Relevant bibliographies by topics / Grassmannian varieties

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  2. Dissertations / Theses
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Academic literature on the topic 'Grassmannian varieties'

Author: Grafiati
Published: 4 June 2021
Last updated: 1 February 2022

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Journal articles on the topic "Grassmannian varieties"

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MARTÍN, FRANCISCO J. PLAZA. "PRYM VARIETIES AND THE INFINITE GRASSMANNIAN." International Journal of Mathematics 09, no. 01 (February 1998): 75–93. http://dx.doi.org/10.1142/s0129167x98000051.

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In this paper we study Prym varieties and their moduli space using the well-known techniques of the infinite Grassmannian. There are three main results of this paper: a new definition of the BKP hierarchy over an arbitrary base field (that generalizes the classical one over [Formula: see text]); a characterization of Prym varieties in terms of dynamical systems, and explicit equations for the moduli space of (certain) Prym varieties. For all of these problems the language of the infinte Grassmannian, in its algebro-geometric version, allows us to deal with these problems from the same point of view.
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Draisma, Jan, and Rob H. Eggermont. "Plücker varieties and higher secants of Sato’s Grassmannian." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 737 (April 1, 2018): 189–215. http://dx.doi.org/10.1515/crelle-2015-0035.

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AbstractEvery Grassmannian, in its Plücker embedding, is defined by quadratic polynomials. We prove a vast, qualitative, generalisation of this fact to what we callPlücker varieties. A Plücker variety is in fact a family of varieties in exterior powers of vector spaces that, like the Grassmannian, is functorial in the vector space and behaves well under duals. A special case of our result says that for each fixed natural numberk, thek-th secant variety ofanyPlücker-embedded Grassmannian is defined in bounded degree independent of the Grassmannian. Our approach is to take the limit of a Plücker variety in the dual of a highly symmetric space known as theinfinite wedge, and to prove that up to symmetry the limit is defined by finitely many polynomial equations. For this we prove the auxiliary result that for every natural numberpthe space ofp-tuples of infinite-by-infinite matrices is Noetherian modulo row and column operations. Our results have algorithmic counterparts: every bounded Plücker variety has a polynomial-time membership test, and the same holds for Zariski-closed, basis-independent properties ofp-tuples of matrices.
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Shimada, Ichiro. "Zariski Hyperplane Section Theorem for Grassmannian Varieties." Canadian Journal of Mathematics 55, no. 1 (February 1, 2003): 157–80. http://dx.doi.org/10.4153/cjm-2003-007-9.

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AbstractLet ϕ: X → M be a morphism from a smooth irreducible complex quasi-projective variety X to a Grassmannian variety M such that the image is of dimension ≥ 2. Let D be a reduced hypersurface in M, and γ a general linear automorphism of M. We show that, under a certain differentialgeometric condition on ϕ(X) and D, the fundamental group π1((γ ○ ϕ)−1 (M \ D)) is isomorphic to a central extension of π1(M \ D) × π1(X) by the cokernel of π2(ϕ) : π2(X) → π2(M).
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Braun, Lukas. "Hilbert series of the Grassmannian and k-Narayana numbers." Communications in Mathematics 27, no. 1 (June 1, 2019): 27–41. http://dx.doi.org/10.2478/cm-2019-0003.

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AbstractWe compute the Hilbert series of the complex Grassmannian using invariant theoretic methods. This is made possible by showing that the denominator of the q-Hilbert series is a Vandermonde-like determinant. We show that the h-polynomial of the Grassmannian coincides with the k-Narayana polynomial. A simplified formula for the h-polynomial of Schubert varieties is given. Finally, we use a generalized hypergeometric Euler transform to find simplified formulae for the k-Narayana numbers, i.e. the h-polynomial of the Grassmannian.
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LENAGAN, T. H., and L. RIGAL. "QUANTUM ANALOGUES OF SCHUBERT VARIETIES IN THE GRASSMANNIAN." Glasgow Mathematical Journal 50, no. 1 (January 2008): 55–70. http://dx.doi.org/10.1017/s0017089507003928.

