# Research Seminars

We are currently running two research seminars:

- Monday afternoon:
**Arrangements and Symmetries** - Tuesday afternoon:
**Combinatorics**

Below you can find all talks, presentations, and lectures of past seminars.

## summer term 2024

**Daniel Bath** (KU Leuven)

Seminar: Arrangements and Symmetries

Tuesday, April 29, 2024, 16:15, Room IA 1/135

**Title:** Bernstein—Sato polynomials of Hyperplane Arrangements in C^3

**Abstract:** The roots of Bernstein—Sato polynomial of a hypersurface manages to simultaneously contain most classical singularity invariants. Alas, computing these roots is mostly infeasible. For hyperplane arrangements in C^3, one can hope the roots of its Bernstein—Sato polynomial are combinatorially determined. Alas, things are more subtle. Walther demonstrated two arrangements with the same intersection lattice but whose respective Bernstein—Sato polynomials differ by exactly one root. We will show this is the only pathology possible. For arrangements in C^3, we prove that all but one root are (easily) combinatorially determined. We also give several equivalent criterion for the outlier, -2 + (2/deg), to in fact be a root of the Bernstein—Sato polynomial. These involve local cohomology data of the Milnor algebra and the non-formality of the arrangement.

This is an application of a study of Bernstein—Sato polynomials for a larger class of C^3 divisors than just arrangements. We will discuss Bernstein—Sato polynomials at large, our general strategy for divisor class, our main results, and how the promised formula for hyperplane arrangements appears.

**Amandine Favre** (University of Lausanne)

Seminar: Combinatorics

Tuesday, April 23, 2024, 14:15, Room IA 1/177

**Title:** Around the notion of atomic length on Weyl groups

**Abstract:** In this talk, we investigate the notion of atomic length on Weyl groups, which was first introduced by Nathan Chapelier-Laget and Thomas Gerber in 2022. We will see how this statistic permits to generalise some classical results on the combinatorics of core partitions. Moreover, we will introduce a balanced version of the atomic length, and present some interesting properties.

## winter term 2023/2024

**Johannes Schmitt** (Universität Siegen)

Seminar: Arrangements and Symmetries

Monday, January 22, 2024, 14:00, Room IA 1/177

**Title:** Symplectic reflections and quotient singularities

**Abstract:** Let V be a finite-dimensional vector space over the complex numbers and let G < GL(V) be a finite group. It is classically known that the reflections contained in G control aspects of the geometry of the linear quotient V/G, namely its smoothness (Chevalley- Shephard-Todd) and the class group (Benson). In this talk, we restrict to symplectic vector spaces V and groups G < Sp(V) which leave the symplectic form invariant and study partial symplectic resolutions of V/G. Here we find similar phenomena. We describe how the existence of a (smooth) symplectic resolution is related to aptly named symplectic reflections in G by a theorem of Verbitsky, but also how this relationship is still not completely understood. We further show that the class group of a partial symplectic resolution is controlled by the symplectic reflections in G.

**Takuro Abe** (Rikkyo University)

Seminar: Arrangements and Symmetries

Monday, January 15, 2024, 14:00, Room IA 1/177

**Title:** Multi-Euler derivations

**Abstract:** In the study of hyperplane arrangements, the research of the logarithmic derivation modules of them is one of the most important topics. Based on Kyoji Saito's primitive derivation, Terao's investigation of the multi-Coxeter arrangements and Yoshinaga's solving of the Edelman-Reiner conjecture, the study of logarithmic derivation modules of multi-reflection arrangements is also important. Among these researches, the existence of so called the universal derivations, defined by Wakamiko, has been a key of the research, and investigated by Stump, Roehrle, Terao, Yoshinaga, Wakamiko and so on. In this talk, we generalize this concept of universal derivations to arbitrary arrangements as the multi-Euler derivations, and give a criterion for a derivation to be multi-Euler.

**Vanthana Ganeshalingam**(University of Warwick)

Seminar: Arrangements and Symmetries

Monday, November 20, 2023, 14:15, Room IA 1/177

**Title:** Subgroup Structure of Reductive Groups

**Abstract:** This talk will introduce the concept of G complete-reducibility (G c-r) originally thought of by Serre in the 90s. This idea has important connections to the open problem of classifying the subgroups of a reductive group G. I will explain the methodology of the classification so far and the main obstacle which is understanding the non-G-cr subgroups.

**Lorenzo Giordani** (RUB)

Seminar: Arrangements and Symmetries

Monday, November 13, 2023, 14:45, Room IA 1/177

**Title:** Cohomology Rings of toric wonderful Models

**Abstract:** One of the leading motifs in the theory of arrangements is to understand the interplay of algebraic, geometric and topological properties with combinatorial ones, meant as properties of the intersection structure of the arrangements. An approach in this direction, initiated by DeConcini and Procesi in the nineties, is the introduction of ``wonderful'' compactifications for complement spaces of subspace arrangements. The study of wonderful models has ensured the combinatorial nature of relevant topological invariants of complement spaces, namely: cohomology, rational homotopy type, and mixed Hodge structure. Projective wonderful models were extended to the case of toric arrangements of arbitrary codimension. Their cohomology has then been studied by DeConcini and Gaiffi in the ``well-connected'' case, in which properties holding naturally in the linear case are imposed. I will report on work in progress with Roberto Pagaria and Viola Siconolfi, where we remove the hypothesis of well-connectedness, and offer a different presentation for the cohomology ring of the model. Our approach is closer to the combinatorial picture: we adapt the notions of blowups for semilattices and nested set complex introduced by Feichtner and Kozlov to the poset of layers of toric arrangements.

**Sven Wiesner** (RUB)

Seminar: Arrangements and Symmetries

Monday, November 13, 2023, 14:15, Room IA 1/177

**Title:** Free multiderivations on connected subgraph arrangements

**Abstract:** Recently, Cuntz and Kühne introduced (a way to assign) a hyperplane arrangement A(G) associated with a given connected graph G and they classified all graphs G such that A(G) is a free arrangement. I will present joint work with Gerhard Röhrle and Paul Mücksch in which we generalize this result to the multiarrangement case. We give a complete list of graphs which possess at least one non-trivial multiplicity m such that the corresponding multiarrangement (A(G),m) is free.

**Bernd Sturmfels**(MPI Leipzig)

Seminar: Arrangements and Symmetries

Friday, November 10, 2023, 14:00, Room IB 1/103

**Title:** Algebraic Varieties in Quantum Chemistry

**Abstract:** We discuss the algebraic geometry behind coupled cluster (CC) theory of quantum many-body systems. The high-dimensional eigenvalue problems that encode the electronic Schroedinger equation are approximated by a hierarchy of polynomial systems at various levels of truncation. The exponential parametrization of the eigenstates gives rise to truncation varieties. These generalize Grassmannians in their Pluecker embedding. We explain how to derive Hamiltonians, we offer a detailed study of truncation varieties and their CC degrees, and we present the state of the art in solving the CC equations. This is joint work with Fabian Faulstich and Svala Sverrisdóttir.

