Schubert Calculus and Schur Functions

AbstractWe construct the Schubert basis of the torus-equivariant K-homology of the affine Grassmannian of a simple algebraic group G, using the K-theoretic NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a construction of Peterson in equivariant homology. For the case where G=SLn, the K-homology of the affine Grassmannian is identified with a sub-Hopf algebra of the ring of symmetric functions. The Schubert basis is represented by inhomogeneous symmetric functions, calledK-k-Schur functions, whose highest-degree term is a k-Schur function. The dual basis in K-cohomology is given by the affine stable Grothendieck polynomials, verifying a conjecture of Lam. In addition, we give a Pieri rule in K-homology. Many of our constructions have geometric interpretations by means of Kashiwara’s thick affine flag manifold.

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Combinatorial description of the cohomology of the affine flag variety

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.6326 ◽

2020 ◽

Vol DMTCS Proceedings, 28th... ◽

Author(s):

Seung Jin Lee

Keyword(s):

Symmetric Functions ◽

Schur Functions ◽

Flag Variety ◽

Schubert Calculus ◽

Power Sum ◽

Schubert Polynomials ◽

International Audience ◽

Stanley Symmetric Functions ◽

Affine Flag Variety ◽

Affine Flag

International audience We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigatethe combinatorics of affine Schubert calculus for typeA. We introduce Murnaghan-Nakayama elements and Dunklelements in the affine FK algebra. We show that they are commutative as Bruhat operators, and the commutativealgebra generated by these operators is isomorphic to the cohomology of the affine flag variety. As a byproduct, weobtain Murnaghan-Nakayama rules both for the affine Schubert polynomials and affine Stanley symmetric functions. This enable us to expressk-Schur functions in terms of power sum symmetric functions. We also provide the defi-nition of the affine Schubert polynomials, polynomial representatives of the Schubert basis in the cohomology of theaffine flag variety.

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Optimal Assembly of Systems Using Schur-Functions and Majorization.

10.21236/ada172821 ◽

1986 ◽

Author(s):

Emad El-Neweihi ◽

Frank Proschan ◽

Jayaram Sethuraman

Keyword(s):

Schur Functions ◽

Optimal Assembly

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Matrix Schur functions, permutation matrices, and young operators as inner product spaces

Journal of Physics Conference Series ◽

10.1088/1742-6596/213/1/012010 ◽

2010 ◽

Vol 213 ◽

pp. 012010

Author(s):

James D Louck

Keyword(s):

Schur Functions ◽

Inner Product ◽

Product Spaces ◽

Inner Product Spaces ◽

Permutation Matrices

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Equivariant -theory of Grassmannians II: the Knutson–Vakil conjecture

Compositio Mathematica ◽

10.1112/s0010437x16008186 ◽

2017 ◽

Vol 153 (4) ◽

pp. 667-677 ◽

Cited By ~ 4

Author(s):

Oliver Pechenik ◽

Alexander Yong

Keyword(s):

Schubert Calculus ◽

Equivariant Theory

In 2005, Knutson–Vakil conjectured apuzzlerule for equivariant$K$-theory of Grassmannians. We resolve this conjecture. After giving a correction, we establish a modified rule by combinatorially connecting it to the authors’ recently proved tableau rule for the same Schubert calculus problem.

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