Combinatorics of quantum affine Lie algebra representations

Demazure Crystals and Extended Young Diagrams

“…In the special case k = 1, Corollary 4.5 was proved in [13]. It was also proved in the case n = 2 in [1].…”

“…[13] An extended Young diagram Y = (y i ) i≥0 is a weakly increasing sequence with integer entries such that there exists some fixed integer y ∞ such that y i = y ∞ for all i 0. We call y ∞ = γ the charge of the extended Young diagram Y .…”

On Demazure crystals for $U_{q}(G_{2}^{(1)})$

“…It is known that the mixing index κ is dependent on the choice of the perfect crystal. For example, as seen in [18] for U q (C…”

“…It is conjectured that κ ≤ 2 for other quantum affine algebras. It is known that the mixing index κ is dependent on the perfect crystal B (for example, see [18]). In [13], it is shown that for each g = A (1) n , B (1) n , C (1) n , D (1) n , A (2) 2n−1 , D (2) n+1 , A (2) 2n and λ = lΛ (where Λ is a dominant weight of level 1), there exists a sequence {w (k) } k≥0 of Weyl group elements and a perfect crystal B of level l such that the path realizations of the Demazure crystals B w (k) (λ) have tensor-product-like structures with κ = 1.…”

Raising/Lowering Maps and Modules for the Quantum Affine Algebra

“…Quantum affine algebras were first introduced and studied by Drinfeld (1985Drinfeld ( , 1988 and Jimbo (1985Jimbo ( , 1986 in relation to the Yang-Baxter equation of mathematical physics. Since then quantum affine algebras have played an important role in various areas of mathematics and physics, for example see Beck and Nakajima (2004), Kang and Kwon (2004), Mansour and Zakkari (2004), Misra and Williams (2004). In this article we will be concerned with the quantum affine algebra U q 2 .…”