A description based on Schubert classes of cohomology of flag manifolds

We describe the integral cohomology rings of the flag manifolds of types B_n, D_n, G_2 and F_4 in terms of their Schubert classes. The main tool is the divided difference operators of Bernstein-Gelfand-Gelfand and Demazure. As an application, we compute the Chow rings of the corresponding complex algebraic groups, recovering thereby the results of R. Marlin.Comment: 25 pages, AMS-LaTeX; typos corrected, an error concerning the Schubert classes corrected, Remarks 4.4 and 4.8 adde

The integral cohomology ring of E_8/T

We give a complete description of the integral cohomology ring of the flag manifold E_8/T, where E_8 denotes the compact exceptional Lie group of rank 8 and T its maximal torus, by the method due to Borel and Toda. This completes the computation of the integral cohomology rings of the flag manifolds for all compact connected simple Lie groups.Comment: 5 pages, AMS-LaTeX, Minor errors corrected

Generating functions for the universal Hall-Littlewood PP- and QQ-functions

Recently, P. Pragacz described the ordinary Hall-Littlewood $P$-polynomials by means of push-forwards (Gysin maps) from flag bundles in the ordinary cohomology theory. Together with L. Darondeau, he also gave push-forward formulas (Gysin formulas) for all flag bundles of types $A$, $B$, $C$ and $D$ in the ordinary cohomology theory. In this paper, we introduce a generalization of the ordinary Hall-Littlewood $P$- and $Q$-polynomials, which we call the {\it universal $($factorial$)$ Hall-Littlewood $P$- and $Q$-functions}, and characterize them in terms of Gysin maps from flag bundles in the complex cobordism theory. We also generalize the (type $A$) push-forward formula due to Darondeau-Pragacz to the complex cobordism theory. As an application of our Gysin formulas in complex cobordism, we give generating functions for the universal Hall-Littlewood $P$- and $Q$-functions and their factorial analogues. Using our generating functions, classical determinantal and Pfaffian formulas for Schur $S$- and $Q$-polynomials, and their $K$-theoretic or factorial analogues can be obtained in a simple and unified manner.Comment: 46 pages, AMSLaTeX; Section 6 added, An error of the generating function for the universal factorial Hall-Littlewood $P$-functions was correcte

Flux quench in a system of interacting spinless fermions in one dimension

We study a quantum quench in a one-dimensional spinless fermion model (equivalent to the XXZ spin chain), where a magnetic flux is suddenly switched off. This quench is equivalent to imposing a pulse of electric field and therefore generates an initial particle current. This current is not a conserved quantity in presence of a lattice and interactions and we investigate numerically its time-evolution after the quench, using the infinite time-evolving block decimation method. For repulsive interactions or large initial flux, we find oscillations that are governed by excitations deep inside the Fermi sea. At long times we observe that the current remains non-vanishing in the gapless cases, whereas it decays to zero in the gapped cases. Although the linear response theory (valid for a weak flux) predicts the same long-time limit of the current for repulsive and attractive interactions (relation with the zero-temperature Drude weight), larger nonlinearities are observed in the case of repulsive interactions compared with that of the attractive case.Comment: 10 pages, 10 figures; v2: Added references. Figures are refined and animations are added. Corrected typos. Published versio