Description
Let $F$ a free module with homogeneous basis $\{g_1,g_2,\ldots,g_r\}.$ If $M$ is a graded submodule of
F, as a consequence of a generalization of the Kruskal-Katona theorem, then there exists a unique lex submodule of
F with the same Hilbert function as
M. If
M is a monomial submodule of
F, we denote by $M^\mathrm{lex}$ the unique lex submodule of
F with the same Hilbert function as
M. $M^\mathrm{lex}$ is called the lex submodule associated to
M. This construction uses the generalization of the Kruskal-Katona theorem.
Example:
i1 : E=QQ[e_1..e_4,SkewCommutative=>true]
o1 = E
o1 : PolynomialRing, 4 skew commutative variable(s)
|
i2 : F=E^{0,0}
2
o2 = E
o2 : E-module, free
|
i3 : lexModule({2,8,3,1,0},F)
o3 = image | e_3e_4 e_2e_4 e_1e_4 e_2e_3 e_1e_3 e_1e_2 0 0 0 |
| 0 0 0 0 0 0 e_1e_4 e_1e_3 e_1e_2 |
2
o3 : E-module, submodule of E
|
i4 : I_1=ideal(e_1*e_2,e_1*e_3,e_2*e_3)
o4 = ideal (e e , e e , e e )
1 2 1 3 2 3
o4 : Ideal of E
|
i5 : I_2=ideal(e_1*e_2,e_1*e_3)
o5 = ideal (e e , e e )
1 2 1 3
o5 : Ideal of E
|
i6 : M=createModule({I_1,I_2},F)
o6 = image | e_2e_3 e_1e_3 e_1e_2 0 0 |
| 0 0 0 e_1e_3 e_1e_2 |
2
o6 : E-module, submodule of E
|
i7 : Mlex=lexModule M
o7 = image | e_2e_4 e_1e_4 e_2e_3 e_1e_3 e_1e_2 0 0 0 |
| 0 0 0 0 0 e_1e_3e_4 e_1e_2e_4 e_1e_2e_3 |
2
o7 : E-module, submodule of E
|