Known Results
- The essential dimension of a linear algebraic group G is always finite and bounded by the minimal dimension of a generically free representation minus the dimension of G.
- The essential dimension of a finite algebraic p-group over k equals the minimal dimension of a faithful representation, provided that the base field k contains a primitive p-th root of unity.
- The essential dimension of the symmetric group Sn (viewed as algebraic group over k) is known for n≤5 (for every base field k), for n=6 (for k of characteristic not 2) and for n=7 (in characteristic 0).
- Let T be an algebraic torus admitting a Galois splitting field L/k of degree a power of a prime p. Then the essential dimension of T equals the least rank of the kernel of a homomorphism of Gal(L/k)-lattices P → X(T) with cokernel finite and of order prime to p, where P is a permutation lattice.
Read more about this topic: Essential Dimension
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