**Graham Denham**(Western University Ontario)

Seminar: Arrangements and Symmetries

Friday, October 23, 2023, 14:00, Room IA 1/177

**Title:** -

**Abstract:** -

## summer term 2023

**Sarah Brauner** (Leipzig)

Seminar: Combinatorics

Thursday, August 31, 2023, 16:15, Room IB 3/73

**Title:** The Eulerian representations of the symmetric group and generalizations

**Abstract:** The Eulerian representations are a well-known family of symmetric group representations that decompose the regular representation. In this talk, I will focus on two of their most intriguing features. First, they are isomorphic (as representations) to the graded pieces of the cohomology of the configuration space of n ordered points in R^3. Second, they “lift” from representations of S_n to the Whitehouse representations of S_{n+1}, which also have a cohomological interpretation. I will discuss my work generalizing both of these properties, the former to real reflection groups of coincidental type—that is, reflection groups whose exponents form an arithmetic progression—and the latter to the hyperoctahedral group.

**Nathan Chapelier-Laget** (University of Sydney)

Seminar: Arrangements and Symmetries

Monday, August 14, 2023, 16:15, Room IA 1/135

**Title:** The Shi variety of an affine Weyl group

**Abstract:** In this talk I will introduce a geometrical object associated to an affine Weyl group W, called the Shi variety, that has the property of having its set of integral points in bijection with W. Then we will discuss some consequences obtained with the set of the irreducible components of this variety, in particular its connections in type A with a certain conjugacy class and Young's lattice.

**Thomas Gerber** (RUB)

Seminar: Arrangements and Symmetries

Monday, April 24, 2023, 16:15, Room IA 1/135

**Title:** Atomic length on Weyl groups II: Combinatorics

**Abstract:** In recent joint work with Nathan Chapelier-Laget, we introduced the notion of atomic length for (finite and affine) Weyl groups, as a variant of the usual Coxeter length function. In this second talk, I will present various properties and interpretations of this statistic using root system combinatorics. We will also see how this gives natural extensions of the results of the previous talk.

**Thomas Gerber** (RUB)

Seminar: Arrangements and Symmetries

Monday, April 17, 2023, 16:15, Room IA 1/135

**Title:** Atomic length on Weyl groups I: Representation Theory

**Abstract:** In recent joint work with Nathan Chapelier-Laget, we introduced the notion of atomic length for (finite and affine) Weyl groups, as a variant of the usual Coxeter length function. In this first talk, I will present the representation-theoretic motivations for studying this statistic. More precisely, we will review some fundamental problems in modular representation theory of symmetric groups and Hecke algebras, which can be tackled by investigating partition combinatorics.

**Álvaro Gutiérrez** (Bonn)

Seminar: Combinatorics

Tuesday, April 11, 2023, 16:15, Room IB 3/73

**Title:** The coefficients of plethysm

**Abstract:** The composition of two Schur functors is known as the plethysm of the respective representations. Understanding the coefficients of plethysm (that is, the decomposition into irreducible representations of the plethysm of two irreducible representations) is a notoriously hard problem, featured in Stanley's list of important problems in algebraic combinatorics. In this talk, we present the definition of plethysm and the coefficients of plethysm in the language of symmetric functions, and one or two recent results. The first one, a condition for the positivity of the coefficients. If time permits, a second one, which is a structural property of some coefficients of iterated plethysms. We also present some further conjectures on other structural behaviours.

## winter term 2022/2023

**Susanna Fishel** (Arizona)

Seminar: Combinatorics

Tuesday, February 21, 2023, 15:15, Room IB 3/73

**Title:** Shi arrangements and low elements in Coxeter groups

**Abstract:** The m-Shi arrangement for an arbitrary Coxeter system (W, S) and a nonnegative integer m is a refinement of the Coxeter hyperplane arrangement of the system. The classical Shi arrangement (m = 0) was introduced in the case of affine Weyl groups by Shi to study Kazhdan-Lusztig cells for W. In two key results, Shi showed that each region of the classical Shi arrangement contains exactly one element of minimal length in W and that the union of their inverses form a convex subset of the Coxeter complex. In this talk, we will discuss generalizations of Shi's results to arbitrary Coxeter systems. This is joint work with Dyer, Hohlweg, and Mark.

**Clément Chenevière** (IRMA Strasbourg and Ruhr-Universität Bochum)

Seminar: Combinatorics

Tuesday, February 14, 2023, 16:15, Room IB3/73

**Title:** The linear intervals in the alt v-Tamari lattices

**Abstract:** The v-Tamari lattices are partial orders on the set of paths weakly above a lattice path v. They generalize the classical Tamari lattices which can be defined on Dyck paths. In this talk, I will first introduce these lattices and then a new family of posets that further generalizes these lattices, namely the alt v-Tamari lattices. This family is very rich and promising as it also contains the Dyck lattices, another very natural and interesting order on Dyck paths. I will then focus on the linear intervals in these posets and especially the main result stating that for a fixed path v, all the alt v-Tamari lattices have the same number of linear intervals.

**Christopher Schmidt** (RUB)

Seminar: Combinatorics

Tuesday, February 7, 2023, 16:15, Room IB 3/73

**Title:** Graphs, Determinants and Formal Proof Assistants

**Abstract:** Proof assistants have had a subordinate role in mathematics. While this may still be the case, Peter Scholze has recently brought some awareness to them by underlining their importance. This talk aims to give an insight into Lean, an upcoming interactive theorem prover, in the context of the well- known lemma by Lindström-Gessel-Viennot. We investigate that lemma from a theoretical point of view before presenting a general introduction to Lean and its key components, such as its dependent type theory. Afterwards, we examine, how the statement of the Lindström-Gessel-Viennot lemma and the relevant definitions can be implemented.

**Lorenzo Giordani** (RUB)

Seminar: Arrangements and Symmetries

Monday, February 6, 2023, 14:15, Room IA 1/177

**Title:** On the Combinatorics and Cohomology of Wonderful models for subspace arrangements

**Abstract:** A classical problem in the theory of hyperplane arrangements is to understand to what extent the combinatorial information of the arrangement, encoded in the associated matroid or lattice of intersections, determines geometric proprieties of the complement space. P. Orlik, L. Solomon, E. Brieskorn et al. proved that the cohomology ring of the complement space is isomorphic to the so called Orlik-Solomon algebra, which is defined entirely in terms of the underlying matroid. In this seminar, we recall the results on the Orlik-Solomon algebra and present some constructions by C. De Concini and P. Procesi, including their "Wonderful model" and its proprieties. Using the model, they proved that the cohomology ring of the complement space is still determined by the combinatorial data when hyperplanes are substituted by subspaces of arbitrary codimension.

