## Oberseminar "Arrangements and Symmetries"

an der Fakultät für Mathematik der Ruhr-Universität Bochum (KVV 150926)

Organisatoren:
Galen Dorpalen-Barry,
Georges Neaime,
Gerhard Röhrle,
Christian Stump

Abstracts werden durch **Klick auf den Titel** sichtbar. Die Vorträge können auch als **Kalender** aboniert werden.

## Vorträge Sommersemester 2020

**Tan Nhat Tran**, RUB, Montag, 19. Juli 2021, 16:15

**Arrangements arising from digraphs and freeness of arrangements between Shi and Ish**

**Abstract:**
To a given vertex-weighted digraph (directed graph) we associate an arrangement analogous to the notion of Stanley's $\psi$-graphical arrangements and study it from perspectives of combinatorics and freeness. Our arrangement unifies several arrangements in literature including the Catalan arrangement, the Shi arrangement, the Ish arrangement, and especially the arrangements interpolating between Shi and Ish recently introduced by Duarte and Guedes de Oliveira.

It was shown that the arrangements between Shi and Ish all share the same characteristic polynomial with all nonnegative integer roots, thus raising the natural question of their freeness. We introduce two operations on the vertex-weighted digraphs and prove that subject to certain conditions on the weight $\psi$, the operations preserve the characteristic polynomials and freeness of the associated arrangements. In particular, by applying a sequence of these operations to the Shi arrangement, we affirmatively prove that the arrangements between Shi and Ish all are free, and among them only the Ish arrangement has supersolvable cone. Notably, all of the arrangements between Shi and Ish appear as the members in the operation sequence, thus giving a new insight into how they naturally arise and interpolate between Shi and Ish.

This is joint work with T. Abe (Kyushu) and S. Tsujie (Hokkaido)

**Sven Wiesner**, RUB, Montag, 12. Juli 2021, 16:15

**Takuro Abe**, Kyushu University, Montag, 12. Juli 2021, 12:00

**Logarithmic vector fields and differential forms revisited**

**Abstract:**
Logarithmic vector fields and logarithmic differential forms are known to be dual to each other, so their behaviors are similar. For example, it is free if the other is free. However, though they are similar, they are very different too. For example, if we delete one hyperplane from a free arrangement, then the projective dimension of the logarithmic vector field is at most one, but that of logarithmic differential forms can be larger as we want. We give a way to understand these differences in a uniform way, and give several applications of this viewpoint by solving several problems. This is a joint work with Graham Denham.

**Henri Mühle**, TU Dresden, Montag, 5. Juli 2021, 16:30-17:45

**Connectivity Properties of Factorization Posets**

**Abstract:**
Let G be a group generated by a finite set A. A factorization poset of a group element g is a graded partially ordered set whose maximal chains are in bijection with the reduced A-factorizations of g. If the reduced A-factorizations of g have length n, then the braid group on n strands acts naturally on these factorizations by so-called Hurwitz moves.
We consider three different notions of connectivity in factorization posets. 1) Chain-connectivity is satisfied if any maximal chain can be reached from a given one by a sequence of one-element substitutions. 2) Hurwitz-connectivity is satisfied if any reduced A-factorization of g can be reached from a given one by a sequence of Hurwitz moves. 3) Shellability is satisfied if the order complex of the proper part of the factorization poset is (topologically) shellable.
We explain how these three types of connectivity can be interpreted in terms of factorization posets and discuss connections, implications and non-implications among them. We exploit the recursive structure of factorization posets to give local (rank-2) criteria implying some of these connectivity types.
This talk is based on joint work with Vivien Ripoll, who is currently running a puzzle hunt business (https://solving-fun.com/).

**Georges Neaime**, RUB, Montag, 14. Juni 2021, 16:00-17:30

**Tan Nhat Tran**, RUB, Montag, 3./10./17. Mai 2021, 16:15-17:45

**Characteristic quasi-polynomials of integral hyperplane arrangements**

## Vorträge Wintersemester 2020/2021

**Misha Feigin**, Glasgow, Montag, 8. März 2021, 16:00-17:30

**Quasi-invariants and free multiarrangements**

**Abstract:** Quasi-invariant polynomials are associated to a reflection arrangement and an invariant multiplicity function on it. They first appeared in the work of Chalykh and Veselov on quantum integrable systems in 1990 in the Coxeter case. Freeness results for quasi-invariants can be related with freeness of modules of logarithmic vector fields for the reflection arrangements. This is due to the observation that components of invariant logarithmic vector fields are given by certain quasi-invariants. This has useful consequences both for quasi-invariants and for logarithmic vector fields. This relation can also be extended to finite complex reflection groups and to affine settings in which cases it gives new free (multi-)arrangements. The talk is based on joint work with T. Abe, N. Enomoto and M. Yoshinaga.

