Mirković-Vilonen cycles and polytopes

Kamnitzer, Joel

doi:10.4007/annals.2010.171.245

Cited by 68 publications

(138 citation statements)

References 25 publications

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“…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].…”

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

Lee¹,

Salisbury²

2014

Journal of the Korean Mathematical Society

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“…In the finite type, rank 2 MV polytopes were characterized combinatorially on a case-by-case basis in [9] using the tropical Plucker relations. In the present paper, we define MV polytopes for sl 2 and A (2) 2 using diagonals (Definition 3.4).…”

Section: (Iv) For a Dominant Weight λ The Lowest Weight Crystal B(−λmentioning

confidence: 99%

Rank 2 affine MV polytopes

et al. 2013

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Abstract. We give a realization of the crystal B(−∞) for sl 2 using decorated polygons. The construction and proof are combinatorial, making use of Kashiwara and Saito's characterization of B(−∞), in terms of the * involution. The polygons we use have combinatorial properties suggesting they are the sl 2 analogues of the Mirković-Vilonen polytopes defined by Anderson and the third author in finite type. Using Kashiwara's similarity of crystals we also give MV polytopes for A (2) 2 , the other rank 2 affine Kac-Moody algebra.

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“…The action is hamiltonian and its image by the moment map is a polytope. The so obtained Mirković-Vilonen polytopes are investigated in, for instance, [2,110].…”

Section: Further Developments and Applicationsmentioning

confidence: 99%

The decomposition theorem, perverse sheaves and the topology of algebraic maps

Andrea

Migliorini

2009

Bull. Amer. Math. Soc.

162

152

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We give a motivated introduction to the theory of perverse sheaves, culminating in the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties. While most proofs are omitted, we discuss several approaches to the decomposition theorem, indicate some important applications and examples.

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Mirković-Vilonen cycles and polytopes

Cited by 68 publications

References 25 publications

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

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

Rank 2 affine MV polytopes

The decomposition theorem, perverse sheaves and the topology of algebraic maps

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