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The skew Schubert polynomials

William Y.C. Chen, Guo-Guang Yan, Arthur L.B. Yang
2004 European journal of combinatorics (Print)  
We obtain a tableau definition of the skew Schubert polynomials named by Lascoux, which are defined as flagged double skew Schur functions.  ...  The lattice path explanation immediately leads to the determinantal definition and the tableau definition of the skew Schubert polynomials.  ...  Acknowledgements This work was done under the auspices of the 973 Project on Mathematical Mechanization of the Ministry of Science and Technology, and the National Science Foundation of China.  ... 
doi:10.1016/j.ejc.2003.11.004 fatcat:dfpcdzgzmncx5g4e36rewwot7m

Skew Schubert polynomials

Cristian Lenart, Frank Sottile
2002 arXiv   pre-print
These skew Schubert polynomials expand in the basis of Schubert polynomials with nonnegative integer coefficients that are precisely the structure constants of the cohomology of the complex flag variety  ...  We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold.  ...  We introduce skew Schubert polynomials, which are the Schubert polynomial analogs of skew Schur polynomials, in the sense that they generalize the two properties mentioned above.  ... 
arXiv:math/0202090v2 fatcat:wxb5yeh6nraipcj3vpekfuq2l4

Schubert polynomials and skew Schur functions

Axel Kohnert
1992 Journal of symbolic computation  
A brief example will illustrate the definiton of skew Schur polynomials : S1(A4) S3(A4) S5(A4) S(o,1,2,3)/(0,0,1,2)(A4) = 0 S, (A4) S3(A4) 0 0 S1(A4) One fundamental property of the Schubert polynomials  ...  Above example was the Schubert polynomial Y .i,o,j-1,0,k-2, . . . .  ... 
doi:10.1016/0747-7171(92)90036-4 fatcat:2kwwqbtilffihpbw6dzewqgwya

Skew Divided Difference Operators and Schubert Polynomials

Anatol N. Kirillov
2007 Symmetry, Integrability and Geometry: Methods and Applications  
We study an action of the skew divided difference operators on the Schubert polynomials and give an explicit formula for structural constants for the Schubert polynomials in terms of certain weighted paths  ...  We also prove that, under certain assumptions, the skew divided difference operators transform the Schubert polynomials into polynomials with positive integer coefficients.  ...  Acknowledgements This note is based on the lectures "Schubert polynomials" delivered in the Spring 1995 at the University of Minneapolis and in the Spring 1996 at the University of Tokyo.  ... 
doi:10.3842/sigma.2007.072 fatcat:kq7wtnji45hlti6k35lu5d2zp4

Skew divided difference operators and Schubert polynomials [article]

Anatol N. Kirillov
1997 arXiv   pre-print
We study an action of the skew divided difference operators on the Schubert polynomials and give an explicit formula for structural constants for the Schubert polynomials in terms of certain weighted paths  ...  We also prove that, under certain assumptions, the skew divided difference operators transform Schubert polynomials into polynomials with positive integer coefficients.  ...  This note is based on the lectures "Schubert polynomials" delivered in the Spring 1995 at the University of Minneapolis and in the Spring 1996 at the University of Tokyo.  ... 
arXiv:q-alg/9712053v1 fatcat:lln75q7mlrdwdkjzzrvbkztvvq

Sparse multivariate polynomial interpolation on the basis of Schubert polynomials

Priyanka Mukhopadhyay, Youming Qiao
2016 Computational Complexity  
In 2003, Lenart and Sottile introduced skew Schubert polynomials, which generalize skew Schur polynomials, and expand in the Schubert basis with the generalized Littlewood-Richardson coefficients.  ...  We first observe that skew Schubert polynomials, and therefore Schubert polynomials, are in (when evaluating on non-negative integral inputs) and .  ...  Part of the work was done when Youming was visiting the Simons Institute for the program Algorithms and Complexity in Algebraic Geometry.  ... 
doi:10.1007/s00037-016-0142-y fatcat:vtsm7jboqnbudn47jmq3av75cu

Page 5181 of Mathematical Reviews Vol. , Issue 2004g [page]

2004 Mathematical Reviews  
This paper presents a new definition of skew Schubert polynomials.  ...  The authors also provide a monomial expression for the skew Schubert polynomials in terms of re-graphs.  ... 

