# A bijection between cores and dominant Shi regions

Algebra/Geometry/Topology Seminar

#### by Susanna Fishel

Institution: Arizona State
Date: Fri 12th August 2011
Time: 3:15 PM
Location: 213 Richard Berry

Abstract: It is well-known that Catalan numbers $$C_n$$ count the number of dominant regions in the Shi arrangement of type A, and that they also count partitions which are both n-cores as well as $$(n+1)-cores$$. These concepts have natural extensions, which we call here the m-Catalan numbers and m-Shi arrangement. In this paper, we construct a bijection between dominant regions of the m-Shi arrangement and partitions which are both n-cores as well as $$(mn + 1)-cores$$. This is joint work with M. Vazirani.

We also modify our construction to produce a bijection between bounded dominant regions of the m-Shi arrangement and partitions which are both $$n-cores$$ as well as $$(mn − 1)-cores$$. The bijections are natural in the sense that they commute with the action of the afﬁne symmetric group.

If there is time, I will describe how we ﬁx certain hyperplanes in the arrangement and use the bijection above for enumeration of the regions which have that hyperplane as a separating wall. This is joint work with E. Tzanaki and M. Vazirani.