**Sven Wiesner** (RUB)

Seminar: Arrangements and Symmetries

Monday, January 30, 2023, 14:15, Room IA 1/177

**Title:** Techniques from algebraic geometry applied to matroids

**Abstract:** June Huh et al. proved longstanding conjectures about specific sequences associated to matroids which are combinatorial objects. They did so by associating a structure to these matroids on which tools from algebraic geometry can get deployed. In my talk I want to give a short overview about the structures involved and how they derived the results about the matroid from them.

**Chi Ho Yuen** (Oslo)

Seminar: Combinatorics

Tuesday, January 24, 2023, 16:15, via Zoom

**Title:** Filtrations of the Tope Space of an Oriented Matroid

**Abstract:** Oriented matroids are matroids with extra sign data, and they are useful in the tropical study of real algebraic geometry. In order to study the topology of real algebraic/tropical hypersurfaces constructed from patchworking, Renaudineau and Shaw introduced an algebraically defined filtration of the tope spaces of oriented matroids (for a real hyperplane arrangement, the tope space is the homology of its complement) based on Quillen filtration. We will prove the equivalence of their filtration and the topologically defined Kalinin filtration, as well as the combinatorially defined Gelfand-Varchenko dual degree filtration. If time permits, we will also discuss a potential globalization. This is joint work with Kris Shaw.

**Tim Strunkheide** (RUB)

Seminar: Combinatorics

Tuesday, January 17, 2023, 16:15, Room IB 3/73

**Title:** Symplectic Hecke insertion

**Abstract:** The K-theoretic Schur P-function for a given strict partition is the generating function of all semistandard set-valued shifted tableaus with the shape of this partition. We want to check some pieces in the proof, that these functions have a ring structure. Therefore, we will look at some properties of a bijection, the symplectic Hecke insertion of special kinds of Hecke words and pairs of tableaus, where the second tableau is such a set-valued shifted tableau. After we are convinced by this, we want to risk a look at an unproved conjecture of another property of this insertion and let a computer check a few examples with the algorithm for us. This is an expository talk based on the paper "A symplectic refinement of shifted Hecke insertion" by Eric Marberg.

**Michael Cuntz** (Hannover)

Seminar: Arrangements and Symmetries

Monday, January 9, 2023, 16:15, via Zoom

**Title:** On arrangements of hyperplanes from connected subgraphs

**Abstract:** We investigate arrangements of hyperplanes whose normal vectors are given by connected subgraphs of a fixed graph. These include the resonance arrangement and certain ideal subarrangements of Weyl arrangements. We characterize those which are free, simplicial, factored, or supersolvable. In particular, such an arrangement is free if and only if the graph is a cycle, a path, an almost path, or a path with a triangle attached to it. This is joint work with Lukas Kühne.

**Theo Douvropoulos** (UMass Amherst)

Seminar: Combinatorics

Tuesday, December 20, 2022, 17:15, Room IB 3/73

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

**Abstract:** The collection of parking functions under a natural Sn-action (which has Catalan-many orbits, indexed by Dyck paths) has been a central object in Algebraic Combinatorics since the work of Haiman more than 30 years ago. One of the lines of research spawned around it was towards defining and studying analogous objects for real and complex reflection groups W; the main candidates are known as the W-non-nesting and W-non-crossing parking functions. The W-non-nesting parking functions are relatively well understood; they form the so called algebraic W-parking-space which has a concrete interpretation as a quotient ring. The W-non-crossing ones on the other hand have defied unified explanations while simultaneously proving themselves central in the study of Coxeter and Artin groups (their geometric group theory, representation theory, and combinatorics). One of the main open problems in the field since the early 2000's has been to give a type-independent proof of the W-isomorphism between the algebraic and the non-crossing W-parking spaces. In this talk, I will present such a proof, solving the more general Fuss version of the problem, that proceeds by comparing a collection of recursions that are shown to be satisfied by both objects. This relies on a variety of recent techniques we introduced, in particular the enumeration of certain flats of full support via Crapo's beta invariant, the W-Laplacian matrices for reflection arrangements and, in collaboration with Matthieu Josuat-Verges, the refined f- to h- transformation between the cluster complex and the non-crossing lattice of W.

**David Stewart** (University of Newcastle)

Seminar: Arrangements and Symmetries

Monday, December 5, 2022, 16:15, Room IA 1/109

**Title:** A Prolog-assisted search for simple Lie algebras

**Abstract:** Prolog is a very unusual programming language, developed by Alain Colmerauer in one of the buildings on the way to the CIRM in Luminy. It is not fundamentally iterative in the way that, for example, GAP and Magma are. Instead it operates by taking a list of axioms as input, and responds at the command line to queries asking the language to achieve particular goals. It gained some notoriety by beating contestants on the game show Jeopardy in 2011. It is also the worlds fastest sudoku solver. I will describe some recent Prolog investigations to search for new simple Lie algebras over the field GF(2). We were able to discover some new examples in dimensions 15 and 31 and extrapolate from these to construct two new infinite families of simple Lie algebras.

**Laura Voggesberger** (RUB)

Seminar: Combinatorics

Tuesday, November 29, 2022, 16:15, Room IB 3/73

**Title:** Working with algebraic groups in Magma

**Abstract:** In group theory, there is a large family of infinite groups called algebraic groups. For connected reductive groups there is a rather good understanding of the structure which can be described by combinatorial data such as root systems. In order to solve certain questions about algebraic groups, we often need to do a case-by-case analysis. Together with the structure of these groups, this often leads to a computational approach. For instance, it is possible to compute the action of a connected reductive group on its Lie algebra by using computer algebra systems such as Magma. For this talk, we are taking a look at parametrizing the group elements such that we can implement an action of the group on the Lie algebra in Magma.

**Frédéric Chapoton** (Strasbourg)

Seminar: Combinatorics

Tuesday, November 22, 2022, 16:15, Room IB 3/73

**Title:** Some reasons why intervals matter

**Abstract:** The notion of interval in partially ordered sets is a fundamental one, and there are many aspects to their study. I will explain some of the good reasons to look at them, ranging from pure combinatorics to more algebraic ones. This will be illustrated with various examples of combinatorial posets of current interest.

**Laura Voggesberger** (RUB)

Seminar: Arrangements and Symmetries

Monday, November 21, 2022, 14:15, Room IA 1/177

**Title:** Nilpotent Pieces in Lie Algebras of Exceptional Type in Bad Characteristic

**Abstract:** This talk will be a trial run for my defense concerning certain structures in algebraic groups and their Lie algebras. In group theory, a big and important family of infinite groups is given by the algebraic groups. These groups and their structures are already well-understood. In representation theory, the study of the unipotent variety in algebraic groups — and by extension the study of the nilpotent variety in the associated Lie algebra — is of particular interest. Let G be a connected reductive algebraic group over an algebraically closed field k, and let Lie(G) be its associated Lie algebra. By now, the orbits in the nilpotent and unipotent variety under the action of G are completely known and can be found for example in a book of Liebeck and Seitz. There exists, however, no uniform description of these orbits that holds in both good and bad characteristic. With this in mind, Lusztig defined a partition of the unipotent variety of G in 2011. Equivalently, one can consider certain subsets of the nilpotent variety of Lie(G) called the nilpotent pieces. This approach appears in the same paper by Lusztig in which he explicitly determines the nilpotent pieces for simple algebraic groups of classical type. The nilpotent pieces for the exceptional groups of type G2 , F4 , E6 , E7 , and E8 in bad characteristic have not yet been determined. In my thesis, I have explored the cases for G2 , F4 , and E6, and will present them in this talk.

