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- Essential dimension and linear codes
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Essential dimension and linear codes Cernele, Shane
Abstract
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.
Item Metadata
Title |
Essential dimension and linear codes
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
2014
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Description |
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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Genre | |
Type | |
Language |
eng
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Date Available |
2014-04-24
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivs 2.5 Canada
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DOI |
10.14288/1.0167327
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2014-09
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Campus | |
Scholarly Level |
Graduate
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivs 2.5 Canada