English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Each matrix representation \pi\!: G \longrightarrow {\rm GL}_n({\cal K})of a finite Group G over a field {\cal K} induces an action of G on the module {\cal A}^n over the polynomial algebra {\cal A}={\cal K}[x_1, \ldots , x_n] The graded {\cal A}\hbox{-submodule} M(Π) of {\cal A}^n 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.

Hilbert Series of Group Representations and Gröbner Bases for Generic Modules