Geometric and Combinatorial Realizations of Crystal Graphs

Savage, Alistair

doi:10.1007/s10468-005-0565-7

Cited by 25 publications

(43 citation statements)

References 17 publications

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“…It is also worth noting that there are well-known bijections between the Lusztig parametrization, string parametrization, and semistandard Young tableaux in type A r . More details may be found in [27,29].…”

Section: Introductionmentioning

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: Introductionmentioning

confidence: 99%

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

Lee¹,

Salisbury²

2014

Journal of the Korean Mathematical Society

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“…There are other, purely combinatorial, constructions of these crystals as well. In [17], the author described an explicit isomorphism between the geometric construction from quiver varieties and the combinatorial construction via Young tableaux and Young walls. One limitation of the quiver variety approach to Kac-Moody Lie algebras and their representations is that the geometric Lie algebra action is only defined for those algebras whose generalized Cartan matrix is symmetric.…”

Section: Introductionmentioning

confidence: 99%

A geometric construction of crystal graphs using quiver varieties: extension to the non-simply laced case

Savage¹

2005

Infinite-Dimensional Aspects of Representation Theory and Applications

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Abstract. We consider a generalization of the quiver varieties of Lusztig and Nakajima to the case of all symmetrizable Kac-Moody Lie algebras. To deal with the non-simply laced case one considers admissible automorphisms of a quiver and the irreducible components of the quiver varieties fixed by this automorphism. We define a crystal structure on these irreducible components and show that the crystals obtained are isomorphic to those associated to the crystal bases of the lower half of the universal enveloping algebra and the irreducible highest weight representations of the non-simply laced Kac-Moody Lie algebra. As an application, we realize the crystal of the spin representation of so 2n+1 on the set of self-conjugate Young diagrams that fit inside an n × n box.

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“…We now undertake the task of giving an explicit description of this action in terms of the combinatorial data enumerating irreducible components. The arguments are similar to those used in [11]. 4.1.…”

Section: Combinatorial Crystal Graphsmentioning

confidence: 93%

“…In [11], an enumeration of the irreducible components of Nakajima's quiver varieties for finite and affine types A and D were given in terms of Young tableaux, Young walls and new objects called Young pyramids. In this section, we describe a natural enumeration of the irreducible components of Lusztig's quiver varieties in finite and affine type A.…”

Section: Enumeration Of Componentsmentioning

confidence: 99%

“…Note that we could have avoided the appearance of the map σ by choosing the opposite orientation of our quiver. However, we have chosen the orientation to agree with [1,11]. We also note that while the proof that Y(∞) ∼ = B(∞) found in [7] involves the Kyoto path realization of B(∞), the alternative geometric proof presented here is direct in the sense that it avoids any reference to this path realization.…”

Section: Typementioning

confidence: 99%

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Geometric and combinatorial realizations of crystals of enveloping algebras

Savage¹

2007

Lie Algebras, Vertex Operator Algebras and Their Applications

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Abstract. Kashiwara and Saito have defined a crystal structure on the set of irreducible components of Lusztig's quiver varieties. This gives a geometric realization of the crystal graph of the lower half of the quantum group associated to a simply-laced Kac-Moody algebra. Using an enumeration of the irreducible components of Lusztig's quiver varieties in finite and affine type A by combinatorial data, we compute the geometrically defined crystal structure in terms of this combinatorics. We conclude by comparing the combinatorial realization of the crystal graph thus obtained with other combinatorial models involving Young tableaux and Young walls.

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Geometric and Combinatorial Realizations of Crystal Graphs

Cited by 25 publications

References 17 publications

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

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

A geometric construction of crystal graphs using quiver varieties: extension to the non-simply laced case

Geometric and combinatorial realizations of crystals of enveloping algebras

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