1,050 research outputs found
Crystal duality and Littlewood-Richardson rule of extremal weight crystals
We consider a category of \gl_\infty-crystals, whose objects are disjoint
unions of extremal weight crystals of non-negative level with certain finite
conditions on the multiplicity of connected components. We show that it is a
monoidal category under tensor product of crystals and the associated
Grothendieck ring is anti-isomorphic to an Ore extension of the character ring
of integrable lowest weight \gl_\infty-modules with respect to derivations
shifting the characters of fundamental modules. A Littlewood-Richardson rule of
extremal weight crystals with non-negative level is described explicitly in
terms of classical Littlewood-Richardson coefficients
Littlewood identity and Crystal bases
We give a new combinatorial model for the crystals of integrable highest
weight modules over the classical Lie algebras of type and in terms of
classical Young tableux. We then obtain a new description of its
Littlewood-Richardson rule and a maximal Levi branching rule in terms of
classical Littlewood-Richardson tableaux, which extends in a bijective way the
well-known stable formulas at large ranks. We also show that this tableau model
admits a natural superization and it produces the characters of irreducible
highest weight modules over orthosymplectic Lie superalgebras, which correspond
to the integrable highest weight modules over the classical Lie algebras of
type and under the Cheng-Lam-Wang's super duality.Comment: 51 page
Crystal bases of modified quantized enveloping algebras and a double RSK correspondence
The crystal base of the modified quantized enveloping algebras of type
or is realized as a set of integral bimatrices. It is
obtained by describing the decomposition of the tensor product of a highest
weight crystal and a lowest weight crystal into extremal weight crystals, and
taking its limit using a tableaux model of extremal weight crystals. This
realization induces in a purely combinatorial way a bicrystal structure of the
crystal base of the modified quantized enveloping algebras and hence its
Peter-Weyl type decomposition generalizing the classical RSK correspondence.Comment: 30 page
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