An antichain of monomial ideals in a twisted commutative algebra

Abstract

We resolve an open question posed by Nagpal, Sam and Snowden [Selecta. Math. (N.S.) 22 (2016), pp. 913–937] in 2015 concerning a Gröbner theoretic approach to the noetherianity of the twisted commutative algebra S y m ( S y m 2 ( C ∞ ) ) Sym(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.