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Minimal generators for symmetric ideals

Authors: Christopher J. Hillar and Troels Windfeldt
Journal: Proc. Amer. Math. Soc. 136 (2008), 4135-4137
MSC (2000): Primary 13E05, 13E15, 20B30, 06A07
Published electronically: June 11, 2008
MathSciNet review: 2431024
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Abstract: Let $ R = K[X]$ be the polynomial ring in infinitely many indeterminates $ X$ over a field $ K$, and let $ {\mathfrak{S}}_{X}$ be the symmetric group of $ X$. The group $ {\mathfrak{S}}_{X}$ acts naturally on $ R$, and this in turn gives $ R$ the structure of a module over the group ring $ R[{\mathfrak{S}}_{X}]$. A recent theorem of Aschenbrenner and Hillar states that the module $ R$ is Noetherian. We address whether submodules of $ R$ can have any number of minimal generators, answering this question positively.

References [Enhancements On Off] (What's this?)

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Additional Information

Christopher J. Hillar
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

Troels Windfeldt
Affiliation: Department of Mathematical Sciences, University of Copenhagen, DK-1165 Copenhagen, Denmark

Keywords: Invariant ideal, symmetric group, Gr\"obner basis, minimal generators
Received by editor(s): September 6, 2006
Received by editor(s) in revised form: October 25, 2007
Published electronically: June 11, 2008
Additional Notes: The work of the first author was supported under an NSF Postdoctoral Fellowship.
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2008 American Mathematical Society

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