Equivariant oriented cohomology of flag varieties

In this chapter basic ingredients of the theory of linear algebraic groups are introduced: maximal tori, Borel groups, parabolic subgroups. Fundamental results are the conjugacy theorems for Borel groups and maximal tori (6.2.7 and6.4.1). The structure theory of connected solvable groups is also treated. The chapter begins with a brief discussion of complete varieties.

Geometric representations of the formal affine Hecke algebra

Let G be a group and let R be a commutative ring. Let T rP icR(RG) be the group of isomorphism classes of standard self-equivalences of the derived category of bounded complexes of RG-modules. The subgroup HDR(G) of T rP icR(RG) consisting of self-equivalences xing the trivial RG-module acts on the cohomology ring H (G; R). The action is functorial with respect to R. The self-equivalences which are 'splendid' in a sense dened by J. Rickard act natural with respect to transfer and restriction to centralizers of p-subgroups in case R is a eld of characteristic p. In the present paper we prove that this action of self-equivalences on H (G; R) commutes with the action of the Steenrod algebra and study the behaviour of the action of splendid self-equivalences with respect to Lannes' T -functor. Introduction In an earlier joint paper (9) with RaphaRouquier I dened the group T rP icR(A) of standard self-equivalences of a derived module category D b (A) for an R-algebra A which is projective as an R-module. For any A-module M let HDM(A) be the subgroup of T rP icR(A) which is formed by the self-equivalences mapping M to an isomorphic copy. Then, in an earlier paper (11) I showed that, under some hypothesis on M, the group HDM(A) acts in a natural way on the Ext-algebra Ext A (M; M). In case of A being a group algebra RG, with R being a eld of characteristic p and G being a nite group, then J. Rickard denes in (8) what is a splendid equivalence by some technical conditions basically by asking that the homogeneous components of a tilting complex are p-permutation modules induced from diagonal p-subgroups, and by some invertibility condition in the homotopy category. These splendid equivalences induce self-equivalences of the derived categories of centralizers of p-subgroups by the Brauer construction. In (12) I showed that then, for M = R being the trivial module, the action of those splendid equivalences commute with transfer and restriction from and to the cohomology rings of centralizers of p-subgroups. In the present paper we enlarge these properties still further. The action of self-equivalences of the bounded derived category D b (RG) on H (G; R) commutes with the action of the Steenrod algebra on H (G; R) for any prime p. As consequence of the above statements, any cohomology ring H (G; Fp) denes a functor H (G; Fp) p HD p (B0( p G)) from the modules over the group of derived self-equivalences of the principal block of the group ring Fp HD p (FpG) xing the trivial module to the category of unstable modules Up and similarly in the opposite direction. Obviously, we may restrict to splendid self-equivalences. We shall describe the composition of Lannes' T -functor with the rst functor and the im- age of free unstable modules by the second. Moreover, by a result of Lannes TV (H (G; Fp)) decomposes into direct product of cohomology rings as unstable modules. We shall prove that this decomposition is also a decomposition as modules over the action of splendid self- equivalences. This will give evidence that splendid equivalences are the correct objects to study in this context. The paper is organized as follows. Section 1 recalls the necessary denitions and properties of Even's norm map as it is used here and the denition of the Steenrod operation. In Section 2 it is shown that the normalized part of the outer automorphism group of the

For a linear algebraic group G over an algebraically closed field k and a parabolic subgroup P of G the modality of P is defined to be the maximal number of parameters upon which a family of G-orbits on Lie P u depends and it is denoted by mod P, where P u is the unipotent radical of P. The principal aim of this note is a generalization of two basic “monotonicity” results from [19] to positive characteristic: (1) If Θ is a semisimple automorphism of G and P is Θ-stable, then mod P \Θ≤ mod P. (2) If G is reductive, char k is a good prime for G, and H is a closed reductive subgroup of G normalized by a maximal torus T⊂P of G, then mod (P∩H)≤ mod P.

Let ${\mathfrak o}$ be the ring of integers in a finite extension field of ${\mathbb Q}_p$, let $k$ be its residue field. Let $G$ be a split reductive group over ${\mathbb Q}_p$, let ${\mathcal H}(G,I_0)$ be its pro-$p$-Iwahori Hecke ${\mathfrak o}$-algebra. In \cite{dfun} we introduced a general principle how to assign to a certain additionally chosen datum $(C^{(\bullet)},\phi,\tau)$ an exact functor $M\mapsto{\bf D}(\Theta_*{\mathcal V}_M)$ from finite length ${\mathcal H}(G,I_0)$-modules to $(\varphi^r,\Gamma)$-modules. In the present paper we concretely work out such data $(C^{(\bullet)},\phi,\tau)$ for the classical matrix groups. We show that the corresponding functor identifies the set of (standard) supersingular ${\mathcal H}(G,I_0)\otimes_{{\mathfrak o}}k$-modules with the set of $(\varphi^r,\Gamma)$-modules satisfying a certain symmetry condition.