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AbstractWe study quantum Schubert varieties from the point of view of regularity conditions. More precisely, we show that these rings are domains that are maximal orders and are AS-Cohen-Macaulay and we determine which of them are AS-Gorenstein. One key fact that enables us to prove these results is that quantum Schubert varieties are quantum graded algebras with a straightening law that have a unique minimal element in the defining poset. We prove a general result showing when such quantum graded algebras are maximal orders. Finally, we exploit these results to show that quantum determinantal rings are maximal orders.
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Serhiyenko, K., M. Sherman‐Bennett, and L. Williams. "Cluster structures in Schubert varieties in the Grassmannian." Proceedings of the London Mathematical Society 119, no. 6 (July 29, 2019): 1694–744. http://dx.doi.org/10.1112/plms.12281.

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Bakshi, Sarjick, S. Senthamarai Kannan, and Subrahmanyam K. Venkata. "Torus quotients of Richardson varieties in the Grassmannian." Communications in Algebra 48, no. 2 (September 27, 2019): 891–914. http://dx.doi.org/10.1080/00927872.2019.1668005.

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SCHILLEWAERT, J., and H. VAN MALDEGHEM. "A COMBINATORIAL CHARACTERIZATION OF THE LAGRANGIAN GRASSMANNIAN LG(3,6)()." Glasgow Mathematical Journal 58, no. 2 (July 21, 2015): 293–311. http://dx.doi.org/10.1017/s0017089515000208.

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AbstractWe provide a combinatorial characterization of LG(3,6)(${\mathbb{K}}$) using an axiom set which is the natural continuation of the Mazzocca–Melone set we used to characterize Severi varieties over arbitrary fields (Schillewaert and Van Maldeghem, Severi varieties over arbitrary fields,Preprint). This fits within a large project aiming at constructing and characterizing the varieties related to the Freudenthal–Tits magic square.
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Boe, Brian D., and Joseph H. G. Fu. "Characteristic Cycles in Hermitian Symmetric Spaces." Canadian Journal of Mathematics 49, no. 3 (June 1, 1997): 417–67. http://dx.doi.org/10.4153/cjm-1997-021-7.

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AbstractWe give explicit combinatorial expresssions for the characteristic cycles associated to certain canonical sheaves on Schubert varieties X in the classical Hermitian symmetric spaces: namely the intersection homology sheaves IHX and the constant sheaves ℂX. The three main cases of interest are the Hermitian symmetric spaces for groups of type An (the standard Grassmannian), Cn (the Lagrangian Grassmannian) and Dn. In particular we find that CC(IHX) is irreducible for all Schubert varieties X if and only if the associated Dynkin diagramis simply laced. The result for Schubert varieties in the standard Grassmannian had been established earlier by Bressler, Finkelberg and Lunts, while the computations in the Cn and Dn cases are new.Our approach is to compute CC(ℂX) by a direct geometric method, then to use the combinatorics of the Kazhdan-Lusztig polynomials (simplified for Hermitian symmetric spaces) to compute CC(IHX). The geometric method is based on the fundamental formula where the Xr ↓ X constitute a family of tubes around the variety X. This formula leads at once to an expression for the coefficients of CC(ℂX) as the degrees of certain singular maps between spheres.
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Upadhyay, Shyamashree. "Initial Ideals of Tangent Cones to the Richardson Varieties in the Orthogonal Grassmannian." International Journal of Combinatorics 2013 (March 26, 2013): 1–19. http://dx.doi.org/10.1155/2013/392437.

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A Richardson variety in the Orthogonal Grassmannian is defined to be the intersection of a Schubert variety in the Orthogonal Grassmannian and an opposite Schubert variety therein. We give an explicit description of the initial ideal (with respect to certain conveniently chosen term order) for the ideal of the tangent cone at any T-fixed point of , thus generalizing a result of Raghavan and Upadhyay (2009). Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the Orthogonal-bounded-RSK (OBRSK).
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Dissertations / Theses on the topic "Grassmannian varieties"

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Brunson, Jason Cory. "Matrix Schubert varieties for the affine Grassmannian." Diss., Virginia Tech, 2014. http://hdl.handle.net/10919/25286.

