# BettiCharacters Example 3 -- Klein configuration of points

In this example, we identify the Betti characters of the defining ideal of the Klein configuration of points in the projective plane and its square. The defining ideal of the Klein configuration is explicitly constructed in Proposition 7.3 of T. Bauer, S. Di Rocco, B. Harbourne, J. Huizenga, A. Seceleanu, T. Szemberg - Negative Curves on Symmetric Blowups of the Projective Plane, Resurgences, and Waldschmidt Constants. We start by constructing the ideal, its square, and both their resolutions and Betti tables. In order to later use characters, we work over the cyclotomic field obtained by adjoining a primitive 7th root of unity to $\mathbb{Q}$.

 i1 : kk = toField(QQ[a]/ideal(sum apply(7,i->a^i))) o1 = kk o1 : PolynomialRing i2 : R = kk[x,y,z] o2 = R o2 : PolynomialRing i3 : f4 = x^3*y+y^3*z+z^3*x 3 3 3 o3 = x y + y z + x*z o3 : R i4 : H = jacobian transpose jacobian f4 o4 = {-3} | 6xy 3x2 3z2 | {-3} | 3x2 6yz 3y2 | {-3} | 3z2 3y2 6xz | 3 3 o4 : Matrix R <--- R i5 : f6 = -1/54*det(H) 5 5 2 2 2 5 o5 = x*y + x z - 5x y z + y*z o5 : R i6 : I = minors(2,jacobian matrix{{f4,f6}}) 3 5 7 7 4 2 2 4 3 2 5 8 7 o6 = ideal (14x y - 5x z - 3y z - 35x y z + 35x*y z - 7x y*z + z , 3x y - ------------------------------------------------------------------------ 8 4 3 5 2 5 3 2 2 4 7 8 7 5 2 y - 35x y z + 7x*y z - 14x z + 35x y z + 5y*z , x - 5x*y - 7x y z ------------------------------------------------------------------------ 2 4 2 3 4 3 5 7 - 35x y z + 35x y*z + 14y z - 3x*z ) o6 : Ideal of R i7 : RI = res I 1 3 2 o7 = R <-- R <-- R <-- 0 0 1 2 3 o7 : ChainComplex i8 : betti RI 0 1 2 o8 = total: 1 3 2 0: 1 . . 1: . . . 2: . . . 3: . . . 4: . . . 5: . . . 6: . . . 7: . 3 . 8: . . . 9: . . 1 10: . . . 11: . . 1 o8 : BettiTally i9 : I2 = I^2; o9 : Ideal of R i10 : RI2 = res I2 1 6 6 1 o10 = R <-- R <-- R <-- R <-- 0 0 1 2 3 4 o10 : ChainComplex i11 : betti RI2 0 1 2 3 o11 = total: 1 6 6 1 0: 1 . . . 1: . . . . 2: . . . . 3: . . . . 4: . . . . 5: . . . . 6: . . . . 7: . . . . 8: . . . . 9: . . . . 10: . . . . 11: . . . . 12: . . . . 13: . . . . 14: . . . . 15: . 6 . . 16: . . . . 17: . . 3 . 18: . . . . 19: . . 3 . 20: . . . . 21: . . . 1 o11 : BettiTally

The unique simple group of order 168 acts as described in §2.2 of BDHHSS. In particular, the group is generated by the elements g of order 7, h of order 3, and i of order 2, and is minimally defined over the 7th cyclotomic field. In addition, we consider the identity, the inverse of g, and another element j of order 4 as representatives of the conjugacy classes of the group. The action of the group on the resolution of both ideals is described in the second proof of Proposition 8.1.

 i12 : g = matrix{{a^4,0,0},{0,a^2,0},{0,0,a}} o12 = | a4 0 0 | | 0 a2 0 | | 0 0 a | 3 3 o12 : Matrix kk <--- kk i13 : h = matrix{{0,1,0},{0,0,1},{1,0,0}} o13 = | 0 1 0 | | 0 0 1 | | 1 0 0 | 3 3 o13 : Matrix ZZ <--- ZZ i14 : i = (2*a^4+2*a^2+2*a+1)/7 * matrix{ {a-a^6,a^2-a^5,a^4-a^3}, {a^2-a^5,a^4-a^3,a-a^6}, {a^4-a^3,a-a^6,a^2-a^5} } o14 = | 3/7a5+1/7a4+1/7a3+3/7a2-1/7 -1/7a5+2/7a4+2/7a3-1/7a2-2/7 | -1/7a5+2/7a4+2/7a3-1/7a2-2/7 -2/7a5-3/7a4-3/7a3-2/7a2-4/7 | -2/7a5-3/7a4-3/7a3-2/7a2-4/7 3/7a5+1/7a4+1/7a3+3/7a2-1/7 ----------------------------------------------------------------------- -2/7a5-3/7a4-3/7a3-2/7a2-4/7 | 3/7a5+1/7a4+1/7a3+3/7a2-1/7 | -1/7a5+2/7a4+2/7a3-1/7a2-2/7 | 3 3 o14 : Matrix kk <--- kk i15 : j = -1/(2*a^4+2*a^2+2*a+1) * matrix{ {a^5-a^4,1-a^5,1-a^3}, {1-a^5,a^6-a^2,1-a^6}, {1-a^3,1-a^6,a^3-a} } o15 = | -1/7a5-1/7a4+2/7a2-2/7a+2/7 1/7a5+4/7a4+2/7a3+2/7a2+4/7a+1/7 | 1/7a5+4/7a4+2/7a3+2/7a2+4/7a+1/7 1/7a5-1/7a4+1/7a3+3/7a+3/7 | -2/7a5-1/7a3+2/7a2+2/7a-1/7 1/7a5+3/7a4-1/7a3+3/7a2+1/7a ----------------------------------------------------------------------- -2/7a5-1/7a3+2/7a2+2/7a-1/7 | 1/7a5+3/7a4-1/7a3+3/7a2+1/7a | 2/7a4-1/7a3-2/7a2-1/7a+2/7 | 3 3 o15 : Matrix kk <--- kk i16 : G = {id_(R^3),i,h,j,g,inverse g};

