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Essential dimension and linear codes Cernele, Shane


The essential dimension of an algebraic group G is a measure of the complexity of G-torsors. One of the central open problems in the theory of essential dimension is to compute the essential dimension of PGL_n, whose torsors correspond to central simple algebras up to isomorphism. In this thesis, we study the essential dimension of groups of the form G/μ, where G is a reductive algebraic group satisfying certain properties, and μ is a central subgroup of G. In particular, we consider the case G=GL_(n₁) × ⋯ × GL_(n_r ) where each n_i is a power of a single prime p, which is a generalization of the group PGL_(p^a )=GL_(p^a )/G_m. We will see that torsors for G/μ correspond to tuples of central simple algebras satisfying certain properties. Surprisingly, computing the essential dimension of G/μ becomes easier when r≥3. Using techniques from Galois cohomology, representation theory and the essential dimension of stacks, we give upper and lower bounds for the essential dimension of G/μ. To do this, we first attach a linear ‘code’ to the central subgroup μ, and define a weight function on this code. Our upper and lower bounds are given in terms of a minimal weight generator matrix for the code. In some cases we can determine the exact value of the essential dimension of G/μ.

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