COSY INFINITY is a DA‐based code that allows the computation of high order maps including parameter dependencies for systems consisting of electric, magnetic, and circular elements, including their fringe fields. It is written in standard FORTRAN with dedicated versions for a variety of computers and supports several different graphics drivers. It has a very intuitive user interface based on a programming language with nonlinear optimization features built in and an interface to MAD input. One example for the power and generality of this interface is the analytical search for four cell third order achromats described in a companion paper. Currently the code is used by about 100 registered users in about 60 laboratories. Recently several new features have been added to the code, including the fast fringe field method and high precision fringe field descriptions described in companion papers. For the study of repetitive systems, the conventional DA‐Lie algebraic normal form methods were replaced with a purely DA formulation, which is rather direct and also allows the analysis of non symplectic maps. Using the optimization features of COSY, these methods can readily be used for nonlinear optimization in general, and in particular the suppression of nonlinear resonances. Furthermore, the use of parameters allows a very direct and efficient computation of high‐order chromaticities even without normal form theory. For the high‐order correction of spectrographs, a new method of trajectory reconstruction allowing a computational correction of otherwise uncorrectable aberrations has been developed. This method will be used for the S800, a large acceptance high resolution magnetic spectrograph now under construction at NSCL.

This study aims to describe the ability to think creatively in solving arithmetic operations on algebraic forms in terms of gender differences in seventh-grade students of SMP Negeri 3 Sungguminasa. This type of research is a descriptive study with a qualitative approach designed to determine students' creative thinking skills in the subject of arithmetic operations in algebraic forms in terms of gender differences. Data collection techniques in the form of tests and interviews. The instrument used is in the form of a creative thinking ability test, which consists of 2 questions. Interviews were conducted to further explore students' creative thinking skills. The research subjects consisted of 1 female student and one male student. The results showed that the creative thinking ability of female students in solving all algebraic arithmetic operations questions was able to meet two indicators of creative thinking including fluency and flexibility. The ability to think creatively in male students in solving all arithmetic operations questions in algebraic forms is also only able to fulfill two indicators of creative thinking including fluency and flexibility.

Let G be a semisimple, simply connected algebraic group over an algebraically closed field k and let T ⊂ G be a maximal torus in G and B ⊂ G a Borel subgroup containing T . In two recent papers ([K1] and [K2] Senthamarai Kannan classified all parabolic subgroups G ⊃ P ⊃ B with the property that there exists an ample line bundle L on G/P such that, with respect to the T linearization of L induced by the uniqueG linearization, the setG/P (T )ss of semistable points coincides with the set G/P (T )s of stable points. In this note, we give a general characterization of those ample line bundlesL onG/P .We then show how to recover in a very simple wayKannan’s result from ours. To state our result, we need to introduce some notations and recall a few facts. X∗(T ) will denote the character lattice of T and X∗(T ) its dual lattice, i.e. the lattice of one parameter subgroups in T .We shall denote by

Solvability of the rational quantum integrable systems related to exceptional root spaces $G_2, F_4$ is re-examined and for $E_{6,7,8}$ is established in the framework of a unified approach. It is shown the Hamiltonians take algebraic form being written in a certain Weyl-invariant variables. It is demonstrated that for each Hamiltonian the finite-dimensional invariant subspaces are made from polynomials and they form an infinite flag. A notion of minimal flag is introduced and minimal flag for each Hamiltonian is found. Corresponding eigenvalues are calculated explicitly while the eigenfunctions can be computed by pure linear algebra means for {\it arbitrary} values of the coupling constants. The Hamiltonian of each model can be expressed in the algebraic form as a second degree polynomial in the generators of some infinite-dimensional but finitely-generated Lie algebra of differential operators, taken in a finite-dimensional representation.

In the present paper and the companion paper (Berman, Kähler–Einstein metrics, canonical random point processes and birational geometry. arXiv:1307.3634, 2015) a probabilistic (statistical-mechanical) approach to the construction of canonical metrics on complex algebraic varieties X is introduced by sampling “temperature deformed” determinantal point processes. The main new ingredient is a large deviation principle for Gibbs measures with singular Hamiltonians, which is proved in the present paper. As an application we show that the unique Kähler–Einstein metric with negative Ricci curvature on a canonically polarized algebraic manifold X emerges in the many particle limit of the canonical point processes on X. In the companion paper (Berman in 2015) the extension to algebraic varieties X with positive Kodaira dimension is given and a conjectural picture relating negative temperature states to the existence problem for Kähler–Einstein metrics with positive Ricci curvature is developed.

In the present paper we continue the project of systematic explicit construction of invariant differential operators. On the example of the non-compact group SO(p, 9 − p) (for p = 5, 6) we give the classification multiplets of indecomposable elementary representations induced from a new (relative to earlier considerations) choice of parabolic subgroup MAN so that the factor-group G/M is an Einstein manifold. This classification includes the data for the relevant invariant differential operators.

Residual stresses in press-braked stainless steel sections, II: Press-braking operations

Planetary perturbations for Oort cloud comets: III. Evolution of the cloud and production of centaurs and Halley type comets

Lipschitz-Killing curvatures of angular partially ordered sets

It is shown that the F4 rational and trigonometric integrable systems are exactly-solvable for arbitrary values of the coupling constants. Their spectra are found explicitly while eigenfunctions are by pure algebraic means. For both systems new variables are introduced in which the Hamiltonian has an algebraic form being also (block)-triangular. These variables are invariant with respect to the Weyl group of F4 root system and can be obtained by averaging over an orbit of the Weyl group. An alternative way of finding these variables exploiting a property of duality of the F4 model is presented. It is demonstrated that in these variables the Hamiltonian of each model can be expressed as a quadratic polynomial in the generators of some infinite-dimensional Lie algebra of differential operators in a finite-dimensional representation. Both Hamiltonians preserve the same flag of spaces of polynomials and each subspace of the flag coincides with the finite-dimensional representation space of this algebra. Quasi-exactly-solvable generalization of the rational F4 model depending on two continuous and one discrete parameters is found.