Combinatorics Reading Seminar: Domain-wall boundary conditions and the Macdonald polynomials

Seminar | November 13 | 11:10 a.m.-12:10 p.m. | 748 Evans Hall

 Aaron Brookner, UC Berkeley

 Department of Mathematics

For a certain choice of boundary conditions, the 6-vertex model's partition function has an elegant formula in terms of determinants. We associate a variable to each row and each column of the grid, which can be interpreted as a physical "inhomogeneity" factor. It follows from the YBE that the partition function is symmetric in the row variables, and in the column variables, separately; we will find that when a certain parameter, q, is a primitive 3rd root of unity, the partition function is actually a Schur polynomial, and is thus symmetric in the union of these variables. Finally, all of the above has natural generalization to the "higher spin 6-vertex model", whereby the Schur polynomial is replaced by a Macdonald polynomial.