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Weyl groupoids with at most three objects
Finite Weyl groupoids
Using previous results concerning the rank two and rank three cases, all connected simply connected Cartan schemes for which the real roots form a finite irreducible root system of arbitrary rank are…
Simplicial Arrangements with up to 27 Lines
- M. Cuntz
- MathematicsDiscret. Comput. Geom.
- 15 August 2011
It is proved that Grünbaum’s conjecture about unstretchable arrangements with at most 16 lines is false, and the conjecture that any simplicial arrangement with at least 14 pseudolines is stretchable is false.
Finite Weyl groupoids of rank three
We continue our study of Cartan schemes and their Weyl groupoids. The results in this paper provide an algorithm to determine connected simply connected Cartan schemes of rank three, where the real…
A combinatorial model for tame frieze patterns
- M. Cuntz
- 27 November 2017
Let $R$ be an arbitrary subset of a commutative ring. We introduce a combinatorial model for the set of tame frieze patterns with entries in $R$ based on a notion of irreducibility of frieze…
Coxeter and crystallographic arrangements are inductively free
The freeness of ideal subarrangements of Weyl arrangements
AbstractA Weyl arrangement is the arrangement deﬁned by the root sys-tem of a ﬁnite Weyl group. When a set of positive roots is an ideal inthe root poset, we call the corresponding arrangement an…
Frieze patterns over integers and other subsets of the complex numbers
We study (tame) frieze patterns over subsets of the complex numbers, with particular emphasis on the corresponding quiddity cycles. We provide new general transformations for quiddity cycles of…
On Wild Frieze Patterns
- M. Cuntz
- Mathematics, PhysicsExp. Math.
- 27 April 2015
There is an -frieze pattern whose entries are positive integers and with infinitely many different entries, and there are non-periodic -f Frieze patterns whose entries is positive integers.
Free but not recursively free arrangements
We construct counterexamples to the conjecture that every free arrangement is recursively free in characteristic zero. The intersection lattice of our smallest example has a realization over a finite…