Daisuke Sagaki

Path model for a level-zero extremal weight module over a quantum affine algebra

We give a path model for a level zero extremal weight module over a quantum affine algebra. By using this result, we prove a branching rule for an extremal weight module with respect to a Levi subalgebra. Furthermore, we also show a decomposition rule of Littelmann type for the concatenation of path models for an integrable highest weight module and a level zero extremal weight module in the case where the extremal weight is minuscule.

A Uniform Model for Kirillov–Reshetikhin Crystals I: Lifting the Parabolic Quantum Bruhat Graph

© 2014

Construction of Perfect Crystals Conjecturally Corresponding to Kirillov-Reshetikhin Modules over Twisted Quantum Affine Algebras

Assuming the existence of the perfect crystal bases of Kirillov-Reshetikhin modules over simply-laced quantum affine algebras, we construct certain perfect crystals for twisted quantum affine algebras, and also provide compelling evidence that the constructed crystals are isomorphic to the conjectural crystal bases of Kirillov-Reshetikhin modules over twisted quantum affine algebras.

A UNIFORM MODEL FOR KIRILLOV–RESHETIKHIN CRYSTALS III: NONSYMMETRICMACDONALD POLYNOMIALS AT t = 0 AND DEMAZURE CHARACTERS

We establish the equality of the specialization Ewλ(x ; q; 0) of the nonsymmetric Macdonald polynomial Ewλ(x ; q; t) at t = 0 with the graded character gch Uw+(λ) of a certain Demazure-type submodule Uw+(λ) of a tensor product of “single-column” Kirillov–Reshetikhin modules for an untwisted affine Lie algebra, where λ is a dominant integral weight and w is a (finite) Weyl group element; this generalizes our previous result, that is, the equality between the specialization Pλ(x ; q; 0) of the symmetric Macdonald polynomial Pλ(x ; q; t) at t = 0 and the graded character of a tensor product of single-column Kirillov–Reshetikhin modules. We also give two combinatorial formulas for the mentioned specialization of nonsymmetric Macdonald polynomials: one in terms of quantum Lakshmibai–Seshadri paths and the other in terms of the quantum alcove model.

Lakshmibai–Seshadri paths of level-zero shape and one-dimensional sums associated to level-zero fundamental representations

We give an interpretation of the energy function and classically restricted one-dimensional sums associated to tensor products of level-zero fundamental representations of quantum affine algebras in terms of Lakshmibai–Seshadri paths of level-zero shape.

Posets, tensor products and Schur positivity

Let g be a complex finite-dimensional simple Lie algebra. Given a positive integer k and a dominant weight \lambda, we define a preorder on the set

Three kinds of extremal weight vectors fixed by a diagram automorphism

P(\lambda, k)

Simplicity of a Vertex Operator Algebra Whose Griess Algebra is the Jordan Algebra of Symmetric Matrices

of k-tuples of dominant weights which add up to \lambda. Let

CERTAIN MODULES WITH TWINING MAPS AND DECOMPOSITION RULES OF LITTELMANN TYPE

P(\lambda, k)/\sim

Toward Berenstein-Zelevinsky Data in Affine Type A, Part III: Proof of the Connectedness

be the corresponding poset of equivalence classes defined by the preorder. We show that if \lambda is a multiple of a fundamental weight (and k is general) or if k=2 (and \lambda is general), then