**Benjamin Schröter** (KTH/Frankfurt)

Seminar: Combinatorics

Tuesday, November 15, 2022, 16:15, Room IB 3/73

**Title:** Valuative Invariants for Matroids

**Abstract:** Valuations on polytopes are maps that combine the geometry of polytopes with relations in a group. It turns out that many important invariants of matroids are valuative on the collection of matroid base polytopes, e.g., the Tutte polynomial and its specializations or the Hilbert–Poincaré series of the Chow ring of a matroid. In this talk I will present a framework that allows us to compute such invariants on large classes of matroids, e.g., sparse paving and elementary split matroids, explicitly. The concept of split matroids introduced by Joswig and myself is relatively new. However, this class appears naturally in this context. Moreover, (sparse) paving matroids are split. I will demonstrate the framework by looking at Ehrhart polynomials, relations in Chow rings of combinatorial geometries, and further examples.

**Markus Reineke** (RUB)

Seminar: Combinatorics

Tuesday, November 8, 2022, 16:15, Room IB3/73

**Title:** Zonotopal algebras, orbit harmonics, and Donaldson-Thomas invariants of symmetric quivers

**Abstract:** Donaldson-Thomas invariants of quivers were introduced by Kontsevich and Soibelman as an abstraction/precision of string-theoretic BPS state counts. In exploring their integrality and positivity properties, various geometric and algebraic interpretations of these invariants were developed. Very recently, an almost general purely combinatorial interpretation could be given. In the talk, after recalling the background on quivers and their DT invariants, we will discuss the main combinatorial players (break divisors and zonotopal algebras), state thecombinatorial interpretation of DT invariants, and consider examples and consequences.

**Galen Dorpalen-Barry** (RUB)

Seminar: Combinatorics

Tuesday, October 25, 2022, 16:15, Room IB 3/73

**Title:** Balanced Flag Poincaré Polynomials

**Abstract:** A brief introduction to balanced flag Poincaré polynomials and some of their properties. This talk contains ongoing work with Christian Stump and Josh Maglione.

**Paul Mücksch** (Kyushu University)

Seminar: Arrangements and Symmetries

Monday, October 24, 2022, 14:15, via Zoom

**Title:** Topology of supersolvable oriented matroids

**Abstract:** A central result in the topology of complex hyperplane arrangements, due to Falk, Randell and Terao, states that supersolvability of the intersection lattice of such arrangements implies that their complements are K(pi,1)-spaces. The homotopy type of the complement of a complexified real hyperplane arrangement can be modeled by a nice regular CW-complex introduced by Salvetti. The Salvetti complex can be constructed for any oriented matroid -- a combinatorial abstraction of a real hyperplane arrangement. In my talk, I will present a novel combinatorial way to prove that supersolvability of the geometric lattice of an oriented matroid implies the asphericity of its Salvetti complex. In particular, this extends to the non-realizable case.

**Avi Steiner** (Mannheim)

Seminar: Arrangements and Symmetries

Monday, October 17, 2022, 14:15, Room IA 1/177

**Title:** Falling Powers and the Algebra of Descents

**Abstract:** Of interest to people who study both hyperplane arrangements and commutative algebra are the homological properties of the module of logarithmic derivations of a hyperplane arrangement A. I will introduce the "ideal of pairs", which is a sort of "symmetrization" of this module of logarithmic derivations with respect to matroid duality. This is an ideal which simultaneously "sees" many of the homological properties of both the arrangement and its dual.

**Elena Hoster** (RUB)

Seminar: Combinatorics

Tuesday, September 20, 2022, 16:15, Room IB 3/73

**Title:** On combinatorial formulas for Schur and Macdonald polynomials

**Abstract:** Macdonald symmetric functions are an orthogonal basis for the ring of symmetric functions. They are best-studied in Type A, where they generalize Schur functions, although they can be defined for any Weyl group, and they form an orthogonal basis for an appropriately-defined ring of W-invariant functions. We give a combinatorial description, from Ram and Yip, for Macdonald polynomials via alcove walks on the affine Weyl arrangement. We then turn our attention to Type A, where Lenart gives another description of Macdonald polynomials via nonattacking tableaux fillings.

**Joris Köfler** (RUB)

Seminar: Combinatorics

Tuesday, September 13, 2022, 16:00, Room IB 3/73

**Title:** Combinatorial Models of the Grassmannian and the Amplituhedron

**Abstract:** The Grassmannian is the space of k-dimensional subspaces of a n-dimensional vector space. In this expository talk, we survey some of the models for the totally nonnegative part of the Grassmannian and show that they stratify this part of the Grassmannian. Time permitting, we will turn our attention toward the amplitudehedron, which is the image of the totally nonnegative part of the Grassmannian under a certain map, and present the Karp-Williams cell decomposition of the the m = 1 amplituhedron.

## summer term 2022

**Götz Pfeiffer** (Galway)

Seminar: Arrangements and Symmetries

Monday, July 4, 2022, 14:15, Room IC 03/647

**Title:** Falling Powers and the Algebra of Descents

**Abstract:** A finite Coxeter group of classical type A, B or D contains a chain of subgroups of the same type. We show that intersections of conjugates of these subgroups are again of the same type, and make precise in which sense and to what extent this property is exclusive to the classical types of Coxeter groups. As the main tool for the proof we use Solomon’s descent algebra. Using Stirling numbers, we express certain basis elements of the descent algebra as polynomials and derive explicit multiplication formulas for a commutative subalgebra of the descent algebra. This is joint work with Linus Hellebrandt.

**Darij Grinberg** (Drexel)

Seminar: Combinatorics

Tuesday, June 21, 2022, 17:15, Room IB 3/73

**Title:** The one-sided cycle shuffles in the symmetric group algebra

**Abstract:** Elements in the group algebra of a symmetric group S_n are known to have an interpretation in terms of card shuffling. I will discuss a new family of such elements, recently constructed by Nadia Lafrenière: Given a positive integer n, we define n elements t_1, t_2, ..., t_n in the group algebra of S_n by t_i = the sum of the cycles (i), (i, i+1), (i, i+1, i+2), ..., (i, i+1, ..., n), where the cycle (i) is the identity permutation. The first of them, t_1, is known as the top-to-random shuffle and has been studied by Diaconis, Fill, Pitman (among others). The n elements t_1, t_2, ..., t_n do not commute. However, we show that they can be simultaneously triangularized in an appropriate basis of the group algebra (the "descent-destroying basis"). As a consequence, any rational linear combination of these n elements has rational eigenvalues. The maximum number of possible distinct eigenvalues turns out to be the Fibonacci number f_{n+1}, and underlying this fact is a filtration of the group algebra connected to "lacunar subsets" (i.e., subsets containing no consecutive integers). This talk will include an overview of other families (both well-known and exotic) of elements of these group algebras. I will also briefly discuss the probabilistic meaning of these elements as well as some tempting conjectures. This is joint work with Nadia Lafrenière.

