scholarly journals An Inhomogeneous Lambda-Determinant

10.37236/2756 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Philippe Di Francesco
Keyword(s):  
Cluster Algebra ◽  
Weighted Sum ◽  
Alternating Sign Matrices ◽  
T System

We introduce a multi-parameter generalization of the Lambda-determinant of Robbins and Rumsey, based on the cluster algebra with coefficients attached to a T-system recurrence. We express the result as a weighted sum over alternating sign matrices.

scholarly journals Positivity of the T-System Cluster Algebra

10.37236/229 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Philippe Di Francesco ◽  
Rinat Kedem
Keyword(s):  
Continued Fraction ◽  
Cluster Algebra ◽  
Path Model ◽  
Partition Functions ◽  
Weighted Graphs ◽  
Generic Boundary ◽  
Seed Data ◽  
Q System ◽  
Extra Parameter ◽  
T System

We give the path model solution for the cluster algebra variables of the $T$-system of type $A_r$ 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 $Q$-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 $T$-system from the $Q$-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.

scholarly journals $(q,t)$-Characters of Kirillov-Reshetikhin Modules of Type $A_r$ as Quantum Cluster Variables

10.37236/7188 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Bolor Turmunkh
Keyword(s):  
Dynamical System ◽  
Discrete Dynamical System ◽  
Cluster Algebra ◽  
Quiver Varieties ◽  
Quantum Cluster ◽  
T System

Nakajima (2003) introduced a $t$-deformation of $q$-characters, $(q,t)$-characters for short, and their twisted multiplication through the geometry of quiver varieties. The Nakajima $(q,t)$-characters of Kirillov-Reshetikhin modules satisfy a $t$-deformed $T$-system. The $T$-system is a discrete dynamical system that can be interpreted as a mutation relation in a cluster algebra in two different ways, depending on the choice of direction of evolution. In this paper, we show that the Nakajima $t$-deformed $T$-system of type $A_r$ forms a quantum mutation relation in a quantization of exactly one of the cluster algebra structures attached to the $T$-system.

scholarly journals Solutions to the T-Systems with Principal Coefficients

10.37236/5698 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Panupong Vichitkunakorn
Keyword(s):  
Recurrence Relation ◽  
Cluster Algebra ◽  
Perfect Matchings ◽  
Partition Functions ◽  
Initial Condition ◽  
Free Cluster ◽  
Octahedron Recurrence ◽  
T System

The $A_\infty$ T-system, also called the octahedron recurrence, is a dynamical recurrence relation. It can be realized as mutation in a coefficient-free cluster algebra (Kedem 2008, Di Francesco and Kedem 2009). We define T-systems with principal coefficients from cluster algebra aspect, and give combinatorial solutions with respect to any valid initial condition in terms of partition functions of perfect matchings, non-intersecting paths and networks. This also provides a solution to other systems with various choices of coefficients on T-systems including Speyer's octahedron recurrence (Speyer 2007), generalized lambda-determinants (Di Francesco 2013) and (higher) pentagram maps (Schwartz 1992, Ovsienko et al. 2010, Glick 2011, Gekhtman et al. 2016).

scholarly journals Periodicities of T-systems and Y-systems

2010 ◽  
Vol 197 ◽  
pp. 59-174 ◽  
Author(s):  
Rei Inoue ◽  
Osamu Iyama ◽  
Atsuo Kuniba ◽  
Tomoki Nakanishi ◽  
Junji Suzuki
Keyword(s):  
Lie Algebra ◽  
Direct Method ◽  
Cluster Algebra ◽  
Quantum Affine Algebra ◽  
Cluster Category ◽  
Affine Algebra ◽  
Grothendieck Ring ◽  
Finite Dimensional ◽  
Level 2 ◽  
T System

The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of Yangian or quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for types A and C, and the direct method for types A, D, and B (level 2).

scholarly journals T-systems with Boundaries from Network Solutions

10.37236/2645 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Philippe Di Francesco ◽  
Rinat Kedem
Keyword(s):  
Initial Data ◽  
Boundary Conditions ◽  
Cluster Algebra ◽  
Simple Explanation ◽  
Combinatorial Interpretation ◽  
Network Solution ◽  
Various Boundary Conditions ◽  
Periodicity Property ◽  
T System

