Low-degree representations determining a unitary shuffle's gap
Let $n \ge 2$ and assign nonnegative weights $w_B$ to the nonempty subsets $B \subseteq [n] = \{1, \dots, n\}$, such that the graph on $[n]$ joining two vertices whenever they lie in a common subset of positive weight is connected. For each $B$ let $U_B \le U(n)$ be the subgroup of unitary matrices that act as the identity outside the coordinate subspace indexed by $B$, with Haar probability measure $m_B$. For a nontrivial irreducible unitary representation $\rho$ of $U(n)$ let $$A_\rho = \sum_B w_B \Bigl( I - \int_{U_B} \rho(V)\, dm_B(V) \Bigr),$$ a positive semidefinite matrix, and let $\gamma = \inf_{\rho \ne \mathbf{1}} \lambda_{\min}(A_\rho)$: the spectral gap of the process that, at rate $w_B$, multiplies its current unitary matrix by an independent Haar element of $U_B$. Let $\rho_1$ and $\rho_2$ be the irreducible representations with highest weights $(1, 0, \dots, 0, -1)$ and $(2, 0, \dots, 0, -2)$; for $n = 2$ these are $(1, -1)$ and $(2, -2)$. This is an analogue for $U(n)$ of Aldous's spectral gap conjecture for the symmetric group, proved by Caputo, Liggett and Richthammer. Alon and Puder proved the identity below when $w_B$ depends only on $|B|$, in which case $\gamma = \lambda_{\min}(A_{\rho_2})$, and when the weights are supported on subsets of size at least $n - 1$, in which case each of $\rho_1$ and $\rho_2$ is needed for some weights. Caputo, Quattropani and Sau proved a consequence of this identity for the discrete KMP (uniform reshuffling) process on the hypergraph: its spectral gap with any number $k \ge 2$ of indistinguishable particles equals its spectral gap with two particles.
Problem. Is $$\gamma = \min\{\lambda_{\min}(A_{\rho_1}), \lambda_{\min}(A_{\rho_2})\}$$ for every $n \ge 2$ and every such choice of weights?
Source: Alon and Puder (2026), Conjecture 1.7; stated for connected hypergraphs by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- GR, RT
- Keywords
- AIM Open Applied Problems, random walks on groups, spectral gaps, unitary groups
References
@misc{alon2026,
author = {Alon, Gil and Puder, Doron},
title = {Aldous-type spectral gaps in unitary groups},
year = {2026},
eprint = {2603.00353},
archivePrefix = {arXiv},
}
@article{caputo2010,
author = {Caputo, Pietro and Liggett, Thomas M. and Richthammer, Thomas},
title = {Proof of {A}ldous' spectral gap conjecture},
journal = {J. Amer. Math. Soc.},
volume = {23},
number = {3},
year = {2010},
pages = {831--851},
doi = {10.1090/S0894-0347-10-00659-4},
}
@misc{caputo2026,
author = {Caputo, Pietro and Quattropani, Matteo and Sau, Federico},
title = {Aldous' spectral gap phenomena in stochastic exchange models},
year = {2026},
eprint = {2609.10450},
archivePrefix = {arXiv},
}
@misc{aim-unitary-hypergraph-gap,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Low-degree representations determining a unitary shuffle's gap}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/609-unitary-hypergraph-gap.md},
note = {Entry 609 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00313, "Low-degree representations determining a unitary shuffle's gap", clopen.solutions.