Integrable systems have recently become an interesting topic for connecting fields such as probability, combinatorics, symmetric functions, and representation theory [BP16], [BW18], [BCG16], [MV14], [HK02]. The vertex model is a powerful tool for understanding the deep structure of the algebraic, probabilistic, and combinatorial sides of the system. The origin of the vertex model comes from diagonalizing the transfer matrix of some R matrix (The eigenfunctions of the transfer matrix are also known as the Bethe ansatz). The six vertex model (also known as the loop model [PZJ09] and the ice-type model) is one of the central objects in the study of vertex models.
The connection between the quantum group U_{q}(sl_2) and the six vertex model makes the structure of the vertex model well-behaved. About 40 years ago, Izergin and Korepin [IK81] proposed a new model in relation to the diluted loop model. Since then, both the coordinate Bethe ansatz [WBB92] and the Algebraic Bethe ansatz [VOT88] have been solved. More recently, Alexandra Garbali [AG] has found a nice determinant formula for the Izergin-Korepin model at the roots of unity (domain wall boundary condition).
In this talk, we are going to give a brief overview of the construction of the six vertex model and discuss some of the rational functions arising from it. Moreover, we will present a Cauchy identity for the Izergin-Korepin model and a symmetrization formula for the rational symmetric functions of the Izergin-Korepin model. This work is a joint effort with Sasha Garbali and Michael Wheeler.
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