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The skew Schubert polynomials

2004
*
European journal of combinatorics (Print)
*

We obtain a tableau definition of

doi:10.1016/j.ejc.2003.11.004
fatcat:dfpcdzgzmncx5g4e36rewwot7m
*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. ...##
###
Skew Schubert polynomials

2002
*
arXiv
*
pre-print

These

arXiv:math/0202090v2
fatcat:wxb5yeh6nraipcj3vpekfuq2l4
*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. ...##
###
Schubert polynomials and skew Schur functions

1992
*
Journal of symbolic computation
*

A brief example will illustrate

doi:10.1016/0747-7171(92)90036-4
fatcat:2kwwqbtilffihpbw6dzewqgwya
*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, . . . . ...##
###
Skew Divided Difference Operators and Schubert Polynomials

2007
*
Symmetry, Integrability and Geometry: Methods and Applications
*

We study an action of

doi:10.3842/sigma.2007.072
fatcat:kq7wtnji45hlti6k35lu5d2zp4
*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. ...##
###
Skew divided difference operators and Schubert polynomials
[article]

1997
*
arXiv
*
pre-print

We study an action of

arXiv:q-alg/9712053v1
fatcat:lln75q7mlrdwdkjzzrvbkztvvq
*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. ...##
###
Sparse multivariate polynomial interpolation on the basis of Schubert polynomials

2016
*
Computational Complexity
*

In 2003, Lenart and Sottile introduced

doi:10.1007/s00037-016-0142-y
fatcat:vtsm7jboqnbudn47jmq3av75cu
*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. ...##
###
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]

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. ...

##
###
Pieri's formula for flag manifolds and Schubert polynomials

1996
*
Annales de l'Institut Fourier
*

For w € 5yi, define

doi:10.5802/aif.1508
fatcat:ukw2o7tcdzbsvdm7vn6aivk2sm
*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 ...##
###
Kohnert polynomials
[article]

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. ...

##
###
Flagged Schur functions, Schubert polynomials, and symmetrizing operators

1985
*
Journal of combinatorial theory. Series A
*

They appear in

doi:10.1016/0097-3165(85)90091-3
fatcat:4nbqycovyrd3hgjn3spgw7m25a
*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,, ,). ...##
###
Pieri's rule for flag manifolds and Schubert polynomials
[article]

1995
*
arXiv
*
pre-print

Thus, we generalize

arXiv:alg-geom/9505001v1
fatcat:hn5r2lm4jnhtpghnvpcpwxcaoe
*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. ...##
###
A dual approach to structure constants for K-theory of Grassmannians

2020
*
Discrete Mathematics & Theoretical Computer Science
*

International audience

doi:10.46298/dmtcs.6361
fatcat:3jd6eztpizhhdlilhdxahlsj3i
*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. ...##
###
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]

1997
*
arXiv
*
pre-print

formulae for

arXiv:q-alg/9701005v3
fatcat:pg6ruets4baqxmurvfeplc23xm
*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 ...
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