99 research outputs found
Functional approach to (2+1) dimensional gravity
We work out the phase-space functional integral of the gravitational field in
2+1 dimensions interacting with N point particles in an open universe.Comment: 3 pages, LaTeX, contribution to the X Marcel Grossmann Meeting on
General Relativity, Rio de Janeiro, July 20-26, 200
Matrix product and sum rule for Macdonald polynomials
We present a new, explicit sum formula for symmetric Macdonald polynomials
and show that they can be written as a trace over a product of
(infinite dimensional) matrices. These matrices satisfy the
Zamolodchikov--Faddeev (ZF) algebra. We construct solutions of the ZF algebra
from a rank-reduced version of the Yang--Baxter algebra. As a corollary, we
find that the normalization of the stationary measure of the multi-species
asymmetric exclusion process is a Macdonald polynomial with all variables set
equal to one.Comment: 11 pages, extended abstract submission to FPSA
Matrix product formula for Macdonald polynomials
We derive a matrix product formula for symmetric Macdonald polynomials. Our
results are obtained by constructing polynomial solutions of deformed
Knizhnik--Zamolodchikov equations, which arise by considering representations
of the Zamolodchikov--Faddeev and Yang--Baxter algebras in terms of
-deformed bosonic operators. These solutions form a basis of the ring of
polynomials in variables, whose elements are indexed by compositions. For
weakly increasing compositions (anti-dominant weights), these basis elements
coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural
combinatorial interpretation in terms of solvable lattice models. They also
imply that normalisations of stationary states of multi-species exclusion
processes are obtained as Macdonald polynomials at .Comment: 27 pages; typos corrected, references added and some better
conventions adopted in v
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