Symmetric decomposition of the Hilbert function of an ideal
Meghana Bhat, Saipriya Dubey, Shreedevi K. Masuti
TL;DR
This work generalizes Iarrobino's symmetric decomposition to the Hilbert function of G(I) for ideals in AG k-algebras, establishing that HF(G(I))_i decomposes as a sum of symmetric pieces HF(G(I))_i = ∑_{a=0}^{u} H(a)_i with each H(a) symmetric about (u−a)/2. It introduces the end-degree framework and the modules Q(a) to prove symmetry of H(a) and that Q(0) is a Gorenstein quotient of G(I); it also links HF(G(I)) symmetry to G(I) being Gorenstein. The paper then explores b-admissible sequences and I-Gorenstein sequences, providing both open problems and, in the codimension two case, a complete enumeration of 2-admissible sequences with h0=2 and end degree ≤3 that occur as HF(G(I)). These results offer a structural toolkit for classifying Hilbert functions of ideals in AG k-algebras and raise questions about the uniqueness of symmetric decompositions in broader settings.
Abstract
Let $(R, \mathcal{M})$ be a local ring over a field $k$ with $k = R/\mathcal M$ and $J$ an ideal in $R$ such that $A =R/J$ is an Artinian Gorenstein (AG) $k$-algebra. In 1989, A. Iarrobino introduced the symmetric decomposition of the Hilbert function of $A$. This became a very powerful tool for classifying the Hilbert functions of AG $k$-algebras. In this article, we introduce the symmetric decomposition of the Hilbert function of any ideal $I$ in $A.$ Our hope is that this result will be useful in classifying the possible Hilbert function of an ideal in an AG $k$-algebra. We illustrate this by giving a complete list of $2$-admissible sequences of length at most $3$ and with $h_0=2$ that are realizable by an ideal in an AG $k$-algebra.