Products and powers of principal symmetric ideals
arxiv(2024)
摘要
Principal symmetric ideals were recently introduced by Harada, Seceleanu, and
Sega, with a focus on their homological properties. They are ideals generated
by the orbit of a single polynomial under permutations of variables in a
polynomial ring. In this paper we seek to determine when a product of two
principal symmetric ideals is principal symmetric and when all the powers of a
principal symmetric ideal are again principal symmetric ideals. We characterize
the ideals that have the latter property as being generated by polynomials
invariant up to a scalar multiple under permutation of variables. Recognizing
principal symmetric ideals is an open question for the purpose of which we
produce certain obstructions. We also demonstrate that the Hilbert functions of
symmetric monomial ideals are not all given by symmetric monomial ideals, in
contrast to the non-symmetric case.
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