Products and powers of principal symmetric ideals
Abstract
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.
Keywords
Cite
@article{arxiv.2402.16214,
title = {Products and powers of principal symmetric ideals},
author = {Eric Dannetun and Riccardo Formenti and Bo Y. Gao and Juliann Geraci and Ross Kogel and Yuelin Li and Shreya Mandal and Vinuge Rupasinghe and Alexandra Seceleanu and Duc Van Khank Tran and Noah Walker},
journal= {arXiv preprint arXiv:2402.16214},
year = {2024}
}
Comments
A flaw in the proof of Theorem 3.5 in version 1 = Theorem 3.6 in version 2 was remedied