**Katharina Jochemko** (KTH)

Seminar: Combinatorics

Tuesday, June 21, 2022, 16:15, via Zoom (also streamed to IB3/73)

**Title:** The Eulerian transformation

**Abstract:** Many polynomials arising in combinatorics are known or conjectured to have only real roots. One approach to these questions is to study transformations that preserve the real-rootedness property. This talk is centered around the Eulerian transformation which is the linear transformation that sends the i-th standard monomial to the i-th Eulerian polynomial. Eulerian polynomials appear in various guises in enumerative and geometric combinatorics and have many favorable properties, in particular, they are real-rooted and symmetric. We discuss how these properties carry over to the Eulerian transformation. In particular, we give a counterexample to a conjecture by Brenti (1989) concerning the preservation of real roots, extend recent results on binomial Eulerian polynomials and provide enumerative and geometric interpretations. This is joint work with Petter Brändén.

**Gerhard Röhrle** (Bochum)

Seminar: Arrangements and Symmetries

Monday, June 20, 2022, 14:15, Room IC 03/647

**Title:** Inductive Freeness of Ziegler's Canonical Multiderivations

**Abstract:** Let A be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction A'' of A to any hyperplane endowed with the natural multiplicity k is then a free multiarrangement (A'',k), alo known as the Ziegler restriction. I'll report on recent joint work with Torsten Hoge where we prove an analogue of Ziegler's theorem for the stronger notion of inductive freeness. Namely, if A is inductively free, then so is the multiarrangement (A'',k). In a related result we derive that if a deletion A' of A is free and the corresponding restriction A'' is inductively free, then so is (A'',k) -- irrespective of the freeness of A. I shall discuss several consequences of the theorem for natural classes of inductively free arrangements. Time permitting I shall explain counterparts of the latter kind for the notion of additive and recursive freeness.

**John Machacek** (Oregon)

Seminar: Arrangements and Symmetries

Monday, June 13, 2022, 16:15, via Zoom

**Title:** Braid cones and the Gorenstein property

**Abstract:** To a poset we associate a convex cone called a braid cone. This cone is a union of some regions in the braid arrangement. We aim to determine when the toric variety associated with the cone is Gorenstein. For toric varieties, being Gorenstein is equivalent to the existence of a certain affine hyperplane. We show in our setting that the existence of some vertex labeling of the Hasse diagram is equivalent to being Gorenstein. In the case that the poset has a maximum element we find that the labeling is determined by a Möbius function. This will give us a recursive algorithm to check the Gorenstein property. This is joint work with Josh Hallam. No background knowledge on varieties will be assumed.

**Christian Lange** (München)

Seminar: Arrangements and Symmetries

Monday, May 30, 2022, 14:15, Room IC 03/647

**Title:** From polytopes to reflection-rotation groups

**Abstract:** In the talk I will first explain a characterization of alcoves among convex polytopes in terms of billiard flows, and an analogous characterization of isosceles tetrahedra among (boundaries) of convex polytopes. These spaces are examples of constant curvature Riemannian orbifold metrics on simply connected manifolds, which arise as quotients of constant curvature model spaces by discrete groups generated by reflections and rotations, i.e. isometries with codimension one or two fixed point subspace. In the spherical case such groups are finite and completely classified. Examples are given by (orientation preserving subgroups of) finite real and complex reflection groups, but there are also many other examples. I will sketch this classification and discuss some properties which these groups (seem to) share with (complex) reflection groups, for instance concerning their isotropy groups and the asphericity of arrangement complements. Depending on time and interest I may also say something more about the flat and the hyperbolic case.

**Lorenzo Venturello** (KTH)

Seminar: Combinatorics

Tuesday, May 24, 2022, 17:15, via Zoom

**Title:** Gorenstein algebras from simplicial complexes

**Abstract:** Gorenstein algebras are intriguing objects which often show up in combinatorics and geometry. In this talk I will present a construction which associates to every pure simplicial complex a standard graded Gorenstein algebra, which can be presented as a polynomial ring modulo an ideal generated by monomials and pure binomials defined from the combinatorial data of the complex. When the simplicial complex is flag, i.e., it is the clique complex of its graph, our main results establish equivalences between well studied properties of the complex (being S_2, Cohen-Macaulay, Shellable) with those of the algebra (being quadratic, Koszul, having a quadratic GB). I will provide all the necessary definitions during the talk. Finally, we study the h-vector of the Gorenstein algebras in our construction and answer a question of Peeva and Stillman by showing that it is very often not gamma-positive. This is joint work with Alessio D'Alì.

**Luis Ferroni** (KTH)

Seminar: Combinatorics

Tuesday, May 24, 2022, 16:15, via Zoom

**Title:** Valuative invariants for large classes of matroids

**Abstract:** There are classes of matroids that are very large. For example, Mayhew et al. conjecture that asymptotically almost all matroids are sparse paving. The class of paving matroids and split matroids are even larger. We will explain how to compute any nice valuation for split matroids. In particular, for an arbitrary split matroid we can describe explicitly the volume, Tutte polynomial, the Kazhdan-Lusztig polynomial, the Ehrhart polynomial, the Speyer's invariant, the spectrum polynomial (and much more!). As an application we can show that for a large class of matroids the Speyer's invariant has positive coefficients; also we will show how using this machinery one can easily produce a matroid polytope with some negative Ehrhart coefficients.

**Andrés R. Vindas Meléndez** (MSRI/Berkeley)

Seminar: Combinatorics

Tuesday, May 17, 2022, 16:15, via Zoom

**Title:** Ehrhart Theory of Paving and Panhandle Matroids

**Abstract:** Ehrhart theory is a topic in geometric combinatorics which involves the enumeration of lattice points in integral dilates of polytopes. We show that the base polytope P_M of any paving matroid M can be obtained from a hypersimplex by slicing off subpolytopes. The pieces removed are base polytopes of lattice path matroids corresponding to panhandle-shaped Ferrers diagrams, whose Ehrhart polynomials we can calculate explicitly. Consequently, we can write down the Ehrhart polynomial of P_M. Combinatorially, our construction corresponds to constructing a uniform matroid from a paving matroid by iterating the operation of stressed-hyperplane relaxation introduced by Ferroni, Nasr, and Vecchi, which generalizes the standard matroid-theoretic notion of circuit-hyperplane relaxation. We present evidence that panhandle matroids are Ehrhart positive and describe a conjectured combinatorial formula involving chain gangs and Eulerian numbers from which Ehrhart positivity of panhandle matroids will follow. As an application of the main result, we calculate the Ehrhart polynomials of matroids associated with Steiner systems and finite projective planes, and show that they depend only on their design-theoretic parameters. (This is joint work with D. Hanely, J. Martin, D. McGinnis, D. Miyata, G. Nasr, and, M. Yin).

