scholarly journals Double Grothendieck Polynomials and Colored Lattice Models

Author(s):  
Valentin Buciumas ◽  
Travis Scrimshaw

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.

10.37236/4971 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Angèle M. Hamel ◽  
Ronald C. King

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.


2011 ◽  
Vol 844 (2) ◽  
pp. 289-307 ◽  
Author(s):  
Wen-Li Yang ◽  
Xi Chen ◽  
Jun Feng ◽  
Kun Hao ◽  
Bo-Yu Hou ◽  
...  

2011 ◽  
Vol 226 (1) ◽  
pp. 840-886 ◽  
Author(s):  
Takeshi Ikeda ◽  
Leonardo C. Mihalcea ◽  
Hiroshi Naruse

1992 ◽  
Vol 07 (30) ◽  
pp. 2799-2810 ◽  
Author(s):  
HIROSI OOGURI

We define a lattice statistical model on a triangulated manifold in four dimensions associated to a group G. When G= SU (2), the statistical weight is constructed from the 15j-symbol as well as the 6j-symbol for recombination of angular momenta, and the model may be regarded as the four-dimensional version of the Ponzano-Regge model. We show that the partition function of the model is invariant under the Alexander moves of the simplicial complex, thus it depends only on the piecewise linear topology of the manifold. For an orientable manifold, the model is related to the so-called BF model. The q-analog of the model is also constructed, and it is argued that its partition function is invariant under the Alexander moves. It is discussed how to realize the 't Hooft operator in these models associated to a closed surface in four dimensions as well as the Wilson operator associated to a closed loop. Correlation functions of these operators in the q-deformed version of the model would define a new type of invariants of knots and links in four dimensions.


1996 ◽  
Vol 153 (1-3) ◽  
pp. 189-198 ◽  
Author(s):  
Christian Krattenthaler ◽  
Robert A. Sulanke

1992 ◽  
Vol 07 (25) ◽  
pp. 6385-6403
Author(s):  
Y.K. ZHOU

A method to find continuum quantum integrable systems from two-dimensional vertex models is presented. We explain the method with the example where the quantum sine-Gordon model is obtained from an inhomogeneous six-vertex model in its scaling limit. We also show that the method can be applied to other models.


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

2000 ◽  
Vol 100 (2) ◽  
pp. 2141-2146 ◽  
Author(s):  
A. G. Izergin ◽  
E. Karjalainen ◽  
N. A. Kitanin

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