Crystallographic arrangements - Finite Weyl groupoids


This page is a collection of material about crystallographic arrangements. It is still under construction, much more data will be here soon (for example a GAP-package to access the root sets).

For the classification of crystallographic arrangements see Finite Weyl groupoids.

Hasse diagrams

These are "Hasse diagrams" for the crystallographic arrangements of rank three to seven. More precisely: Let C, C' be irreducible connected Cartan schemes for which the real roots are finite root systems. If there exist objects a in C and b in C' such that Ra is a subset of Rb (up to permutations of the coordinates), then in the diagram we draw an arrow from C to C'. To simplify the diagrams, if there is an arrow from C to C' and from C' to C'', then we omit the arrow from C to C''. A label "x (y)" means the arrangement x, it has y hyperplanes. Notice that in these diagrams, arrangements which only differ by an automorphism may have different labels since we have one label for each Cartan scheme up to equivalence.

Rank three Rank four Rank five Rank six Rank seven

Inductive freeness

Crystallographic arrangements are inductively free, i.e. their freeness may be proved via the Addition-Theorem. See here for more details and for certificates of inductive freeness.

Pictures

Here are some pictures (object change diagrams of rank three Cartan schemes).

50
A B C DC1 DC2
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
26 27 28 29 30
31 32 33 34 35
36 37 38 39 40
41 42 43 44 45
46 47 48 49 50




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Last updated: November 2010