The partition problem for equifacetal simplices

Allan L. Edmonds

Abstract: Associated with any equifacetal $d$-simplex, which necessarily has a vertex transitive isometry group, there is a well-defined partition of $d$ that counts the number of edges of each possible length incident at a given vertex. The partition problem asks for a characterization of those partitions that arise from equifacetal simplices. The partition problem is resolved by proving that a partition of the number $d$ arises this way if and only if the number of odd entries in the partition is at most $\iota(d+1)$, the maximum number of involutions in a finite group of order $d+1$. When $n$ is even the number $\iota(n)$ is shown to be $n/2+n_{2}/2-1$, where $n_{2}$ denotes the $2$-part of $n$. Those extremal equifacetal $d$-simplices for which the number of odd entries of the associated partition is exactly $\iota(d+1)$ are characterized.

Keywords: equifacetal simplex, isometry group, partition, involution

Classification (MSC2000): 52B12; 52B11, 52B15, 20D60, 51N20

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