Tensor Symmetry
Exercise PLAN-310
primary source nl only. Current content-bound mathematical second pass accepted. No new Lean verification is claimed for this NL view.
Assumptions and conventions
- Diagrams with more than three rows are excluded for C^3.
- Hook length of a box counts boxes to its right, boxes below it, and the box itself; the Weyl formula is the product over row pairs of the row-length differences adjusted by the index gap, divided by the index gap.
Question
For the fourth tensor power of C^3, determine all Young diagrams that index irreducible pieces in the Schur-Weyl decomposition, compute for each the dimension of the symmetric-group piece using hook lengths and the dimension of the unitary-group piece using the Weyl dimension formula for dimension three, and verify that eighty-one equals the sum over diagrams of the product of these two dimensions.
Hint: mathematical route
Explicit Schur-Weyl dimension count for the fourth power of C^3 verifying eighty-one equals the sum of products of the two piece dimensions over the four admissible diagrams.
- The contributing diagrams are the partitions of 4 with at most 3 rows.
- Hook-length computations give symmetric-group dimensions one, three, two, three.
- The Weyl dimension formula gives unitary dimensions fifteen, fifteen, six, three.
- Summing the products gives fifteen plus forty-five plus twelve plus nine equal to eighty-one.
Reference solution
The relevant diagrams are the partitions of 4 into at most 3 parts: (4), (3,1), (2,2), and (2,1,1). Compute each pair of dimensions. For (4): there is one standard tableau so the symmetric-group dimension is 1; the Weyl formula gives the unitary dimension 15 (the dimension of the degree-four symmetric power of C^3, which is six choose two). Product 15. For (3,1): hook lengths are four, two, one across the first row and one in the second, product eight, so the symmetric-group dimension is twenty-four over eight equal to 3; the Weyl formula with parts three, one, zero gives three times five halves times two equal to 15. Product 45. For (2,2): hook lengths are three and two across the first row and two and one across the second, product twelve, so the symmetric-group dimension is twenty-four over twelve equal to 2; the Weyl formula with parts two, two, zero gives one times two times three equal to 6. Product 12. For (2,1,1): hook lengths are four and one across the first row and two then one down the first column, product eight, so the symmetric-group dimension is twenty-four over eight equal to 3; the Weyl formula with parts two, one, one gives two times three halves times one equal to 3. Product 9. Summing the products fifteen, forty-five, twelve, nine gives eighty-one, which equals three to the fourth, confirming the identity.
Assessment rubric
- Lists exactly the partitions of 4 into at most 3 parts: (4), (3,1), (2,2), (2,1,1) (1 weight)
- Computes each symmetric-group dimension correctly from hook lengths as one, three, two, three (1 weight)
- Computes each unitary dimension correctly from the Weyl formula as fifteen, fifteen, six, three (1 weight)
- Forms the products and sums them to eighty-one, verifying three to the fourth equals the sum of the products (1 weight)
Graph-grounded prerequisites
These are support/source anchors, not an automatically extracted theorem-dependency proof.
- Tensor networks (frozen curriculum anchor; see graph registry)
- Matrix analysis (frozen curriculum anchor; see graph registry)
Lean view and evidence boundary
No aligned Lean view has been admitted for this native core in this textbook checkpoint. This is not a claim that the underlying mathematics lacks a formal proof. Contributions must preserve this exact target contract and split.
Sources and provenance
- website/research/sources.json
leditzky-2025-repth-qit
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Authored support steps: 4. This is not measured graph distance or model difficulty.