Gauged permutation invariant tensor quantum mechanics, least common multiples and the inclusion-exclusion principle

We derive the canonical ensemble partition functions for gauged permutation invariant tensor quantum harmonic oscillator thermodynamics, finding surprisingly simple expressions with number-theoretic characteristics. These systems have a gauged symmetry of $S_N$, the symmetric group of all permutations of a set of $N$ objects. The symmetric group acts on tensor variables $\Phi_{ i_1, \cdots , i_s }$, where the $s$ indices each range over $\{ 1, 2, \cdots , N \}$ and have the standard $S_N$ action of permutations. The result is a sum over partitions of $N$ and the summand is a product admitting simple expressions, which depend on the least common multiples (LCMs) of subsets of the parts of the partition. The inclusion-exclusion principle of combinatorics plays a central role in the derivation of these expressions. The behaviour of these partition functions under inversion of the Boltzmann factor $x = e^{ - \beta }$ is governed by universal sequences associated with invariants of symmetric groups and alternating groups. The partition functions allow the development of a high temperature expansion analogous to the $s=2$ matrix case. The calculation of an $s$-dependent breakdown point leads to a critical Boltzmann factor $x_c = { \log N \over sN^{ s-1}}$ as the leading large $N$ approximation.