**Eirini Chavli** (Stuttgart)

Seminar: Arrangements and Symmetries

Monday, May 9, 2022, 14:15, via Zoom

**Title:** Complex Hecke algebras are real

**Abstract:** Iwahori Hecke algebras associated with real reflection groups appear in the study of finite reductive groups. In 1998 Broué, Malle, and Rouquier generalized in a natural way the definition of these algebras to complex case. However, some basic properties of the real case are also true for Hecke algebras in the complex case. In this talk we will talk about these properties and their state of the art.

**Giovanni Paolini** (Amazon Web Services/Caltech)

Seminar: Arrangements and Symmetries

Monday, May 2, 2022, 16:15, via Zoom

**Title:** Dual Coxeter groups of rank three

**Abstract:** In this talk, I will present ongoing work aimed at understanding the noncrossing partition posets associated with Coxeter groups of rank three. In particular, I will describe the combinatorial and geometric techniques used to prove the lattice property and lexicographic shellability. These properties can then be used to solve several problems on the corresponding Artin groups, such as the K(π,1) conjecture, the word problem, the center problem, and the isomorphism between standard and dual Artin groups. Joint work with Emanuele Delucchi and Mario Salvetti.

**Paul Mücksch** (Bonn)

Seminar: Arrangements and Symmetries

Monday, April 25, 2022, 14:15, Room IC 03/647

**Title:** On formality for hyperplane arrangements

**Abstract:** An arrangement of hyperplanes is called formal provided all linear dependencies among the defining linear forms of the hyperplanes are generated by ones corresponding to intersections of codimension two. This notion turns out to be necessary at the one hand for the apshericity of the complement of a complex arrangement due to work by Falk and Randell. One the other hand it is also necessary for the freeness of the module of logarithmic vector fields thanks to a result by Yuzvinsky. In joint work with T. Möller and G. Röhrle we extend the above line of results by showing that the combinatorial property of factoredness implies formality. Furthermore, we study formality with respect to the standard arrangement constructions of restriction and localization and comment on the behavior of the stronger property of k-formality introduced by Brand and Terao.

**Shuhei Tsujie** (Hokkaido)

Seminar: Arrangements and Symmetries

Monday, April 11, 2022, 13:15, via Zoom

**Title:** MAT-free graphic arrangements and strongly chordal graphs.

**Abstract:** Recently Cuntz and Mücksch introduced MAT-free arrangements based on the Multiple Addition Theorem (MAT) by Abe, Barakat, Cuntz, Hoge, and Terao. In this talk, we will focus on graphic arrangements. Stanley showed that a graphic arrangement is free if and only if the graph is chordal. We will show that a graphic arrangement is MAT-free if and only if it is strongly chordal. This is joint work with Tan Nhat Tran.

## winter term 2021/2022

**Gaku Liu** (University of Washington (Seattle)

Seminar: Combinatorics

Tuesday, February 8, 2022, 17:15, via Zoom

**Title:** Unimodular triangulations of sufficiently large dilations

**Abstract:** An integral polytope is a polytope whose vertices have integer coordinates. A unimodular triangulation of an integral polytope in R^d is a triangulation in which all simplices are integral with volume 1/d!. A classic result of Kempf, Mumford, and Waterman states that for every integral polytope P, there exists a positive integer c such that cP has a unimodular triangulation. We strengthen this result by showing that for every integral polytope P, there exists c such that for every positive integer c' < c, c'P admits a unimodular triangulation.

**Lukas Kühne** (Bielefeld)

Seminar: Combinatorics

Tuesday, February 1, 2022, 16:15, via Zoom

**Title:** Geometry of Flag Hilbert-Poincaré series

**Abstract:** Flag Hilbert–Poincaré series arise in the context of local Igusa zeta functions associated to hyperplane arrangements and are connected to other seemingly different enumeration problems in algebra and geometry. We study a coarsening of these series in the case of real hyperplane arrangements by applying geometric and combinatorial tools related to their chambers. In this case, we prove that the numerators of these series are coefficient-wise bounded below by the Eulerian polynomial and equality holds if and only if all chambers are simplicial. This generalizes a recent theorem of Maglione–Voll. This is joint work with Joshua Maglione.

**Jacob Matherne** (Bonn/Max Planck)

Seminar: Arrangements and Symmetries

Monday, January 31, 2022, 14:00, via Zoom

**Title:** Singular Hodge theory for combinatorial geometries

**Abstract:** If you take a collection of planes in R^3, then the number of lines you get by intersecting the planes is at least the number of planes. This is an example of a more general statement, called the "Top-Heavy Conjecture", that Dowling and Wilson conjectured for all matroids in 1974. On the other hand, given a hyperplane arrangement (or more generally a matroid), I will explain how to uniquely associate to it a certain polynomial, called its Kazhdan–Lusztig (KL) polynomial. I will spend some portion of the talk comparing and contrasting these KL polynomials with the classical ones in Lie theory. The problems of proving the "Top-Heavy Conjecture" and the non-negativity of the coefficients of these KL polynomials are related, and they are controlled by the Hodge theory of a certain singular projective variety, called the Schubert variety of the arrangement. The "Top-Heavy Conjecture" was proven for hyperplane arrangements by Huh and Wang in 2017, and the non-negativity was proven by Elias, Proudfoot, and Wakefield in 2016. I will discuss joint work with Tom Braden, June Huh, Nicholas Proudfoot, and Botong Wang which resolves these two problems for arbitrary matroids. If time permits, I will mention the equivariant versions of these problems, which arise from taking into account the symmetries of the matroid.

Philip Dörr(Magdeburg)

Seminar: Combinatorics

Tuesday, January 25, 2022, 16:15, via Zoom

Title:Extreme Values of Permutation Statistics in Suitable Triangular Arrays

Abstract:Two important statistics on random permutations are the number of inversions and the number of descents. These statistics can be generalized to Coxeter groups, where inversions and descents are defined with respect to a specific system of generators. The question of under which conditions these statistics satisfy a central limit theorem is well-understood. In contrast to that, the extremal type behavior of such statistics is rather unexplored yet. As the considered random permutation statistics only take finitely many values, we need to consider a suitably scaled triangular array of these statistics on groups of growing rank and tackle the dependencies therein.

Norihiro Nakashima(Nagoya)

Seminar: Arrangements and Symmetries

Monday, January 24, 2022, 13:15, via Zoom

Title:Dimensions for extended Shi and Catalan arrangements to be hereditarily free

Abstract:A central arrangement is said to be hereditarily free if all restriction arrangements are free. Several investigations are interested in hereditarily free arrangements. Recently, Hoge and Röhrle proved that the finite complex reflection arrangements are hereditarily free. In this talk, we show that the cone of the extended Catalan arrangement of type A is always hereditarily free, while we determine the dimension that the cone of the extended Shi arrangement of type A is hereditarily free. For this purpose, using digraphs, we define a class of arrangements which contains the extended Shi and Catalan arrangements, and we characterize the freeness for the cone of this arrangement by graphical conditions. We also define contraction to prove that the class of arrangements are closed under restriction. The contraction is different from ordinary vertex contraction on digraphs. This is a joint work with Shuhei Tsujie.

