# genericSkewMultidimensionalMatrix -- make a generic skew symmetric multidimensional matrix of variables

## Synopsis

• Usage:
genericSkewMultidimensionalMatrix(n,d)
• Inputs:
• (n,d), two positive integers.
• Optional inputs:
• CoefficientRing => ..., default value QQ
• Variable => ..., default value a
• Outputs:

## Description

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, 1 0 1 2 0 2 ------------------------------------------------------------------------ 0, a , a }, {0, 0, 0, 0}, {-a , 0, 0, -a }, {-a , 0, a , 0}}, {{0, -a , 1 0 1 3 0 3 1 ------------------------------------------------------------------------ 0, a }, {a , 0, 0, a }, {0, 0, 0, 0}, {-a , -a , 0, 0}}, {{0, -a , -a , 2 1 3 2 3 0 2 ------------------------------------------------------------------------ 0}, {a , 0, -a , 0}, {a , a , 0, 0}, {0, 0, 0, 0}}} 0 3 2 3 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, 1 0 1 2 0 2 ------------------------------------------------------------------------ 0, a , a }, {0, 0, 0, 0}, {-a , 0, 0, -a }, {-a , 0, a , 0}}, {{0, -a , 1 0 1 3 0 3 1 ------------------------------------------------------------------------ 0, a }, {a , 0, 0, a }, {0, 0, 0, 0}, {-a , -a , 0, 0}}, {{0, -a , -a , 2 1 3 2 3 0 2 ------------------------------------------------------------------------ 0}, {a , 0, -a , 0}, {a , a , 0, 0}, {0, 0, 0, 0}}} 0 3 2 3 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, 3 2 3 0 2 0 ------------------------------------------------------------------------ 0, b , b }, {0, 0, 0, 0}, {-b , 0, 0, -b }, {-b , 0, b , 0}}, {{0, -b , 3 2 3 1 2 1 3 ------------------------------------------------------------------------ 0, b }, {b , 0, 0, b }, {0, 0, 0, 0}, {-b , -b , 0, 0}}, {{0, -b , -b , 0 3 1 0 1 2 0 ------------------------------------------------------------------------ 0}, {b , 0, -b , 0}, {b , b , 0, 0}, {0, 0, 0, 0}}} 2 1 0 1 ZZ o3 : 3-dimensional matrix of shape 4 x 4 x 4 over ---[b ..b ] 101 0 3