Writing Down Polynomials via Representation Theory II

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" summary

Constructs explicit highest-weight tensors for Sym^d(Sym^n(C^{n^2})^) and writes polynomials as symmetric tensor contraction so evaluation becomes a sum of products of small determinants rather than a huge symmetrization. Gives a combinatorial recipe via semi-standard tableaux, proves a pigeonhole vanishing criterion (if the first row is too short the polynomial vanishes on the padded permanent), and shows a power-sum evaluation* that certifies nonzero plethysm coefficients for even partitions. Methods: GL/S_n representation theory, tableaux, determinant building blocks; it works and speeds up many checks.

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" chapters (11)

1. 0:00Polynomials via tensors
2. 3:55Polynomials and symmetric tensors
3. 11:03Padded permanent and invariant spaces
4. 15:20Highest weight vectors for polynomials
5. 23:13Highest weight vector as determinant
6. 28:57Tensor powers into irreducible representations
7. 36:33Semi-standard tableaux and basis relations
8. 42:40Evaluating tableaux via tensor contraction
9. 53:27Vanishing obstructions in geometric complexity
10. 1:00:50Plethysm coefficients positivity
11. 1:07:05VNP computational limits