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A numerical invariant for linear representations of finite groups
, 2014
"... We study the notion of essential dimension for a linear representation of a finite group. In characteristic zero we relate it to the canonical dimension of certain products of Weil transfers of generalized SeveriBrauer varieties. We then proceed to compute the canonical dimension of a broad class ..."
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We study the notion of essential dimension for a linear representation of a finite group. In characteristic zero we relate it to the canonical dimension of certain products of Weil transfers of generalized SeveriBrauer varieties. We then proceed to compute the canonical dimension of a broad class of varieties of this type, extending earlier results of the first author. As a consequence, we prove analogues of classical theorems of R. Brauer and O. Schilling about the Schur index, where the Schur index of a representation is replaced by its essential dimension. In the last section we show that essential dimension of representations can behave in rather unexpected ways in the modular setting.
ESSENTIAL pDIMENSION OF ALGEBRAIC GROUPS WHOSE CONNECTED COMPONENT IS A TORUS
"... Following upon our earlier work and the work of N. Karpenko and A. Merkurjev, we study the essential pdimension of linear algebraic groups G whose connected component G 0 is a torus. ..."
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Following upon our earlier work and the work of N. Karpenko and A. Merkurjev, we study the essential pdimension of linear algebraic groups G whose connected component G 0 is a torus.
PSEUDOREFLECTION GROUPS AND ESSENTIAL DIMENSION
"... Abstract. Given a pseudoreflection group, we give a simple formula for the essential dimension at a prime p. Additionally, we determine the absolute essential dimension in most cases. We also study the “poor man’s essential dimension ” of an arbitrary finite group, an intermediate notion between th ..."
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Abstract. Given a pseudoreflection group, we give a simple formula for the essential dimension at a prime p. Additionally, we determine the absolute essential dimension in most cases. We also study the “poor man’s essential dimension ” of an arbitrary finite group, an intermediate notion between the absolute essential dimension and the essential dimension at a prime p. 1.
ESSENTIAL DIMENSION AND CANONICAL DIMENSION OF GERBES BANDED BY GROUPS OF MULTIPLICATIVE TYPE
"... We prove the formula ed(X) = cdim(X)+ed(A) for any gerbe X banded by an algebraic group A which is the kernel of a homomorphism of algebraic tori Q → S with Q invertible and S split. This result is applied to prove new results on the essential dimension of algebraic groups. ..."
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We prove the formula ed(X) = cdim(X)+ed(A) for any gerbe X banded by an algebraic group A which is the kernel of a homomorphism of algebraic tori Q → S with Q invertible and S split. This result is applied to prove new results on the essential dimension of algebraic groups.
ESSENTIAL DIMENSION OF SEPARABLE ALGEBRAS EMBEDDING IN A FIXED CENTRAL SIMPLE ALGEBRA
"... Abstract. One of the key problems in noncommutative algebra is the classification of central simple algebras and more generally of separable algebras over fields, i.e., Azumayaalgebras whose center is étale over the given field. In this paper we fix a central simple Falgebra A of prime power degr ..."
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Abstract. One of the key problems in noncommutative algebra is the classification of central simple algebras and more generally of separable algebras over fields, i.e., Azumayaalgebras whose center is étale over the given field. In this paper we fix a central simple Falgebra A of prime power degree and study seperable algebras over extensions K/F, which embed in AK. The type of such an embedding is a discrete invariant indicating the structure of the image of the embedding and of its centralizer over an algebraic closure. For fixed type we study the minimal number of independent parameters, called essential dimension, needed to define the separable Kalgebras embedding in AK for extensions K/F. We find a remarkable dichotomy between the case where the index of A exceeds a certain bound and the opposite case. In the second case the task is equivalent to the problem of computing the essential dimension of the algebraic groups (PGLd) m ⋊Sm, which is extremely difficult in general. In the first case, however, we manage to compute the exact value of the essential dimension, except in one special case, where we provide lower and upper bounds on the essential dimension. 1.
LOWER BOUNDS FOR ESSENTIAL DIMENSIONS IN CHARACTERISTIC 2 VIA ORTHOGONAL REPRESENTATIONS
"... Abstract. We give a lower bound for the essential dimension of a split simple algebraic group of “adjoint ” type over a field of characteristic 2. We also compute the essential dimension of orthogonal and special orthogonal groups in characteristic 2. ..."
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Abstract. We give a lower bound for the essential dimension of a split simple algebraic group of “adjoint ” type over a field of characteristic 2. We also compute the essential dimension of orthogonal and special orthogonal groups in characteristic 2.
2 ESSENTIAL pDIMENSION OF ALGEBRAIC GROUPS WHOSE CONNECTED COMPONENT IS A TORUS
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