$$k$$ -Double Schur functions and equivariant (co)homology of the affine Grassmannian

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.

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Skew Littlewood―Richardson rules from Hopf algebras

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2853 ◽

2010 ◽

Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽

Author(s):

Thomas Lam ◽

Aaron Lauve ◽

Frank Sottile

Keyword(s):

Hopf Algebras ◽

Symplectic Group ◽

Schur Functions ◽

Affine Grassmannian ◽

International Audience ◽

Pieri Rules ◽

Skew Schur Functions ◽

Nous Utilisons

International audience We use Hopf algebras to prove a version of the Littlewood―Richardson rule for skew Schur functions, which implies a conjecture of Assaf and McNamara. We also establish skew Littlewood―Richardson rules for Schur $P-$ and $Q-$functions and noncommutative ribbon Schur functions, as well as skew Pieri rules for k-Schur functions, dual k-Schur functions, and for the homology of the affine Grassmannian of the symplectic group. Nous utilisons des algèbres de Hopf pour prouver une version de la règle de Littlewood―Richardson pour les fonctions de Schur gauches, qui implique une conjecture d'Assaf et McNamara. Nous établissons également des règles de Littlewood―Richardson gauches pour les $P-$ et $Q-$fonctions de Schur et les fonctions de Schur rubbans non commutatives, ainsi que des règles de Pieri gauches pour les $k-$fonctions de Schur, les $k-$fonctions de Schur duales, et pour l'homologie de la Grassmannienne affine du groupe symplectique.

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K-theory Schubert calculus of the affine Grassmannian

Compositio Mathematica ◽

10.1112/s0010437x09004539 ◽

2010 ◽

Vol 146 (4) ◽

pp. 811-852 ◽

Cited By ~ 10

Author(s):

Thomas Lam ◽

Anne Schilling ◽

Mark Shimozono

Keyword(s):

Symmetric Functions ◽

Schur Functions ◽

Flag Manifold ◽

Schubert Calculus ◽

Schur Function ◽

Dual Basis ◽

Simple Algebraic Group ◽

Affine Grassmannian ◽

Equivariant Homology ◽

Degree Term

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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