rankTableToASM -- to find the a partial ASM associated to a given rank table

Usage:

rankTableToASM T
Inputs:
- T, a matrix,
Outputs:
- a matrix,

Description

Given a matrix that is a valid minimal rank table, returns the unique partial ASM of the same size associated to it. This algorithm follows

[Wei, Equation (21)]: Weigandt, Prism tableaux for alternating sign matrix varieties (see arXiv:1708.07236).

i1 : T = matrix {{0,0,1,1},{0,1,1,2},{1,2,2,3},{1,2,3,4}}

o1 = | 0 0 1 1 |
     | 0 1 1 2 |
     | 1 2 2 3 |
     | 1 2 3 4 |

              4       4
o1 : Matrix ZZ  <-- ZZ

i2 : rankTableToASM T

o2 = | 0 0 1  0 |
     | 0 1 -1 1 |
     | 1 0 0  0 |
     | 0 0 1  0 |

              4       4
o2 : Matrix ZZ  <-- ZZ

i3 : U = matrix {{0,0,1,1,1},{1,1,1,2,2},{1,2,2,3,3},{1,2,3,4,4},{1,2,3,4,5}}

o3 = | 0 0 1 1 1 |
     | 1 1 1 2 2 |
     | 1 2 2 3 3 |
     | 1 2 3 4 4 |
     | 1 2 3 4 5 |

              5       5
o3 : Matrix ZZ  <-- ZZ

i4 : rankTableToASM U

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

              5       5
o4 : Matrix ZZ  <-- ZZ

Ways to use rankTableToASM:

rankTableToASM(Matrix)

For the programmer

The object rankTableToASM is a method function.

The source of this document is in MatrixSchubert/MatrixSchubertConstructionsDOC.m2:1206:0.