SharedExample computations.sagewsOpen in CoCalc
Author: John Wiltshire-Gordon
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load("coker.sage") #We compute the dimension of the second exterior power of an n-dimensional vector space print coker([2], "[[ab+ba]]", [2])
Taking M to be the matrix [2] -----> [2] given by M=[[ab+ba]], the cokernel decomposes into simples as follows: [coker(M*)] = 1*[1, 1] + 1*[1, 1, 1]. In particular, the dimension is given as a function of n > 0: (1/2) * (n - 1) * n. Total computation time: 0.0955851078033 seconds
load("coker.sage") #Let V be an n-dimensional vector space equipped with some chosen basis, and suppose that each of the variables a, b, and c range freely over this basis. #We compute the quotient of the third tensor power of V by the relations that impose antisymmetry in the first two tensor factors and the Jacobi law across all three print coker([3], "[[abc-bac, abc+bca+cab]]", [3,3])
Taking M to be the matrix [3] -----> [3, 3] given by M=[[abc-bac, abc+bca+cab]], the cokernel decomposes into simples as follows: [coker(M*)] = 1*[1, 1] + 1*[2] + 1*[1, 1, 1] + 1*[2, 1]. In particular, the dimension is given as a function of n > 0: (1/3) * (n - 1) * n * (n + 1). Total computation time: 0.023894071579 seconds
load("coker.sage") #We compute the vector space of degree-five polynomials in n variables modulo the span of all polynomials of the form a^2 * d^2 * e - b^2 * c^2 * e where each of a, b, c, d, and e range over the n variables print coker([5],"[[abcde-bacde,abcde-bcdea,aadde-bbcce]]",[5,5,5])
Taking M to be the matrix [5] -----> [5, 5, 5] given by M=[[abcde-bacde,abcde-bcdea,aadde-bbcce]], the cokernel decomposes into simples as follows: [coker(M*)] = 1*[1] + 1*[1, 1] + 1*[2, 1] + 1*[3] + 1*[3, 1] + 1*[4] + 1*[5]. In particular, the dimension is given as a function of n > 0: (1/120) * n * (n + 2) * (n^3 + 8*n^2 - 41*n + 72). Total computation time: 0.449846029282 seconds
load("coker.sage") #We compute the number of n-colorings of a pentagon print coker([5],"[[aabcd,abbcd,abccd,abcdd,abcda]]",[4,4,4,4,4])
Taking M to be the matrix [5] -----> [4, 4, 4, 4, 4] given by M=[[aabcd,abbcd,abccd,abcdd,abcda]], the cokernel decomposes into simples as follows: [coker(M*)] = 5*[1, 1, 1] + 10*[2, 1] + 5*[3] + 10*[1, 1, 1, 1] + 15*[2, 1, 1] + 10*[2, 2] + 15*[3, 1] + 5*[4] + 6*[1, 1, 1, 1, 1] + 4*[2, 1, 1, 1] + 5*[2, 2, 1] + 6*[3, 1, 1] + 5*[3, 2] + 4*[4, 1] + 1*[5] + 1*[1, 1, 1, 1, 1, 1]. In particular, the dimension is given as a function of n > 0: (n - 2) * (n - 1) * n * (n^2 - 2*n + 2). Total computation time: 0.235414981842 seconds
load("coker.sage") #We compute the number of n-colorings of a pentagon up to rotation and reflection print coker([5],"[[aabcd,abbcd,abccd,abcdd,abcda,abcde-bcdea,abcde-edcba]]",[4,4,4,4,4,5,5])
Taking M to be the matrix [5] -----> [4, 4, 4, 4, 4, 5, 5] given by M=[[aabcd,abbcd,abccd,abcdd,abcda,abcde-bcdea,abcde-edcba]], the cokernel decomposes into simples as follows: [coker(M*)] = 1*[2, 1] + 1*[3] + 1*[2, 1, 1] + 1*[2, 2] + 2*[3, 1] + 1*[4] + 1*[1, 1, 1, 1, 1] + 1*[2, 2, 1] + 1*[3, 2] + 1*[5] + 1*[1, 1, 1, 1, 1, 1]. In particular, the dimension is given as a function of n > 0: (1/10) * (n - 2) * (n - 1) * n * (n^2 - 2*n + 2). Total computation time: 0.601917982101 seconds
load("coker.sage") #We compute the second cohomology of the configuration space of n points in the plane #using the classical presentation of Arnol'd print coker([4],"[[aabc,abcd-bacd,abcd+cdab,abbc+bcca+caab]]",[3,4,4,3])
Taking M to be the matrix [4] -----> [3, 4, 4, 3] given by M=[[aabc,abcd-bacd,abcd+cdab,abbc+bcca+caab]], the cokernel decomposes into simples as follows: [coker(M*)] = 1*[2, 1] + 1*[3, 1]. In particular, the dimension is given as a function of n > 0: (1/8) * (n - 2) * (n - 1) * (n - 1/3) * n. Total computation time: 0.17354798317 seconds
load("coker.sage") #We compute the first cohomology of the real locus of the Deligne-Mumford compactification of the moduli space of n marked points on a genus zero curve #using the presentation computed in https://arxiv.org/abs/math/0507514 print coker([4],"[[abcd+bacd,abcd+bcda,abcd+bcde+cdea+deab+eabc]]",[4,4,5])
Taking M to be the matrix [4] -----> [4, 4, 5] given by M=[[abcd+bacd,abcd+bcda,abcd+bcde+cdea+deab+eabc]], the cokernel decomposes into simples as follows: [coker(M*)] = 1*[1, 1, 1, 1]. In particular, the dimension is given as a function of n > 0: (1/6) * (n - 3) * (n - 2) * (n - 1). Total computation time: 1.2256231308 seconds