Induced operators on symmetry classes of tensors
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Bibliographic record
Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional Hilbert space. Suppose <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a subgroup of the symmetric group of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi colon upper H right-arrow double-struck upper C"> <mml:semantics> <mml:mrow> <mml:mi> χ </mml:mi> <mml:mo>:</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\chi : H \rightarrow \mathbb C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a character of degree 1 on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Consider the symmetrizer on the tensor space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="circled-times Overscript m Endscripts upper V"> <mml:semantics> <mml:mrow> <mml:mover> <mml:mo> ⨂ </mml:mo> <mml:mi>m</mml:mi> </mml:mover> <mml:mi>V</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\bigotimes ^m V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S left-parenthesis v 1 circled-times midline-horizontal-ellipsis circled-times v Subscript m Baseline right-parenthesis equals StartFraction 1 Over StartAbsoluteValue upper H EndAbsoluteValue EndFraction sigma-summation Underscript sigma element-of upper H Endscripts chi left-parenthesis sigma right-parenthesis v Subscript sigma Sub Superscript negative 1 Subscript left-parenthesis 1 right-parenthesis Baseline circled-times midline-horizontal-ellipsis circled-times v Subscript sigma Sub Superscript negative 1 Subscript left-parenthesis m right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>v</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo> ⊗ </mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo> ⊗ </mml:mo> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:mrow> </mml:mfrac> </mml:mrow> <mml:munder> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> σ </mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>H</mml:mi> </mml:mrow> </mml:munder> <mml:mi> χ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi> σ </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mi>v</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi> σ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> <mml:mo> ⊗ </mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo> ⊗ </mml:mo> <mml:msub> <mml:mi>v</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi> σ </mml:mi> <mml:mrow c
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it