In recent work ([9],[10]), Kostant and Wallach construct an action of a simply connected Lie group A ≃ C( n 2 ) on gl(n) using a completely integrable system derived from the Poisson analogue of the Gelfand-Zeitlin subalgebra of the enveloping algebra. In [9], the authors show that A-orbits of dimension ( n 2 ) form Lagrangian submanifolds of regular adjoint orbits in gl(n). They describe the o...

We show that a modiication of the combined Phase I-Phase II interior-point algorithm for linear programming, due to Anstreicher, de Ghellinck and Vial, Fra-ley, and Todd, terminates in O(p nL) iterations from a suitable initial (interior but infeasible) solution. The algorithm either detects infeasibility, or approaches feasibility and optimality simultaneously, or generates a feasible primal-d...

We give a recursive algorithm for computing the Orlik-Terao algebra of the Coxeter arrangement of type An−1 as a graded representation of Sn, and we give a conjectural description of this representation in terms of the cohomology of the configuration space of n points in SU(2) modulo translation. We also give a version of this conjecture for more general graphical arrangements.

In this paper we prove that every finitely generated Coxeter group has a finite index subgroup that is the fundamental group of a special cube complex. Some consequences include: Every f.g. Coxeter group is virtually a subgroup of a right-angled Coxeter group. Every word-hyperbolic Coxeter group has separable quasiconvex subgroups. © 2010 Elsevier Inc. All rights reserved. MSC: 53C23; 20F36; 20...

We show that all groups in a very large class of Coxeter groups are locally quasiconvex and have uniform membership problem solvable in quadratic time. If a group in the class satisfies a further hypothesis it is subgroup separable and relevant homomorphisms are also calculable in quadratic time. The algorithm also decides if a finitely generated subgroup has finite index.

A Coxeter system is an ordered pair (W,S) where S is the generating set in a particular type of presentation for the Coxeter group W . A subgroup of W is called special if it is generated by a subset of S. Amalgamated product decompositions of a Coxeter group having special factors and special amalgamated subgroup are easily recognized from the presentation of the Coxeter group. If a Coxeter gr...

In this paper, we present a predictor-corrector path-following interior-point algorithm for symmetric cone optimization based on Darvay's technique. Each iteration of the algorithm contains a predictor step and a corrector step based on a modification of the Nesterov and Todd directions. Moreover, we show that the algorithm is well defined and that the obtained iteration bound is √ log , where ...

These are the notes of a mini-course given at the conference “Arrangements in Pyrénées”, held in Pau (France) from 11th to 15th of June, 2012. Definition. Let S be a finite set. A Coxeter matrix on S is a square matrix M = (ms,t)s,t∈S indexed by the elements of S and satisfying: (a) ms,s = 1 for all s ∈ S; (b) ms,t = mt,s ∈ {2, 3, 4, . . . } ∪ {∞} for all s, t ∈ S, s 6= t. A Coxeter matrix is u...

To any complex reflection group W ⊂ GL(V ), one may attach a braid group B(W ), defined as the fundamental group of the space of regular orbits for the action of W on V [Broué et al. 98]. The “ordinary” braid group on n strings, introduced by [Artin 47], corresponds to the case of the symmetric group Sn, in its monomial reflection representation in GLn(C). More generally, any Coxeter group can ...