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Permutation representations on Schubert varieties
[article]

2007
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arXiv
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pre-print

This paper defines and studies permutation representations on the equivariant cohomology of

arXiv:math/0604578v2
fatcat:ypok3li6kbhlzk2nzqlqzaogl4
*Schubert*varieties, as representations both over C and over C[t_1, t_2,...,t_n]. ... i+1v u ) u∈Sn for the corresponding*Schubert*classes. Then (1) , . . . , t u(n) ). Corollary 3.4. Let v,*s*i,i+1 ∈*S*n satisfy*s*i,i+1 v > v. ... Every*Schubert*variety is T -equivariant and is equivariantly formal with respect to this action. Proof. Each*Schubert*variety has a cell decomposition as a union of*Schubert*cells. ...##
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Gröbner Bases for Schubert Codes
[article]

2017
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arXiv
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pre-print

We also use them to study the decoding of binary

arXiv:1707.02199v2
fatcat:sn3o22ap55c3xfhwiqx7gexaoq
*Schubert*codes. ... In this paper, Gr\"obner bases of linear codes associated to Grassmann varieties and*Schubert*varieties over a binary field have been obtained. ... For u, v ∈ F n q , one can easily prove that*S*(u + v) =*S*(u) +*S*(v), and*S*(u) = 0 ⇔ u ∈ C,*S*(u) =*S*(v) ⇔ u and v are in the same coset of C. ...##
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F. Schubert, Zur Theorie des stationären Verdichtungsstoßes. Z. angew. Math. Mech. Bd. 23 (1943), S. 129 bis 138

1944
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Zeitschrift für angewandte Mathematik und Mechanik
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*Schubert*. 558 H. Karl,‚Biegung gekrümmter, dünnwandiger Rohre. Z. angew. Math. Mech. Bd. 23 (1943),

*S*. 331 bis 345. ...

*Schubert*genau überein- stimmen. ...

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On the automorphism of a smooth Schubert variety
[article]

2015
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arXiv
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pre-print

Let w be an element of the Weyl group W and let X(w) be the

arXiv:1312.7066v3
fatcat:capme2x4afgl7ilwvt3s6tikcy
*Schubert*variety in G/B corresponding to w. Let α_0 denote the highest root of G with respect to T and B. ...*S*. Seshadri, A. J. Parameswaran and D.*S*. Nagaraj for useful discussions. ... For w ∈ W , let X(w) := BwB/B denote the*Schubert*variety in G/B corresponding to w. ...##
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Torus Quotients of Schubert Varieties

2020
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International Journal of Mathematics
*

In this paper, we consider the GIT quotients of

doi:10.1142/s0129167x20501232
fatcat:iojqvsqlnrfk5gfxeik735qtui
*Schubert*varieties for the action of a maximal torus. ... As a consequence, we study the smoothness of torus quotients of*Schubert*varieties in the Grassmannian. ... The partial flag variety G/P ω is called minuscule and the*Schubert*varieties in G/P ω are called minuscule*Schubert*varieties. ...##
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Towards Generalizing Schubert Calculus in the Symplectic Category
[article]

2009
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arXiv
*
pre-print

When they exist, canonical classes form a natural basis of the equivariant cohomology of M; in particular, when M is a flag variety, these classes are the equivariant

arXiv:0904.1245v1
fatcat:zuzqm7cudrcmtbr7ox2e37o73m
*Schubert*classes. ... The main purpose of this article is to extend some of the ideas from*Schubert*calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points ... In the case that M = G/B, where G is a complex semi-simple Lie group (of any type) and B is a Borel subgroup, the equivariant*Schubert*classes are canonical classes. ...##
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Smooth torus quotients of Schubert varieties in the Grassmannian
[article]

2019
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arXiv
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pre-print

We study the GIT quotient of

arXiv:1912.08618v2
fatcat:5uh64vz53rfbpgklsuca2ga7ni
*Schubert*varieties X(w) in the Grassmannian G_r,n, admitting semistable points for the action of T with respect to the T-linearized line bundle L(nω_r). ...*s*4*s*3*s*8*s*7*s*6*s*5*s*4 , the element in W P corresponding to the*Schubert*variety 3, 5, 6, 7), respectively. ... Smooth locus of*Schubert*varieties in G r,n The singular loci of*Schubert*varieties in miniscule G/P were determined in [11] . ...##
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A family of reductions for Schubert intersection problems
[article]

2009
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arXiv
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pre-print

We produce a family of reductions for

arXiv:0909.0908v1
fatcat:y63xrpqravcvdhk6zm7fkoqati
*Schubert*intersection problems whose applicability is checked by calculating a linear combination of the dimensions involved. ... E ′ , I ′ ) ∩*S*(F ′ , J ′ ) ∩*S*(G ′ , K ′ ) ⊂*S*(E, I) ∩*S*(F , J) ∩*S*(G, K). ... Assume that we want to solve the*Schubert*problem associated to a measure m ∈ M r . ...##
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Free resolutions of some Schubert singularities
[article]

