scholarly journals Solutions of the Yang–Baxter equation: Descendants of the six-vertex model from the Drinfeld doubles of dihedral group algebras

2011 ◽  
Vol 847 (2) ◽  
pp. 387-412 ◽  
Author(s):  
P.E. Finch ◽  
K.A. Dancer ◽  
P.S. Isaac ◽  
J. Links
Keyword(s):  
Dihedral Group ◽  
Group Algebras ◽  
Baxter Equation ◽  
Vertex Model

scholarly journals On the Yang–Baxter equation for the six-vertex model

2014 ◽  
Vol 882 ◽  
pp. 70-96 ◽  
Author(s):  
Vladimir V. Mangazeev
Keyword(s):  
Baxter Equation ◽  
Vertex Model

A FAMILY OF FACE MODELS RELATED TO ZN-SYMMETRIC VERTEX MODEL OF BELAVIN

2004 ◽  
Vol 19 (supp02) ◽  
pp. 478-509
Author(s):  
Y. YAMADA
Keyword(s):  
Baxter Equation ◽  
Vertex Model ◽  
Face Type ◽  
The Face ◽  
Elliptic Solutions ◽  
Intertwining Relation

We present face-type elliptic solutions to the Yang-Baxter equation. They have 2N-2 real parameters. When specializing them to definite values, we recover the various models so far known. The intertwining relation between the face models above and the ZN-symmetric vertex model of Belavin is also given.

scholarly journals Littlewood–Richardson coefficients for Grothendieck polynomials from integrability

2019 ◽  
Vol 2019 (757) ◽  
pp. 159-195 ◽  
Author(s):  
Michael Wheeler ◽  
Paul Zinn-Justin
Keyword(s):  
Baxter Equation ◽  
Partition Functions ◽  
Vertex Model ◽  
Structure Constants ◽  
Product Rules ◽  
Grothendieck Polynomials

AbstractWe study the Littlewood–Richardson coefficients of double Grothendieck polynomials indexed by Grassmannian permutations. Geometrically, these are the structure constants of the equivariant K-theory ring of Grassmannians. Representing the double Grothendieck polynomials as partition functions of an integrable vertex model, we use its Yang–Baxter equation to derive a series of product rules for the former polynomials and their duals. The Littlewood–Richardson coefficients that arise can all be expressed in terms of puzzles without gashes, which generalize previous puzzles obtained by Knutson–Tao and Vakil.

Solutions of Yang–Baxter equation in the vertex model and the face model for octet representation

1991 ◽  
Vol 32 (8) ◽  
pp. 2210-2218 ◽  
Author(s):  
Bo‐Yu Hou ◽  
Bo‐Yuan Hou ◽  
Zhong‐Qi Ma ◽  
Yu‐Dong Yin
Keyword(s):  
Baxter Equation ◽  
Vertex Model ◽  
Face Model ◽  
The Face

scholarly journals Integrable probability: stochastic vertex models and symmetric functions

2018 ◽  
Author(s):  
Alexei Borodin ◽  
Leonid Petrov
Keyword(s):  
Symmetric Functions ◽  
Height Function ◽  
Rational Functions ◽  
Universality Class ◽  
Integral Representations ◽  
Baxter Equation ◽  
Partition Functions ◽  
Vertex Model ◽  
Schur Polynomials ◽  
Higher Spin

This chapter presents the study of a homogeneous stochastic higher spin six-vertex model in a quadrant. For this model concise integral representations for multipoint q-moments of the height function and for the q-correlation functions are derived. At least in the case of the step initial condition, these formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1+1)d KPZ universality class, including the stochastic six-vertex model, ASEP, various q-TASEPs, and associated zero-range processes. The arguments are largely based on properties of a family of symmetric rational functions that can be defined as partition functions of the higher spin six-vertex model for suitable domains; they generalize classical Hall–Littlewood and Schur polynomials. A key role is played by Cauchy-like summation identities for these functions, which are obtained as a direct corollary of the Yang–Baxter equation for the higher spin six-vertex model.

scholarly journals Quantum double finite group algebras and their representations

1993 ◽  
Vol 48 (2) ◽  
pp. 275-301 ◽  
Author(s):  
M.D. Gould
Keyword(s):  
Finite Group ◽  
Explicit Construction ◽  
Unitary Representations ◽  
Group Algebras ◽  
Baxter Equation ◽  
Irreducible Matrix ◽  
Matrix Representations ◽  
Quantum Double ◽  
Induced Representations ◽  
A Finite Group

The quantum double construction is applied to the group algebra of a finite group. Such algebras are shown to be semi-simple and a complete theory of characters is developed. The irreducible matrix representations are classified and applied to the explicit construction of R-matrices: this affords solutions to the Yang-Baxter equation associated with certain induced representations of a finite group. These results are applied in the second paper of the series to construct unitary representations of the Braid group and corresponding link polynomials.

Group algebras of free products of finite groups which are CS-rings

2018 ◽  
pp. 1850224
Author(s):  
Shoichi Kondo
Keyword(s):  
Free Product ◽  
Group Algebra ◽  
Finite Groups ◽  
Dihedral Group ◽  
Group Algebras ◽  
Free Products ◽  
Cyclic Groups ◽  
Algebra Over A Field ◽  
Cs Rings

This paper proves that the infinite dihedral group is only such a free product of two finite groups that its group algebra over a field [Formula: see text] is a CS-ring in case the orders of two groups are not zero in [Formula: see text]. Furthermore, it is shown that the group algebra of any free product of two finite cyclic groups does not satisfy the condition [Formula: see text].

scholarly journals YANG–BAXTER FIELD FOR SPIN HALL–LITTLEWOOD SYMMETRIC FUNCTIONS

2019 ◽  
Vol 7 ◽  
Author(s):  
ALEXEY BUFETOV ◽  
LEONID PETROV
Keyword(s):  
Probability Distribution ◽  
Symmetric Functions ◽  
Exclusion Process ◽  
Baxter Equation ◽  
Two Dimensional ◽  
Vertex Model ◽  
Joint Distributions ◽  
Simple Exclusion Process ◽  
Asymmetric Simple Exclusion ◽  
Simple Exclusion

Employing bijectivization of summation identities, we introduce local stochastic moves based on the Yang–Baxter equation for $U_{q}(\widehat{\mathfrak{sl}_{2}})$ . Combining these moves leads to a new object which we call the spin Hall–Littlewood Yang–Baxter field—a probability distribution on two-dimensional arrays of particle configurations on the discrete line. We identify joint distributions along down-right paths in the Yang–Baxter field with spin Hall–Littlewood processes, a generalization of Schur processes. We consider various degenerations of the Yang–Baxter field leading to new dynamic versions of the stochastic six-vertex model and of the Asymmetric Simple Exclusion Process.

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