Stirling decomposition of graph homology in genus \(1\) (Q6607605)

Stirling complexes \(\mathcal{S}_{n,k}\) appeared in [\textit{B. C. Ward}, Int. Math. Res. Not. 2022, No. 11, 8086--8161 (2022; Zbl 1493.18020)] as certain subcomplexes of the Feynman transform [\textit{E. Getzler} and \textit{M. M. Kapranov}, Compos. Math. 110, No. 1, 65--126 (1998; Zbl 0894.18005)] of Lie graph homology in genus 1.\N\NThe goal of the paper is to provide a complete computation of the rational homology of Stirling complexes. In particular, it is shown that the Betti numbers \(\beta_i(\mathcal{S}_{n,k})\) of the complexes \(\mathcal{S}_{n,k}\) are the (absolute values of the) signed Stirling numbers \(s_{n,k}\) of the first kind, if \(i=n\), and that they are \(0\) otherwise. These values count the number of permutations of the set \(\{1,\dots, n\}\) which can be written as a product of \(k\) disjoint cycles, and the homology of Stirling complexes yields a categorification of the combinatorial formula\N\[\Ns_{n,k}=\sum_{m=k+1}^{n+1}\binom{m-1}{k}s_{n+1,m} \ .\N\]\NThe computation of the homology of Stirling complexes is then used to show that commutative graph homology in genus \(g=1\) with \(n\geq 3\) markings has a direct sum decomposition whose summands have ranks given by Stirling numbers of the first kind.\N\NAmong other applications to graph homology, the paper also deals with decompositions in irreducibles (with respect to the symmetric group action). In particular, it is conjectured -- see Conjecture 5.7 -- that the sequences \(H_n(\mathcal{S}_{n,n-i})\) are representation stable (after tensoring with the alternating representation).\N\NFor the entire collection see [Zbl 1545.18002].