ISSN:
1572-9192
Keywords:
Gröbner basis
;
linear representation
;
generic module
;
computational algebra
;
finite group
;
Hilbert series
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Each matrix representation π:G → GLn(ϰ) of a finite Group G over a field ϰ induces an action of G on the module Αn over the polynomial algebra $${\rm A} = \kappa [x_1 , \ldots ,x_n ]$$ The graded Α-submodule M(Π) of Αn generated by the orbit of $$(x_1 , \ldots ,x_n )$$ is studied. A decomposition of M(Π) into generic modules is given. Relations between the numerical invariants of Π and those of M(Π), the latter being efficiently computable by Gröbner bases methods, are examined. It is shown that if Π is multiplicity-free, then the dimensions of the irreducible constituents of Π can be read off from the Hilbert series of M(Pi;). It is proved that determinantal relations form Gröbner bases for the syzygies on generic matrices with respect to any lexicographic order. Gröbner bases for generic modules are also constructed, and their Hilbert series are derived. Consequently, the Hilbert series of M(Pi;) is obtained for an arbitrary representation.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1022445607540
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