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Principal symmetric ideals were recently introduced by Harada et al. in [The minimal free resolution of a general principal symmetric ideal, preprint (2023), arXiv:2308.03141], where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper, we determine when a product of two principal symmetric ideals is principal symmetric and when 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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This content will become publicly available on January 6, 2026
The combinatorial structure of symmetric strongly shifted ideals
Symmetric strongly shifted ideals constitute a class of monomial ideals which are equipped with an action of the symmetric group and are analogous to the well-studied class of strongly stable monomial ideals. In this paper, we focus on algebraic and combinatorial properties of symmetric strongly shifted ideals. On the algebraic side, we elucidate properties that pertain to behavior under ideal operations, primary decomposition, and the structure of their Rees algebra. On the combinatorial side, we develop a notion of partition Borel generators which leads to connections to discrete polymatroids, convex polytopes, and permutohedral toric varieties.
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- PAR ID:
- 10582916
- Publisher / Repository:
- EMS Press
- Date Published:
- Journal Name:
- Journal of Combinatorial Algebra
- ISSN:
- 2415-6302
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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