Degenerate Schubert varieties

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Flag₄
Flag₅
L. Bossinger
M. Lanini

Mathematisches Institut
University of Cologne
Weyertal 86 - 90
50931 Cologne
Germany

Room 211
Tel: +49 (0) 221 470 3722
Email: lbossing (at) math.uni-koeln.de

Flag₅

The ideal corresponding to the embedding of Flag₅ into a product of Grassmannians Gr(1,5)xGr(2,5)xGr(3,5)xGr(4,5) and further with respect to the Plücker embedding of each Grassmannian into a product of projective spaces is generated by

p_3,4,5p_1,2,4,5-p_2,4,5p_1,3,4,5+p_1,4,5p_2,3,4,5,
p_3,4,5p_1,2,3,5-p_2,3,5p_1,3,4,5+p_1,3,5p_2,3,4,5,
p_2,4,5p_1,2,3,5-p_2,3,5p_1,2,4,5+p_1,2,5p_2,3,4,5,
p_1,4,5p_1,2,3,5-p_1,3,5p_1,2,4,5+p_1,2,5p_1,3,4,5,
p_4,5p_1,2,3,5-p_3,5p_1,2,4,5+p_2,5p_1,3,4,5-p_1,5p_2,3,4,5,
p_3,4,5p_1,2,3,4-p_2,3,4p_1,3,4,5+p_1,3,4p_2,3,4,5,
p_2,4,5p_1,2,3,4-p_2,3,4p_1,2,4,5+p_1,2,4p_2,3,4,5,
p_1,4,5p_1,2,3,4-p_1,3,4p_1,2,4,5+p_1,2,4p_1,3,4,5,
p_2,3,5p_1,2,3,4-p_2,3,4p_1,2,3,5+p_1,2,3p_2,3,4,5,
p_1,3,5p_1,2,3,4-p_1,3,4p_1,2,3,5+p_1,2,3p_1,3,4,5,
p_1,2,5p_1,2,3,4-p_1,2,4p_1,2,3,5+p_1,2,3p_1,2,4,5,
p_4,5p_1,2,3,4-p_3,4p_1,2,4,5+p_2,4p_1,3,4,5-p_1,4p_2,3,4,5,
p_3,5p_1,2,3,4-p_3,4p_1,2,3,5+p_2,3p_1,3,4,5-p_1,3p_2,3,4,5,
p_2,5p_1,2,3,4-p_2,4p_1,2,3,5+p_2,3p_1,2,4,5-p_1,2p_2,3,4,5,
p_1,5p_1,2,3,4-p_1,4p_1,2,3,5+p_1,3p_1,2,4,5-p_1,2p_1,3,4,5,
p₅p_1,2,3,4-p₄p_1,2,3,5+p₃p_1,2,4,5-p₂p_1,3,4,5+p₁p_2,3,4,5,
p_2,3,5p_1,4,5-p_1,3,5p_2,4,5+p_1,2,5p_3,4,5,
p_2,3,4p_1,4,5-p_1,3,4p_2,4,5+p_1,2,4p_3,4,5,
p_4,5p_2,3,5-p_3,5p_2,4,5+p_2,5p_3,4,5,
p_2,3,4p_1,3,5-p_1,3,4p_2,3,5+p_1,2,3p_3,4,5,
p_4,5p_1,3,5-p_3,5p_1,4,5+p_1,5p_3,4,5,
p_2,3,4p_1,2,5-p_1,2,4p_2,3,5+p_1,2,3p_2,4,5,
p_1,3,4p_1,2,5-p_1,2,4p_1,3,5+p_1,2,3p_1,4,5,
p_4,5p_1,2,5-p_2,5p_1,4,5+p_1,5p_2,4,5,
p_3,5p_1,2,5-p_2,5p_1,3,5+p_1,5p_2,3,5,
p_3,4p_1,2,5-p_2,4p_1,3,5+p_1,4p_2,3,5+p_2,3p_1,4,5-p_1,3p_2,4,5+p_1,2p_3,4,5,
p_4,5p_2,3,4-p_3,4p_2,4,5+p_2,4p_3,4,5,
p_3,5p_2,3,4-p_3,4p_2,3,5+p_2,3p_3,4,5,
p_2,5p_2,3,4-p_2,4p_2,3,5+p_2,3p_2,4,5,
p_1,5p_2,3,4-p_1,4p_2,3,5+p_1,3p_2,4,5-p_1,2p_3,4,5,
p₅p_2,3,4-p₄p_2,3,5+p₃p_2,4,5-p₂p_3,4,5,
p_4,5p_1,3,4-p_3,4p_1,4,5+p_1,4p_3,4,5,
p_3,5p_1,3,4-p_3,4p_1,3,5+p_1,3p_3,4,5,
