The boundary of the stacked 4-polytope on 6 vertices. Algebraic shifting preserves the f-vector.
An empty triangle is a shifted complex.
The multigraded algebraic shifting does not preserve the Betti numbers.
i10 : grading = {{1,0,0},{1,0,0},{1,0,0},{0,1,0},{0,0,1}};
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i11 : R=QQ[x_{1,1},x_{1,2},x_{1,3},x_{2,1},x_{3,1}, Degrees=>grading];
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i12 : delta = simplicialComplex({x_{1,3}*x_{2,1}*x_{3,1},x_{1,1}*x_{2,1},x_{1,2}*x_{3,1}})
o12 = | x_{1, 3}x_{2, 1}x_{3, 1} x_{1, 2}x_{3, 1} x_{1, 1}x_{2, 1} |
o12 : SimplicialComplex
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i13 : shifted = algebraicShifting(delta, Multigrading => true)
o13 = | x_{1, 3}x_{2, 1}x_{3, 1} x_{1, 2}x_{3, 1} x_{1, 2}x_{2, 1} x_{1, 1} |
o13 : SimplicialComplex
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i14 : prune (homology(delta))_1
o14 = 0
o14 : QQ-module
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i15 : prune (homology(shifted))_1
1
o15 = QQ
o15 : QQ-module, free
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G. Kalai, Algebraic Shifting, Computational Commutative Algebra and Combinatorics, 2001;
S. Murai, Betti numbers of strongly color-stable ideals and squarefree strongly color-stable ideals, Journal of Algebraic Combinatorics.