50 research outputs found
On a conjecture by Naito-Sagaki: Littelmann paths and Littlewood-Richardson Sundaram tableaux
We prove a special case of a conjecture of Naito-Sagaki about a branching
rule for the restriction of irreducible representations of
to . The
conjecture is in terms of certain Littelmann paths, with the embedding given by
the folding of the type Dynkin diagram. We propose and motivate an
approach to the conjecture in general, in terms of Littlewood-Richardson
Sundaram tableaux.Comment: 13 pages. Comments welcom
Word reading is a crystal morphism
We observe that word reading is a crystal morphism. This leads us to prove
that, in the case of the complex special linear group, the map from all
galleries to MV cycles is a morphism of crystals.Comment: 15 pages, comments welcom
Atoms and charge in type
We construct atomic decompositions for crystals of type and define a
charge statistic on them, thus providing positive combinatorial formulas for
Kostka-Foulkes polynomials associated to them together with a natural geometric
interpretation.Comment: 69 pages, comments welcom
Symplectic cacti, virtualization and Berenstein-Kirillov groups
We explicitly realize an internal action of the symplectic cactus group,
recently defined by Halacheva for any complex, reductive, finite-dimensional
Lie algebra, on crystals of Kashiwara-Nakashima tableaux. Our methods include a
symplectic version of jeu de taquin due to Sheats and Lecouvey, symplectic
reversal, and virtualization due to Baker. As an application, we define and
study a symplectic version of the Berenstein-Kirillov group and show that it is
a quotient of the symplectic cactus group. In addition two relations for
symplectic Berenstein-Kirillov group are given that do not follow from the
defining relations of the symplectic cactus group.Comment: 71 pages, comments welcom