We present a uniform construction of level 1 perfect crystals B for all affine Lie algebras. We also introduce the notion of a crystal algebra and give an explicit description of its multiplication. This allows us to determine the energy function on B ⊗ B completely and thereby give a path realization of the basic representations at q = 0 in the homogeneous picture.
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 multiplicity information for small ranks to be transferred to arbitrary ranks.
Abstract. We give a realization of crystal graphs for basic representations of the quantum affine algebra Uq(C (1) n ) using combinatorics of Young walls. The notion of splitting blocks plays a crucial role in the construction of crystal graphs.
Using combinatorics of Young walls, we give a new realization of arbitrary level irreducible highest weight crystals B(λ) for quantum affine algebras of type2n , and D (2) n+1 . The irreducible highest weight crystals are realized as the affine crystals consisting of reduced proper Young walls. The notion of slices and splitting of blocks plays a crucial role in the construction of crystals.
Mathematics Subject Classifications (2000) 81R50 · 17B37
Abstract. Nakajima introduced a certain set of monomials realizing the irreducible highest weight crystals B(λ). The monomial set can be extended so that it contains crystal B(∞) in addition to B(λ). We present explicit descriptions of the crystals B(∞) and B(λ) over special linear Lie algebras in the language of extended Nakajima monomials. There is a natural correspondence between the monomial description and Young tableau realization, which is another realization of crystals B(∞) and B(λ). §1. IntroductionThe theory of Nakajima monomials is a combinatorial scheme for realizing crystal bases of quantum groups. Nakajima introduced a certain set of monomials realizing the irreducible highest weight crystals in [16]. Kashiwara and Nakajima independently defined a crystal structure on the set of Nakajima monomials and also gave a realization of irreducible highest weight crystal B(λ) in terms of Nakajima monomials, as the connected component of the monomial set containing a maximal vector of dominant integral weight λ [9], [17]. This has lead to the belief that it should be possible to give a similar realization for B(∞), which is the crystal base of the negative part U − q (g) of a quantum group over symmetrizable Kac-Moody algebra g, also.Much effort has been made to give realization of B(∞) over various Kac-Moody algebras. In addition to these works, in our recent works [3], [14], we gave new realization of B(∞) for the finite simple Lie algebras, in terms of Young tableaux.Starting from the realization theorem of Kashiwara and Nakajima [9], [17], we can argue that it is not possible to find the crystal B(∞) within the set of Nakajima monomials with their given crystal structure. Hence,
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