Tutorial 2: Algorithmic Invariant Theory
summary
Defines when a group is reductive by the existence of a Reynolds operator and proves Hilbert's 1890 finite-generation for reductive matrix groups; gives the Molien formula for finite groups and a practical crude algorithm (Molien + Reynolds averages + Hilbert series check) and a second method using Derksen elimination (introduce group variables, eliminate, average). Presents concrete invariants: ternary cubic generators S (degree 4, 26 terms) and T (degree 6, 103 terms) with discriminant Δ = T^2 − 64S^3 (degree 12, 2040 terms); shows an explicit Cayley–Bacharach bracket formula for the ninth point; exhibits Nagata/Mukai-style nonreductive counterexample (additive subgroup action, blowups) where the invariant ring fails to be finitely generated when M − D = 3 and M ≥ 9.
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chapters (16)
- 0:00Intro
- 0:12Who am I
- 0:52Personal History
- 2:44Invariant Theory
- 4:30Ring of symmetric polynomials
- 5:40Invariants
- 12:07The Reynolds Operator
- 15:16reductive groups
- 19:51Hilberts theorem
- 22:11Eminerto
- 23:03The Hilbert Series
- 26:13Crude Algorithm
- 32:47Representation
- 39:26Orbits
- 43:17Brackets
- 46:13Geometric Theory
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