We compute the action of this group on the two resolutions above. Notice how the group action is passed as a list of square matrices (instead of one-row substitution matrices as in BettiCharacters Example 1 and BettiCharacters Example 2); to enable this, we set the option Sub to false.

 i17 : A1 = action(RI,G,Sub=>false) o17 = ChainComplex with 6 actors o17 : ActionOnComplex i18 : A2 = action(RI2,G,Sub=>false) o18 = ChainComplex with 6 actors o18 : ActionOnComplex i19 : elapsedTime a1 = character A1 -- 3.68979 seconds elapsed o19 = Character over R (0, {0}) => | 1 1 1 1 1 1 | (1, {8}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 | (2, {11}) => | 1 1 1 1 1 1 | (2, {13}) => | 1 1 1 1 1 1 | o19 : Character i20 : elapsedTime a2 = character A2 -- 151.247 seconds elapsed o20 = Character over R (0, {0}) => | 1 1 1 1 1 1 | (1, {16}) => | 6 2 0 0 -1 -1 | (2, {19}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 | (2, {21}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 | (3, {24}) => | 1 1 1 1 1 1 | o20 : Character

Next we set up the character table of the group and decompose the Betti characters of the resolutions. The arguments are: a list with the cardinality of the conjugacy classes, a matrix with the values of the irreducible characters, the base polynomial ring, and the complex conjugation map restricted to the field of coefficients. See characterTable for more details.

 i21 : s = {1,21,56,42,24,24} o21 = {1, 21, 56, 42, 24, 24} o21 : List i22 : m = matrix{{1,1,1,1,1,1}, {3,-1,0,1,a^4+a^2+a,-a^4-a^2-a-1}, {3,-1,0,1,-a^4-a^2-a-1,a^4+a^2+a}, {6,2,0,0,-1,-1}, {7,-1,1,-1,0,0}, {8,0,-1,0,1,1}}; 6 6 o22 : Matrix kk <--- kk i23 : conj = map(kk,kk,{a^6}) 5 4 3 2 o23 = map (kk, kk, {- a - a - a - a - a - 1}) o23 : RingMap kk <--- kk i24 : T = characterTable(s,m,R,conj) o24 = Character table over R | 1 21 56 42 24 24 ----+----------------------------------------------------- X0 | 1 1 1 1 1 1 | 4 2 4 2 X1 | 3 -1 0 1 a + a + a - a - a - a - 1 | 4 2 4 2 X2 | 3 -1 0 1 - a - a - a - 1 a + a + a X3 | 6 2 0 0 -1 -1 X4 | 7 -1 1 -1 0 0 X5 | 8 0 -1 0 1 1 o24 : CharacterTable i25 : a1/T o25 = Decomposition table | X0 X1 -----------+-------- (0, {0}) | 1 0 (1, {8}) | 0 1 (2, {11}) | 1 0 (2, {13}) | 1 0 o25 : CharacterDecomposition i26 : a2/T o26 = Decomposition table | X0 X1 X3 -----------+------------ (0, {0}) | 1 0 0 (1, {16}) | 0 0 1 (2, {19}) | 0 1 0 (2, {21}) | 0 1 0 (3, {24}) | 1 0 0 o26 : CharacterDecomposition

Since X0 is the trivial character, this computation shows that the free module in homological degree two in the resolution of the defining ideal of the Klein configuration is a direct sum of two trivial representations, one in degree 11 and one in degree 13. It follows that its second exterior power is a trivial representation concentrated in degree 24. As observed in the second proof of Proposition 8.1 in BDHHSS, the free module in homological degree 3 in the resolution of the square of the ideal is exactly this second exterior power (and a trivial representation).

Alternatively, we can compute the symbolic square of the ideal modulo the ordinary square. The component of degree 21 of this quotient matches the generators of the last module in the resolution of the ordinary square in degree 24 (by local duality); in particular, it is a trivial representation. We can verify this directly.

 i27 : needsPackage "SymbolicPowers" o27 = SymbolicPowers o27 : Package i28 : Is2 = symbolicPower(I,2); o28 : Ideal of R i29 : M = Is2 / I2; i30 : B = action(M,G,Sub=>false) o30 = Module with 6 actors o30 : ActionOnGradedModule i31 : elapsedTime b = character(B,21) -- 69.0019 seconds elapsed o31 = Character over R (0, {21}) => | 1 1 1 1 1 1 | o31 : Character i32 : b/T o32 = Decomposition table | X0 -----------+------ (0, {21}) | 1 o32 : CharacterDecomposition