45 research outputs found
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
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
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
On the Spectra of Real and Complex Lam\'e Operators
We study Lam\'e operators of the form with and a
half-period of . For rectangular period lattices, we can choose
and such that the potential is real, periodic and regular. It is
known after Ince that the spectrum of the corresponding Lam\'e operator has a
band structure with not more than gaps. In the first part of the paper, we
prove that the opened gaps are precisely the first ones. In the second
part, we study the Lam\'e spectrum for a generic period lattice when the
potential is complex-valued. We concentrate on the case, when the
spectrum consists of two regular analytic arcs, one of which extends to
infinity, and briefly discuss the case, paying particular attention to
the rhombic lattices