The Chow ring of M is the ring R := QQ[x_F]/(I1 + I2), where $I1 = (\sum_{i_1\in F} x_F - \sum_{i_2\in F} x_F : i_1, i_2 \in M)$ and $I2 = (x_Fx_{F'} : F, F' incomparable)$, as $F$ runs over all proper nonempty flats of $M$. This is the same as the Chow ring of the toric variety associated to the Bergman fan of M. This ring is an Artinian standard graded Gorenstein ring, by a result of Adiprasito, Katz, and Huh: cf. https://arxiv.org/abs/1511.02888, Theorem 6.19.
This method returns the defining ideal of the Chow ring, which lives in a polynomial ring with variable indices equal to the flats of M. To work with these subscripts, use "last baseName v" to get the index of a variable v. For more information, cf. Working with Chow rings of matroids.
i1 : M = matroid completeGraph 4 o1 = a matroid of rank 3 on 6 elements o1 : Matroid |
i2 : I = idealChowRing M
o2 = ideal (x x , x x , x x , x x , x x , x x ,
{5} {4} {5} {3} {4} {3} {5} {2} {4} {2} {3} {2}
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x x , x x , x x , x x , x x , x x , x x ,
{5} {1} {4} {1} {3} {1} {2} {1} {5} {0} {4} {0} {3} {0}
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x x , x x , x x , x x , x x , x x ,
{2} {0} {1} {0} {4} {0, 5} {3} {0, 5} {2} {0, 5} {1} {0, 5}
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x x , x x , x x , x x , x x ,
{5} {1, 4} {3} {1, 4} {2} {1, 4} {0} {1, 4} {0, 5} {1, 4}
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x x , x x , x x , x x , x x , x
{5} {2, 3} {4} {2, 3} {1} {2, 3} {0} {2, 3} {0, 5} {2, 3} {1,
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x , x x , x x , x x , x x
4} {2, 3} {2} {3, 4, 5} {1} {3, 4, 5} {0} {3, 4, 5} {0, 5} {3,
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, x x , x x , x x , x x
4, 5} {1, 4} {3, 4, 5} {2, 3} {3, 4, 5} {4} {1, 2, 5} {3} {1, 2,
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, x x , x x , x x , x x
5} {0} {1, 2, 5} {0, 5} {1, 2, 5} {1, 4} {1, 2, 5} {2, 3} {1, 2,
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, x x , x x , x x , x x
5} {3, 4, 5} {1, 2, 5} {5} {0, 2, 4} {3} {0, 2, 4} {1} {0, 2,
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, x x , x x , x x , x
4} {0, 5} {0, 2, 4} {1, 4} {0, 2, 4} {2, 3} {0, 2, 4} {3, 4,
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x , x x , x x , x x ,
5} {0, 2, 4} {1, 2, 5} {0, 2, 4} {5} {0, 1, 3} {4} {0, 1, 3}
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x x , x x , x x , x x ,
{2} {0, 1, 3} {0, 5} {0, 1, 3} {1, 4} {0, 1, 3} {2, 3} {0, 1, 3}
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x x , x x , x x , x -
{3, 4, 5} {0, 1, 3} {1, 2, 5} {0, 1, 3} {0, 2, 4} {0, 1, 3} {1}
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x - x + x + x - x , x - x - x
{0} {0, 5} {1, 4} {1, 2, 5} {0, 2, 4} {2} {0} {0,
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+ x + x - x , x - x - x + x
5} {2, 3} {1, 2, 5} {0, 1, 3} {3} {0} {0, 5} {2, 3}
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+ x - x , x - x - x + x + x
{3, 4, 5} {0, 2, 4} {4} {0} {0, 5} {1, 4} {3, 4, 5}
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- x , x - x + x + x - x - x
{0, 1, 3} {5} {0} {3, 4, 5} {1, 2, 5} {0, 2, 4} {0,
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)
1, 3}
o2 : Ideal of QQ[x , x , x , x , x , x , x , x , x , x , x , x , x ]
{5} {4} {3} {2} {1} {0} {0, 5} {1, 4} {2, 3} {3, 4, 5} {1, 2, 5} {0, 2, 4} {0, 1, 3}
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i3 : basis comodule I
o3 = | 1 x_{0} x_{0, 5} x_{1, 4} x_{2, 3} x_{3, 4, 5} x_{1, 2, 5} x_{0, 2, 4}
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x_{0, 1, 3} x_{0, 1, 3}^2 |
o3 : Matrix
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i4 : (0..<rank M)/(i -> hilbertFunction(i, I)) o4 = (1, 8, 1) o4 : Sequence |
i5 : betti res minimalPresentation I
0 1 2 3 4 5 6 7 8
o5 = total: 1 35 160 350 448 350 160 35 1
0: 1 . . . . . . . .
1: . 35 160 350 448 350 160 35 .
2: . . . . . . . . 1
o5 : BettiTally
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i6 : apply(gens ring I, v -> last baseName v)
o6 = {{5}, {4}, {3}, {2}, {1}, {0}, {0, 5}, {1, 4}, {2, 3}, {3, 4, 5}, {1, 2,
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5}, {0, 2, 4}, {0, 1, 3}}
o6 : List
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The object idealChowRing is a method function.