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Macaulay2Doc :: Schubert(ZZ,ZZ,VisibleList)

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

Synopsis

Description

Given natural numbers k ≤ n, this routine finds the ideal of the Schubert variety indexed by sigma in the Grassmannian of projective k-planes in Pn.

For example, the ideal of the Schubert variety indexed by {1,2,4} in the Grassmannian of projective planes in P4 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

See also