Haar graphical representations of finite groups and an application to poset representations
Bibliographic record
Abstract
Let R be a group and let S be a subset of R . The Haar graph Haar ( R , S ) of R with connection set S is the graph having vertex set R × { − 1 , 1 } , where two distinct vertices ( x , − 1 ) and ( y , 1 ) are declared to be adjacent if and only if y x − 1 ∈ S . The name Haar graph was coined by Tomaž Pisanski in one of the first investigations on this class of graphs. For every g ∈ R , the mapping ρ g : ( x , ε ) ↦ ( x g , ε ) , ∀ ( x , ε ) ∈ R × { − 1 , 1 } , is an automorphism of Haar ( R , S ) . In particular, the set R ˆ : = { ρ g | g ∈ R } is a subgroup of the automorphism group of Haar ( R , S ) isomorphic to R . In the case that the automorphism group of Haar ( R , S ) equals R ˆ , the Haar graph Haar ( R , S ) is said to be a Haar graphical representation of the group R . Answering a question of Feng, Kovács, Wang, and Yang, we classify the finite groups admitting a Haar graphical representation. Specifically, we show that every finite group admits a Haar graphical representation, with abelian groups and ten other small groups as the only exceptions. Our work on Haar graphs allows us to improve a 1980 result of Babai concerning representations of groups on posets, achieving the best possible result in this direction. An improvement to Babai's related result on representations of groups on distributive lattices follows.
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.004 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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".