Spectra of Symmetrized Shuffling Operators
Bibliographic record
Abstract
(Abridged abstract) For a finite real reflection group W and a W-orbit O of flats in its reflection arrangement---or equivalently a conjugacy class of its parabolic subgroups---we introduce a statistic on elements of W. We then study the operator of right-multiplication within the group algebra of W by the element whose coefficients are given by this statistic. We reinterpret the operators geometrically in terms of the arrangement of reflecting hyperplanes for W. We show that they are self-adjoint and positive semidefinite. via two explicit factorizations into a symmetrized form A^t A. In one such factorization, A is a generalization of the projection of a simplex onto the linear ordering polytope. In the other factorization, A is the transition matrix for one of the well-studied Bidigare-Hanlon-Rockmore random walks on the chambers of an arrangement. We study the family of operators in which O is the conjugacy classes of Young subgroups of type (k,1^{n-k}). A special case within this family is the operator corresponding to random-to-random shuffling. We show in a purely enumerative fashion that these operators pairwise commute. We furthermore conjecture that they have integer spectrum, generalizing a conjecture of Uyemura-Reyes for the case k=n-1. We use representation theory to show that if O is a conjugacy class of rank one parabolics in W, the corresponding operator has integer spectrum. Our proof makes use of an (apparently) new family of twisted Gelfand pairs for W. We also study the family of operators in which O is the conjugacy classes of Young subgroups of type (2^k,1^{n-2k}). Here the construction of a Gelfand model for the symmetric group shows that these operators pairwise commute and that they have integer spectrum. For the symmetric group, we conjecture that apart from the two commuting families above, no other pair of operators of this form commutes.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".