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[Solved] Unexpected asymmetrical regular subdivision obtained from a symmetrical input

Posted: 25 Mar 2024, 11:59
Hi everyone,
I am pretty new to polymake and have been wanting to understand the combinatorial structure of some regular subdivision I am studying for my PhD.
It is a subdivision of the vertices of the (twice dilated) second hypersimplex as the one studied by De Loera, Sturmfels, and Thomas in http://link.springer.com/10.1007/BF01299745, to which I have added as many additionnal points as the dimension of the ambient space (all contained in the hyperplane $$\sum_{i=1}^n x_i = 4$$).
The vertices of the second hypersimplex are indexed from 0 to 14 and the last 6 vertices are indexed from 15 to 20.
I have entered the following commands in Polymake (v4.11):

Code: Select all

$M = new Matrix<Rational>([[2,2,0,0,0,0],[0,2,2,0,0,0],[0,0,2,2,0,0],[0,0,0,2,2,0],[0,0,0,0,2,2],[2,0,0,0,0,2],[2,0,2,0,0,0],[0,2,0,2,0,0],[0,0,2,0,2,0],[0,0,0,2,0,2],[2,0,0,0,2,0],[0,2,0,0,0,2],[2,0,0,2,0,0],[0,2,0,0,2,0],[0,0,2,0,0,2],[3/2,1/2,1/2,1/2,1/2,1/2],[1/2,3/2,1/2,1/2,1/2,1/2],[1/2,1/2,3/2,1/2,1/2,1/2],[1/2,1/2,1/2,3/2,1/2,1/2],[1/2,1/2,1/2,1/2,3/2,1/2],[1/2,1/2,1/2,1/2,1/2,3/2]]);  Code: Select all $w = new Vector<Rational>([6,6,6,6,6,6,3,3,3,3,3,3,2,2,2,10,10,10,10,10,10]); 

Code: Select all

$S = new fan::SubdivisionOfPoints(POINTS=>$M,WEIGHTS=>$w);  Code: Select all print$S->MAXIMAL_CELLS;
and obtain the following:

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{6 7 12 13 14 16 17} {1 6 7 13 14 16 17} {9 10 12 13 14 19 20} {6 8 10 12 13 14} {0 5 6 10 11 12} {3 7 8 12 13 18} {6 7 8 12 13 17} {5 6 10 11 12 14} {6 10 11 12 13 14} {1 6 7 8 13 17} {0 6 10 11 12 13} {5 10 11 12 14 20} {4 5 9 11 14 20} {5 9 10 12 14 20} {4 5 9 10 14 20} {3 4 8 10 13 19} {3 4 8 9 10 19} {7 8 12 13 14 17 18} {7 9 11 12 13 14} {1 2 6 8 14 17} {2 3 7 8 12 18} {8 9 12 13 14 18 19} {3 8 9 10 12 19} {3 8 9 12 13 18 19} {9 11 12 13 14 20} {10 11 12 13 14 20} {4 9 11 13 14 20} {5 9 11 12 14 20} {4 10 11 13 14 20} {4 5 10 11 14 20} {9 10 11 12 13 20} {4 9 10 11 13 20} {4 5 9 10 11 20} {5 9 10 11 12 20} {3 4 8 9 13 19} {4 8 9 13 14 19} {3 8 10 12 13 19} {4 9 10 13 14 19 20} {4 8 10 13 14 19} {3 9 10 12 13 19} {3 4 9 10 13 19} {8 10 12 13 14 19} {8 9 10 12 14 19} {4 8 9 10 14 19} {2 7 8 9 14 18} {7 8 9 13 14 18} {3 7 8 9 13 18} {2 3 7 8 9 18} {2 7 9 12 14 18} {3 7 9 12 13 18} {2 3 8 9 12 18} {2 8 9 12 14 18} {7 9 12 13 14 18} {2 3 7 9 12 18} {1 2 7 8 14 17} {1 7 8 13 14 17} {1 2 6 7 8 17} {2 6 7 8 12 17} {2 6 7 12 14 17} {1 2 6 7 14 17} {1 6 8 13 14 17} {6 8 12 13 14 17} {2 6 8 12 14 17} {2 7 8 12 14 17 18} {1 7 11 13 14 16} {0 1 6 7 13 16} {0 6 7 12 13 16} {0 1 7 11 13 16} {6 7 11 12 14 16} {0 6 7 11 12 16} {0 1 6 7 11 16} {1 6 7 11 14 16} {0 1 6 11 13 16} {1 6 11 13 14 16} {0 7 11 12 13 16} {6 11 12 13 14 16} {0 6 11 12 13 16} {7 11 12 13 14 16} 
I am very surprised that from the symmetry of the weights and point configuration I have provided, I obtain a completely asymmetrical result, as the vertex of index 15 does not even appear in the subdivision, while playing the same role as all my other additional points (of indexes 16,...,20).

Can someone enlighten me?

Best,
Mathieu V.

P.S.: I have tried the subdivision without my additional vertices and it outputed the mathematically expected result.

Re: Unexpected asymmetrical regular subdivision obtained from a symmetrical input

Posted: 25 Mar 2024, 12:35

Code: Select all

polytope > $M = new Matrix<Rational>([[2,2,0,0,0,0],[0,2,2,0,0,0],[0,0,2,2,0,0],[0,0,0,2,2,0],[0,0,0,0,2,2],[2,0,0,0,0,2],[2,0,2,0,0,0],[0,2,0,2,0,0],[0,0,2,0,2,0],[0,0,0,2,0,2],[2,0,0,0,2,0],[0,2,0,0,0,2],[2,0,0,2,0,0],[0,2,0,0,2,0],[0,0,2,0,0,2],[3/2,1/2,1/2,1/2,1/2,1/2],[1/2,3/2,1/2,1/2,1/2,1/2],[1/2,1/2,3/2,1/2,1/2,1/2],[1/2,1/2,1/2,3/2,1/2,1/2],[1/2,1/2,1/2,1/2,3/2,1/2],[1/2,1/2,1/2,1/2,1/2,3/2]]); polytope >$w = new Vector<Rational>([6,6,6,6,6,6,3,3,3,3,3,3,2,2,2,10,10,10,10,10,10]); polytope > $S = new fan::SubdivisionOfPoints(POINTS=>ones_vector |$M,WEIGHTS=>$w); polytope > print$S->MAXIMAL_CELLS; {1 6 7 11 13 14} {4 5 9 10 11 14} {5 9 10 11 12 14} {4 8 9 10 13 14} {4 9 10 11 13 14} {3 4 8 9 10 13} {3 8 9 10 12 13} {9 10 11 12 13 14} {8 9 10 12 13 14} {2 3 7 8 9 12} {3 7 8 9 12 13} {7 8 9 12 13 14} {2 7 8 9 12 14} {7 9 11 12 13 14} {6 7 11 12 13 14} {2 6 7 8 12 14} {6 7 8 12 13 14} {6 8 10 12 13 14} {6 10 11 12 13 14} {5 6 10 11 12 14} {1 6 7 8 13 14} {1 2 6 7 8 14} {0 5 6 10 11 12} {0 1 6 7 11 13} {0 6 7 11 12 13} {0 6 10 11 12 13} 
Similar to the corresponding properties for Polytope, PointConfiguration and PolyhedralComplex; the POINTS of a SubdivisionOfPoints need to be homogenized explicitly.

Best
Benjamin

Re: Unexpected asymmetrical regular subdivision obtained from a symmetrical input

Posted: 25 Mar 2024, 13:22
polytope > $S = new fan::SubdivisionOfPoints(POINTS=>ones_vector |$M,WEIGHTS=>\$w);