Narrow Results

We propose a potential function $W$ for the cohomology ring of partial flag manifolds. We prove a formula expressing integrals over partial flag manifolds by residues, which generalizes [E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, in Geometry, Topology, Physics (International Press, 1995), pp. 357–422]. Using this formula, we prove a Landweber–Stong type vanishing theorem for generalized ${\mathrm{\text{string}}}^c$ complete intersections in flag manifolds, which serves as evidence for the ${\mathrm{\text{string}}}^c$ version of Stolz conjecture [Q. Chen, F. Han and W. Zhang, Generalized Witten genus and vanishing theorems, J. Differential Geom.88(1) (2011) 1–39].

The Landau problem on the flag manifold F₂ = SU(3)/U(1)×U(1) is analyzed from an algebraic point of view. The involved magnetic background is induced by two U(1) Abelian connections. In quantizing the theory, we show that the wave functions, of a nonrelativistic particle living on F₂, are the SU(3) Wigner -functions satisfying two constraints. Using the F₂ algebraic and geometrical structures, we derive the Landau Hamiltonian as well as its energy levels. The lowest Landau level (LLL) wave functions coincide with the coherent states for the mixed SU(3) representations. We discuss the quantum Hall effect for a filling factor ν = 1, where the obtained particle density is constant and finite for a strong magnetic field. In this limit, we also show that the system behaves like an incompressible fluid. We study the semiclassical properties of the system confined in LLL. These will be used to discuss the edge excitations and construct the corresponding Wess–Zumino–Witten action.

We introduce inversions for classical Weyl group elements and relate them, by counting, to the length function, root systems and Schubert cells in flag manifolds. Special inversions are those that only change signs in the Weyl groups of types B_n, C_n and D_n. Their counting is related to the (only) generator of the Weyl group that changes signs, to the corresponding roots, and to a special subvariety in the flag manifold fixed by a finite group.

This paper takes a detailed look at a subject that occurs in various contexts in mathematics, the fixed-point sets of torus actions on flag manifolds, and considers it from the (perhaps nontraditional) perspective of moment maps and length functions on Weyl groups. The approach comes from earlier work of the author where it is shown that certain singular flows in the Hamiltonian system known as the Toda lattice generate the action of a group A on a flag manifold, where A is a direct product of a non-maximal torus and unipotent group. As a first step in understanding the orbits of A in connection with the Toda lattice, this paper seeks to understand the fixed points of the non-maximal tori in this setting.

There is a unipotent subgroup of Sl(n, C) whose action on the flag manifold of Sl(n, C) completes the flows of the complex Kostant–Toda lattice (a Hamiltonian system in Lax form) through initial conditions where all the eigenvalues coincide. The action preserves the Bruhat cells, which are in one-to-one correspondence with the elements of the permutation group Σ_n. A generic orbit in a given cell is homeomorphic to C^m, where m is determined by the "gap sequence" of the permutation, which lists the number inversions of each degree.

A flag manifold is a homogeneous space M = G/K, where G is a compact semisimple Lie group, and K the centralizer of a torus in G. Equivalently, M can be identified with the adjoint orbit Ad(G)w of an element w in the Lie algebra of G. We present several aspects of flag manifolds, such as their classification in terms of painted Dynkin diagrams, T-roots and G-invariant metrics, and Kähler metrics. We give a Lie-theoretic expression of the Ricci tensor in M, hence reducing the Einstein equation on flag manifolds into an algebraic system of equations, which can be solved in several cases. A flag manifold is also a complex manifold, and this dual representation as a real and a complex manifold is related to a similar property of an infinite-dimensional manifold, the loop space, which in fact can be viewed as a "universal" flag manifold.

We associate to a homogeneous manifold M = G/H, with a simple spectrum of the isotropy representation, a compact convex polytope P_M which is the Newton polytope of the rational function s(t) and that to each invariant metric t of M associates its scalar curvature. We estimate the number of isolate invariant holomorphic Einstein metrics (up to homothety) on M^ℂ = G^ℂ/H^ℂ. Using the results of A. G. Kouchnirenko and D. N. Bernstein, we prove that , where ν(M) is the integer volume of P_M, and give conditions when the defect . In case when G is a compact semisimple Lie group, the positiveness of d(M) is related with the existence of Ricci-flat holomorphic metric on a complexification of a noncompact homogeneous space M_γ = G_γ/H_P which is a contraction of M and is associated with a proper face γ of P_M.

In this paper we study the evolution operator of a time-dependent Hamiltonian in the three level system. The evolution operator is based on SU(3) and its dimension is eight, so we obtain three complex Riccati differential equations interacting with one another (which have been obtained by Fujii and Oike) and two real phase equations. This is a canonical form of the evolution operator.

Going back to Kirwan and others, there is an established theory that uses moment map techniques to study actions by complex reductive groups on Kähler manifolds. Work of P. Heinzner, G. Schwarz, and H. Stötzel has extended this theory to actions by real reductive groups. In this paper, we apply these techniques to actions of the real group SU(1,1) ⊂ SL(2, ℂ) on a certain complex manifold of dimension two. More precisely, because of the SU(2)-invariant hyperkähler structure on this manifold, we are able to study a family of actions which includes and “interpolates” two well-known actions of SL(2, ℂ): the adjoint action on the orbit of a semisimple element of 𝔰𝔩(2, ℂ), and the action of SL(2, ℂ) on the cotangent bundle of the flag variety of SL(2, ℂ).

The Grassmannian has a natural action by an algebraic torus. In this survey we describe the diagonal action of this torus on subvarieties of products of Grassmannians. From this action we describe how to construct an associated moment polytope. The main varieties considered are Schubert varieties, Richardson varieties, and their desingularizations. The moment polytopes of these varieties include the permutahedron and associahedron. We also look at the Barbasch-Evens-Magyar varieties, which are desingularizations of symmetric orbit closures in the flag manifold.