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# Schubert(ZZ,ZZ,VisibleList) -- find the Plücker ideal of a Schubert variety

## Synopsis

• Function: Schubert
• Usage:
Schubert(k,n,sigma)
• Inputs:
• k, an integer
• n, an integer
• sigma, , a subset of 0..n of size k+1 that indexes the Schubert variety
• Optional inputs:
• CoefficientRing => a ring, default value ZZ, the coefficient ring for the polynomial ring to be made
• Variable => , default value "p", the base symbol for the indexed variables to be used. The subscripts are the elements of subsets(n+1,k+1), converted to sequences and, if k is 0, converted to integers.
• Outputs:
• an ideal, the ideal of the Schubert variety indexed by sigma

## Description

Given natural numbers $k \le{} n$, this routine finds the ideal of the Schubert variety indexed by sigma in the Grassmannian of projective $k$-planes in $P^n$.

For example, the ideal of the Schubert variety indexed by $\{1,2,4\}$ in the Grassmannian of projective planes in $P^4$ is displayed in the following example.

 i1 : I = Schubert(2,4,{1,2,4},CoefficientRing => QQ) o1 = ideal (p , p , p , p p - p p , p p 2,3,4 1,3,4 0,3,4 1,2,3 0,2,4 0,2,3 1,2,4 1,2,3 0,1,4 ------------------------------------------------------------------------ - p p , p p - p p ) 0,1,3 1,2,4 0,2,3 0,1,4 0,1,3 0,2,4 o1 : Ideal of QQ[p ..p , p , p , p , p , p , p , p , p ] 0,1,2 0,1,3 0,2,3 1,2,3 0,1,4 0,2,4 1,2,4 0,3,4 1,3,4 2,3,4 i2 : R = ring I; i3 : C = res I 1 6 14 16 9 2 o3 = R <-- R <-- R <-- R <-- R <-- R <-- 0 0 1 2 3 4 5 6 o3 : ChainComplex i4 : betti C 0 1 2 3 4 5 o4 = total: 1 6 14 16 9 2 0: 1 3 3 1 . . 1: . 3 11 15 9 2 o4 : BettiTally