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arrangement(String,Ring) -- access a database of classic hyperplane arrangements

Synopsis

• Function: arrangement
• Usage:
arrangement(s, R)
arrangement s
• Inputs:
• s, , corresponding to the name of a hyperplane arrangement in the database
• R, a ring, that determines the coefficient ring of the hyperplane arrangement or that determines the ambient ring
• Outputs:
• , from the database

Description

A hyperplane is an affine-linear subspace of codimension one. An arrangement is a finite set of hyperplanes. This method allows convenient access to the hyperplane arrangements with the following names

 i1 : sort keys arrangementLibrary o1 = {(9_3)_2, bracelet, braid, Desargues, Hessian, MacLane, nonFano, ------------------------------------------------------------------------ notTame, Pappus, prism, X2, X3, Ziegler1, Ziegler2} o1 : List

We illustrate various ways to specify the ambient ring for some classic hyperplane arrangements.

 i2 : A0 = arrangement "(9_3)_2" o2 = {x , x , x , x + x , x + x , x + 3x , x + 2x + x , x + 2x + 3x , 4x + 6x + 6x } 1 2 3 1 2 2 3 1 3 1 2 3 1 2 3 1 2 3 o2 : Hyperplane Arrangement  i3 : ring A0 o3 = QQ[x ..x ] 1 3 o3 : PolynomialRing i4 : A1 = arrangement("bracelet", ZZ) o4 = {x , x , x , x + x , x + x , x + x , x + x + x , x + x + x , x + x + x } 1 2 3 1 4 2 4 3 4 1 2 4 1 3 4 2 3 4 o4 : Hyperplane Arrangement  i5 : ring A1 o5 = ZZ[x ..x ] 1 4 o5 : PolynomialRing i6 : A2 = arrangement("braid", ZZ/101) o6 = {x , x , x , x - x , x - x , x - x } 1 2 3 1 2 1 3 2 3 o6 : Hyperplane Arrangement  i7 : ring A2 ZZ o7 = ---[x ..x ] 101 1 3 o7 : PolynomialRing i8 : A3 = arrangement("Desargues", ZZ[vars(0..2)]) o8 = {a, b, c, a + b + c, 2a - 3c, 2a + b - 3c, - 3a - 2b + 2c, a + 2b + c, 3a + 2b + c, 2a + b} o8 : Hyperplane Arrangement  i9 : ring A3 o9 = ZZ[a..c] o9 : PolynomialRing i10 : A4 = arrangement("nonFano", QQ[a..c]) o10 = {a, b, c, b - c, a - c, a - b, a + b - c} o10 : Hyperplane Arrangement  i11 : ring A4 o11 = QQ[a..c] o11 : PolynomialRing i12 : A5 = arrangement("notTame", ZZ/32003[w,x,y,z]) o12 = {w, x, y, z, w + x, w + y, w + z, x + y, x + z, y + z, w + x + y, w + x + z, w + y + z, x + y + z, w + x + y + z} o12 : Hyperplane Arrangement  i13 : ring A5 ZZ o13 = -----[w..z] 32003 o13 : PolynomialRing

Two of the entries in the database are defined over the finite field with $31627$ elements where $6419$ is a cube root of unity.

 i14 : A6 = arrangement "MacLane" o14 = {x , x , x , x - x , x - x , x - 6420x , x - 6420x - x , x - 6420x + 6419x } 1 2 3 1 2 1 3 2 3 1 2 3 1 2 3 o14 : Hyperplane Arrangement  i15 : ring A6 ZZ o15 = -----[x ..x ] 31627 1 3 o15 : PolynomialRing i16 : A7 = arrangement("Hessian", ZZ/31627[a,b,c]) o16 = {a, b, c, a + b + c, a + b + 6419c, a + b - 6420c, a + 6419b + c, a + 6419b + 6419c, a + 6419b - 6420c, a - 6420b + c, a - 6420b + 6419c, a - 6420b - 6420c} o16 : Hyperplane Arrangement  i17 : ring A7 ZZ o17 = -----[a..c] 31627 o17 : PolynomialRing

Every entry in this database determines a central hyperplane arrangement.

 i18 : assert all(keys arrangementLibrary, s -> isCentral arrangement s)

The following two examples have the property that the six triple points lie on a conic in the one arrangement, but not in the other. The difference is not reflected in the matroid. However, Hal Schenck's and Ştefan O. Tohǎneanu's paper "The Orlik-Terao algebra and 2-formality" Mathematical Research Letters 16 (2009) 171-182 arXiv:0901.0253 observes a difference between their respective Orlik-Terao algebras.

 i19 : Z1 = arrangement "Ziegler1" o19 = {x , x , x , x + x + x , 2x + x + x , 2x + 3x + x , 2x + 3x + 4x , 3x + 5x , 3x + 4x + 5x } 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 3 1 2 3 o19 : Hyperplane Arrangement  i20 : Z2 = arrangement "Ziegler2" o20 = {x , x , x , x + x + x , 2x + x + x , 2x + 3x + x , 2x + 3x + 4x , x + 3x , x + 2x + 3x } 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 3 1 2 3 o20 : Hyperplane Arrangement  i21 : assert(matroid Z1 == matroid Z2) -- same underlying matroid i22 : I1 = orlikTerao Z1; o22 : Ideal of QQ[y ..y ] 1 9 i23 : I2 = orlikTerao Z2; o23 : Ideal of QQ[y ..y ] 1 9 i24 : assert(hilbertPolynomial I1 == hilbertPolynomial I2) -- same Hilbert polynomial i25 : hilbertPolynomial ideal super basis(2,I1) o25 = 240*P - 192*P + 64*P 0 1 2 o25 : ProjectiveHilbertPolynomial i26 : hilbertPolynomial ideal super basis(2,I2) -- but not (graded) isomorphic o26 = - 5*P + 24*P - 38*P + 20*P 0 1 2 3 o26 : ProjectiveHilbertPolynomial

• typeA -- make the hyperplane arrangement defined by a type $A_n$ root system
• typeB -- make the hyperplane arrangement defined by a type $B_n$ root system
• typeD -- make the hyperplane arrangement defined by a type $D_n$ root system