A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras

Littelmann, Peter

doi:10.1007/bf01231564

Cited by 301 publications

(346 citation statements)

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“…Moreover, in this isomorphism, the tensor product can be defined in terms of concatenation of paths [15]. Through this isomorphism, the following propositions are direct consequences of [16, Theorem 10.1].…”

Section: 7mentioning

confidence: 99%

Adapted algebras for the Berenstein-Zelevinsky conjecture

Caldero

2003

Transformation Groups

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Let G be a simply connected semi-simple complex Lie group and fix a maximal unipotent subgroup U − of G. Let q be an indeterminate and let's denote by B * the dual canonical basis, [18], of the quantized algebra C q [U − ] of regular functions on U − . Following [19], fix a Z N ≥0 -parametrization of this basis, where N =dimU − . In [2], A. Berenstein and A. Zelevinsky conjecture that two elements of B * q-commute if and only if they are multiplicative, i.e. their product is an element of B * up to a power of q. For all reduced decomposition w0 of the longest element of the Weyl group of g, we associate a subalgebra A w0 , called adapted algebra, of C q [U − ] such that 1) A w0 is a q-polynomial algebra which equals C q [U − ] up to localization, 2) A w0 is spanned by a subset of B * , 3) the Berenstein-Zelevinsky conjecture is true on A w0 . Then, we test the conjecture when one element belongs to the q-center of C q [U − ].

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Section: 7mentioning

confidence: 99%

Adapted algebras for the Berenstein-Zelevinsky conjecture

Caldero

2003

Transformation Groups

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“…This leads to many combinatorial models, discovered in a series of papers ( [KN94], [Lit94], [KS97]). Since this paper has the goal of determining the crystal graph of KR modules, we will only mention in the following remark how we could compute crystal bases for V (λ) using tensor products of polytopes.…”

Section: Tensor Products and Nakajima Monomialsmentioning

confidence: 99%

Realization of affine type A Kirillov–Reshetikhin crystals via polytopes

Kus

2013

Journal of Combinatorial Theory, Series A

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Abstract. On the polytope defined in [FFL11], associated to any rectangle highest weight, we define a structure of an type An-crystal. We show, by using the Stembridge axioms, that this crystal is isomorphic to the one obtained from Kashiwara's crystal bases theory. Further we define on this polytope a bijective map and show that this map satisfies the properties of a weak promotion operator. This implies in particular that we provide an explicit realization of Kirillov-Reshetikhin crystals for the affine type A (1) n via polytopes. IntroductionLet g be a affine Lie algebra and U ′ q (g) be the corresponding quantum algebra without derivation. A promotion operator pr on a crystal B of type A n is defined to be a map satisfying several conditions, namely that pr shifts the content, pr •ẽ j =ẽ j+1 • pr, pr •f j =f j+1 • pr for all j ∈ {1, · · · , n − 1} and pr n+1 = id, whereẽ j andf j respectively are the Kashiwara operators. If the latter condition is not satisfied, but pr is still bijective, then the map pr is called a weak promotion operator (see also [BST10]). The advantage of such (weak) promotion oparators are that we can associate to a given crystal B of type A n a (weak) affine crystal by settingf 0 := prOn the set of all semi-standard Young tableaux of rectangle shape, which is a realization of B(mω i ) the U q (A n )-crystal associated to the irreducible module of highest weight mω i , Schützenberger defined a promotion operator pr, called the Schützenberger's promotion operator [Sch72], which is the analogue of the cyclic Dynkin diagram automorphism i → i + 1 mod (n + 1) on the level of crystals, by using jeu-de-taquin. Given a tableaux T over the alphabet 1 ≺ 2 · · · ≺ n+1, pr(T ) is obtained from T by removing all letters n+1, adding one to each entry in the remaining tableaux, using jeu-de-taquin to slide all letters up and finally filling the holes with 1's (see also Section 6). One of the combinatorial descriptions of KR m,i in the affine A was shown that, as a {1, · · · , n}-crystal, KR m,i is isomorphic to B(mω i ) and the affine crystal constructed from B(mω i ) using Schützenberger's promotion operator is isomorphic to the Kirillov-Reshetikhin crystal KR m,i . The two ways of computing the affine crystal structure, one given by [KKM + 92] and the other by [Shi02], are shown to be equivalent in [OSS03]. Another combinatorial model in this type without using a promotion operator is described in [Kwo]. In this paper, we introduce a new realization of Kirillov-Reshetikhin crystals of type A (1) n . In [FFL11] the authors have constructed for all dominant integral A n weights λ a polytope in R n n−1 2 and a basis of the irreducible A n module of highest weight λ and have shown that the basis elements are parametrized by the integral points. For λ = mω i we can understand this polytope in R i(n−i+1) and denote the intersection of this polytope with Z i(n−i+1) + by B m,i . We define certain maps on B m,i and show that this becomes a crystal of type A n . As a set, we can identify B m,i with certain ...

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“…Follows immediately from the Demazure formula above, and the combinatorial properties of root operators described in [19] Sec. 5.…”

Section: Then the Schubert Polynomial S(w) Is The Generating Functionmentioning

confidence: 99%

“…The work of Lascoux-Schutzenberger [17] and Littlemann [19] allows us to "quantize" our Demazure formula, realizing the terms of the polynomial by certain tableaux endowed with a crystal graph structure. Reiner and Shimozono have shown that our construction gives the same non-commutative Schubert polynomials as those in [16].…”

Section: Young Tableauxmentioning

confidence: 99%

Schubert polynomials and Bott-Samelson varieties

Magyar¹

1998

Comment. Math. Helv.

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Abstract. Schubert polynomials generalize Schur polynomials, but it is not clear how to generalize several classical formulas: the Weyl character formula, the Demazure character formula, and the generating series of semistandard tableaux. We produce these missing formulas and obtain several surprising expressions for Schubert polynomials.The above results arise naturally from a new geometric model of Schubert polynomials in terms of Bott-Samelson varieties. Our analysis includes a new, explicit construction for a BottSamelson variety Z as the closure of a B-orbit in a product of flag varieties. This construction works for an arbitrary reductive group G, and for G = GL(n) it realizes Z as the representations of a certain partially ordered set.This poset unifies several well-known combinatorial structures: generalized Young diagrams with their associated Schur modules; reduced decompositions of permutations; and the chamber sets of Berenstein-Fomin-Zelevinsky, which are crucial in the combinatorics of canonical bases and matrix factorizations. On the other hand, our embedding of Z gives an elementary construction of its coordinate ring, and allows us to specify a basis indexed by tableaux.Mathematics Subject Classification (1991). 14M15, 16G20.

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A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras

Cited by 301 publications

References 7 publications

Adapted algebras for the Berenstein-Zelevinsky conjecture

Adapted algebras for the Berenstein-Zelevinsky conjecture

Realization of affine type A Kirillov–Reshetikhin crystals via polytopes

Schubert polynomials and Bott-Samelson varieties

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