p_2,5p_1,3,4-p_2,4p_1,3,5+p_2,3p_1,4,5+p_1,2p_3,4,5,
p_1,5p_1,3,4-p_1,4p_1,3,5+p_1,3p_1,4,5,
p₅p_1,3,4-p₄p_1,3,5+p₃p_1,4,5-p₁p_3,4,5,
p_4,5p_1,2,4-p_2,4p_1,4,5+p_1,4p_2,4,5,
p_3,5p_1,2,4-p_2,4p_1,3,5+p_1,4p_2,3,5+p_1,2p_3,4,5,
p_2,5p_1,2,4-p_2,4p_1,2,5+p_1,2p_2,4,5,
p_1,5p_1,2,4-p_1,4p_1,2,5+p_1,2p_1,4,5,
p_3,4p_1,2,4-p_2,4p_1,3,4+p_1,4p_2,3,4,
p₅p_1,2,4-p₄p_1,2,5+p₂p_1,4,5-p₁p_2,4,5,
p_4,5p_1,2,3-p_2,3p_1,4,5+p_1,3p_2,4,5-p_1,2p_3,4,5,
p_3,5p_1,2,3-p_2,3p_1,3,5+p_1,3p_2,3,5,
p_2,5p_1,2,3-p_2,3p_1,2,5+p_1,2p_2,3,5,
p_1,5p_1,2,3-p_1,3p_1,2,5+p_1,2p_1,3,5,
p_3,4p_1,2,3-p_2,3p_1,3,4+p_1,3p_2,3,4,
p_2,4p_1,2,3-p_2,3p_1,2,4+p_1,2p_2,3,4,
p_1,4p_1,2,3-p_1,3p_1,2,4+p_1,2p_1,3,4,
p₅p_1,2,3-p₃p_1,2,5+p₂p_1,3,5-p₁p_2,3,5,
p₄p_1,2,3-p₃p_1,2,4+p₂p_1,3,4-p₁p_2,3,4,
p_3,4p_2,5-p_2,4p_3,5+p_2,3p_4,5,
p_3,4p_1,5-p_1,4p_3,5+p_1,3p_4,5,
p_2,4p_1,5-p_1,4p_2,5+p_1,2p_4,5,
p_2,3p_1,5-p_1,3p_2,5+p_1,2p_3,5,
p₅p_3,4-p₄p_3,5+p₃p_4,5,
p₅p_2,4-p₄p_2,5+p₂p_4,5,
p_2,3p_1,4-p_1,3p_2,4+p_1,2p_3,4,
p₅p_1,4-p₄p_1,5+p₁p_4,5,
p₅p_2,3-p₃p_2,5+p₂p_3,5,
p₄p_2,3-p₃p_2,4+p₂p_3,4,
p₅p_1,3-p₃p_1,5+p₁p_3,5,
p₄p_1,3-p₃p_1,4+p₁p_3,4,
p₅p_1,2-p₂p_1,5+p₁p_2,5,
p₄p_1,2-p₂p_1,4+p₁p_2,4,
p₃p_1,2-p₂p_1,3+p₁p_2,3

Click here for the ideal in Macaulay2 code. We compute the ideals for degenerate Schubert varieties by taking the initial ideal of the Schubert varieties with respect to the weight vector

w=(1,1,1,1,0,1,1,2,1,2,2,0,1,1,1,1,1,2,2,0,1,1,1,1,2,1,0,1,1,1)

The entries of w are with respect to the order on Plücker coordinates:

p₁, p₂, p₃, p₄, p₅,
p_1,2, p_1,3, p_2,3, p_1,4, p_2,4, p_3,4, p_1,5,p_2,5,p_3,5,p_4,5,
p_1,2,3, p_1,2,4, p_1,3,4, p_2,3,4, p_1,2,5, p_1,3,5, p_2,3,5,p_1,4,5,p_2,4,5,p_3,4,5,
p_1,2,3,4,p_1,2,3,5,p_1,2,4,5,p_1,3,4,5,p_2,3,4,5.

The ideals can be found here.

We compute further the primary decompositions of the initial ideals to see if the degenerate Schubert varieties are irreducible. It turns out that the following observation is true for Flag₅: if none of the Plücker relations degenerates to a monomial when considering their initial form wrt w, then the ideal for the degenerate Schubert variety is prime.

All primary decompositons can be found here.

Links: Macaulay2, sage