Standard monomial theory for Bott-Samelson varieties

Lakshmibai, V.; Magyar, Péter

doi:10.1016/s0764-4442(99)80401-7

Cited by 30 publications

(50 citation statements)

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“…(IV) follows from Proposition 7. (V) follows from the Combinatorial Excellent Filtration Theorem [23,Proposition 12], which implies that λ * j B j +1 is isomorphic to a union of Demazure crystals B y (μ) with y w * j . (VI) Both sides are stable under the f i , e i for i = 1, .…”

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confidence: 99%

Littelmann paths for the basic representation of an affine Lie algebra

Magyar

2006

Journal of Algebra

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A highest-weight representation of an affine Lie algebraĝ can be modeled combinatorially in several ways, notably by the semi-infinite paths of the Kyoto school and by Littelmann's finite paths. In this paper, we unify these two models in the case of the basic representation of an untwisted affine algebra, provided the underlying finite-dimensional algebra g possesses a minuscule representation (i.e., g is of classical or E 6 , E 7 type).We apply our "coil model" to prove that the basic representation ofĝ, when restricted to g, is a semiinfinite tensor product of fundamental representations, and certain of its Demazure modules are finite tensor products.

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confidence: 99%

Littelmann paths for the basic representation of an affine Lie algebra

Magyar

2006

Journal of Algebra

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“…The first part ( §1) introduces Bott-Samelson varieties for a general reductive group, and shows the isomorphism between our new construction and the classical one. This lays the groundwork for our papers [13], [14] with V. Lakshmibai, giving a Standard Monomial Theory for Bott-Samelson varieties.…”

Section: Introductionmentioning

confidence: 79%

“…[23]. [13], [14] and §4.4 below, we make a general definition of "standard tableaux" giving bases of the Weyl modules for strongly separated families. (b) We briefly indicate the equivalence between our definition of the Weyl modules and the tensor product definition given in [1], [23], [21].…”

Section:  mentioning

confidence: 99%

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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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“…While the skew shapes have a long history, closely related to the semistandard Young tableaux (see, e.g., [Sa91]) these generalized shapes were introduced by Akin, Buchsbaum, and Weyman in [ABW82]. Recently, they have received considerable attention from the point of view of combinatorics and representation theory by Reiner and Shimozono [RS95,RS96,RS96c,S96] and from a more geometric viewpoint by Magyar and Lakshmibai [M94,LM96].…”

Section: Introductionmentioning

confidence: 99%

Compressed Straight Tableaux and a Distributive Lattice of Representations

Taylor

2000

Journal of Combinatorial Theory, Series A

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dedicated to the memory of gian-carlo rota A new``compressed straight'' basis for the polynomial ring Z[t i, j ] is constructed. This basis is then employed to study the lattice of representations generated by the representations associated with row-convex shapes under the operations of intersection and linear span. Applications to ascertaining the Cohen Macaulay property for rings associated to elements of this lattice are also given. Academic Press

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Standard monomial theory for Bott-Samelson varieties

Cited by 30 publications

References 5 publications

Littelmann paths for the basic representation of an affine Lie algebra

Littelmann paths for the basic representation of an affine Lie algebra

Schubert polynomials and Bott-Samelson varieties

Compressed Straight Tableaux and a Distributive Lattice of Representations

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