Edge partitions of the complete graph and a determinant like function

Abstract

In this paper we prove the case $dim(V_3)=3$ of a conjecture about the exterior operad ${\Lambda }_{V_d}^{S^2}$. For this we introduce a collection of natural involutions on the set of homogeneous cycle-free $d$-partitions of the complete graph $K_{2d}$, and show that these involutions correspond to the relations in ${\Lambda }_{V_d}^{S^2}(2d+1)$. When $d=3$ this correspondence allows us to give an explicit description of a determinant-like map and to settle the above mentioned conjecture.

33 pages, 35 figures, all comments are welcome