Degenerate Schubert varieties

Flag₄

The ideal corresponing the embedding of Flag₄ into a product of Grassmannians Gr(1,4)xGr(2,4)xGr(3,4) and further with respect to the Plücker embedding of each Grassmannian into a product of projective spaces, is generated by

p_3,4p_1,2,4-p_2,4p_1,3,4+p_1,4p_2,3,4,
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,
p₄p_1,2,3-p₃p_1,2,4+p₂p_1,3,4-p₁p_2,3,4,
p_1,4p_2,3-p_1,3p_2,4+p_1,2p_3,4,
p₄p_2,3-p₃p_2,4+p₂p_3,4,
p₄p_1,3-p₃p_1,4+p₁p_3,4,
p₄p_1,2-p₂p_1,4+p₁p_2,4,
p₃p_1,2-p₂p_1,3+p₁p_2,3

A more copy-paste friendly version in Macaulay2 code can be found here.

We compute the ideals for degenerate Schubert varieties by taking the initial ideal of the Schubert varieties with respect to the weight vector

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

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

p₁, p₂, p₃, p₄, p_1,2, p_1,3, p_1,4, p_2,3, p_2,4, p_3,4, p_1,2,3, p_1,2,4, p_1,3,4, p_2,3,4

The ideals can be found here. They are all equal to their radical ideal.

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 Flag₄: 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. An overview is given in the table below:

Sym. group elem.	monomial deg. Plücker relation	no. of components
id	no	1
s₁	no	1
s₂	no	1
s₃	no	1
s₂s₁	no	1
s₃s₁	no	1
s₁s₂	yes	2
s₃s₂	no	1
s₂s₃	yes	2
s₂s₁s₂	no	1
s₂s₃s₂	no	1
s₁s₂s₃	yes	2
s₃s₂s₁	no	1
s₂s₁s₃	yes	2
s₁s₃s₂	yes	2
s₂s₁s₃s₂	yes	5
s₁s₃s₂s₁	no	1
s₁s₂s₁s₃	yes	3
s₁s₂s₃s₂	yes	2
s₂s₃s₂s₁	no	1
s₂s₃s₁s₂s₃	yes	3
s₁s₃s₂s₁s₃	yes	3
s₂s₃s₁s₂s₁	yes	2
s₁s₂s₁s₃s₂s₁	no	1

Degenerate Schubert varieties

Flag4

Flag₄