Tensor product multiplicities, canonical bases and totally positive varieties

Berenstein, Arkady; Zelevinsky, Andrei

doi:10.1007/s002220000102

Cited by 278 publications

(537 citation statements)

References 21 publications

(71 reference statements)

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“…In particular, this holds for all i ∈ I and all i-trails π from ω i to w 0 s i ω i in V (ω i ). By Theorem 3.10 in [6], this implies c ∈ C i . [23] of the canonical basis on a combinatorial level.…”

Section: Proof Let Us Consider An I-trailmentioning

confidence: 73%

“…The truly new points are the inequality µ + (Z) λ in Statement (ii) and the fact that the equality µ + (Z) = λ holds only if c ∈ C i . We will ground our proof on the notion of i-trail in Berenstein and Zelevinsky's work [6]. We first recall what it is about.…”

Section: Proposition 45 (Ii) Thus Asserts That the Map (P Z)mentioning

confidence: 99%

“…It turns out that the closureỸ i,c is always an MV cycle, even when c does not belong to the string cone in direction i. Proposition 4.7 presents a necessary and sufficient condition onỸ i,c in order that c may belong to the string cone; its proof relies on Berenstein and Zelevinsky's characterization of the string cone in terms of i-trails [6].…”

Section: Crystal Structure and String Parametrizationsmentioning

confidence: 99%

“…, i N ) be a reduced decomposition of w 0 , let γ and δ be two weights in X, let V be a rational G-module, and write V = η∈X V η for its decomposition in weight subspaces. According to Definition 2.1 in [6], an i-trail from γ to δ in V is a sequence of weights π = (γ = γ 0 , γ 1 , . .…”

Section: Proposition 45 (Ii) Thus Asserts That the Map (P Z)mentioning

confidence: 99%

“…For each i ∈ I, we can thus speak of the simple rational G-module with highest weight ω i , which we denote by V (ω i ). Then by Theorem 3.10 in [6], the string cone C i is the set of all (c 1 , . .…”

Section: Proposition 45 (Ii) Thus Asserts That the Map (P Z)mentioning

confidence: 99%

See 4 more Smart Citations

On Mirković-Vilonen cycles and crystal combinatorics

Baumann

Gaussent

2008

Represent. Theory

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Abstract. Let G be a complex connected reductive group and let G ∨ be its Langlands dual. Let us choose a triangular decomposition n −,∨ ⊕ h ∨ ⊕ n +,∨ of the Lie algebra of G ∨ . Braverman, Finkelberg and Gaitsgory show that the set of all Mirković-Vilonen cycles in the affine Grassmannianis a crystal isomorphic to the crystal of the canonical basis of U (n +,∨ ). Starting from the string parameter of an element of the canonical basis, we give an explicit description of a dense subset of the associated MV cycle. As a corollary, we show that the varieties involved in Lusztig's algebraic-geometric parametrization of the canonical basis are closely related to MV cycles. In addition, we prove that the bijection between LS paths and MV cycles constructed by Gaussent and Littelmann is an isomorphism of crystals.

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Section: Proof Let Us Consider An I-trailmentioning

confidence: 73%

Section: Proposition 45 (Ii) Thus Asserts That the Map (P Z)mentioning

confidence: 99%

Section: Crystal Structure and String Parametrizationsmentioning

confidence: 99%

Section: Proposition 45 (Ii) Thus Asserts That the Map (P Z)mentioning

confidence: 99%

Section: Proposition 45 (Ii) Thus Asserts That the Map (P Z)mentioning

confidence: 99%

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On Mirković-Vilonen cycles and crystal combinatorics

Baumann

Gaussent

2008

Represent. Theory

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On Harmonic Elements for Semi-simple Lie Algebras

Caldero

2002

Advances in Mathematics

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Let g be a semi-simple complex Lie algebra and g=n − À h À n its triangular decomposition. Let U(g), resp. U q (g), be its enveloping algebra, resp. its quantized enveloping algebra. This article gives a quantum approach to the combinatorics of (classical) harmonic elements and Kostant's generalized exponents for g. A quantum analogue of the space of harmonic elements has been given by A. Joseph and G. Letzter (1994, Amer. J. Math. 116, 127-177). On the one hand, we give specialization results concerning harmonic elements, central elements of U q (g), and the decomposition of Joseph and Letzter (cited above). For g=sl n+1 , we describe the specialization of quantum harmonic space in the N-filtered algebra U(sl n+1 ) as the materialization of a theorem of A. Lascoux et al. (1995, Lett. Math. Phys. 35, 359-374). This enables us to study a Joseph-Letzter decomposition in the algebra U(sl n+1 ). On the other hand, we prove that highest weight harmonic elements can be calculated in terms of the dual of Lusztig's canonical basis. In the simply laced case, we parametrize a basis of n-invariants of minimal primitive quotients by the set C 0 of integral points of a convex cone. © 2002 Elsevier Science (USA) Key Words: quantum groups; enveloping algebras; canonical basis; harmonic space.0. INTRODUCTION 0.1. Let V be a complex finite dimensional space and G a Lie subgroup of GL(V). Assume that V is completely reducible as a G-module. Then G acts semi-simply on the symmetric C-algebra S(V). Let J be the ideal generated by the non-constant homogeneous G-invariant elements of S(V). Let V g be the dual of V. There exists a natural non-degenerate duality between S(V) and S(V) be the orthogonal of J and H g be the space of G-harmonic polynomials of V.

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Regions of Linearity, Lusztig Cones, and Canonical Basis Elements for the Quantized Enveloping Algebra of Type A4

Carter

Marsh

2000

Journal of Algebra

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to professor helmut wielandt on his 90th birthday Let U q be the quantum group associated to a Lie algebra g of rank n. The negative part U − of U has a canonical basis B with favourable properties (see M. Kashiwara (1991, Duke Math. J. 63, 465-516) and G. Lusztig (1993. "Introduction to Quantum Groups," Sect. 14.4.6, Birkhäuser, Boston)). The approaches of Lusztig and Kashiwara lead to a set of alternative parametrizations of the canonical basis, one for each reduced expression for the longest word in the Weyl group of g. We show that if g is of type A 4 there are close relationships between the Lusztig cones, canonical basis elements, and the regions of linearity of reparametrization functions arising from the above parametrizations. A graph can be defined on the set of simplicial regions of linearity with respect to adjacency, and we further show that this graph is isomorphic to the graph with vertices given by the reduced expressions of the longest word of the Weyl group modulo commutation and edges given by long braid relations.

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Tensor product multiplicities, canonical bases and totally positive varieties

Cited by 278 publications

References 21 publications

On Mirković-Vilonen cycles and crystal combinatorics

On Mirković-Vilonen cycles and crystal combinatorics

On Harmonic Elements for Semi-simple Lie Algebras

Regions of Linearity, Lusztig Cones, and Canonical Basis Elements for the Quantized Enveloping Algebra of Type A4

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