Action of the symmetric groups on the homology of the hypertree posets

TL;DR

This paper provides a new proof for the homology dimension of hypertree posets and determines the symmetric group's action on this homology using species theory, linking it to the Prelie operad's anti-cyclic structure.

Contribution

It introduces a novel proof for the homology dimension and applies species theory to analyze the symmetric group's action on hypertree homology, connecting it with operad structures.

Findings

Confirmed the dimension of the unique non-zero homology group.

Determined the symmetric group's action on the homology group.

Linked the homology action to the anti-cyclic structure of the Prelie operad.

Abstract

The set of hypertrees on $n$ vertices can be endowed with a poset structure. J. McCammond and J. Meier computed the dimension of the unique non zero homology group of the hypertree poset. We give another proof of their result and use the theory of species to determine the action of the symmetric group on this homology group, which is linked with the anti-cyclic structure of the $\operatorname{Prelie}$ operad. We also compute the action on the Whitney homology of the poset. ----- L'ensemble des hyperarbres \`a $n$ sommets peut \^etre muni d'un ordre partiel. J. McCammond et J. Meier ont calcul\'e la dimension de l'unique groupe d'homologie non trivial du poset des hyperarbres. Apr\`es avoir donn\'e une autre preuve de ce r\'esultat, nous utilisons la th\'eorie des esp\`eces pour d\'eterminer l'action du groupe sym\'etrique sur ce groupe, que nous relions \`a la structure anti-cyclique de l'op\'erade $\operatorname{Prelie}$. Nous calculons aussi l'action du groupe sym\'etrique sur l'homologie de Whitney du poset.