Giovanni Paolini(Amazon/Caltech)

Seminar: Arrangements and Symmetries

Monday, January 17, 2022, 16:15, via Zoom [postponed, see]

Title:tba

Abstract:tba

Shuhei Tsujie(Hokkaido)

Seminar: Arrangements and Symmetries

Monday, December 20, 2021, 13:15, via Zoom [cancelled]

Title:MAT-free graphic arrangements

Abstract:Recently Cuntz and Mücksch introduced MAT-free arrangements based on the Multiple Addition Theorem by Abe, Brakat, Cuntz, Hoge, and Terao. In this talk, we will focus on graphic arrangements. Stanley showed that a graphic arrangement is free if and only if the graph is chordal. We will show that every MAT-free graphic arrangement is strongly chordal. This is joint work with Tan Nhat Tran.

Mariel Supina(KTH)

Seminar: Combinatorics

Tuesday, December 7, 2021, 16:15, via Zoom

Title:The Universal Valuation for Coxeter Matroids

Abstract:Matroids are combinatorial objects that generalize the notion of independence, and their subdivisions have rich connections to geometry. Thus we are often interested in functions on matroids that behave nicely with respect to subdivisions, which are called valuations. Matroids are naturally linked to the symmetric group; generalizing to other finite reflection groups gives rise to Coxeter matroids. I will give an overview of these ideas and then present some work with Chris Eur and Mario Sanchez on constructing the universal valuative invariant of Coxeter matroids. Since matroids and their Coxeter analogues can be understood as families of polytopes with special combinatorial properties, I will present these results from a polytopal perspective.

Georg Loho(Universität Twente)

Seminar: Combinatorics

Tuesday, November 30, 2021, 16:15, Room IB 3/73

Title:Generalized permutahedra and complete classes of valuated matroids

Abstract:Generalized permutahedra form an important class of polytopes occurring explicitly or implicitly in several branches of mathematics and beyond. I start with an overview of concepts related with generalized permutahedra from optimization and Discrete Convex Analysis. Valuated generalized matroids give rise to a particularly interesting subclass. I show some fundamental constructions for these objects. This leads to an answer to two open questions from auction theory and discrete convex analysis.

Laura Escobar(Washington University in St. Louis)

Seminar: Combinatorics

Tuesday, November 23, 2021, 16:15, via Zoom

Title:Determining the complexity of Kazhdan-Lusztig varieties

Abstract:Kazhdan-Lusztig varieties are defined by ideals generated by certain minors of a matrix, which are chosen using a combinatorial rule. These varieties are of interest in commutative algebra and the study of Schubert varieties. Each Kazhdan-Lusztig variety has a natural torus action from which one can construct a polyhedral cone. The complexity of this torus action can be computed from the dimension of the cone and, in some sense, indicates how close the variety is to the toric variety of the cone. In joint work with Maria Donten-Bury and Irem Portakal we address the problem of classifying which Kazhdan-Lusztig varieties have a given complexity. We do so by utilizing the rich combinatorics of Kazhdan-Lusztig varieties.

Nicholas Proudfoot(University of Oregon)

Seminar: Arrangements and Symmetries

Monday, November 22, 2021, 15:00, via Zoom

Title:Equivariant log concavity in the cohomology of configuration spaces

Abstract:June Huh proved in 2012 that the Betti numbers of the complement of a complex hyperplane arrangement form a log concave sequence. But what if the arrangement has symmetries, and we regard the cohomology as a representation of the symmetry group? The motivating example is the braid arrangement, where the complement is the configuration space of n points in the plane, and the symmetric group acts by permuting the points. I will present an equivariant log concavity conjecture, and show that one can use the theory of representation stability to prove infinitely many cases of this conjecture for configuration spaces.

Jesus de Loera(UC Davis)

Seminar: Combinatorics

Tuesday, November 16, 2021, 16:15, via Zoom

Title:Stochastic Tverberg-type theorems and their relevance in Machine Learning and Statistical Inference

Abstract:Discrete geometry can play a role in foundations of data science. Here I present concrete examples. In statistical inference we wish to find the properties or parameters of a distribution or model through sufficiently many samples. A famous example is logistic regression, a popular non-linear model in multivariate statistics and supervised learning. Users often rely on optimizing of maximum likelihood estimation, but how much training data do we need, as a function of the dimension of the covariates of the data, before we expect an MLE to exist with high probability? Similarly, for unsupervised learning and non-parametric statistics, one wishes to uncover the shape and patterns from samples of a measure or measures. We use only the intrinsic geometry and topology of the sample. A famous example of this type of method is the $k$-means clustering algorithm. A fascinating challenge is to explain the variability of behavior of $k$-means algorithms with distinct random initializations and the shapes of the clusters. In this talk we present new stochastic combinatorial theorems, direct new variations of Tverberg’s theorem, that give bounds on the probability of existence of maximum likelihood estimators in multinomial logistic regression and also quantify to the variability of clustering initializations. Along the way we will see fascinating connections to the coupon collector problem, topological data analysis, and to the computation of Tukey centerpoints of data clouds (a high-dimensional generalization of median). This is joint work with (in various papers) with T. Hogan, R. D. Oliveros, E. Jaramillo-Rodriguez, A. Torres-Hernandez, and Dominic Yang.

Galen Dorpalen-Barry(RUB)

Seminar: Arrangements and Symmetries

Monday, November 15, 2021, 14:00, Room IA 03/449

Title:A Short Introduction to Cones of Hyperplane Arrangements (Part II)

Abstract:In this two-part series we introduce some material relating to cones of hyperplane arrangements. This is the second talk of this series. In the first half of this talk, we will introduce some useful tools from commutative algebra (initial forms, filtered rings, Gröbner bases, etc). In the second half, we will use everything we’ve learned so far to introduce the Varchenko-Gel’fand ring and use it to study hyperplane arrangements and their cones.

Clément Chenevière(IRMA Strasbourg and Ruhr-Universität Bochum)

Seminar: Combinatorics

Tuesday, November 9, 2021, 16:15, Room IB3/73

Title:Intervals in the m-Tamari and m-Cambrian lattices

Abstract:The Tamari lattice can be described on Dyck paths with covering relations described as a swap of a down step with the excursion following it. This lattice can also be embedded on triangulations of a polygon with increasing flips of diagonals as covering relations. Both descriptions lead to a genaralization of this lattice, the former by changing Dyck paths into m-Dyck paths, and the latter by changing triangles into (m+2)-gons. This raises the m-Tamari and the (linear) m-Cambrian lattices, which are not isomorphic in general but we conjecture that they would have the same number of intervals, even with refined countings.

