## Oberseminar "Arrangements and Symmetries"

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

Organisatoren:
Paul Mücksch,
Gerhard Röhrle,
Christian Stump

## Vorträge Sommersemester 2021

Montag, 3. Mai 2021, 16:15-17:45

**Tan Nhat Tran**, RUB

**CHARACTERISTIC QUASI-POLYNOMIALS OF INTEGRAL HYPERPLANE
ARRANGEMENTS (1/3)**

## Vorträge Wintersemester 2020/2021

Montag, 8. März 2021, 16:00-17:30

**Misha Feigin**, Glasgow

**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.

Montag, 8. Februar 2021, 16:15-17:45, per Zoom

**Volkmar Welker**, Philipps-Universität Marburg

**tba**

Montag, 18. Januar 2021, 16:15-17:45

**Paul Mücksch**, Ruhr-Universität Bochum

**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.

Montag, 18. Januar 2021, 12:15-13:45

**Masahiko Yoshinaga**, Hokkaido University Sapporo

**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.

Montag, 7. Dezember 2020, 16:15-17:45

**René Marczinzik**, Universität Stuttgart

**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.

Montag, 30. November 2020, 16:15-17:45, per Zoom

**Lukas Kühne**, Max-Planck-Institut Leipzig

**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.

Montag, 23. November 2020, 16:15-17:45, per Zoom

**Theo Douvropoulos**, University of Massachusetts Amherst

**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. November 2020, 16:15-17:45, Gebäude HIB

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

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

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

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

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

**Georges Neaime**, Ruhr-Universität Bochum

**Garside Theory I - VI**