A counterexample to a Gröbner approach for noetherianity of the twisted commutative algebra $\mathrm{Sym}({\mathrm{Sym}}^2({\mathbf{C}}^{\infty }))$

Abstract

We resolve an open question posed by the authors of arXiv:1501.06925v2 in 2015 concerning a Gröbner theoretic approach for the noetherianity of the twisted commutative algebra $\mathrm{Sym}({\mathrm{Sym}}^2({\mathbf{C}}^{\infty }))$. We provide a negative answer to their question by producing an explicit antichain. In doing so, we establish a connection to well studied posets of graphs under the subgraph and induced subgraph relation. We then analyze this connection to suggest future paths of investigation, for example a deeper study of initial ideals in twisted commutative algebras.

arXiv comment22 pages, comments welcome

References in corpus (8)

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• GL-equivariant modules over polynomial rings in infinitely many variables
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• Small permutation classes
• Noetherianity of some degree two twisted commutative algebras
• Noetherianity of some degree two twisted skew-commutative algebras
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