Mark Shimozono
From double quantum Schubert polynomials to k-double Schur functions via the Toda lattice.
Tue, 09/13/2011 - 15:01 — SomNambulAuthors: Thomas Lam, Mark Shimozono
Subjects: Combinatorics
AbstractWe show that the k-double Schur functions defined by the authors, and the
quantum double Schubert polynomials studied by Kirillov and Maeno and by
Ciocan-Fontanine and Fulton, can be obtained from each other by an explicit
rational substitution. The main new ingredient is an explicit computation of
Kostant's solution to the Toda lattice in terms of equivariant Schubert
classes.Equivariant Pieri Rule for the homology of the affine Grassmannian.
Fri, 05/27/2011 - 15:01 — SomNambulAuthors: Thomas Lam, Mark Shimozono
Subjects: Combinatorics
AbstractAn explicit rule is given for the product of the degree two class with an
arbitrary Schubert class in the torus-equivariant homology of the affine
Grassmannian. In addition a Pieri rule (the Schubert expansion of the product
of a special Schubert class with an arbitrary one) is established for the
equivariant homology of the affine Grassmannians of SL_n and a similar formula
is conjectured for Sp_{2n} and SO_{2n+1}. For SL_n the formula is explicit and
positive.k-shape poset and branching of k-Schur functions.
Mon, 08/02/2010 - 15:01 — SomNambulAuthors: Thomas Lam, Mark Shimozono, Luc Lapointe, Jennifer Morse
Subjects: Combinatorics
AbstractWe give a combinatorial expansion of a Schubert homology class in the affine
Grassmannian Gr_{SL_k} into Schubert homology classes in Gr_{SL_{k+1}}. This is
achieved by studying the combinatorics of a new class of partitions called
k-shapes, which interpolates between k-cores and k+1-cores. We define a
symmetric function for each k-shape, and show that they expand positively in
terms of dual k-Schur functions. We obtain an explicit combinatorial
description of the expansion of an ungraded k-Schur function into k+1-Schur
functions.Affine crystals, one-dimensional sums and parabolic Lusztig q-analogues.
Mon, 02/22/2010 - 16:01 — SomNambulAuthors: Mark Shimozono, Masato Okado, Cedric Lecouvey
Subjects: Quantum Algebra
AbstractThis paper is concerned with one-dimensional sums in classical affine types.
We prove a conjecture of the third author by showing they all decompose in
terms of one-dimensional sums related to affine type A provided the rank of the
root system considered is sufficiently large. As a consequence, any
one-dimensional sum associated to a classical affine root system with
sufficiently large rank can be regarded as a parabolic Lusztig q-analogue.K-theory Schubert calculus of the affine Grassmannian.
Wed, 09/02/2009 - 12:02 — SomNambulAuthors: Anne Schilling, Thomas Lam, Mark Shimozono
Subjects: Combinatorics
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.K-theory Schubert calculus of the affine Grassmannian.
Wed, 09/02/2009 - 12:02 — SomNambulAuthors: Anne Schilling, Thomas Lam, Mark Shimozono
Subjects: Combinatorics
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.
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