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Schubert calculus has become an indispensable tool for enumerative geometry. It concerns the multiplication of Schubert classes in the cohomology of flag varieties, and is typically conducted using algebraic combinatorics by way of a polynomial ring presentation of the cohomology ring. The polynomials that represent the Schubert classes are called Schubert polynomials. An ongoing project in Schubert calculus has been to provide geometric foundations for the combinatorics. An example is the recovery by Knutson and Miller of the Schubert polynomials for finite flag varieties as the equivariant cohomology classes of matrix Schubert varieties. The present thesis is the start of a project to recover Schubert polynomials for the Borel-Moore homology of the (special linear) affine Grassmannian by an analogous process. This requires finitizing an affine Schubert variety to produce a matrix affine Schubert variety. This involves a choice of ``window'', so one must then identify a class representative that is independent of this choice. Examples lead us to conjecture that this representative is a k-Schur function. Concluding the discussion is a preliminary investigation into the combinatorics of Gröbner degenerations of matrix affine Schubert varieties, which should lead to a combinatorial proof of the conjecture.
Ph. D.
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Kioulos, Charalambos. "From Flag Manifolds to Severi-Brauer Varieties: Intersection Theory, Algebraic Cycles and Motives." Thesis, Université d'Ottawa / University of Ottawa, 2020. http://hdl.handle.net/10393/40716.

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The study of algebraic varieties originates from the study of smooth manifolds. One of the focal points is the theory of differential forms and de Rham cohomology. It’s algebraic counterparts are given by algebraic cycles and Chow groups. Linearizing and taking the pseudo-abelian envelope of the category of smooth projective varieties, one obtains the category of pure motives. In this thesis, we concentrate on studying the pure Chow motives of Severi-Brauer varieties. This has been a subject of intensive investigation for the past twenty years, with major contributions done by Karpenko, [Kar1], [Kar2], [Kar3], [Kar4]; Panin, [Pan1], [Pan2]; Brosnan, [Bro1], [Bro2]; Chernousov, Merkurjev, [Che1], [Che2]; Petrov, Semenov, Zainoulline, [Pet]; Calmès, [Cal]; Nikolenko, [Nik]; Nenashev, [Nen]; Smirnov, [Smi]; Auel, [Aue]; Krashen, [Kra]; and others. The main theorem of the thesis, presented in sections 4.3 and 4.4, extends the result of Zainoulline et al. in the paper [Cal] by providing new examples of motivic decompositions of generalized Severi-Brauer varieties.
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Sullca, Alberth John Nuñez. "Decomposição celular de variedades Grassmannianas via teoria de Morse." Universidade Federal de Juiz de Fora (UFJF), 2017. https://repositorio.ufjf.br/jspui/handle/ufjf/4076.

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Apresentamos neste trabalho uma decomposição celular CW das variedades Grassmannianas via teoria de Morse. Isto é feito de duas maneiras distintas por meio de representações matriciais das Grassmannianas chamadas modelo projeção e modelo reflexão. Definimos funções de Morse, a saber, uma função do tipo altura e uma função do tipo “distância ao quadrado”, respectivamente, para cada um dos modelos projeção e reflexão. Estudamos os seus pontos críticos e os índices dos mesmos, obtendo assim duas formas para calcular a decomposição celular CW. Em particular, no modelo projeção, isto é feito exibindo-se as curvas integrais associadas ao campo gradiente da função altura.
We present in this work a CW cellular decomposition of Grassmannian varieties via Morse theory. This is done in two different ways. By means of matrix representations of Grassmannian called model projection and reflection model. We define Morse functions, namely a height-type function and a "square-distance" function, respectively, for each of the projection and reflection models. We study their critical points and their indices, thus obtaining two ways to calculate the CW cellular decomposition. In particular, in the projection model, this is done by displaying the integral curves associated with the gradient field of the height function.
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Pütz, Alexander [Verfasser], Markus [Gutachter] Reineke, Evgeny [Gutachter] Feigin, and Deniz [Gutachter] Kus. "Degenerate affine flag varieties and quiver grassmannians / Alexander Pütz ; Gutachter: Markus Reineke, Evgeny Feigin, Deniz Kus ; Fakultät für Mathematik." Bochum : Ruhr-Universität Bochum, 2019. http://d-nb.info/1195221487/34.