**Galen Dorpalen-Barry** (RUB)

Seminar: Arrangements and Symmetries

Monday, November 8, 2021, 14:00, Room IA 03/449

**Title:** A Short Introduction to Cones of Hyperplane Arrangements (Part I)

**Abstract:** In this two-part series we introduce some material relating to cones of hyperplane arrangements. This is the first talk of this series. In this talk we will introduce (cones of) hyperplane arrangements and use them to motivate the study of oriented matroids. Along the way, we will point out some recent results related to cones of hyperplane arrangements and pick up tools for proving the theorems we will encounter during the second talk of this series.

Christian Krattenthaler(Universität Wien)

Seminar: Combinatorics

Tuesday, October 26, 2021, 17:15, Room IB 3/73

Title:Reciprocity between Dyck paths and alternating sequences - heaps, orthogonal polynomials, ...

Abstract:Reciprocity is a much used (and misused) term in mathematics. The meaning of "reciprocity" in our context is the one introduced by Richard Stanley: if a sequence $(a_n)_{n\ge0}$ is extended to negative indices $n$, and if there is a combinatorial meaning for these "negative" terms of the sequence, then he speaks of a "combinatorial reciprocity law". The theme of this talk is a previously unobserved "reciprocity law" between Dyck paths and alternating sequences. As it turns out, this "reciprocity" extends to (certain) families of Dyck paths and alternating sequences, and it relates to several other (combinatorial and non-combinatorial) objects, including Viennot's theory of heaps, orthogonal polynomials, determinants, ...

Alex Black(UC Davis)

Seminar: Combinatorics

Tuesday, October 26, 2021, 17:15, via Zoom

Title:Polyhedral Geometry of Pivot Rules

Abstract:A fundamental open question in the theory of linear programming is whether there exists a pivot rule such that the Simplex Method with that rule runs in polynomial time. Part of the difficulty in studying this question is that the space of all pivot rules is not easy to define. Instead, we introduce a tractable yet robust subclass called normalized weight pivot rules. The normalization serves a role similar to the norm in the steepest edge pivot rule, and the weight is a vector that works as an auxiliary objective function as with the shadow vertex pivot rule. For a fixed LP, a normalized weight pivot rule determines a unique choice of outgoing edge for each vertex on the graph of the associated polytope and thus an arborescence. Our main results are finding two polytopes that describe how these arborescences vary as we vary either the weight or objective function. We relate these polytopes to other known constructions and show these polytopes yield new realizations of polytopes in algebraic combinatorics including the Stanley-Pitman polytope, associahedron, and permutahedra of types A and B.

Sarah Rees(University of Newcastle)

Seminar: Arrangements and Symmetries

Monday, October 18, 2021, 14:00, via Zoom

Title:Rewriting in Artin groups and their relations

Abstract:A fundamental open question in the theory of linear programming is whether there exists a pivot rule such that the Simplex Method with that rule runs in polynomial time. Part of the difficulty in studying this question is that the The family of Artin groups contains a variety of groups with apparently quite different properties. But despite that some of the techniques developed to deal with particular subfamilies reveal a lot of commonality in their approaches. I’ll discuss what is known about rewrite systems for Artin groups, surveying work of a number of different authors for various types, from Artin and Garside to very recent work of Dehornoy and others. In particular I’ll examine Dehornoy’s search for a common approach. I’ll finish with a discussion of interval groups associated with Coxeter groups of type Dn. Joint work with Baumeister and Neaime, and now Holt, derives presentations for these groups as quotients of various Artin groups, and we hope these may lead to effective rewriting techniques for them.

## summer term 2021

**Tan Nhat Tran** (RUB)

Seminar: Arrangements and Symmetries

Monday, July 19, 2021, 16:15, via Zoom

**Title:** 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)

Seminar: Arrangements and Symmetries

Monday, July 12, 2021, 16:15, via Zoom

**Title:** On inductively free and additionally free arrangements

**Abstract:**pdf

**Takuro Abe** (Kyushu University)

Seminar: Arrangements and Symmetries

Monday, July 12, 2021, 12:00, via Zoom

**Title:** 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)

Seminar: Arrangements and Symmetries

Monday, July 5, 2021, 16:30, via Zoom

**Title:** 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)

Seminar: Arrangements and Symmetries

Monday, June 14, 2021, 16:00, via Zoom

**Title:** Towards the Linearity of Complex Braid Groups

**Abstract:**pdf

**Tan Nhat Tran** (RUB)

Seminar: Arrangements and Symmetries

Monday, May 3/10/17, 2021, 16:15, via Zoom

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

**Abstract:**pdf

## winter term 2020/2021

**Misha Feigin** (University of Glasgow)

Seminar: Arrangements and Symmetries

Monday, March 8, 2021, 16:00, via Zoom

**Title:** 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)

Seminar: Arrangements and Symmetries

Monday, February 8, 2021, 16:15, via Zoom

**Title:** tba

**Abstract:**

**Paul Mücksch** (RUB)

Seminar: Arrangements and Symmetries

Monday, January 18, 2021, 16:15, via Zoom

**Title:** 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)

Seminar: Arrangements and Symmetries

Monday, January 18, 2021, 12:15, via Zoom

**Title:** 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)

Seminar: Arrangements and Symmetries

Monday, December 7, 2020, 16:15, via Zoom

**Title:** 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)

Seminar: Arrangements and Symmetries

Monday, November 30, 2020, 16:15, via Zoom

**Title:** 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)

Seminar: Arrangements and Symmetries

Monday, November 23, 2020, 16:15, via Zoom

**Title:** 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.

**Georges Neaime** (RUB)

Seminar: Arrangements and Symmetries

Monday, November 2/9/16, 2020, 16:15-17:45, HIB

Monday, November 2/9/16, 2020, 16:15-17:45, HIB

Monday, December 14/21, 2020, 16:15-17:45, via Zoom

Monday, January 11, 2021, 16:15-17:45, via Zoom

**Title:** Garside Theory I - VI

**Abstract:**pdf

## winter term 2019/2020

**Jonathan Kliem** (Berlin)

Seminar: Combinatorics

Wednesday, December 4, 2019, 12:15, Room IB 2/141

**Title:** A face iterator for polyhedra

**Abstract:** We discuss a new algorithm that iterates over all elements of a meet-semilattice, where every interval is coatomic.

**Robert Löwe** (Berlin)

Seminar: Combinatorics

Wednesday, November 6, 2019, 12:15, Room IB 2/141

**Title:** Computing the discriminant of a quaternary cubic form

**Abstract:** We determine the 166104 extremal monomials of the discriminant of a quaternary cubic form. These are in bijection with D-equivalence classes of regular triangulations of the 3-dilated tetrahedron. We describe how to compute these triangulations and their D-equivalence classes in order to arrive at our main result.

The computation poses several challenges, such as dealing with the sheer number of triangulations effectively, as well as devising a suitably fast algorithm for computation of a D-equivalence class. This is joint work with Lars Kastner.