11,789 research outputs found
Baxterization, dynamical systems, and the symmetries of integrability
We resolve the `baxterization' problem with the help of the automorphism
group of the Yang-Baxter (resp. star-triangle, tetrahedron, \dots) equations.
This infinite group of symmetries is realized as a non-linear (birational)
Coxeter group acting on matrices, and exists as such, {\em beyond the narrow
context of strict integrability}. It yields among other things an unexpected
elliptic parametrization of the non-integrable sixteen-vertex model. It
provides us with a class of discrete dynamical systems, and we address some
related problems, such as characterizing the complexity of iterations.Comment: 25 pages, Latex file (epsf style). WARNING: Postscript figures are
BIG (600kB compressed, 4.3MB uncompressed). If necessary request hardcopy to
[email protected] and give your postal mail addres
Multiplicative Invariants of Root Lattices
We describe the multiplicative invariant algebras of the root lattices of all
irreducible root systems under the action of the Weyl group. In each case, a
finite system of fundamental invariants is determined and the class group of
the invariant algebra is calculated. In some cases, a presentation and a
Hironaka decomposition of the invariant algebra is given
Integration over matrix spaces with unique invariant measures
We present a method to calculate integrals over monomials of matrix elements
with invariant measures in terms of Wick contractions. The method gives exact
results for monomials of low order. For higher--order monomials, it leads to an
error of order 1/N^alpha where N is the dimension of the matrix and where alpha
is independent of the degree of the monomial. We give a lower bound on the
integer alpha and show how alpha can be increased systematically. The method is
particularly suited for symbolic computer calculation. Explicit results are
given for O(N), U(N) and for the circular orthogonal ensemble.Comment: 12 pages in revtex, no figure
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