Sami Assaf
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View article: Corrigendum: A Demazure crystal construction for Schubert polynomials
Corrigendum: A Demazure crystal construction for Schubert polynomials Open
We give a corrigendum to our paper [4] entitled “A Demazure crystal construction for Schubert polynomials”.
View article: Extremal Tensor Products of Demazure Crystals
Extremal Tensor Products of Demazure Crystals Open
Demazure crystals are subcrystals of highest weight irreducible $$\mathfrak {g}$$ -crystals. In this article, we study tensor products of a larger class of subcrystals, called extremal, and give a local characterization for exactly when …
View article: Monk's Rule for Demazure Characters of the General Linear Group
Monk's Rule for Demazure Characters of the General Linear Group Open
Key polynomials are characters of Demazure modules for the general linear group that generalize the Schur polynomials. We prove a nonsymmetric generalization of Monk's rule by giving a cancellation-free, multiplicity-free formula for the k…
View article: A Littlewood-Richardson rule for Grassmannian Schubert varieties
A Littlewood-Richardson rule for Grassmannian Schubert varieties Open
We propose a combinatorial model for the Schubert structure constants of the complete flag manifold when one of the factors is Grassmannian.
View article: Extremal tensor products of Demazure crystals
Extremal tensor products of Demazure crystals Open
Demazure crystals are subcrystals of highest weight irreducible $\mathfrak{g}$-crystals. In this article, we study tensor products of a larger class of subcrystals, called extremal, and give a local characterization for exactly when the te…
View article: A bijective proof of Kohnert's rule for Schubert polynomials
A bijective proof of Kohnert's rule for Schubert polynomials Open
Kohnert proposed the first monomial positive formula for Schubert polynomials as the generating polynomial for certain unit cell diagrams obtained from the Rothe diagram of a permutation. Billey, Jockusch and Stanley gave the first proven …
View article: Affine Demazure crystals for specialized nonsymmetric Macdonald polynomials
Affine Demazure crystals for specialized nonsymmetric Macdonald polynomials Open
We give a crystal-theoretic proof that nonsymmetric Macdonald polynomials specialized to are affine Demazure characters. We explicitly construct an affine Demazure crystal on semistandard key tabloids such that removing the affine edges r…
View article: Demazure crystals for Kohnert polynomials
Demazure crystals for Kohnert polynomials Open
Kohnert polynomials are polynomials indexed by unit cell diagrams in the first quadrant defined earlier by the author and Searles that give a common generalization of Schubert polynomials and Demazure characters for the general linear grou…
View article: An insertion algorithm for multiplying Demazure characters by Schur polynomials
An insertion algorithm for multiplying Demazure characters by Schur polynomials Open
We introduce an insertion algorithm on Kohnert's combinatorial model for Demazure characters, generalizing Robinson--Schensted--Knuth insertion on tableaux. Our new insertion yields an explicit, nonnegative formula expressing the product o…
View article: An insertion algorithm for multiplying Demazure characters by Schur\n polynomials
An insertion algorithm for multiplying Demazure characters by Schur\n polynomials Open
We introduce an insertion algorithm on Kohnert's combinatorial model for\nDemazure characters, generalizing Robinson--Schensted--Knuth insertion on\ntableaux. Our new insertion yields an explicit, nonnegative formula expressing\nthe produc…
View article: A generalization of Edelman–Greene insertion for Schubert polynomials
A generalization of Edelman–Greene insertion for Schubert polynomials Open
Edelman and Greene generalized the Robinson–Schensted–Knuth correspondence to reduced words in order to give a bijective proof of the Schur positivity of Stanley symmetric functions. Stanley symmetric functions may be regarded as the stabl…
View article: Queer dual equivalence graphs
Queer dual equivalence graphs Open
We introduce a new paradigm for proving the Schur $P$-positivity. Generalizing dual equivalence, we give an axiomatic definition for a family of involutions on a set of objects to be a queer dual equivalence, and we prove whenever such a f…
View article: Kohnert's rule for flagged Schur modules
Kohnert's rule for flagged Schur modules Open
