93 research outputs found
A lower bound on the essential dimension of a connected linear group
Let G be a connected linear algebraic group defined over an algebraically
closed field k and H be a finite abelian subgroup of G whose order is prime to
char(k). We show that the essential dimension of G is bounded from below by
rank(H) - rank C_G(H)^0, where rank C_G(H)^0 denotes the rank of the maximal
torus in the centralizer C_G(H). This inequality, conjectured by J.-P. Serre,
generalizes previous results of Reichstein -- Youssin (where char(k) is assumed
to be 0 and C_G(H) to be finite) and Chernousov -- Serre (where H is assumed to
be a 2-group).Comment: 21 page
Group actions and invariants in algebras of generic matrices
We show that the fixed elements for the natural GL_m-action on the universal
division algebra UD(m,n) of m generic n x n matrices form a division subalgebra
of degree n, assuming n >= 3 and 2 <= m <= n^2 - 2. This allows us to describe
the asymptotic behavior of the dimension of the space of SL_m-invariant
homogeneous central polynomials p(X_1,...,X_m) for n x n matrices. Here the
base field is assumed to be of characteristic zero.Comment: 22 pages. Final version, to appear in Advances in Applied Mathematics
(Amitai Regev issue). Theorem 1.3 has been strengthene
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