In this paper, we use the network solution of the $A_r$ $T$-system to derive that of the unrestricted $A_\infty$ $T$-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 $T$-systems are solved exactly in terms of networks. This gives a simple explanation for phenomena such as the Zamolodchikov periodicity property for $T$-systems (corresponding to the case $A_\ell\times A_r$) and a combinatorial interpretation for the positive Laurent property for the variables of the associated cluster algebra. We also explain the relation between the $T$-system wrapped on a torus and the higher pentagram maps of Gekhtman et al.

scholarly journals Periodicities of T-systems and Y-systems

2010 ◽  
Vol 197 ◽  
pp. 59-174 ◽  
Author(s):  
Rei Inoue ◽  
Osamu Iyama ◽  
Atsuo Kuniba ◽  
Tomoki Nakanishi ◽  
Junji Suzuki
Keyword(s):  
Lie Algebra ◽  
Direct Method ◽  
Cluster Algebra ◽  
Quantum Affine Algebra ◽  
Cluster Category ◽  
Affine Algebra ◽  
Grothendieck Ring ◽  
Finite Dimensional ◽  
Level 2 ◽  
T System

The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of Yangian or quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for typesAandC, and the direct method for typesA, D, andB(level 2).

Three-Dimensional Visualization of the T-System of Frog Muscle Using High Voltage Electron Microscopy and a Lanthanum Stain

1977 ◽  
Vol 35 ◽  
pp. 570-571 ◽  
Author(s):  
Lee D. Peachey ◽  
Clara Franzini-Armstrong
Keyword(s):  
Electron Microscopy ◽  
High Voltage ◽  
Three Dimensional ◽  
Biological Tissues ◽  
High Voltage Electron Microscopy ◽  
Transverse Tubular System ◽  
Peroxidase Reaction ◽  
Voltage Electron ◽  
High Voltage Electron ◽  
T System

The effective study of biological tissues in thick slices of embedded material by high voltage electron microscopy (HVEM) requires highly selective staining of those structures to be visualized so that they are not hidden or obscured by other structures in the image. A tilt pair of micrographs with subsequent stereoscopic viewing can be an important aid in three-dimensional visualization of these images, once an appropriate stain has been found. The peroxidase reaction has been used for this purpose in visualizing the T-system (transverse tubular system) of frog skeletal muscle by HVEM (1). We have found infiltration with lanthanum hydroxide to be particularly useful for three-dimensional visualization of certain aspects of the structure of the T- system in skeletal muscles of the frog. Specifically, lanthanum more completely fills the lumen of the tubules and is denser than the peroxidase reaction product.

The Network Parameters Of The T-System In Frog Muscle Measured With The High Voltage Electron Microscope.

1975 ◽  
Vol 33 ◽  
pp. 550-551 ◽  
Author(s):  
Brenda R. Eisenberg ◽  
Lee D. Peachey
Keyword(s):  
Electron Microscopy ◽  
Electron Microscope ◽  
High Voltage ◽  
Sartorius Muscle ◽  
High Voltage Electron Microscopy ◽  
Network Parameters ◽  
Voltage Electron ◽  
High Voltage Electron Microscope ◽  
High Voltage Electron ◽  
T System

Analysis of the electrical properties of the t-system requires knowledge of the geometry of the t-system network. It is now possible to determine the network parameters experimentally by use of high voltage electron microscopy. The t-system was marked with exogenous peroxidase. Conventional methods of electron microscopy were used to fix and embed the sartorius muscle from four frogs. Transverse slices 0.5-1.0 μm thick were viewed at an accelerating voltage of 1000 kV using the JEM-1000 high voltage electron microscope at Boulder, Colorado and prints at x5000 were used for analysis.The length of a t-branch (t) from node to node (Fig. 1a) was measured with a magnifier; at least 150 t-branches around 30 myofibrils were measured from each frog. The mean length of t is 0.90 ± 0.11 μm and the number of branches per myofibril is 5.4 ± 0.2 (mean ± SD, n = 4 frogs).

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