27 research outputs found
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
The solution of the quantum T-system for arbitrary boundary
We solve the quantum version of the -system by use of quantum
networks. The system is interpreted as a particular set of mutations of a
suitable (infinite-rank) quantum cluster algebra, and Laurent positivity
follows from our solution. As an application we re-derive the corresponding
quantum network solution to the quantum -system and generalize it to
the fully non-commutative case. We give the relation between the quantum
-system and the quantum lattice Liouville equation, which is the quantized
-system.Comment: 24 pages, 18 figure
Discrete integrable systems, positivity, and continued fraction rearrangements
In this review article, we present a unified approach to solving discrete,
integrable, possibly non-commutative, dynamical systems, including the - and
-systems based on . The initial data of the systems are seen as cluster
variables in a suitable cluster algebra, and may evolve by local mutations. We
show that the solutions are always expressed as Laurent polynomials of the
initial data with non-negative integer coefficients. This is done by
reformulating the mutations of initial data as local rearrangements of
continued fractions generating some particular solutions, that preserve
manifest positivity. We also show how these techniques apply as well to
non-commutative settings.Comment: 24 pages, 2 figure
T-systems, Y-systems, and cluster algebras: Tamely laced case
The T-systems and Y-systems are classes of algebraic relations originally
associated with quantum affine algebras and Yangians. Recently they were
generalized to quantum affinizations of quantum Kac-Moody algebras associated
with a wide class of generalized Cartan matrices which we say tamely laced.
Furthermore, in the simply laced case, and also in the nonsimply laced case of
finite type, they were identified with relations arising from cluster algebras.
In this note we generalize such an identification to any tamely laced Cartan
matrices, especially to the nonsimply laced ones of nonfinite type.Comment: 31 pages, final version to appear in the festschrift volume for
Tetsuji Miwa, "Infinite Analysis 09: New Trends in Quantum Integrable
Systems
Kirillov-Reshetikhin crystals for using Nakajima monomials
We give a realization of the Kirillov--Reshetikhin crystal using
Nakajima monomials for using the crystal structure
given by Kashiwara. We describe the tensor product in terms of a shift of indices, allowing us to recover the Kyoto
path model. Additionally, we give a model for the KR crystals using
Nakajima monomials.Comment: 24 pages, 6 figures; v2 improved introduction, added more figures,
and other misc improvements; v3 changes from referee report
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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