On Hilbert series of Koszul operads and a classification result for set-operads
Paul Laubie
TL;DR
This work classifies all Koszul set-operads generated by a single arity-2 element and proves a sharp conjecture about their Hilbert series. By reducing to quotients of the Magmatic operad via $\mathfrak{S}_3$-equivariant quadratic relations on $\mathrm{Mag}(3)$, the authors enumerate 57 candidate operads, identify 11 that are Koszul (7 known and 4 new), and compute explicit Hilbert series for each. Non-Koszul cases are detected through the Ginzburg–Kapranov criterion and Koszul-dual analyses, leaving four Koszul operads whose series are explicitly described. Overall, the Hilbert series of these 11 Koszul set-operads are shown to be differential algebraic of order 1 over $\mathbb{Z}[t]$, supporting the broader conjecture for this family and enriching the catalog of Koszul operads with new constructions such as $\mathcal{P}_{10}$, $\mathcal{P}_{2;2}$, $\mathcal{P}_{11}$, and $\mathcal{P}_{2;10}$.
Abstract
Motivated by numerous examples in the literature, we state a conjecture on the Hilbert series of Koszul symmetric operads generated by one element of arity $2$. We prove this conjecture for all Koszul symmetric set-operads generated by one element of arity $2$ by explicitly classifying those. There are $11$ such operads; $4$ of them are new.