When computing the homology of Grassmannian manifolds, the first step is usually to look at the Schubert cell decomposition, and the chain complex associated with it. In the complex case and the real unoriented case with Z₂ coefficients the additive structure is obtained immediately (i.e., generated by the homology classes represented by the Schubert cells) because the boundary map is trivial. In the real unoriented case (with Z₂ coefficients) and the real oriented case, finding the additive structure is more complicated since the boundary map is nontrivial. In this paper, this boundary map is computed by cell orientation comparisons, using graph coordinates where the cells are linear, to simplify the comparisons. The integral homology groups for some low dimensional oriented and unoriented Grassmannians are determined directly from the chain complex (with the boundary map as computed). The integral cohomology ring structure for complex Grassmannians has been completely determined mainly using Schubert cell intersections (what is known as Schubert Calculus).. In this paper, a method using Schubert cell intersections to describe the Z₂ cohomology ring structure of the real Grassmannians is sketched. The results are identical to those for the complex Grassmannians (with coefficients), but the notation used for the cohomology generators is not the usual one. It indicates that the products are to a certain degree independent of the Grassmannian. / Science, Faculty of / Mathematics, Department of / Graduate

No description available.

Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2006. / Includes bibliographical references (leaves 57-59). Also available in electronic version.

In this thesis we introduce and study the (i) Grassmannian, (ii) Lagrangian Grassmannian, and (iii) double Lagrangian Grassmannian of subspaces in ( A ⊗ B )n, where A and B are normed division algebras, i.e. R,C,H or O . / This gives a simple and uniform description of all symmetric spaces. This is analogous to Tits magic square description for simple Lie algebras. / We show that every irreducible compact Riemannian symmetric space X must be one of these Grassmannian spaces (up to a finite cover) or a compact simple Lie group. Furthermore, its noncompact dual symmetric space is the open sub-manifold of X consisting of spacelike linear subspaces, at least in the classical cases. / Huang, Yongdong. / "July 2007." / Adviser: Naichung Conan Leung. / Source: Dissertation Abstracts International, Volume: 69-01, Section: B, page: 0353. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 64-65). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

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Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

Thesis (Ph.D.)--Tufts University, 2004. / Adviser: Fulton B. Gonzalez. Submitted to the Dept. of Mathematics. Includes bibliographical references (leaves 72-73). Access restricted to members of the Tufts University community. Also available via the World Wide Web;

The representation theory of infinite dimensional groups is in its infancy. This paper is an attempt to apply the orbit method to a particular infinite dimensional group, the spin extension of the restricted unitary group. Our main contribution is in showing that various homogeneous spaces for this group admit measures which can be used to realize the unitary structure for the standard modules.

In this work, the connection between quantum K-theory and quantum integrable systems is studied. Using quasimap spaces the quantum equivariant K-theory of Naka- jima quiver varieties is defined. For every tautological bundle in the K-theory there exists its one-parametric deformation, referred to as quantum tautological bundle. For specific cases of cotangent bundles to Grassmannians and flag varieties it is proved that the spectrum of operators of quantum multiplication by these quantum classes is governed by the Bethe ansatz equations for the inhomogeneous XXZ spin chain. It is also proved that each such operator corresponds to the universal elements of quantum group U􏰁(sln). In particular, the Baxter operator for the XXZ spin chain is identified with the operator of quantum multiplication by the exterior algebra of the tautological bundle. An explicit universal combinatorial formula for this operator is found in the case of U􏰁(sl2). The relation between quantum line bundles and quantum dynamical Weyl group is shown. This thesis is based on works [37] and [24].

<p> This thesis develops the Riemannian Geometry of the real and complex Grassmann Manifolds in a notationally accessible way. The canonical volume form is related to explicit Jacobi Field calculations. The implementation of a packing algorithm based on repulsive forces is proposed. Standard packing bounds and bounds on the volumes of geodesic balls are used to test the performance of the algorithm.</p> / Thesis / Master of Science (MSc)