[Paper Review] Operads for n-ary algebras - calculations and conjectures

Martin Markl, Elisabeth Remm|arXiv (Cornell University)|Mar 21, 2011

Advanced Topics in Algebra8 references3 citations

TL;DR

This paper investigates the Koszulity of the operad $\mathsf{t\mathcal{A}ss}^n_d$ governing degree $d$ totally associative $n$-ary algebras. Using numerical computations of minimal model generating series, the authors provide strong evidence that $\mathsf{t\mathcal{A}ss}^n_d$ is non-Koszul when $d$ is odd and $n \geq 8$, conjecturing a gap of length $n-1$ in the minimal model of its Koszul dual $\mathsf{p\mathcal{A}ss}^n_d$ when $n \not\equiv d \pmod{2}$, which would imply non-Koszulity for $d$ odd and $n \geq 8$. The results suggest $\mathsf{t\mathcal{A}ss}^8_1$ may be the first such non-Koszul operad not detectable by the Ginzburg-Kapranov criterion.

ABSTRACT

In our earlier work we studied Koszulity of a family of operads depending on a natural number n and on the degree d of the generating operation. While we proved that, for n < 8, this operad is Koszul if and only if d is even, and while it follows from a work of Hoffbeck that it is Koszul for d even and arbitrary n, its (non)Koszulity for d odd and n > 7 remains an open problem. In this note, we describe some related numerical experiments, and formulate a conjecture suggested by the results of these computations.

Motivation & Objective

• To investigate the Koszulity of the operad $\mathsf{t\mathcal{A}ss}^n_d$ for odd $d$ and $n \geq 8$, where the status was previously open.
• To compute the generating series of the minimal models of the Koszul dual operads $\mathsf{p\mathcal{A}ss}^n_d$ for small $n$ and various $d$.
• To detect structural patterns in the minimal model generators, particularly the presence of vanishing coefficients, to infer non-Koszulity of $\mathsf{t\mathcal{A}ss}^n_d$.
• To formulate a conjecture on the length of the gap in the minimal model of $\mathsf{p\mathcal{A}ss}^n_d$ when $n \not\equiv d \pmod{2}$, based on computational results.
• To provide evidence that $\mathsf{t\mathcal{A}ss}^8_1$ may be the first non-Koszul operad not detected by the Ginzburg-Kapranov criterion.

Proposed method

• Computing the generating series of the operads $\mathsf{p\mathcal{A}ss}^n_d$, the Koszul duals of $\mathsf{t\mathcal{A}ss}^n_d$, for small $n$ and various $d$.
• Calculating the formal inverse of the generating series to determine the Poincaré series of the minimal model generators.
• Identifying vanishing coefficients in the inverse series, particularly in degrees $n-1, 2(n-1), \dots$, as indicators of structural gaps in the minimal model.
• Using the vanishing of Euler characteristics of certain components to infer that the minimal model has a gap of length $n-1$ in the generating collection.
• Applying Proposition 3.4, which links vanishing coefficients to the absence of generators in specific degrees.
• Extending the pattern observed in $n=2$ to $n=5$ to conjecture a general gap structure for $\mathsf{p\mathcal{A}ss}^n_d$ when $n \not\equiv d \pmod{2}$.

Experimental results

Research questions

• RQ1 Is the operad $\mathsf{t\mathcal{A}ss}^n_d$ Koszul when $d$ is odd and $n \geq 8$?
• RQ2 What structural features emerge in the minimal model of the Koszul dual operad $\mathsf{p\mathcal{A}ss}^n_d$ for $n \geq 4$ and odd $d$?
• RQ3 Does the minimal model of $\mathsf{p\mathcal{A}ss}^n_d$ exhibit a gap of length $n-1$ in the generating collection when $n \not\equiv d \pmod{2}$?
• RQ4 Can the vanishing of coefficients in the inverse generating series of $\mathsf{p\mathcal{A}ss}^n_d$ be used to infer non-Koszulity of $\mathsf{t\mathcal{A}ss}^n_d$?
• RQ5 Is $\mathsf{t\mathcal{A}ss}^8_1$ the first example of a non-Koszul operad not detectable by the Ginzburg-Kapranov criterion?

Key findings

• The generating series of $\mathsf{p\mathcal{A}ss}^2_1$ has a zero coefficient at degree 4, indicating a gap of length 3 in its minimal model.
• The generating series of $\mathsf{p\mathcal{A}ss}^3_0$ has zero coefficients at degrees 7 and 9, suggesting a gap of length 6 in the minimal model.
• The generating series of $\mathsf{p\mathcal{A}ss}^4_1$ has zero coefficients at degrees 10, 13, and 16, indicating a gap of length 9 in the minimal model.
• The generating series of $\mathsf{p\mathcal{A}ss}^5_0$ has zero coefficients at degrees 13, 17, 21, and an unknown term at degree 25, suggesting a gap of length 12 in the minimal model.
• The vanishing of coefficients in the inverse series implies that the Euler characteristic of the corresponding component in the minimal model is zero, indicating no generators in those degrees.
• Based on the observed pattern, the authors conjecture that the minimal model of $\mathsf{p\mathcal{A}ss}^n_d$ has a gap of length $n-1$ when $n \not\equiv d \pmod{2}$, which would imply that $\mathsf{t\mathcal{A}ss}^n_d$ is non-Koszul for odd $d$ and $n \geq 8$.