Tableau formulas for skew Schubert polynomials [article]

Harry Tamvakis
2020 arXiv   pre-print
The skew Schubert polynomials are those which are indexed by skew elements of the Weyl group, in the sense of arXiv:0812.0639.  ...  These are the first such theorems for symplectic and orthogonal Schubert polynomials, even in the single case.  ...  The aim here is to prove tableau formulas for the skew Schubert polynomials, in a type uniform manner.  ... 
arXiv:2008.07034v1 fatcat:2hdijotiwvddrewlk74tdjmik4

Pieri's formula for flag manifolds and Schubert polynomials

Frank Sottile
1996 Annales de l'Institut Fourier  
For w € 5yi, define the Schubert polynomial ©w by © -/) i ^/y 71 " 1^"2 . .  ...  The set {©^ [w 6 Sn} of Schubert polynomials is a basis for Z{a*^ "-x^Z[ \ij < n-j'}, a transversal to S m Rn' Thus Schubert polynomials are explicit polynomial representatives of an integral basis for  ... 
doi:10.5802/aif.1508 fatcat:ukw2o7tcdzbsvdm7vn6aivk2sm

Kohnert polynomials [article]

Sami Assaf, Dominic Searles
2018 arXiv   pre-print
The elements of one basis are conjecturally Schubert-positive and stabilize to the skew-Schur functions; the elements of the other basis stabilize to a new basis of quasisymmetric functions that contains  ...  These Kohnert bases provide a simultaneous generalization of Schubert polynomials and Demazure characters for the general linear group.  ...  The authors thank Per Alexandersson, Nantel Bergeron, and Vic Reiner for helpful comments and illuminating discussions.  ... 
arXiv:1711.09498v2 fatcat:3d2t43godjdxtfwmebgmuvolry

Flagged Schur functions, Schubert polynomials, and symmetrizing operators

Michelle L Wachs
1985 Journal of combinatorial theory. Series A  
They appear in the work of Lascoux and Schutzenberger [2] in their study of Schubert polynomials.  ...  For each row of these tableaux there is an upper bound (flag) on the entries. The Schubert polynomials are obtained by applying certain symmetrizing operators to a monomial.  ...  For w E Z5, define the Schubert polynomial F,,=a,,,o~(x;-'x;~2...x,, ,).  ... 
doi:10.1016/0097-3165(85)90091-3 fatcat:4nbqycovyrd3hgjn3spgw7m25a

Pieri's rule for flag manifolds and Schubert polynomials [article]

Frank Sottile
1995 arXiv   pre-print
Thus, we generalize the classical Pieri's rule for symmetric polynomials/Grassmann varieties to Schubert polynomials/flag manifolds.  ...  This formula also describes the multiplication of a Schubert polynomial by either an elementary symmetric polynomial or a complete homogeneous symmetric polynomial.  ...  We use the term Schubert polynomial for both the polynomial and the associated cohomology class.  ... 
arXiv:alg-geom/9505001v1 fatcat:hn5r2lm4jnhtpghnvpcpwxcaoe

A dual approach to structure constants for K-theory of Grassmannians

Huilan Li, Jennifer Morse, Pat Shields
2020 Discrete Mathematics & Theoretical Computer Science  
International audience The problem of computing products of Schubert classes in the cohomology ring can be formulated as theproblem of expanding skew Schur polynomial into the basis of ordinary Schur polynomials  ...  problem of expanding the generating functions forskew reverse-plane partitions into the basis of polynomials which are Hall-dual to stable Grothendieck polynomials.  ...  Acknowledgements The authors thank Ryan Kaliszewski for many discussions about related work.  ... 
doi:10.46298/dmtcs.6361 fatcat:3jd6eztpizhhdlilhdxahlsj3i

Page 5192 of Mathematical Reviews Vol. , Issue 96i [page]

1996 Mathematical Reviews  
Then they were used in a series of papers by Lascoux and Schiitzenberger as a tool in the investigation of Schubert polynomials.  ...  In Section 4, they pro- vide applications to counting: the number of reduced words, the Schubert polynomials, and the decompositions of certain Specht modules.  ... 

Quantum Schubert polynomials and quantum Schur functions [article]

Anatol N. Kirillov
1997 arXiv   pre-print
formulae for the quantum double Schubert polynomials in the case of Grassmannian permutations.  ...  We prove, also, an analog of the Billey-Jockusch-Stanley formula for quantum Schubert polynomials.  ...  The author would like to thank N. Bergeron and N.A. Liskova for fruitful discussions. This work was initiated during my stay at the University of Tokyo  ... 
arXiv:q-alg/9701005v3 fatcat:pg6ruets4baqxmurvfeplc23xm
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