# algebraicShifting -- the algebraic shifting of a simplicial complex

## Synopsis

• Usage:
A = algebraicShifting D
• Inputs:
• Optional inputs:
• Multigrading => , default value false, If true it returns the colored algebraic shifting w.r.t. the multigrading of the underlying ring.
• Outputs:
• A, The algebraic shifting of the simplicial complex D. If Multigrading => 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 = | 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 = | 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 = HashTable{-1 => 1} 0 => 6 1 => 14 2 => 16 3 => 8 o5 : HashTable i6 : fVector shifted o6 = HashTable{-1 => 1} 0 => 6 1 => 14 2 => 16 3 => 8 o6 : HashTable
An empty triangle is a shifted complex.
 i7 : R=QQ[a,b,c]; i8 : triangle = simplicialComplex{a*b,b*c,a*c}; 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 = | x_{1, 3}x_{2, 1}x_{3, 1} x_{1, 2}x_{3, 1} x_{1, 1}x_{2, 1} | o12 : SimplicialComplex 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 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 .