On General Principal Symmetric Ideals
Noah Walker
TL;DR
This work provides an explicit, effective bound on the number of variables $n$ needed for the results of Harada, Seceleanu, and Şega on general principal symmetric ideals to hold, showing that the Hilbert function, Betti table, and $S_n$-equivariant minimal free resolution are determined once $n \ge 1 + \sum_{i=0}^{d-1} P(i)$, where $P(i)$ are partition numbers. It builds a representation-theoretic framework around maximal $r$-generated submodules and Kostka numbers, giving an algorithmic characterization of the minimal $r$ and a precise decomposition of $R_d$ in terms of Specht modules $S^\lambda$. A key construction, the ideal $J$ generated by a monomial-symmetric module, realizes a psi and proves the principal-symmetric structure under the same bound, enabling a strengthened version of HSS that holds for Char$(K)=0$ or Char$(K)>n$. Overall, the paper links general symmetric ideals to explicit combinatorial data (Kostka numbers, Young tableaux) to yield concrete, scalable criteria for the structure of $I_d$ and the associated minimal free resolutions.
Abstract
In a recent paper by Harada, Seceleanu, and Şega, the Hilbert function, betti table, and graded minimal free resolution of a general principal symmetric ideal are determined when the number of variables in the polynomial ring is sufficiently large. In this paper, we strengthen that result by giving a effective bound on the number of variables needed for their conclusion to hold. The bound is related to a well-known integer sequence involving partition numbers (OEIS A000070). Along the way, we prove a recognition theorem for principal symmetric ideals. We also introduce the class of maximal $r$-generated submodules, determine their structure, and connect them to general symmetric ideals.