bibkey: oconnorramgoolam2025gaugedtensor
authors: Denjoe O'Connor and Sanjaye Ramgoolam
year: 2025
title: "Gauged permutation invariant tensor quantum mechanics, least common multiples and the inclusion-exclusion principle"
doi: 10.48550/arXiv.2506.18813
url: "https://arxiv.org/abs/2506.18813v2"
claim: "The paper expresses the canonical partition functions of gauged permutation invariant tensor quantum mechanics as sums over partitions of N with factors built from least common multiples of parts, conjectures that the identity and transposition classes give the two leading high-temperature contributions for every tensor rank s, and asks for a proof."
strata_touched:
  - D5/S3/StatisticalMechanics/TensorHighTemperatureLeadingClasses
license: citation-only
triage: anchor

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

D. O’Connor and S. Ramgoolam, arXiv:2506.18813v2 (2025). Quotations are from the v2 TeX source.

The rank-$s$ partition function is a sum over the conjugacy classes $p=[a_1^{p_1},\dots ,a_K^{p_K}]$ of $S_N$, with distinct parts $a_i$ and multiplicities $p_i$. The pole order of the class $p$ at $x\to 1$ is

$$\mathrm{Degree}([p,N,s])=\sum _{i_1,\dots ,i_s=1}^K\frac{a_{i_1}\cdots a_{i_s}{p}_{i_1}\cdots p_{i_s}}{L(a_{i_1},\dots ,a_{i_s})},$$

where $L$ is the least common multiple. Section 7.2 states:

Computations with Mathematica and the analysis of special families give strong evidence that the leading contributions in the high temperature limit $x\to 1$ come from $p=[1^N]$ followed by $p=[2,1^{N-2}]$.

and Section 9 asks: “It would be very interesting to find the proof of the general $s$ conjecture for the leading two terms.” The case $s=2$ is proved in the authors’ earlier paper (arXiv:2405.13150, Appendix B).

Verified locator

Source: https://arxiv.org/abs/2506.18813v2

• DOI: https://doi.org/10.48550/arXiv.2506.18813 (the arXiv-issued DataCite DOI; title, authors and year from DataCite).
• URL: https://arxiv.org/abs/2506.18813v2. The e-print member Perm-Invt-Tensor-Counting-Functions-Journal.tex has 1674 lines and md5 202b33a8e55dec1592ee90b7e5fbd8d9. Locations:
  • Section 7.2 and the conjecture, l. 1066–1068;
  • the per-class product and the degree function (7.21), l. 1070–1087;
  • the proof request, Section 9, l. 1322.