reducedSlackMatrix -- a reduced slack matrix of a polytope

Synopsis

• Usage:

  RedS = reducedSlackMatrix(d, S)

  RedS = reducedSlackMatrix S

  RedS = reducedSlackMatrix V

• Inputs:
  • d, an integer, the dimension of the polytope or 1 less than the rank of the matroid
  • S, a matrix, a (symbolic) slack matrix of a polytope or of a matroid
  • V, a list, list of vertex coordinates, matroid vectors, facet labels, or cone generators
• Optional inputs:
  • CoefficientRing => ..., default value QQ, specifies the coefficient ring of the underlying ring of the ideal
  • FlagIndices => ..., default value {}, a list of facet labels that form a flag of facets of given polytope or matroid
  • Object => ..., default value "polytope", specify combinatorial object
  • Vars => ..., default value {}, specifies the variables to use to create the underlying ring of the ideal
• Outputs:
  • RedS, a matrix, a reduced slack matrix

Description

i1 : V = {{0,0,0},{0,0,1},{1,0,0},{1,0,1},{0,1,0},{0,1,1}};

i2 : S = slackMatrix V;

Order of vertices is 
{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {1, 0, 1}, {0, 1, 1}}

              6       5
o2 : Matrix QQ  <-- QQ

i3 : reducedSlackMatrix V

Order of vertices is 
{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {1, 0, 1}, {0, 1, 1}}

o3 = | 0 0 1 1 |
     | 1 0 0 1 |
     | 0 1 0 1 |
     | 0 0 1 0 |
     | 1 0 0 0 |
     | 0 1 0 0 |

              6       4
o3 : Matrix QQ  <-- QQ

i4 : reducedSlackMatrix(3, S)

o4 = | 0 0 1 1 |
     | 1 0 0 1 |
     | 0 1 0 1 |
     | 0 0 1 0 |
     | 1 0 0 0 |
     | 0 1 0 0 |

              6       4
o4 : Matrix QQ  <-- QQ

i5 : R = QQ[x_1..x_30];

i6 : S = matrix {{0, 0, x_1, x_2, x_3, 0, x_4}, {x_5, x_6, 0, 0, x_7, x_8, 0}, {0, x_9, 0, x_10, x_11, 0, 0}, {x_12, 0, x_13, 0, x_14, 0, 0}, {0, 0, x_15, x_16, 0, x_17, x_18}, {x_19, x_20, 0, 0, 0, x_21, x_22}, {0, x_23, 0, x_24, 0, x_25, x_26}, {x_27, 0, x_28, 0, 0, x_29, x_30}};

             8      7
o6 : Matrix R  <-- R

i7 : redS = reducedSlackMatrix(3, S)

o7 = | 0    0    x_1  x_2  x_3  |
     | x_5  x_6  0    0    x_7  |
     | 0    x_9  0    x_10 x_11 |
     | x_12 0    x_13 0    x_14 |
     | 0    0    x_15 x_16 0    |
     | x_19 x_20 0    0    0    |
     | 0    x_23 0    x_24 0    |
     | x_27 0    x_28 0    0    |

             8      5
o7 : Matrix R  <-- R

See also

• findFlag -- computes a list of facet labels that make up a flag in a polytope
• reconstructSlackMatrix -- a list of facet labels that make up a flag in a polytope