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# algebraicShifting -- the algebraic shifting of a simplicial complex

## Synopsis

• Usage:
A = algebraicShifting D
• Inputs:
• Optional inputs:
• Multigrading => , default value false, which, if true, returns the colored algebraic shifting with respect to the multigrading of the underlying ring.
• Outputs:
• A, , which is the algebraic shifting of D. If Multigrading is true, then it returns the so-called colored shifted complex.

## Description

The boundary of the stacked 4-polytope on 6 vertices. Algebraic shifting preserves the f-vector.

 i1 : R = QQ[x_1..x_6]; i2 : I = monomialIdeal(x_2*x_3*x_4*x_5, x_1*x_6); o2 : MonomialIdeal of R i3 : stacked = simplicialComplex I o3 = simplicialComplex | x_3x_4x_5x_6 x_2x_4x_5x_6 x_2x_3x_5x_6 x_2x_3x_4x_6 x_1x_3x_4x_5 x_1x_2x_4x_5 x_1x_2x_3x_5 x_1x_2x_3x_4 | o3 : SimplicialComplex i4 : shifted = algebraicShifting stacked o4 = simplicialComplex | x_3x_4x_5x_6 x_2x_4x_5x_6 x_1x_4x_5x_6 x_2x_3x_5x_6 x_1x_3x_5x_6 x_2x_3x_4x_6 x_1x_3x_4x_6 x_2x_3x_4x_5 | o4 : SimplicialComplex i5 : fVector stacked o5 = {1, 6, 14, 16, 8} o5 : List i6 : fVector shifted o6 = {1, 6, 14, 16, 8} o6 : List

An empty triangle is a shifted complex.

 i7 : R' = QQ[a,b,c]; i8 : triangle = simplicialComplex {a*b, b*c, a*c} o8 = simplicialComplex | bc ac ab | o8 : SimplicialComplex i9 : algebraicShifting triangle === triangle o9 = true

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}}; i11 : R = QQ[x_{1,1}, x_{1,2}, x_{1,3}, x_{2,1}, x_{3,1}, Degrees => grading]; 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 = simplicialComplex | x_{1, 2}x_{3, 1} x_{1, 1}x_{2, 1} x_{1, 3}x_{2, 1}x_{3, 1} | o12 : SimplicialComplex i13 : shifted = algebraicShifting(delta, Multigrading => true) o13 = simplicialComplex | x_{1, 1} x_{1, 2}x_{3, 1} x_{1, 2}x_{2, 1} x_{1, 3}x_{2, 1}x_{3, 1} | o13 : SimplicialComplex i14 : prune (homology(delta))_1 o14 = 0 o14 : QQ-module i15 : prune (homology(shifted))_1 1 o15 = QQ o15 : QQ-module, free

References:

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.

## Ways to use algebraicShifting :

• algebraicShifting(SimplicialComplex)

## For the programmer

The object algebraicShifting is .