Alexander ⋅ R ⋅ Miller

I'm a student of Victor Reiner studying algebraic combinatorics at the University of Minnesota, and expect to graduate in the Spring of 2013. Some topics that I'm particularly interested in are complex reflection groups, complex polytopes, subspace arrangements, classical "type A" combinatorics, representation theory, invariant theory, geometry, and topology. To be even more specific, I like when these topics are mixed together. Here is some more information: my email alexmiller@umn.edu, some of my papers on the arXiv, and my CV.

Reflection arrangements and ribbon representations

Ehrenborg and Jung recently related the order complex for the lattice of d-divisible partitions with the simplicial complex of pointed ordered set partitions via a homotopy equivalence. The latter has top homology naturally identified as a Specht module. Their work unifies that of Calderbank, Hanlon, Robinson, and Wachs. By focusing on the underlying geometry, we strengthen and extend these results from type A to all real reflection groups and the complex reflection groups known as Shephard groups.

(34 pages. Submitted. Preprint arXiv:1108.1429)

Differential posets have strict rank growth: a conjecture of Stanley

We establish strict growth for the rank function of an r-differential poset. We do so by exploiting the representation theoretic techniques developed by Reiner and the author for studying related Smith forms.

(5 pages. Submitted. Preprint arXiv:1202.3006)

Differential posets and Smith normal forms (with V. Reiner)

We conjecture a strong property for the up and down maps U and D in an r-differential poset: DU+tI and UD+tI have Smith normal forms over Z[t]. In particular, this would determine the integral structure of the maps U, D, UD, DU, including their ranks in any characteristic. As evidence, we prove the conjecture for the Young-Fibonacci lattice YF studied by Okada and its r-differential generalizations Z(r), as well as verifying many of its consequences for Young's lattice Y and the r-differential Cartesian products Y^r.

(32 pages. Appeared. Order, 26 (2009), no. 3, 197-228. Preprint arXiv:0811.1983)