A bijection between cores and dominant Shi regions
by Susanna Fishel
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.