplicities given by a subset R of {1,..,n-1}. Their inverses are ordered partitions of {1,..,n} whose block sizes derive from R. These "R-permutations" depict the min length coset reps for the quotient of S_n by the parabolic subgroup W_J, with J the complement of R. We refer to those that blockwise avoid the pattern 312 as "312-avoiding R-permutations" and define the "parabolic R-Catalan number" to be the number of them. Let lambda be a partition of N with at most n parts whose set of shape column lengths less than n is R. We show that the number…
We construct colored lattice models whose partition functions represent symplectic and odd orthogonal Demazure characters and atoms. We show that our lattice models are not solvable, but we are able to show the existence of sufficiently many solutions of the Yang--Baxter equation that allows us to compute functional equations for the corresponding partition functions. From these functional equations, we determine that the partition function of our models are the Demazure atoms and characters for the symplectic and odd orthogonal Lie groups. We coin our lattice models as quasi-solvable. We use …
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…
Stanley symmetric functions are the stable limits of Schubert polynomials. In this paper, we show that, conversely, Schubert polynomials are Demazure truncations of Stanley symmetric functions. This parallels the relationship between Schur functions and Demazure characters for the general linear group. We establish this connection by imposing a Demazure crystal structure on key tableaux, recently introduced by the first author in connection with Demazure characters and Schubert polynomials, and linking this to the type A crystal structure on reduced word factorizations, recently introduced by …
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.
We establish a poset structure on combinatorial bases of multivariate polynomials defined by positive expansions, and study properties common to bases in this poset. Included are the well-studied bases of Schubert polynomials, Demazure characters and Demazure atoms; the quasi-key, fundamental and monomial slide bases introduced in 2017 by Assaf and the author; and a new basis we introduce completing this poset structure. We show the product of a Schur polynomial and an element of a basis in this poset expands positively in that basis; in particular, we give the first Littlewood-Richardson rule…
Lascoux and Schützenberger introduced a notion of key associated to any Young tableau. More recently Lascoux defined the key of an alternating sign matrix by recursively removing all -1's in such matrices. But alternating sign matrices are in bijection with monotone triangles, which form a subclass of Young tableaux. We show that in this case these two notions of keys coincide. Moreover we obtain an elegant and direct way to compute the key of any Young tableau, and discuss consequences of our result.
The right and left key of a semistandard Young tableau were introduced by Lascoux and Schutzenberger in 1990. Most prominently, the right key is a tool used to find Demazure characters for sl(n,C). Previous methods used to compute these keys require introducing other types of combinatorial objects. This paper gives methods to obtain the right and left keys by inspection of the semistandard Young tableau.
Iterating the skew RSK correspondence discovered by Sagan and Stanley in the late '80s, we define a deterministic dynamics on the space of pairs of skew Young tableaux $(P,Q)$. We find that this skew RSK dynamics displays conservation laws which, in the picture of Viennot's shadow line construction, identify generalizations of Greene invariants. The introduction of a novel realization of $0$-th Kashiwara operators reveals that the skew RSK dynamics possesses symmetries induced by an affine bicrystal structure, which, combined with connectedness properties of Demazure crystals, leads to its lin…
