## Principal minors and rhombus tilings

## Richard Kenyon (Brown)

#### Abstract :

The algebraic relations between the principal minors (PMs)
of an nXn matrix are mysterious and complicated.
If we add in "almost" principal minors (APMs),
however, we show that the relations
are generated by a single relation, the so-called hexahedron relation.
Using this relation we give a set of Laurent-polynomial
parameterizations of a matrix using APMs and PMs,
indexed by rhombus tilings of a regular 2n-gon.

This is joint work with Robin Pemantle.