Title: Enumeration in quantum algebras
|Affiliation:||University of Waterloo|
Many of the classical algebras that occur in algebraic geometry and other mathematical fields have natural quantizations; that is, one can deform the multiplication rule using a parameter q, which has the property that when we specialize q at 1 we recover the classical object. As part of the general goal of understanding the representation theory of these rings, one often wants to understand the prime spectra of these algebras, that is, the collection of prime ideals in these rigs.
We explain how in many cases the prime spectrum of such an algebra decomposes naturally into a collection of cells and it becomes an important combinatorial problem to compute the number of cells and their dimensions. We explain this process in the case of the quantized coordinate ring of m x n matrices and show a surprising connection with totally nonnegative matrices via work of Postnikov. This is joint work with Stephane Launois.