# Alexander R. Miller

Universität Wien

Oskar-Morgenstern-Platz 1

A-1090 Wien, Austria

E-mail: alexander.r.miller@univie.ac.at

I am a member of the combinatorics group
in the Fakultät für Mathematik
at the Universität Wien.

You can find some of my papers on
the arxiv,
mathscinet,
and scholar.

#### Publications

### Character restrictions and reflection groups (with E. Giannelli)

Recent results of Ayyer–Prasad–Spallone and Isaacs–Navarro–Olsson–Tiep on odd-degree character restrictions for symmetric groups are extended to reflection groups $G(r,p,n)$.

Preprint pdf

### On parity and characters of symmetric groups

We present a conjectural parity bias in the character values of the symmetric group. The main conjecture says that a character value chosen uniformly at random from the character table of $S_n$ is congruent to $0$ mod $2$ with probability $\to 1$ as $n\to \infty$. A more general conjecture says that the same is true for all primes $p$, not only $p=2$. We relate these conjectures to zeros, give generating functions for computing lower bounds, and present some computational data in support of the main conjecture.

Journal of Combinatorial Theory Series A **162** (2019) 231–240.
pdf

### Walls in Milnor fiber complexes

For a real reflection group the reflecting hyperplanes cut out on the unit sphere a simplicial complex called the Coxeter complex. Abramenko showed that each reflecting hyperplane meets the Coxeter complex in another Coxeter complex if and only if the Coxeter diagram contains no subdiagram of type $D_4$, $F_4$, or $H_4$. The present paper extends Abramenko's result to a wider class of complex reflection groups. These groups have a Coxeter-like presentation and a Coxeter-like complex called the Milnor fiber complex. Our first main theorem classifies the groups whose reflecting hyperplanes meet the Milnor fiber complex in another Milnor fiber complex. To understand better the walls that fail to be Milnor fiber complexes we introduce Milnor walls. Our second main theorem generalizes Abramenko's result in a second way. It says that each wall of a Milnor fiber complex is a Milnor wall if and only if the diagram contains no subdiagram of type $D_4$, $F_4$, or $H_4$.

Documenta Mathematica **23** (2018) 1247–1261.
pdf

### Orthogonal polynomials and Smith normal form (with Dennis Stanton)

Smith normal form evaluations found by Bessenrodt and Stanley for some Hankel matrices of $q$-Catalan numbers are proven in two ways. One argument generalizes the Bessenrodt–Stanley results for the Smith normal form of a certain multivariate matrix that refines one studied by Berlekamp, Carlitz, Roselle, and Scoville. The second argument, which uses orthogonal polynomials, generalizes to a number of other Hankel matrices, Toeplitz matrices, and Gram matrices. It gives new results for $q$-Catalan numbers, $q$-Motzkin numbers, $q$-Schröder numbers, $q$-Stirling numbers, $q$-matching numbers, $q$-factorials, $q$-double factorials, as well as generating functions for permutations with eight statistics.

Monatshefte für Mathematik **187** (2018) 125–145.
arXiv:1704.03539

### Some characters that depend only on length

The author recently introduced Foulkes characters for a wide variety of reflection groups, and the hyperoctahedral ones have attracted some special attention. Diaconis and Fulman connected them to adding random numbers in balanced ternary and other number systems that minimize carries, and Goldstein, Guralnick, and Rains observed experimentally that they also play the role of irreducibles among the hyperoctahedral characters that depend only on length. We prove this conjecture and show that the same is true for a more general family of reflection groups.

Mathematical Research Letters **24** (2017) 879–891.

### 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.

Transactions of the American Mathematical Society **367** (2015) 8543–8578.
pdf

### 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.

Proceedings of the American Mathematical Society **143** (2015) 3281–3293.
pdf

### 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 $n\to\infty$.

Mathematische Zeitschrift **277** (2014) 1011–1015.
pdf

### 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 Journal of Combinatorics **39** (2014) 24–56.
pdf

### 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.

For the unpublished version presented in talks, see
letter to Stanley and Zanello.

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

### Note on 1-crossing partitions

(with M. Bergerson, A. Pliml, V. Reiner, P. Shearer, D. Stanton, and N. Switala)

It is shown that there are $\binom{2n-r-1}{n-r}$ noncrossing partitions of an $n$-set together with a distinguished block of size $r$, and $\binom{n}{k-1}\binom{n-r-1}{k-2}$ of these have $k$ blocks, generalizing a result of Bóna on partitions with one crossing. Furthermore, specializing natural $q$-analogues of these formulae with $q$ equal to certain $d$th roots-of-unity gives the number of such objects having $d$-fold rotational symmetry.

Ars Combinatoria **99** (2011) 83–87.
pdf