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ExteriorModules :: lexModuleBySequences

lexModuleBySequences -- alternative algorithm to 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. The algorithmic construction of the lex submodule is based on the additive property of Hilbert functions and on Kruskal-Katona's 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 : lexModuleBySequences({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

See also

Ways to use lexModuleBySequences :

For the programmer

The object lexModuleBySequences is a method function.