Flagged Schur modules generalize the irreducible representations of the general linear group under the action of the Borel subalgebra. Their characters include many important generalizations of Schur polynomials, such as Demazure character…
View article: A Pieri rule for Demazure characters of the general linear group
A Pieri rule for Demazure characters of the general linear group Open
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 …
View article: Skew key polynomials and a generalized Littlewood-Richardson rule
Skew key polynomials and a generalized Littlewood-Richardson rule Open
Young's lattice is a partial order on integer partitions whose saturated chains correspond to standard Young tableaux, one type of combinatorial object that generates the Schur basis for symmetric functions. Generalizing Young's lattice, w…
View article: Toward the Schur Expansion of Macdonald Polynomials
Toward the Schur Expansion of Macdonald Polynomials Open
We give an explicit combinatorial formula for the Schur expansion of Macdonald polynomials indexed by partitions with second part at most two. This gives a uniform formula for both hook and two column partitions. The proof comes as a corol…
View article: A local characterization of crystals for the quantum queer superalgebra
A local characterization of crystals for the quantum queer superalgebra Open
We define operators on semistandard shifted tableaux and use Stembridge's local characterization for regular graphs to prove they define a crystal structure. This gives a new proof that Schur $P$-polynomials are Schur positive. We define q…
View article: A Demazure crystal construction for Schubert polynomials
A Demazure crystal construction for Schubert polynomials Open
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…
View article: Crystal graphs for shifted tableaux
Crystal graphs for shifted tableaux Open
We define crystal operators on semistandard shifted tableaux, giving a new proof that Schur $P$-functions are Schur positive. We define a queer crystal operator to construct a connected queer crystal on semistandard shifted tableaux of a g…
View article: Kohnert polynomials
Kohnert polynomials Open
We associate a polynomial to any diagram of unit cells in the first quadrant of the plane using Kohnert's algorithm for moving cells down. In this way, for every weak composition one can choose a cell diagram with corresponding row-counts,…
View article: Dual equivalence graphs II: Transformations on locally Schur positive\n graphs
Dual equivalence graphs II: Transformations on locally Schur positive\n graphs Open
Dual equivalence graphs are a powerful tool in symmetric function theory that\nprovide a general framework for proving that a given quasisymmetric function is\nsymmetric and Schur positive. In this paper, we study a larger family of graphs…
View article: Dual equivalence graphs II: Transformations on locally Schur positive graphs
Dual equivalence graphs II: Transformations on locally Schur positive graphs Open
Dual equivalence graphs are a powerful tool in symmetric function theory that provide a general framework for proving that a given quasisymmetric function is symmetric and Schur positive. In this paper, we study a larger family of graphs t…
View article: Combinatorial models for Schubert polynomials
Combinatorial models for Schubert polynomials Open
Schubert polynomials are a basis for the polynomial ring that represent Schubert classes for the flag manifold. In this paper, we introduce and develop several new combinatorial models for Schubert polynomials that relate them to other kno…
View article: Weak dual equivalence for polynomials
Weak dual equivalence for polynomials Open
We use dual equivalence to give a short, combinatorial proof that Stanley symmetric functions are Schur positive. We introduce weak dual equivalence, and use it to give a short, combinatorial proof that Schubert polynomials are key positiv…
View article: Multiplication of a Schubert polynomial by a Stanley symmetric polynomial
Multiplication of a Schubert polynomial by a Stanley symmetric polynomial Open
We prove, combinatorially, that the product of a Schubert polynomial by a Stanley symmetric polynomial is a truncated Schubert polynomial. Using Monk's rule, we derive a nonnegative combinatorial formula for the Schubert polynomial expansi…
View article: The quantile transform of simple walks and Brownian motion
The quantile transform of simple walks and Brownian motion Open
We examine a new path transform on 1-dimensional simple random walks and\nBrownian motion, the quantile transform. This transformation relates to\nidentities in fluctuation theory due to Wendel, Port, Dassios and others, and\nto discrete a…