An $n$-dimensional matrix $M$ is skew symmetric if for every permutation $s$ of the set $\{0,\ldots,n-1\}$ we have permute(M,s) == sign(s)*M.
i1 : genericSkewMultidimensionalMatrix(3,4) o1 = {{{0, 0, 0, 0}, {0, 0, -a , -a }, {0, a , 0, -a }, {0, a , a , 0}}, {{0, 0, a , a }, {0, 0, 0, 0}, {-a , 0, 0, -a }, {-a , 3 2 3 0 2 0 3 2 3 1 2 ---------------------------------------------------------------------------------------------------------------------------- 0, a , 0}}, {{0, -a , 0, a }, {a , 0, 0, a }, {0, 0, 0, 0}, {-a , -a , 0, 0}}, {{0, -a , -a , 0}, {a , 0, -a , 0}, {a , a , 1 3 0 3 1 0 1 2 0 2 1 0 1 ---------------------------------------------------------------------------------------------------------------------------- 0, 0}, {0, 0, 0, 0}}} o1 : 3-dimensional matrix of shape 4 x 4 x 4 over QQ[a ..a ] 0 3 |
i2 : genericSkewMultidimensionalMatrix(3,4,CoefficientRing=>ZZ/101) o2 = {{{0, 0, 0, 0}, {0, 0, -a , -a }, {0, a , 0, -a }, {0, a , a , 0}}, {{0, 0, a , a }, {0, 0, 0, 0}, {-a , 0, 0, -a }, {-a , 3 2 3 0 2 0 3 2 3 1 2 ---------------------------------------------------------------------------------------------------------------------------- 0, a , 0}}, {{0, -a , 0, a }, {a , 0, 0, a }, {0, 0, 0, 0}, {-a , -a , 0, 0}}, {{0, -a , -a , 0}, {a , 0, -a , 0}, {a , a , 1 3 0 3 1 0 1 2 0 2 1 0 1 ---------------------------------------------------------------------------------------------------------------------------- 0, 0}, {0, 0, 0, 0}}} ZZ o2 : 3-dimensional matrix of shape 4 x 4 x 4 over ---[a ..a ] 101 0 3 |
i3 : genericSkewMultidimensionalMatrix(3,4,CoefficientRing=>ZZ/101,Variable=>"b") o3 = {{{0, 0, 0, 0}, {0, 0, -b , -b }, {0, b , 0, -b }, {0, b , b , 0}}, {{0, 0, b , b }, {0, 0, 0, 0}, {-b , 0, 0, -b }, {-b , 1 0 1 2 0 2 1 0 1 3 0 ---------------------------------------------------------------------------------------------------------------------------- 0, b , 0}}, {{0, -b , 0, b }, {b , 0, 0, b }, {0, 0, 0, 0}, {-b , -b , 0, 0}}, {{0, -b , -b , 0}, {b , 0, -b , 0}, {b , b , 3 1 2 1 3 2 3 0 2 0 3 2 3 ---------------------------------------------------------------------------------------------------------------------------- 0, 0}, {0, 0, 0, 0}}} ZZ o3 : 3-dimensional matrix of shape 4 x 4 x 4 over ---[b ..b ] 101 0 3 |
The object genericSkewMultidimensionalMatrix is a method function with options.