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Books on the topic "Grassmannian varieties"

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Real solutions to equations from geometry. Providence, R.I: American Mathematical Society, 2011.

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Clay Mathematics Institute Workshop on Moduli Spaces of Vector Bundles, with a View toward Coherent Sheaves (2006 Cambridge, Mass.). Grassmannians, moduli spaces, and vector bundles: Clay Mathematics Institute Workshop on Moduli Spaces of Vector Bundles, with a View towards Coherent Sheaves, October 6-11, 2006, Cambridge, Massachusetts. Edited by Ellwood D. (David) 1966- and Previato Emma. Providence, RI: American Mathematical Society, 2011.

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1963-, Shapiro Michael, and Vainshtein Alek 1958-, eds. Cluster algebra and Poisson geometry. Providence, R.I: American Mathematical Society, 2010.

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Misra, Kailash C., Milen Yakimov, Pramod N. Achar, and Dijana Jakelic. Recent advances in representation theory, quantum groups, algebraic geometry, and related topics: AMS special sessions on geometric and algebraic aspects of representation theory and quantum groups, and noncommutative algebraic geometry, October 13-14, 2012, Tulane University, New Orleans, Louisiana. Providence, Rhode Island: American Mathematical Society, 2014.

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1938-, Griffiths Phillip, and Kerr Matthew D. 1975-, eds. Hodge theory, complex geometry, and representation theory. Providence, Rhode Island: Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, 2013.

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Book chapters on the topic "Grassmannian varieties"

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Lakshmibai, V., and Justin Brown. "Determinantal Varieties." In The Grassmannian Variety, 143–53. New York, NY: Springer New York, 2015. http://dx.doi.org/10.1007/978-1-4939-3082-1_10.

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Lakshmibai, V., and Justin Brown. "The Grassmannian and Its Schubert Varieties." In The Grassmannian Variety, 51–71. New York, NY: Springer New York, 2015. http://dx.doi.org/10.1007/978-1-4939-3082-1_5.

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Lakshmibai, V., and Justin Brown. "Further Geometric Properties of Schubert Varieties." In The Grassmannian Variety, 73–94. New York, NY: Springer New York, 2015. http://dx.doi.org/10.1007/978-1-4939-3082-1_6.

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Seshadri, C. S. "Schubert Varieties in the Grassmannian." In Texts and Readings in Mathematics, 1–53. Singapore: Springer Singapore, 2016. http://dx.doi.org/10.1007/978-981-10-1813-8_1.

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Lakshmibai, V., and Justin Brown. "Geometry of the Grassmannian, Flag and their Schubert Varieties via Standard Monomial Theory." In Texts and Readings in Mathematics, 197–223. Gurgaon: Hindustan Book Agency, 2009. http://dx.doi.org/10.1007/978-93-86279-41-5_12.

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Lakshmibai, V., and Justin Brown. "Geometry of the Grassmannian, Flag and their Schubert Varieties via Standard Monomial Theory." In Texts and Readings in Mathematics, 165–86. Singapore: Springer Singapore, 2018. http://dx.doi.org/10.1007/978-981-13-1393-6_12.

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Contou-Carrère, Carlos. "Grassmannians and Flag Varieties." In Buildings and Schubert Schemes, 1–19. Boca Raton : Taylor & Francis, 2016. | “A CRC title.”: CRC Press, 2017. http://dx.doi.org/10.1201/9781315367309-1.

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Harris, Joe. "Grassmannians and Related Varieties." In Algebraic Geometry, 63–71. New York, NY: Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4757-2189-8_6.

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Fulton, William, and Piotr Pragacz. "Modern formulation; Grassmannians, flag varieties, schubert varieties." In Schubert Varieties and Degeneracy Loci, 14–25. Berlin, Heidelberg: Springer Berlin Heidelberg, 1998. http://dx.doi.org/10.1007/bfb0096382.

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Contou-Carrère, Carlos. "Schubert Cell Decomposition of Grassmannians and Flag Varieties." In Buildings and Schubert Schemes, 20–48. Boca Raton : Taylor & Francis, 2016. | “A CRC title.”: CRC Press, 2017. http://dx.doi.org/10.1201/9781315367309-2.

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