**Volkmar Welker**, Philipps-Universität Marburg, Montag, 8. Februar 2021, 16:15-17:45

**Paul Mücksch**, Ruhr-Universität Bochum, Montag, 18. Januar 2021, 16:15-17:45

**On Yuzvinsky's lattice sheaf cohomology for hyperplane arrangements**

**Abstract**: In my talk, I will establish the exact relationship between the cohomology of a certain sheaf on the intersection lattice of a hyperplane arrangement introduced by Yuzvinsky and the cohomology of the coherent sheaf on punctured affine space respectively projective space associated to the derivation module of the arrangement. I will derive a Künneth formula connecting the cohomology theories, answering a question posed by Yoshinaga. This, in turn, gives a new proof of Yuzvinsky’s freeness criterion and yields a stronger form of the latter.

**Masahiko Yoshinaga**, Hokkaido University Sapporo, Montag, 18. Januar 2021, 12:15-13:45

**A geometric realization of combinatorial reciprocity of order polynomials**

**Abstract**: The Euler characteristic of topological space can be considered as a generalization of the cardinality of a finite set. In previous work with Hasebe and Miyatani (2017), we generalized Stanley's combinatorial reciprocity for order polynomials to an equality of Euler characteristics of certain spaces of homomorphisms of posets. In this talk, we discuss recent development of geometric realization of the combinatorial reciprocity. The main result asserts that certain spaces of poset homomorphisms are actually homeomorphic which clearly implies the Euler characteristics. The proof is based on the detailed analysis of upper semicontinuous functions on metrizable topological spaces. This is joint work with Taiga Yoshida.

**René Marczinzik**, Universität Stuttgart, Montag, 7. Dezember 2020, 16:15-17:45

**Distributive lattices and Auslander regular algebras**

**Abstract**: We show that the incidence algebra of a finite lattice L is
Auslander regular if and only if L is distributive. As an application
we show that the order dimension of L coincides with the global
dimension of its incidence algebra when L has at least two elements
and we give a categorification of the rowmotion bijection for
distributive lattices. At the end we discuss the Auslander regular
property for other objects coming from combinatorics. This is joint
work with Osamu Iyama.

**Lukas Kühne**, Max-Planck-Institut Leipzig, Montag, 30. November 2020, 16:15-17:45, per Zoom

**The Resonance Arrangement**

**Abstract**: The resonance arrangement is the arrangement of hyperplanes which has all nonzero 0/1-vectors in R^n as normal vectors. It is also called the all-subsets arrangement. Its chambers appear in algebraic geometry, in mathematical physics and as maximal unbalanced families in economics.

In this talk, I will present a universality result of the resonance arrangement. Subsequently, I will report on partial progress on counting its chambers. Along the way, I will review some of the combinatorics of general hyperplane arrangements. If time permits I will also touch upon the related threshold arrangement which encodes Boolean functions that are linearly separable.

**Theo Douvropoulos**, University of Massachusetts Amherst, Montag, 23. November 2020, 16:15-17:45, per Zoom

**Recursions and proofs in Coxeter-Catalan combinatorics**

**Abstract**: The noncrossing partition lattice NC(W) associated to a finite Coxeter
group W has become a central object in Coxeter-Catalan combinatorics
during the last 25 years. We focus on two recursions on the simple
generators of W; the first due to Deligne (and rediscovered by Reading)
determines the chain number of NC(W) and the second, more general, due
to Fomin-Reading recovers the whole zeta polynomial. The resulting
formulas have nice product structures and are key players in the field,
but are still not well understood; in particular, they are derived by
the (case-free) recursions separately for each type.

A uniform derivation of the formulas from these recursions requires
proving certain
identities between the Coxeter numbers and invariant degrees of a
group and those
of its parabolic subgroups. In joint work with Guillaume Chapuy, we
use the W-Laplacian
(for W of rank n, this is an associated nxn matrix that we introduced in earlier
work and which generalizes the usual graph Laplacian) to prove the
required identities
for the chain number of W. We give a second proof by using the theory
of multi-reflection
arrangements and the local-to-global identities for their
characteristic polynomials.
This latter approach is in fact applicable to the study of the whole
zeta polynomial of NC(W)
although it, currently, falls short of giving a uniform derivation of
Chapoton's formula for it.

Montag, 2./9./16. November 2020, 16:15-17:45, Gebäude HIB

Montag, 14./21. Dezember 2020, 16:15-17:45, per Zoom

Montag, 11. Januar 2021, 16:15-17:45, per Zoom

**Georges Neaime**, Ruhr-Universität Bochum