A Study of the Representations Supported by the Orbit Closure of the Determinant

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Compares the GL(E)-orbit closure of the determinant and the GL(E)-orbit closure of a basis-dependent permanent-like polynomial p_{m,n}, defines M̄_n (smallest m with X_perm ⊂ X_det) and MN, and shows known lower bound M̄_n is at least quadratic in n (M̄_n ≥ const·n^2, result of Landsberg–Manivel et al). Proves both cones fail normality with a large nonnormal locus, gives representation-theoretic constraints: occurring partitions vanish past index n^2+1 and have total size a multiple of m, and relates existence of obstructions to the Alon–Tarsi equivalence about Latin squares and the occurrence of the trivial SL(V)-module in S^m(S^m V).

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" chapters (9)

1. 0:00Orbit closures and determinants
2. 10:34Permanent and determinant varieties
3. 21:40Variety containment limits
4. 28:51Landsberg-Manivel-Desalvo bounds
5. 38:06Representation challenges
6. 47:19Latin squares and conjecture
7. 53:35Lando-Tarsi conjecture forms
8. 1:00:09Complexity barriers and open problems
9. 1:05:27Orbit closures with identical representations