Alexander  R.  Miller

Department of Mathematics
University of Illinois at Urbana-Champaign
Urbana, IL 61801

I am a J. L. Doob Research Assistant Professor at the University of Illinois at Urbana-Champaign. My research interests are in combinatorics and representation theory (CV).


The probability that a character value is zero for the symmetric group

We consider random character values $\chi(g)$ of the symmetric group $S_n$, where $\chi$ is chosen at random from the set of irreducible characters and $g$ is chosen at random from the group, and we show that $\chi(g)=0$ with probability $\to 1$ as $n\to\infty$.

(Math. Z. 277 (2014) 1011–1015. Preprint)

Foulkes characters for complex reflection groups

We investigate Foulkes characters for a wide class of reflection groups which contains all finite Coxeter groups. In addition to new results, our general approach unifies, explains, and extends previously known (type A) results due to Foulkes, Kerber–Thürlings, Diaconis–Fulman, and Isaacs.

(Proc. AMS. 143 (2015) 3281–3293. Preprint)

Eigenspace arrangements of reflection groups

The lattice of intersections of reflecting hyperplanes of a complex reflection group $W$ may be considered as the poset of 1-eigenspaces of the elements of $W$. In this paper we replace 1 with an arbitrary eigenvalue and study the topology and homology representation of the resulting poset. After posing the main question of whether this poset is shellable, we show that all its upper intervals are geometric lattices, and then answer the question in the affirmative for the infinite family $G(m,p,n)$ of complex reflection groups, and the first 31 of the 34 exceptional groups, by constructing CL-shellings. In addition, we completely determine when these eigenspaces of $W$ form a $K(\pi,1)$ (resp. free) arrangement.
      For the symmetric group, we also extend the combinatorial model available for its intersection lattice to all other eigenvalues by introducing "balanced partition posets", presented as particular upper order ideals of Dowling lattices, study the representation afforded by the top (co)homology group, and give a simple map to the posets of pointed $d$-divisible partitions.

(Trans. AMS. 367 (2015) 8543–8578. Preprint)

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.

(Ph.D. Thesis. European J. Combin. 39 (2014) 24–56. Preprint)

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.

(Order 30 (2013) 657–662. 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 $\mathbb Z[t]$. 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 ${\mathbf Y}F$ 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 $Y^r$.

(Order 26 (2009) 197–228. Preprint arXiv:0811.1983)