Coxeter group actions on the complement of hyperplanes and special involutions

We consider both standard and twisted action of a (real) Coxeter group G on the complement M_G to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in G and give explicit formulae which describe both actions on the total cohomology H(M_G,C) in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group S_n, the Weyl groups of type D_{2m+1}, E_6 and dihedral groups I_2 (2k+1) and that the standard action has no anti-invariants. We discuss also the relations with the cohomology of generalised braid groups.Comment: 11 page

Elliptic quantum groups and Ruijsenaars models

We construct symmetric and exterior powers of the vector representation of the elliptic quantum groups $E_{\tau,\eta}(gl_N)$. The corresponding transfer matrices give rise to various integrable difference equations which could be solved in principle by the nested Bethe ansatz method. In special cases we recover the Ruijsenaars systems of commuting difference operators.Comment: 15 pages, late

Polynomial solutions of the Knizhnik-Zamolodchikov equations and Schur-Weyl duality

An integral formula for the solutions of Knizhnik-Zamolodchikov (KZ) equation with values in an arbitrary irreducible representation of the symmetric group S_N is presented for integer values of the parameter. The corresponding integrals can be computed effectively as certain iterated residues determined by a given Young diagram and give polynomials with integer coefficients. The derivation is based on Schur-Weyl duality and the results of Matsuo on the original SU(n) KZ equation. The duality between the spaces of solutions with parameters m and -m is discussed in relation with the intersection pairing in the corresponding homology groups.Comment: 14 pages, reference adde

Gaudin subalgebras and wonderful models

Gaudin hamiltonians form families of r-dimensional abelian Lie subalgebras of the holonomy Lie algebra of the arrangement of reflection hyperplanes of a Coxeter group of rank r. We consider the set of principal Gaudin subalgebras, which is the closure in the appropriate Grassmannian of the set of spans of Gaudin hamiltonians. We show that principal Gaudin subalgebras form a smooth projective variety isomorphic to the De Concini-Procesi compactification of the projectivized complement of the arrangement of reflection hyperplanes.Comment: 13 pages, 2 figures; added detailed description of the B_2 and B_3 cases in the new versio

Efficacy and Safety Of Radiation Synovectomy with Yttrium-90

In this long term retrospective study of radiation synovectomy with Yttrium-90 (Y90), we evaluated the results of 164 applications in 82 patients with RA, OA with synovitis, ankylosing spondylitis and psoriatic arthritis. Radiation synovectomy with Y90 has an overall success rate of approximately 50% and is therefore an effective alternative to surgical synovectomy in chronic synovitis which fails to respond to conservative treatment. Elbow and knee responded significantly better than shoulder and ankle joints. Patients with radiological stages from 0 to 2 showed a significantly better success rate than those with stage 3 changes. In responders, repeat therapy for recurrence of symptoms or treatment of a symptomatic corresponding symmetrical joint is advisable. Repeat therapy in a previous non-responder is associated with an unacceptably high failure rate. Therefore, when a joint fails to respond after 6 months, arthroscopy should be performed to evaluate further treatment procedures. A successful result was found in only 11 of 25 joints treated with arthroscopic synovectomy followed by radiation synovectomy within 2 weeks, indicating no benefit of this combination

Quantum Dynamical Rˇ\check{R}- Matrix with Spectral Parameter from Fusion

A quantum dynamical $\check{R}$-matrix with spectral parameter is constructed by fusion procedure. This spin-1 $\check{R}$-matrix is connected with Lie algebra $so(3)$ and does not satisfy the condition of translation invariance.Comment: 6 pages, LaTeX, no figure

Polynomial Solutions of the Knizhnik-Zamolodchikov Equations and Schur-Weyl Duality

An integral formula for the solutions of Knizhnik-Zamolodchikov (KZ) equation with values in an arbitrary irreducible representation of the symmetric group SN is presented for integer values of the parameter. The corresponding integrals can be computed effectively as certain iterated residues determined by a given Young diagram and give polynomials with integer coefficients. The derivation is based on Schur-Weyl duality and the results of Matsuo on the original SU(n) KZ equation. The duality between the spaces of solutions with parameters m and −m is discussed in relation with the intersection pairing in the corresponding homology group

BAKER-AKHIEZER FUNCTION AS ITERATED RESIDUE AND SELBERG-TYPE INTEGRAL

A simple integral formula as an iterated residue is presented for the Baker-Akhiezer function related to An-type root system in both the rational and trigonometric cases. We present also a formula for the Baker-Akhiezer function as a Selberg-type integral and generalise it to the deformed An,1-case. These formulas can be interpreted as new cases of explicit evaluation of Selberg-type integral