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3 Special arrangements

The following describes functions to construct some special arrangements.

3.1 Global methods

3.1-1 AGpql

Returns: A hyperplane arrangement.

Constructs the reflection arrangement associated to the monomial complex reflection group G(p,q,l). The hyperplanes are defined by equations of the form

x_i = \zeta^k x_j

where ζ is a primitive p-th root of unity.

gap> A := AGpql(2,1,3);
 <HyperplaneArrangement: 9 hyperplanes in 3-space>
gap> Roots(A);
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ],
  [ 1, 1, 0 ], [ 1, -1, 0 ], [ 1, 0, 1 ],
  [ 1, 0, -1 ], [ 0, 1, 1 ], [ 0, 1, -1 ] ]

3.1-2 SsS3

Returns: A hyperplane arrangement.

Generates the irreducible supersolvable simplicial arrangement of rank 3 with n hyperplanes. n must either be even oder congruent 1 mod 4. In any case, n\geq 6.

gap> A:=SsS3(9); Roots(A);
<HyperplaneArrangement: 9 hyperplanes in 3-space>
[ [ 0, 0, 1 ], [ 0, 1, 0 ], 
  [ -1/2*E(8)+1/2*E(8)^3, 1/2*E(8)-1/2*E(8)^3, 0 ], 
  [ -1, 0, 0 ], 
  [ -1/2*E(8)+1/2*E(8)^3, -1/2*E(8)+1/2*E(8)^3, 0 ], 
  [ 1/2*E(8)-1/2*E(8)^3, 1/2*E(8)-1/2*E(8)^3, 1 ], 
  [ -1/2*E(8)+1/2*E(8)^3, 1/2*E(8)-1/2*E(8)^3, 1 ], 
  [ -1/2*E(8)+1/2*E(8)^3, -1/2*E(8)+1/2*E(8)^3, 1 ], 
  [ 1/2*E(8)-1/2*E(8)^3, -1/2*E(8)+1/2*E(8)^3, 1 ] ]

3.1-3 ConnectedSubgraphArr

Returns: A hyperplane arrangement.

Generates the connected subgraph arrangement (see [CK22]) of the graph with edges Es.

Let G = (N,E) be an undirected graph with vertex set N = [n] and edge set E. For I \subseteq N, let G[I] be the induced subgraph of G on the set of vertices I. For I \subseteq N, define the hyperplane

H_I := \ker \sum_{i \in I}x_i.

The connected subgraph arrangement consists of hyperplanes \{H_I \mid \varnothing \ne I \subseteq N if G[I] is connected \}.

gap> A:=ConnectedSubgraphArr([[1,2],[1,3],[2,3]]); Roots(A);
<HyperplaneArrangement: 7 hyperplanes in 3-space>
[ [ 1, 1, 1 ], [ 0, 1, 1 ], [ 0, 0, 1 ], [ 0, 1, 0 ], 
  [ 1, 0, 1 ], [ 1, 0, 0 ], [ 1, 1, 0 ] ]

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