Borel–Weil Theorem for Configuration Varieties and Schur Modules

Magyar, Péter

doi:10.1006/aima.1997.1700

Cited by 25 publications

(41 citation statements)

References 26 publications

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“…In this paper we supply several such missing analogs, mainly concerning interpretations 1) and 4): analogs of the Weyl and Demazure character formulas; and a straightforward construction for the mysterious tableaux of Lascoux and Schutzenberger, showing how they "quantize" our Demazure formula. These results also hold for a broad class of Schur-like polynomials associated to generalized Young diagrams, such as skew Schur polynomials [1], [23], [24], [25], [21].…”

Section: Introductionmentioning

confidence: 54%

“…In our paper [21] and Zelevinsky's work [26], there are given other desingularizations of Schubert varieties, all of them expressible as flagged configuration varieties.…”

Section: Examples Of Varietiesmentioning

confidence: 94%

“…(b) The conjecture is known if D satisfies the "northwest condition" (see [21]): that is, the elements of D can be arranged in an order…”

Section: Conjecture 7 For Any Subset Familymentioning

confidence: 99%

“…(We sketch the connection with the symmetrizer picture at the end of §3.1.) Using the same arguments as in [21], our Borel-Weil Theorem is immediate, and we work out a version of Demazure's character formula to get a new expression for generalized Schur polynomials.…”

Section: Schur and Weyl Modulesmentioning

confidence: 99%

“…This method follows our paper [21], but we must do extra geometric work here, giving a precise connection between our B-modules and the Bott-Samelson varieties Z. As a by-product of our analysis, we obtain a new construction of the BottSamelson varieties for an arbitrary reductive group G. In our case G = GL(n), the new construction realizes Z as the variety of representations of a partially ordered set.…”

Section: Introductionmentioning

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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Section: Introductionmentioning

confidence: 54%

“…In our paper [21] and Zelevinsky's work [26], there are given other desingularizations of Schubert varieties, all of them expressible as flagged configuration varieties.…”

Section: Examples Of Varietiesmentioning

confidence: 94%

“…(b) The conjecture is known if D satisfies the "northwest condition" (see [21]): that is, the elements of D can be arranged in an order…”

Section: Conjecture 7 For Any Subset Familymentioning

confidence: 99%

Section: Schur and Weyl Modulesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Schubert polynomials and Bott-Samelson varieties

Magyar¹

1998

Comment. Math. Helv.

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Bioriented flags and resolutions of Schubert varieties

Cibotaru

2020

Mathematische Nachrichten

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We use incidence relations running in two directions in order to construct a Kempf–Laksov type resolution for any Schubert variety of the complete flag manifold but also an embedded resolution for any Schubert variety in the Grassmannian. These constructions are alternatives to the celebrated Bott–Samelson resolutions. The second process led to the introduction of W‐flag varieties, algebro‐geometric objects that interpolate between the standard flag manifolds and products of Grassmannians, but which are singular in general. The surprising simple desingularization of a particular such type of variety produces an embedded resolution of the Schubert variety within the Grassmannian.

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