scholarly journals Tokuyama's Identity for Factorial Schur $P$ and $Q$ Functions

10.37236/4971 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Angèle M. Hamel ◽  
Ronald C. King
Keyword(s):  
Partition Function ◽  
Lattice Paths ◽  
Schur Function ◽  
Vertex Model ◽  
Shifted Tableaux

A recent paper of Bump, McNamara and Nakasuji introduced a factorial version of Tokuyama's identity, expressing the partition function of  six vertex model as the product of a $t$-deformed Vandermonde and a Schur function. Here we provide an extension of their result by exploiting the language of primed shifted tableaux, with its proof based on the use of non-interesecting lattice paths.

scholarly journals Diagonally and antidiagonally symmetric alternating sign matrices of odd order

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Roger Behrend ◽  
Ilse Fischer ◽  
Matjaz Konvalinka
Keyword(s):  
Partition Function ◽  
General Expression ◽  
Schur Function ◽  
Vertex Model ◽  
Grid Graph ◽  
Alternating Sign Matrices ◽  
Odd Order ◽  
International Audience ◽  
Reflection Equations

International audience We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DAS- ASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang– Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbins from the mid 1980's that the total number of (2n + 1) × (2n + 1) DASASMs is∏n (3i)! ,andaconjectureofStroganovfrom2008thattheratiobetweenthenumbersof(2n+1)×(2n+1) i=0 (n+i)! DASASMs with central entry −1 and 1 is n/(n + 1). Among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980's, that for odd-order DASASMs is the last to have been proved.

scholarly journals Double Grothendieck Polynomials and Colored Lattice Models

2020 ◽  
Author(s):  
Valentin Buciumas ◽  
Travis Scrimshaw
Keyword(s):  
Partition Function ◽  
Lattice Models ◽  
Lattice Paths ◽  
Stable Limit ◽  
Vertex Model ◽  
Determinant Formula ◽  
Schubert Polynomials ◽  
Nonintersecting Lattice Paths ◽  
Double Schubert Polynomials ◽  
Grothendieck Polynomials

Abstract We construct an integrable colored six-vertex model whose partition function is a double Grothendieck polynomial. This gives an integrable systems interpretation of bumpless pipe dreams and recent results of Weigandt relating double Grothendieck polynomias with bumpless pipe dreams. For vexillary permutations, we then construct a new model that we call the semidual version model. We use our semidual model and the five-vertex model of Motegi and Sakai to give a new proof that double Grothendieck polynomials for vexillary permutations are equal to flagged factorial Grothendieck polynomials. Taking the stable limit of double Grothendieck polynomials, we obtain a new proof that the stable limit is a factorial Grothendieck polynomial as defined by McNamara. The states of our semidual model naturally correspond to families of nonintersecting lattice paths, where we can then use the Lindström–Gessel–Viennot lemma to give a determinant formula for double Schubert polynomials corresponding to vexillary permutations.

On some representations of the six vertex model partition function

2003 ◽  
Vol 315 (3-4) ◽  
pp. 231-236 ◽  
Author(s):  
F. Colomo ◽  
A.G. Pronko
Keyword(s):  
Partition Function ◽  
Vertex Model

Integrable equations for the partition function of the six-vertex model

2000 ◽  
Vol 100 (2) ◽  
pp. 2141-2146 ◽  
Author(s):  
A. G. Izergin ◽  
E. Karjalainen ◽  
N. A. Kitanin
Keyword(s):  
Partition Function ◽  
Vertex Model ◽  
Integrable Equations

scholarly journals Determinant formula for the partition function of the six-vertex model with a non-diagonal reflecting end

2011 ◽  
Vol 844 (2) ◽  
pp. 289-307 ◽  
Author(s):  
Wen-Li Yang ◽  
Xi Chen ◽  
Jun Feng ◽  
Kun Hao ◽  
Bo-Yu Hou ◽  
...  
Keyword(s):  
Partition Function ◽  
Vertex Model ◽  
Determinant Formula

scholarly journals Domain wall partition function of the eight-vertex model with a non-diagonal reflecting end

2011 ◽  
Vol 847 (2) ◽  
pp. 367-386 ◽  
Author(s):  
Wen-Li Yang ◽  
Xi Chen ◽  
Jun Feng ◽  
Kun Hao ◽  
Kang-Jie Shi ◽  
...  
Keyword(s):  
Partition Function ◽  
Domain Wall ◽  
Vertex Model

GAUGE TRANSFORMATION AND VERTEX MODELS: A POLYNOMIAL FORMULATION

1995 ◽  
Vol 09 (24) ◽  
pp. 3209-3217 ◽  
Author(s):  
G. ROLLET ◽  
F. Y. WU
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Gauge Transformation ◽  
Vertex Model ◽  
Arbitrary Graph ◽  
Vertex Models ◽  
Polynomial Formulation

We consider vertex models on an arbitrary graph and propose a new polynomial formulation for its gauge transformation under which the partition function is invariant. Our formulation recovers in a simple way various, previously obtained results including the condition under which the vertex model is equivalent to an Ising model.

Some exact results for the Ashkin-Teller model

1979 ◽  
Vol 365 (1722) ◽  
pp. 371-380 ◽  
Keyword(s):  
Renormalization Group ◽  
Partition Function ◽  
Potts Model ◽  
Critical Line ◽  
Square Lattice ◽  
Direct Connection ◽  
Exact Results ◽  
Renormalization Group Theory ◽  
Vertex Model ◽  
Special Cases

It is shown that various cases of the Ashkin-Teller model on the square, triangular and hexagonal lattices can be transformed by the dual and star-triangle transformations and, further, that these problems can be reduced to special cases of the eight vertex model on the Kagomé lattice. In general, we can only obtain the partition function of the Ashkin-Teller model if we are on its line of fixed points, and it then turns out that it is reducible to the six vertex model. Since the partition function of the q -state Potts model at its critical point can also be related to the six vertex model, a direct connection between the Ashkin-Teller model and the Potts model can be made. It turns out that moving along the critical line of the Ashkin-Teller model corresponds to varying q for the Potts model. For the square lattice comparison is made with renormalization group calculations, and the agreement found is a satisfactory check of renormalization group theory.

scholarly journals Cylinder partition function of the 6-vertex model from algebraic geometry

2020 ◽  
Vol 2020 (6) ◽  
Author(s):  
Zoltan Bajnok ◽  
Jesper Lykke Jacobsen ◽  
Yunfeng Jiang ◽  
Rafael I. Nepomechie ◽  
Yang Zhang
Keyword(s):  
Algebraic Geometry ◽  
Partition Function ◽  
Vertex Model
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