Geometric Crystals and Set-Theoretical Solutions to the Quantum Yang-Baxter Equation

By Pavel Etingof. 2001

79 citations recorded by the source corpus.

Abstract We study the birational action of S n ×S m on the space of matrices defined by Braverman and Kazhdan using geometric crystals, from the viewpoint of the theory of set-theoretical solutions of the quantum Yang-Baxter equations. In particular, we obtain a direct proof of the well-definedness of this action, which does not use the theory of geometric crystals.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ee1bd0040830b1ef483ce9cb984ab8e567cb7c7b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Total positivity in loop groups I: whirls and curls (2008, background): “and Yamada [NY, Y] in the context of discrete Painlev´e dynamical systems (see also [Ki]). It also occurs as a birational Rmatrix in the Berenstein-Kazhdan [BK] theory of geometric crystals (see also [Et]). The tropicalization of this birational action is the combinatorial R-matrix of aﬃne crystals,”
Yang–Baxter maps and integrable dynamics (2002, background): “Some examples of birational Yang-Baxter maps coming from the theory of the periodic dressing chain and matrix KdV equation are discussed.”
Geometry of the Yang-Baxter Maps: pencils of conics and quadrirational mappings (2003, methodology): “[6] P.”