The following describes functions to construct some special arrangements.
‣ AGpql( p, q, l ) | ( function ) |
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 \(\zeta\) 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 ] ]
‣ SsS3( n ) | ( function ) |
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 ] ]
‣ ConnectedSubgraphArr( Es ) | ( operation ) |
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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