Multiplication of a Schubert polynomial by a Schur polynomial

1997 ◽  
Vol 1 (1) ◽  
pp. 367-375 ◽  
Author(s):  
Axel Kohnert
Keyword(s):  
Schubert Polynomial ◽  
Schur Polynomial

scholarly journals An efficient algorithm for deciding vanishing of Schubert polynomial coefficients

2021 ◽  
Vol 383 ◽  
pp. 107669
Author(s):  
Anshul Adve ◽  
Colleen Robichaux ◽  
Alexander Yong
Keyword(s):  
Efficient Algorithm ◽  
Schubert Polynomial ◽  
Polynomial Coefficients

scholarly journals Schubert Polynomials and $k$-Schur Functions

10.37236/4139 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Carolina Benedetti ◽  
Nantel Bergeron
Keyword(s):  
Finite Type ◽  
Schur Functions ◽  
Type A ◽  
Bruhat Order ◽  
Schur Function ◽  
Quasisymmetric Functions ◽  
Affine Grassmannian ◽  
Schubert Polynomial ◽  
Schubert Polynomials ◽  
General Program

The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type $A$ by a Schur function, which we refer to as Schubert vs. Schur problem, can be understood combinatorially from the multiplication in the space of dual $k$-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the poset given by the Bergeron-Sottile's $r$-Bruhat order, along with certain operators associated to this order. Then, we connect this poset with a graph on dual $k$-Schur functions given by studying the affine grassmannian order of  Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual $k$-Schur functions which are analogous to the ones given for the Schubert vs. Schur problem. This is the first step of our more general program of showing combinatorially  the positivity of the multiplication of a dual $k$-Schur function by a Schur function.

On the stability radius of a Schur polynomial

1993 ◽  
Vol 21 (3) ◽  
pp. 199-205 ◽  
Author(s):  
Q.-H. Wu ◽  
M. Mansour
Keyword(s):  
Stability Radius ◽  
Schur Polynomial ◽  
The Stability

scholarly journals On the stability hyperellipsoid of a Schur polynomial

1994 ◽  
Vol 205-206 ◽  
pp. 1271-1288
Author(s):  
Q.-H. Wu ◽  
M. Mansour
Keyword(s):  
Schur Polynomial ◽  
The Stability

scholarly journals Schur Times Schubert via the Fomin-Kirillov Algebra

10.37236/3659 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Karola Mészáros ◽  
Greta Panova ◽  
Alexander Postnikov
Keyword(s):  
Cohomology Ring ◽  
Quantum Cohomology ◽  
Flag Manifold ◽  
Combinatorial Proof ◽  
Schubert Polynomial ◽  
Schubert Polynomials ◽  
Small Quantum ◽  
Special Cases ◽  
Expansion Coefficients ◽  
Complex Flag

We study multiplication of any Schubert polynomial $\mathfrak{S}_w$ by a Schur polynomial $s_{\lambda}$ (the Schubert polynomial of a Grassmannian permutation) and the expansion of this product in the ring of Schubert polynomials. We derive explicit nonnegative combinatorial expressions for the expansion coefficients for certain special partitions $\lambda$, including hooks and the $2\times 2$ box. We also prove combinatorially the existence of such nonnegative expansion when the Young diagram of $\lambda$ is a hook plus a box at the $(2,2)$ corner. We achieve this by evaluating Schubert polynomials at the Dunkl elements of the Fomin-Kirillov algebra and proving special cases of the nonnegativity conjecture of Fomin and Kirillov.This approach works in the more general setup of the (small) quantum cohomology ring of the complex flag manifold and the corresponding (3-point) Gromov-Witten invariants. We provide an algebro-combinatorial proof of the nonnegativity of the Gromov-Witten invariants in these cases, and present combinatorial expressions for these coefficients.

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 Factorial Characters and Tokuyama's Identity for Classical Groups

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Angèle M. Hamel ◽  
Ronald C. King
Keyword(s):  
Recurrence Relations ◽  
Function Theory ◽  
Lattice Paths ◽  
Schur Function ◽  
Classical Groups ◽  
Orthogonal Groups ◽  
Schubert Polynomial ◽  
International Audience ◽  
Symplectic Case ◽  
Polynomial Theory

International audience In this paper we introduce factorial characters for the classical groups and derive a number of central results. Classically, the factorial Schur function plays a fundamental role in traditional symmetric function theory and also in Schubert polynomial theory. Here we develop a parallel theory for the classical groups, offering combinatorial definitions of the factorial characters for the symplectic and orthogonal groups, and further establish flagged factorial Jacobi-Trudi identities and factorial Tokuyama identities, providing proofs in the symplectic case. These identities are established by manipulating determinants through the use of certain recurrence relations and by using lattice paths.

scholarly journals A Murgnahan-Nakayama rule for Schubert polynomials

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Andrew Morrison
Keyword(s):  
Quantum Cohomology ◽  
Chern Character ◽  
Intersection Theory ◽  
Flag Manifolds ◽  
Nous Obtenons ◽  
Schubert Polynomial ◽  
Power Sum ◽  
Schubert Polynomials ◽  
Small Quantum ◽  
International Audience

International audience We expose a rule for multiplying a general Schubert polynomial with a power sum polynomial in $k$ variables. A signed sum over cyclic permutations replaces the signed sum over rim hooks in the classical Murgnahan-Nakayama rule. In the intersection theory of flag manifolds this computes all intersections of Schubert cycles with tautological classes coming from the Chern character. We also discuss extensions of this rule to small quantum cohomology. Nous écrivons une formule pour multiplier les polynômes de Schubert avec les sommes de Newton. Une somme signée de permutations cycliques remplace la somme signée de rubans dans la formule classique de Murgnahan-Nakayama. Nous obtenons donc des relations dans l’anneau de Chow de la variété de drapeaux. Nous discutons également des extensions de cette formule en cohomologie quantique.

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