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We prove that the multiplicity of an arbitrary dominant weight for an integrable highest weight representation of the affine Kac-Moody algebra A (1) r is a polynomial in the rank r. In the process we show that the degree of this polynomial is less than or equal to the depth of the weight with respect to the highest weight. These results allow weight… (More)

- DONG-UY SHIN
- 2003

We give a 1-1 correspondence with the Young wall realization and the Young tableau realization of the crystal bases for the classical Lie algebras. Introduction Young tableaux and Young walls play important roles in the interplay, which can be explained in a beautiful manner using the crystal base theory for quantum groups, between the fields of… (More)

- DONG-UY SHIN
- 2006

In this paper, we give polyhedral realization of the crystal B(∞) of U− q (g) for the generalized Kac-Moody algebras. As applications, we give explicit descriptions of crystals for the generalized Kac-Moody algebras of rank 2, 3 and Monster Lie algebras. Introduction In his study of Conway and Norton’s Moonshine Conjecture [3] for the infinite dimensional… (More)

- DONG-UY SHIN
- 2008

In this paper, we give a polyhedral realization of the highest weight crystals B(λ) associated with the highest weight modules V (λ) for the generalized Kac-Moody algebras. As applications, we give explicit descriptions of crystals for the generalized Kac-Moody algebras of ranks 2, 3, and Monster algebras.

- Dong-Uy Shin
- 2003

In this paper, we give a new realization of crystal bases for irreducible highest weight modules over Uq(G2) in terms of monomials. We also discuss the natural connection between the monomial realization and tableau realization. Introduction In 1985, the quantum groups Uq(g), which may be thought of as q-deformations of the universal enveloping algebras… (More)

- Jeong-Ah Kim, Dong-Uy Shin
- J. Comb. Theory, Ser. A
- 2013

Article history: Received 27 October 2012

- BY E. BALLICO, C. KEEM, D. SHIN
- 2007

Fix integers q, g, k, d. Set πd,k,q := kd − d − k + kq + 1 and assume q > 0, k ≥ 2, d ≥ 3q + 1, g ≥ kq − k + 1 and πd,k,q − ((⌊d/2⌋ + 1 − q) · (⌊k/2⌋ + 1) ≤ g ≤ πd,k,q. Let Y be a smooth and connected genus q projective curve. Here we prove the existence of a smooth and connected genus g projective curve X, a degree k morphism f : X → Y and a degree d… (More)

- DONG-UY SHIN
- 2003

We give a new realization of crystal bases for finite dimensional irreducible modules over special linear Lie algebras using the monomials introduced by H. Nakajima. We also discuss the connection between this monomial realization and the tableau realization given by Kashiwara and Nakashima. Introduction The quantum groups, which are certain deformations of… (More)

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