2015
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arXiv
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pre-print

Our approach uses the geometry of

arXiv:1504.04415v1
fatcat:njxpcfnsffdy3ffktqtfraewmy
*Schubert*varieties. ... In the first case, Q*s*= B n , so p makes ZP 1 (w) a vector-bundle on a smooth*Schubert*subvariety X B 1 (w ′ ) of GL n /B n . ... #{*s*α | α ∈ R − \ R − P and τ ≥*s*α in W/WP }. (1) (3,1) | O − GL 4 /B = x 31 , p (1) Theorem 3.4. With notation as above, suppose that the*Schubert*variety XP (w) of GL N /P is smooth. ...##
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Maximal singular loci of Schubert varieties in SL(n)/B
[article]

2001
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arXiv
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pre-print

We give an explicit combinatorial description of the irreducible components of the singular locus of the

arXiv:math/0102168v1
fatcat:qxwjb2uimnamteplufmecw27ky
*Schubert*variety X_w for any element w in S_n. ... In the late 1950'*s*, Chevalley [Che94] showed that all*Schubert*varieties in G/B are nonsingular in codimension one. ... of*Schubert*varieties, the classes of*Schubert*varieties form a basis for the cohomology ring of G/B and the*Schubert*varieties correspond to the lower order ideals of a partial order associated to G/ ...##
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Permutation representations on Schubert varieties

2008
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American Journal of Mathematics
*

This paper defines and studies permutation representations on the equivariant cohomology of

doi:10.1353/ajm.0.0018
fatcat:rtbnnxv37zamjkkpvf363thniq
*Schubert*varieties, as representations both over C and over C[t 1 , t 2 , . . . , tn]. ... i+1v u ) u∈Sn for the corresponding*Schubert*classes. Then (1) , . . . , t u(n) ). Corollary 3.4. Let v,*s*i,i+1 ∈*S*n satisfy*s*i,i+1 v > v. ... Every*Schubert*variety is T -equivariant and is equivariantly formal with respect to this action. Proof. Each*Schubert*variety has a cell decomposition as a union of*Schubert*cells. ...##
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Torus quotients of Schubert varieties in the Grassmannian G_2,n
[article]

2021
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arXiv
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pre-print

In this article, we study the GIT quotients of the

arXiv:2103.12621v1
fatcat:3uku4dhc7jdrhgsf6lieb7hhk4
*Schubert*varieties in the Grassmannian G_2,n. ... Also, we prove that the GIT quotients of the*Schubert*varieties in G_2,n have at most finite set of singular points. ... Let w = (*s*6*s*5*s*4*s*3*s*2*s*1 )(*s*9*s*8*s*7*s*6*s*5*s*4*s*3*s*2 ). In one line notation w = (7, 10). We consider the*Schubert*variety X(w) in the Grassmannian G 2, 10 . ...##
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Book Review: Collected papers of C. S. Seshadri. Volume 1. Vector bundles and invariant theory; Collected papers of C. S. Seshadri. Volume 2. Schubert geometry and representation Theory

2013
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Bulletin of the American Mathematical Society
*

*S*. Narasimhan and C.

*S*. Seshadri The seminal work of M.

*S*. Narasimhan and C.

*S*. Seshadri in the late 1960s began a new era in the theory of vector bundles on a compact Riemann surface X. ... Consequently, it has several applications in the study of the geometry of the

*Schubert*variety. ...

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An analogue of Bott's theorem for Schubert varieties-related to torus semistable points
[article]

2013
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arXiv
*
pre-print

We give a necessary and a sufficient condition for a

arXiv:1212.6338v2
fatcat:uzgi2pse3jfyniccmcuxum7ate
*Schubert*variety X(τ) for which all the higher cohomologies H^i(X(τ), E) vanish for the restriction E of the tangent bundle of G/B to X(τ). ... If V ′ is of type (5) , we again use lemma (2.3) to conclude that H 0 (*s*α , V ′ ) must be of type (2) in H 0 (*s*α , H 1 (*s*α τ, V )). ... If V ′ is of type (5) , we again use lemma (2.3) to conclude that H 0 (*s*α , V ′ ) must be of type (2) in H 0 (*s*α , H 1 (*s*α τ, V )). ...##
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Schubert and Salieri

1991
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Journal of the Royal Society of Medicine
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*Schubert*. In The Second Book of the Great Musicians, 6th edn. ...

*S*T GREEN Mozart's last illness I want to express a word of thanks for Wheater's precise discussion of Mozart's last illness (September 1990 JRSM, p 586) and to make a few observations. ...

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