3,854 research outputs found
Integrable Combinatorics
We review various combinatorial problems with underlying classical or quantum
integrable structures. (Plenary talk given at the International Congress of
Mathematical Physics, Aalborg, Denmark, August 10, 2012.)Comment: 21 pages, 16 figures, proceedings of ICMP1
Truncated determinants and the refined enumeration of Alternating Sign Matrices and Descending Plane Partitions
Lecture notes for the proceedings of the workshop "Algebraic Combinatorics
related to Young diagram and statistical physics", Aug. 6-10 2012, I.I.A.S.,
Nara, Japan.Comment: 25 pages, 8 figure
Difference equations for graded characters from quantum cluster algebra
We introduce a new set of -difference operators acting as raising
operators on a family of symmetric polynomials which are characters of graded
tensor products of current algebra KR-modules \cite{FL} for
. These operators are generalizations of the Kirillov-Noumi
\cite{kinoum} Macdonald raising operators, in the dual -Whittaker limit
. They form a representation of the quantum -system of type
\cite{qKR}. This system is a subalgebra of a quantum cluster algebra, and is
also a discrete integrable system whose conserved quantities, analogous to the
Casimirs of , act as difference operators on the
above family of symmetric polynomials. The characters in the special case of
products of fundamental modules are class I -Whittaker functions, or
characters of level-1 Demazure modules or Weyl modules. The action of the
conserved quantities on these characters gives the difference quantum Toda
equations \cite{Etingof}. We obtain a generalization of the latter for
arbitrary tensor products of KR-modules.Comment: 35 page
Positivity of the T-system cluster algebra
We give the path model solution for the cluster algebra variables of the
-system with generic boundary conditions. The solutions are partition
functions of (strongly) non-intersecting paths on weighted graphs. The graphs
are the same as those constructed for the -system in our earlier work, and
depend on the seed or initial data in terms of which the solutions are given.
The weights are "time-dependent" where "time" is the extra parameter which
distinguishes the -system from the -system, usually identified as the
spectral parameter in the context of representation theory. The path model is
alternatively described on a graph with non-commutative weights, and cluster
mutations are interpreted as non-commutative continued fraction rearrangements.
As a consequence, the solution is a positive Laurent polynomial of the seed
data.Comment: 30 pages, 10 figure
T-systems with boundaries from network solutions
In this paper, we use the network solution of the -system to derive
that of the unrestricted -system, equivalent to the octahedron
relation. We then present a method for implementing various boundary conditions
on this system, which consists of picking initial data with suitable
symmetries. The corresponding restricted -systems are solved exactly in
terms of networks. This gives a simple explanation for phenomena such as the
Zamolodchikov periodicity property for -systems (corresponding to the case
) and a combinatorial interpretation for the positive Laurent
property of the variables of the associated cluster algebra. We also explain
the relation between the -system wrapped on a torus and the higher pentagram
maps of Gekhtman et al.Comment: 63 pages, 67 figure
Quantum Q systems: From cluster algebras to quantum current algebras
In this paper, we recall our renormalized quantum Q-system associated with
representations of the Lie algebra , and show that it can be viewed as a
quotient of the quantum current algebra in the Drinfeld presentation. Moreover, we find
the interpretation of the conserved quantities in terms of Cartan currents at
level 0, and the rest of the current algebra, in a non-standard polarization in
terms of generators in the quantum cluster algebra.Comment: 38 pages, 2 figure
Arctic curves of the octahedron equation
We study the octahedron relation (also known as the -system),
obeyed in particular by the partition function for dimer coverings of the Aztec
Diamond graph. For a suitable class of doubly periodic initial conditions, we
find exact solutions with a particularly simple factorized form. For these, we
show that the density function that measures the average dimer occupation of a
face of the Aztec graph, obeys a system of linear recursion relations with
periodic coefficients. This allows us to explore the thermodynamic limit of the
corresponding dimer models and to derive exact "arctic" curves separating the
various phases of the system.Comment: 39 pages, 21 figures; typos fixed, four references and an appendix
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Noncommutative integrability, paths and quasi-determinants
In previous work, we showed that the solution of certain systems of discrete
integrable equations, notably and -systems, is given in terms of
partition functions of positively weighted paths, thereby proving the positive
Laurent phenomenon of Fomin and Zelevinsky for these cases. This method of
solution is amenable to generalization to non-commutative weighted paths. Under
certain circumstances, these describe solutions of discrete evolution equations
in non-commutative variables: Examples are the corresponding quantum cluster
algebras [BZ], the Kontsevich evolution [DFK09b] and the -systems themselves
[DFK09a]. In this paper, we formulate certain non-commutative integrable
evolutions by considering paths with non-commutative weights, together with an
evolution of the weights that reduces to cluster algebra mutations in the
commutative limit. The general weights are expressed as Laurent monomials of
quasi-determinants of path partition functions, allowing for a non-commutative
version of the positive Laurent phenomenon. We apply this construction to the
known systems, and obtain Laurent positivity results for their solutions in
terms of initial data.Comment: 46 pages, minor typos correcte
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