Crystal Bases and Tensor Product Decompositions of U(G2)-Modules

Kang, Seok-Jίn; Misra, Kailash C.

doi:10.1006/jabr.1994.1037

Cited by 34 publications

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“…The G 2 description is due to S.-J. Kang and K. Misra [11]. The Young tableaux description of B(∞) is closely related to that of B(λ) in the sense that the basic building blocks in both characterizations come from B(ω 1 ) for the fundamental weight ω 1 .…”

Section: General Definitionsmentioning

confidence: 99%

Young Tableaux, Canonical Bases, and the Gindikin-Karpelevich Formula

Lee¹,

Salisbury²

2014

Journal of the Korean Mathematical Society

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Abstract. A combinatorial description of the crystal B(∞) for finitedimensional simple Lie algebras in terms of certain Young tableaux was developed by J. Hong and H. Lee. We establish an explicit bijection between these Young tableaux and canonical bases indexed by Lusztig's parametrization, and obtain a combinatorial rule for expressing the Gindikin-Karpelevich formula as a sum over the set of Young tableaux. IntroductionThe Gindikin-Karpelevich formula is a p-adic integration formula proved by Langlands in [18]. He named it the Gindikin-Karpelevich formula after a similar formula originally stated by Gindikin and Karpelevich [5] in the case of real reductive groups. The formula also appears in Macdonald's work [25] on p-adic groups and affine Hecke algebras.Let G be a split semisimple algebraic group over a p-adic field F with ring of integers o F , and suppose the residue field o F /πo F of F has size t, where π is a generator of the unique maximal ideal in o F . Choose a maximal torus T of G contained in a Borel subgroup B with unipotent radical N , and let N − be the opposite group to N . We have B = T N . The group G(F ) has a decomposition G(F ) = B(F )K, whereis the modular character of B and τ is extended to B(F ) to be trivial on N (F ). The function f• is called the standard spherical vector corresponding to τ .Let G ∨ be the Langlands dual of G with the dual torus T ∨ . The set of coroots of G is identified with the set of roots of G ∨ and will be denoted by Φ. Finally, let z be the element of the dual torus T ∨ , corresponding to τ via the Satake isomorphism.Theorem 0.1 (Gindikin-Karpelevich formula, [18]). Given the setting above, we havewhere Φ + is the set of positive roots of G ∨ .Let g be the Lie algebra of G ∨ , and let B(∞) be the crystal basis of the negative part U − q (g) of the quantum group U q (g). Then Φ is the root system of g as well. In recent work, the integral in the Gindikin-Karpelevich formula has been evaluated using Kashiwara's crystal basis or Lusztig's canonical basis. D. Bump and M. Nakasuji where nz(φ i (b)) is the number of nonzero entries in the Lusztig parametrization φ i (b) of b with respect to a reduced expression i of the longest Weyl group element. The purpose of this paper is to describe the sum in (0.2) in a combinatorial way using Young tableaux. Since the canonical basis B is the same as Kashiwara's global crystal basis, we may replace B with B(∞). The associated crystal structure on B may be described in terms of the Lusztig parametrization [2,24] or the string parametrization [1,13,22], and there are formulas relating the two given by Berenstein and Zelevinsky in [2]. Much work has been done on realizations of crystals (e.g., [8,9,15,16,21]). In the case of B(∞) for finitedimensional simple Lie algebras, J. Hong and H. Lee used marginally large semistandard Young tableaux to obtain a realization of crystals [7]. We will use their marginally large semistandard Young tableaux realization of B(∞) to write the right-hand side of (0.2) as a sum over a set T (∞) ...

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Section: General Definitionsmentioning

confidence: 99%

Young Tableaux, Canonical Bases, and the Gindikin-Karpelevich Formula

Lee¹,

Salisbury²

2014

Journal of the Korean Mathematical Society

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“…The authors do not find a description of this crystal structure in the literature (probably because it is not perfect), but one can easily obtain it from the description of its I 0 -crystal structure in [16]. The crystal isomorphism z is given in order.…”

Section: This Follows From Resmentioning

confidence: 99%

Level 0 Monomial Crystals

Hernandez¹,

Nakajima²

2006

Nagoya Mathematical Journal

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Abstract. We study the monomial crystal defined by the second author. We show that each component of the monomial crystal can be embedded into a crystal of an extremal weight module introduced by Kashiwara. And we determine all monomials appearing in the components corresponding to all level 0 fundamental representations of quantum affine algebras except for some nodes of E (2) 6 , E (1) 7 , E (1) 8 . Thus we obtain explicit descriptions of the crystals in these examples. We also give those for the corresponding finite dimensional representations. For classical types, we give them in terms of tableaux. For exceptional types, we list up all monomials.

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“…For each of the finite classical types, we shall use the definitions of semi-standard tableaux as given by Kashiwara and Nakashima [11]. For the G 2 type, we shall take the Young tableau realization of highest weight crystal B(λ) given in [6] as the definition of semi-standard tableaux. Since the first of these two works is a rather well known result, and since the second is very similar in spirit to the first, we refer readers to the original papers and shall not repeat the complicated definitions here.…”

Section: Large Semi-standard Tableauxmentioning

confidence: 99%

“…Since the first of these two works is a rather well known result, and since the second is very similar in spirit to the first, we refer readers to the original papers and shall not repeat the complicated definitions here. The alphabet to be used inside the boxes constituting the Young tableaux for each type will be denoted commonly by J, and it will be equipped with an ordering ≺, as given in [6,11]. For example, in the C n case, it would be…”

Section: Large Semi-standard Tableauxmentioning

confidence: 99%