scholarly journals Double Schubert polynomials and degeneracy loci for the classical groups

2002 ◽  
Vol 52 (6) ◽  
pp. 1681-1727 ◽  
Author(s):  
Andrew Kresch ◽  
Harry Tamvakis
Keyword(s):  
Classical Groups ◽  
Degeneracy Loci ◽  
Schubert Polynomials ◽  
Double Schubert Polynomials

scholarly journals Double Schubert polynomials for the classical groups

2011 ◽  
Vol 226 (1) ◽  
pp. 840-886 ◽  
Author(s):  
Takeshi Ikeda ◽  
Leonardo C. Mihalcea ◽  
Hiroshi Naruse
Keyword(s):  
Classical Groups ◽  
Schubert Polynomials ◽  
Double Schubert Polynomials

QUANTUM SCHUBERT POLYNOMIALS AND QUANTUM SCHUR FUNCTIONS

1999 ◽  
Vol 09 (03n04) ◽  
pp. 385-404
Author(s):  
ANATOL N. KIRILLOV
Keyword(s):  
Schur Functions ◽  
Macdonald Polynomials ◽  
Schur Function ◽  
Quantum Double ◽  
Schubert Polynomial ◽  
Schubert Polynomials ◽  
Quantum Analog ◽  
Double Schubert Polynomials

We introduce the quantum multi–Schur functions, quantum factorial Schur functions and quantum Macdonald polynomials. We prove that for restricted vexillary permutations, the quantum double Schubert polynomial coincides with some quantum multi-Schur function and prove a quantum analog of the Nägelsbach–Kostka and Jacobi–Trudi formulae for the quantum double Schubert polynomials in the case of Grassmannian permutations. We prove also an analog of the Giambelli and the Billey–Jockusch–Stanley formula for quantum Schubert polynomials. Finally we formulate two conjectures about the structure of quantum double and quantum Schubert polynomials for 321–avoiding permutations.

scholarly journals Double Schubert polynomials for the classical Lie groups

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Takeshi Ikeda ◽  
Leonardo Mihalcea ◽  
Hiroshi Naruse
Keyword(s):  
Weyl Group ◽  
Lie Groups ◽  
Equivariant Cohomology ◽  
Natural Extension ◽  
Type B ◽  
Infinite Rank ◽  
Schubert Polynomials ◽  
International Audience ◽  
Schubert Classes ◽  
Double Schubert Polynomials

International audience For each infinite series of the classical Lie groups of type $B$, $C$ or $D$, we introduce a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the corresponding flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. When indexed by maximal Grassmannian elements of the Weyl group, these polynomials are equal to the factorial analogues of Schur $Q$- or $P$-functions defined earlier by Ivanov. Pour chaque série infinie des groupe de Lie classiques de type $B$,$C$ ou $D$, nous présentons une famille de polynômes indexées par de éléments de groupe de Weyl correspondant de rang infini. Ces polynômes représentent des classes de Schubert dans la cohomologie équivariante des variétés de drapeaux. Ils ont une certain propriété de stabilité, et ils étendent naturellement des polynômes Schubert (simples) de Billey et Haiman, que représentent des classes de Schubert dans la cohomologie non-équivariante. Quand ils sont indexées par des éléments Grassmanniennes de groupes de Weyl, ces polynômes sont égaux à des analogues factorielles de fonctions $Q$ et $P$ de Schur, étudiées auparavant par Ivanov.

scholarly journals Double Grothendieck Polynomials and Colored Lattice Models

2020 ◽  
Author(s):  
Valentin Buciumas ◽  
Travis Scrimshaw
Keyword(s):  
Partition Function ◽  
Lattice Models ◽  
Lattice Paths ◽  
Stable Limit ◽  
Vertex Model ◽  
Determinant Formula ◽  
Schubert Polynomials ◽  
Nonintersecting Lattice Paths ◽  
Double Schubert Polynomials ◽  
Grothendieck Polynomials

Abstract We construct an integrable colored six-vertex model whose partition function is a double Grothendieck polynomial. This gives an integrable systems interpretation of bumpless pipe dreams and recent results of Weigandt relating double Grothendieck polynomias with bumpless pipe dreams. For vexillary permutations, we then construct a new model that we call the semidual version model. We use our semidual model and the five-vertex model of Motegi and Sakai to give a new proof that double Grothendieck polynomials for vexillary permutations are equal to flagged factorial Grothendieck polynomials. Taking the stable limit of double Grothendieck polynomials, we obtain a new proof that the stable limit is a factorial Grothendieck polynomial as defined by McNamara. The states of our semidual model naturally correspond to families of nonintersecting lattice paths, where we can then use the Lindström–Gessel–Viennot lemma to give a determinant formula for double Schubert polynomials corresponding to vexillary permutations.

scholarly journals Schubert complexes and degeneracy loci

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Steven V Sam
Keyword(s):  
Euler Characteristic ◽  
Vector Bundles ◽  
Homology Class ◽  
Schur Polynomials ◽  
Degeneracy Loci ◽  
Schubert Polynomials ◽  
Conceptual Proof ◽  
Degeneracy Locus ◽  
Alternating Sums ◽  
International Audience

International audience The classical Thom―Porteous formula expresses the homology class of the degeneracy locus of a generic map between two vector bundles as an alternating sum of Schur polynomials. A proof of this formula was given by Pragacz by expressing this alternating sum as the Euler characteristic of a Schur complex, which gives an explanation for the signs. Fulton later generalized this formula to the situation of flags of vector bundles by using alternating sums of Schubert polynomials. Building on the Schubert functors of Kraśkiewicz and Pragacz, we introduce Schubert complexes and show that Fulton's alternating sum can be realized as the Euler characteristic of this complex, thereby providing a conceptual proof for why an alternating sum appears. \par La formule classique de Thom―Porteous exprime la classe d'homologie du locus de la dégénérescence d'une fonction générique entre deux fibrés vectoriels comme une somme alternée des polynômes de Schur. Un preuve de cette formule a été donnée par Pragacz en exprimant ce alternant somme comme la caractéristique d'Euler d'un complexe de Schur, ce qui donne une explication pour les signes. Fulton puis généralisée cette formule à la situation des drapeaux de fibrés vectoriels à l'aide alternant des sommes de polynômes de Schubert. S'appuyant sur le Schubert foncteurs de Kraśkiewicz et Pragacz, nous introduisons les complexes de Schubert et montrent que la somme alternée de Fulton peuvent être réalisées en tant que Euler caractéristique de ce complexe, fournissant ainsi une preuve conceptuelle pour lesquelles une somme alternée appara\^ıt.

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