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Algebraic and Topological Methods in Discrete Mathematics

Finite reflection groups, hyperplane arrangements,

and (oriented) matroids

(BaM-DAM-k, MaM-DGAK-k)

Prof. Raman Sanyal

Winter term 2024/25



German version

Graphic

What

In this advanced course, we connect combinatorics through geometry and algebra back to combinatorics. The starting point is the observation that the symmetric group Sn (=permutations of an n-element set under composition) describes the symmetries of a regular simplex (segments, triangles, tetrahedra, etc.). The generators of the group, the transpositions (i,j), correspond to reflections in hyperplanes {x:xi=xj}. The complement of these \binom{n}{2} hyperplanes are n! isometric convex cones that bijectively correspond to the elements in \mathfrak{S}_n. This gives a purely geometric perspective on permutations and their combinatorics. This combinatorial-geometric interplay will be generalized in the first part of the lecture to reflection groups, i.e., finite symmetry groups generated by reflections in hyperplanes. It turns out that there are only finitely many reflection groups per dimension, and they are completely described by the well-known Dynkin diagrams.

The mirrors of a reflection group form a hyperplane arrangement. Hyperplane arrangements are natural and recurring configurations of geometric objects, independent of group structure. Given n affine hyperplanes in \mathbb{R}^d, how many components does \mathbb{R}^d break into when the hyperplanes are removed? How many of these components are bounded? For n random lines in the plane, the number of components and bounded components is the evaluation of the polynomial p(t) = t^2 - nt + \binom{n}{2} at t=-1 or t=1, respectively.

Intersections of subsets of hyperplanes yield affine subspaces that are partially ordered by inclusion. The resulting intersection posets (partially ordered sets) contain enough information to answer the above questions. For example, for the arrangement of the n coordinate hyperplanes H_i = \{ x_i = 0\} in \mathbb{R}^d, the intersection poset is isomorphic to the Boolean lattice (2^{[n]},\subseteq). When all hyperplanes are linear (i.e., contain the origin), there is a purely(!) combinatorial characterization of the resulting posets. The defining properties of the resulting posets are cryptomorphic to matroids, which are discrete structures that abstract dependencies in linear algebra. Matroids and their combinatorics are related to virtually all areas of mathematics, and recent advances in understanding their combinatorics make them a very timely topic in discrete mathematics.

Starting from the ubiquitous reflection groups, this lecture introduces the combinatorics of hyperplane arrangements and matroids. Central to the course are the numerous connections to classical discrete structures and the many different perspectives.

Lecture notes
Homework 1
Homework 2
Homework 3
Homework 4
Homework 5
Homework 6

Context and Research Opportunities

The topics of the lecture are fundamental to the research project Simpliciality in Arrangements and Matroids, which is part of the new DFG Priority Program Combinatorial Synergies. The aim of the project is to investigate the geometry of simplicial hyperplane arrangements and the algebraic/geometric combinatorics of simplicial (oriented) matroids. The starting point for the project are the observed phenomena:

The project is led by Prof. Cuntz (Hannover), Prof. Kühne (Bielefeld), and Prof. Sanyal (Frankfurt). The lecture will be (virtually) held at all three locations, involving all project participants. The lecture is open to all interested parties.

The lecture also serves as the basis for a Dives into Research (DiR). DiRs are opportunities for advanced Bachelor and Master students to conduct research on well-curated projects and with guidance provided by senior researchers. Following the lecture, there will be an opportunity for 12 students to work on research projects related to the simpliciality of arrangements and matroids in March 2025. The application deadline is the end of the year. Details on the DiR and the application process will be announced later.

Prerequisites and Rules

Prerequisites for the lecture are Linear Algebra 1+2 and knowledge of Discrete Mathematics (comparable to the required lecture Discrete Mathematics (BaM-DM)). Depending on the participants, the lecture may be held in English.

There will be weekly exercise sheets discussed in tutorials. The final exam is an oral exam. Further details will be announced later.

If you wish to attend the lecture, please register informally by email with Prof. Sanyal (sanyal@math.uni-...).

When and Where

The lecture will take place from October 15 on Tuesdays from 2:15 PM to 3:45 PM in room 902 on Robert-Mayer Str. 10. The lecture will also be streamed online. Participants will receive the Zoom link for the lecture in advance.

There will be a weekly tutorial offered both in person at all three locations and online. Further details will follow.