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Permutation representations on Schubert varieties [article]

Julianna S. Tymoczko
2007 arXiv   pre-print
This paper defines and studies permutation representations on the equivariant cohomology of 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.  ... 
arXiv:math/0604578v2 fatcat:ypok3li6kbhlzk2nzqlqzaogl4

Gröbner Bases for Schubert Codes [article]

Arunkumar R. Patil, Nitin S. Darkunde
2017 arXiv   pre-print
We also use them to study the decoding of binary 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.  ... 
arXiv:1707.02199v2 fatcat:sn3o22ap55c3xfhwiqx7gexaoq

F. Schubert, Zur Theorie des stationären Verdichtungsstoßes. Z. angew. Math. Mech. Bd. 23 (1943), S. 129 bis 138

E. Groth
1944 Zeitschrift für angewandte Mathematik und Mechanik  
Schubert. 558 H. Karl,‚Biegung gekrümmter, dünnwandiger Rohre. Z. angew. Math. Mech. Bd. 23 (1943), S. 331 bis 345.  ...  Schubert genau überein- stimmen.  ... 
doi:10.1002/zamm.19440240215 fatcat:kww2qxqhgrgtre2o5zhdqbyju4

On the automorphism of a smooth Schubert variety [article]

S. Senthamarai Kannan
2015 arXiv   pre-print
Let w be an element of the Weyl group W and let X(w) be the 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.  ... 
arXiv:1312.7066v3 fatcat:capme2x4afgl7ilwvt3s6tikcy

Torus Quotients of Schubert Varieties

B. Narasimha Chary, S. K. Pattanayak
2020 International Journal of Mathematics  
In this paper, we consider the GIT quotients of 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.  ... 
doi:10.1142/s0129167x20501232 fatcat:iojqvsqlnrfk5gfxeik735qtui

Towards Generalizing Schubert Calculus in the Symplectic Category [article]

R. F. Goldin, S. Tolman
2009 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 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.  ... 
arXiv:0904.1245v1 fatcat:zuzqm7cudrcmtbr7ox2e37o73m

Smooth torus quotients of Schubert varieties in the Grassmannian [article]

Sarjick Bakshi, S. Senthamarai Kannan, K.Venkata Subrahmanyam
2019 arXiv   pre-print
We study the GIT quotient of 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] .  ... 
arXiv:1912.08618v2 fatcat:5uh64vz53rfbpgklsuca2ga7ni

A family of reductions for Schubert intersection problems [article]

H. Bercovici, W. S. Li, D. Timotin
2009 arXiv   pre-print
We produce a family of reductions for 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 .  ... 
arXiv:0909.0908v1 fatcat:y63xrpqravcvdhk6zm7fkoqati

Free resolutions of some Schubert singularities [article]

Manoj Kummini and V. Lakshmibai and Pramathanath Sastry and C. S. Seshadri
2015 arXiv   pre-print
Our approach uses the geometry of 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.  ... 
arXiv:1504.04415v1 fatcat:njxpcfnsffdy3ffktqtfraewmy

Maximal singular loci of Schubert varieties in SL(n)/B [article]

Sara C. Billey, Gregory S. Warrington
2001 arXiv   pre-print
We give an explicit combinatorial description of the irreducible components of the singular locus of the 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/  ... 
arXiv:math/0102168v1 fatcat:qxwjb2uimnamteplufmecw27ky

Permutation representations on Schubert varieties

Julianna S. Tymoczko
2008 American Journal of Mathematics  
This paper defines and studies permutation representations on the equivariant cohomology of 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.  ... 
doi:10.1353/ajm.0.0018 fatcat:rtbnnxv37zamjkkpvf363thniq

Torus quotients of Schubert varieties in the Grassmannian G_2,n [article]

S. Senthamarai Kannan, Arpita Nayek, Pinakinath Saha
2021 arXiv   pre-print
In this article, we study the GIT quotients of the 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 .  ... 
arXiv:2103.12621v1 fatcat:3uku4dhc7jdrhgsf6lieb7hhk4

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

Usha N. Bhosle
2013 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.  ... 
doi:10.1090/s0273-0979-2013-01429-0 fatcat:d2ux2ajkxndjtfeszoyay5sj6m

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

S. Senthamarai Kannan
2013 arXiv   pre-print
We give a necessary and a sufficient condition for a 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 )).  ... 
arXiv:1212.6338v2 fatcat:uzgi2pse3jfyniccmcuxum7ate

Schubert and Salieri

S T Green
1991 Journal of the Royal Society of Medicine  
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.  ... 
doi:10.1177/014107689108400539 pmid:2041027 pmcid:PMC1293253 fatcat:7tfxhmj4uzextkokurcc4hfpyq
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