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ExteriorModules :: lexModule

lexModule -- compute the lex submodule with a given Hilbert sequence in a free module

Synopsis

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 variables
 
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

See also

Ways to use lexModule :

For the programmer

The object lexModule is a method function.