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# subArrangement(Arrangement,Flat) -- create the hyperplane arrangement containing a flat

## Synopsis

• Function: subArrangement
• Usage:
subArrangement(A, F)
subArrangement F
A _ F
• Inputs:
• F, , of the hyperplane arrangement $A$
• Outputs:
• , consisting of those hyperplanes in $A$ that contain the linear subspace indexed by the flat $F$

## Description

For any hyperplane arrangement $A$ and any flat $F$ in $A$, this methods creates a new hyperplane arrangement formed by the hyperplanes in $A$ that contain the linear subspace associated to the flat $A$.

We illustrate this method with the Coxeter arrangement of type A.

 i1 : S = QQ[w, x, y, z]; i2 : A3 = typeA(3, S) o2 = {w - x, w - y, w - z, x - y, x - z, y - z} o2 : Hyperplane Arrangement  i3 : F1 = flat(A3, {3,4,5}) o3 = {3, 4, 5} o3 : Flat of {w - x, w - y, w - z, x - y, x - z, y - z} i4 : A3' = subArrangement(A3, F1) o4 = {x - y, x - z, y - z} o4 : Hyperplane Arrangement  i5 : assert(ring A3 === ring A3') i6 : subArrangement flat(A3, {0, 5}) o6 = {w - x, y - z} o6 : Hyperplane Arrangement  i7 : F2 = flat(A3, {0, 1, 3}) o7 = {0, 1, 3} o7 : Flat of {w - x, w - y, w - z, x - y, x - z, y - z} i8 : assert(typeA(2, S) == A3_F2) i9 : assert(A3 === subArrangement flat(A3, {0,1,2,3,4,5}))

An extension of the bracelet arrangement has several subarrangements isomorphic to $A_3$.

 i10 : B = arrangement("bracelet", S); i11 : B' = arrangement({w+x+y+z} | hyperplanes B) o11 = {w + x + y + z, w, x, y, w + z, x + z, y + z, w + x + z, w + y + z, x + y + z} o11 : Hyperplane Arrangement  i12 : subArrangement flat(B', {0,1,2,6,8,9}) o12 = {w + x + y + z, w, x, y + z, w + y + z, x + y + z} o12 : Hyperplane Arrangement  i13 : subArrangement flat(B', {0,1,3,5,7,9}) o13 = {w + x + y + z, w, y, x + z, w + x + z, x + y + z} o13 : Hyperplane Arrangement  i14 : subArrangement flat(B', {0,2,3,4,7,8}) o14 = {w + x + y + z, x, y, w + z, w + x + z, w + y + z} o14 : Hyperplane Arrangement