43 research outputs found
A lecture on the Calogero-Sutherland models
In these lectures, I review some recent results on the Calogero-Sutherland
model and the Haldane Shastry-chain. The list of topics I cover are the
following: 1) The Calogero-Sutherland Hamiltonian and fractional statistics.
The form factor of the density operator. 2) The Dunkl operators and their
relations with monodromy matrices, Yangians and affine-Hecke algebras. 3) The
Haldane-Shastry chain in connection with the Calogero-Sutherland Hamiltonian at
a specific coupling constant.Comment: (2 references added, small modifications
The orbifold transform and its applications
We discuss the notion of the orbifold transform, and illustrate it on simple
examples. The basic properties of the transform are presented, including
transitivity and the exponential formula for symmetric products. The connection
with the theory of permutation orbifolds is addressed, and the general results
illustrated on the example of torus partition functions
Crystal Graphs and -Analogues of Weight Multiplicities for the Root System
We give an expression of the -analogues of the multiplicities of weights
in irreducible \sl_{n+1}-modules in terms of the geometry of the crystal
graph attached to the corresponding U_q(\sl_{n+1})-modules. As an
application, we describe multivariate polynomial analogues of the
multiplicities of the zero weight, refining Kostant's generalized exponents.Comment: LaTeX file with epic, eepic pictures, 17 pages, November 1994, to
appear in Lett. Math. Phy
Hopf algebras and Markov chains: Two examples and a theory
The operation of squaring (coproduct followed by product) in a combinatorial
Hopf algebra is shown to induce a Markov chain in natural bases. Chains
constructed in this way include widely studied methods of card shuffling, a
natural "rock-breaking" process, and Markov chains on simplicial complexes.
Many of these chains can be explictly diagonalized using the primitive elements
of the algebra and the combinatorics of the free Lie algebra. For card
shuffling, this gives an explicit description of the eigenvectors. For
rock-breaking, an explicit description of the quasi-stationary distribution and
sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes
will only appear on the version on Amy Pang's website, the arXiv version will
not be updated.
Root polytopes and abelian ideals
We study the root polytope of a finite irreducible
crystallographic root system using its relation with the abelian ideals
of a Borel subalgebra of a simple Lie algebra with root system . We
determine the hyperplane arrangement corresponding to the faces of codimension
2 of and analyze its relation with the facets of . For of type or , we show that the orbits of some
special subsets of abelian ideals under the action of the Weyl group
parametrize a triangulation of . We show that this
triangulation restricts to a triangulation of the positive root polytope
.Comment: 41 pages, revised version, accepted for publication in Journal of
Algebraic Combinatoric
spectroscopy
In the framework of potential models for heavy quarkonium the mass spectrum
for the system () is considered. Spin-dependent splittings, taking
into account a change of a constant for effective Coulomb interaction between
the quarks, and widths of radiative transitions between the () levels
are calculated. In the framework of QCD sum rules, masses of the lightest
vector and pseudoscalar states are estimated, scaling relation
for leptonic constants of heavy quarkonia is derived, and the leptonic constant
is evaluated.Comment: IHEP 94-51, LATEX, 39 page