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We determine the essential dimension of an arbitrary semisimple group of type B of the form G = (Spin (2 n 1 + 1) × ⋯ × Spin (2 n m + 1)) / μ over a field of characteristic 0, for all n 1 , ... , n m ≥ 7 , and a central subgroup μ of Spin (2 n 1 + 1) × ⋯ × Spin (2 n m + 1) not containing the center of Spin (2 n i + 1) as a direct factor for every i = 1 , ... , m. [ABSTRACT FROM AUTHOR]

The goal of this note is to reduce the existing upper bound for the essential dimension of central simple algebras of degree 8 and exponent 2 over fields of characteristic 2 from 10 to 9. [ABSTRACT FROM AUTHOR]

In this paper, we compute the essential p -dimension of the split finite quasi-simple groups of classical Lie type at the defining prime, specifically the quasi-simple groups arising from the general linear and special linear groups, the symplectic groups, and the orthogonal groups. [ABSTRACT FROM AUTHOR]

In this paper we develop the theory of essential dimension of group schemes over an integral base. Shortly we concentrate over a local base. As a consequence of our theory we give a result of invariance of the essential dimension over a field. The case of group schemes over a discrete valuation ring is discussed. [ABSTRACT FROM AUTHOR]

The notion of a ( G , N ) -slice of a G -variety was introduced by P.I. Katsylo in the early 80's for an algebraically closed base field of characteristic 0. Slices (also known under the name of relative sections) have ever since provided a fundamental tool in invariant theory, allowing reduction of rational or regular invariants of an algebraic group G to invariants of a “simpler” group. We refine this notion for a G -scheme over an arbitrary field, and use it to get reduction of structure group results for G -torsors. Namely we show that any ( G , N ) -slice of a versal G -scheme gives surjective maps H 1 ( L , N ) → H 1 ( L , G ) in fppf-cohomology for infinite fields L containing F . We show that every stabilizer in general position H for a geometrically irreducible G -variety V gives rise to a ( G , N G ( H ) ) -slice in our sense. The combination of these two results is applied in particular to obtain a striking new upper bound on the essential dimension of the simply connected split algebraic group of type E 7 . [ABSTRACT FROM AUTHOR]

In the present paper, we provide a lower bound of the essential dimension over a field of positive characteristic via Kato's cohomology group, defined by cokernel of a general Artin-Schreier operator. Combining this with Tignol's result on the second trace form of simple algebras of degree , we show that over a field of characteristic . [ABSTRACT FROM AUTHOR]

We compute the essential dimension of the functors Formsn;d and Hypersurfn;d of equivalence classes of homogeneous polynomials in n variables and hypersurfaces in Pn1, respectively, over any base field k of characteristic 0. Here two polynomials (or hypersurfaces) over a field K=k are considered equivalent if they are related by a linear change of coordinates with coefficients in K. Our proof is based on a new Genericity Theorem for algebraic stacks, which is of independent interest. As another application of the Genericity Theorem, we prove a new result on the essential dimension of the stack of (not necessarily smooth) local complete intersection curves. [ABSTRACT FROM AUTHOR]

The well-known fiber dimension theorem in algebraic geometry says that for every morphism f:X→Y of integral schemes of finite type the dimension of every fiber of f is at least dim X−dim Y. This has recently been generalized by Brosnan, Reichstein and Vistoli to certain morphisms of algebraic stacks f:풳→풴, where the usual dimension is replaced by essential dimension. We will prove a general version for morphisms of categories fibered in groupoids. Moreover, we will prove a variant of this theorem, where essential dimension and canonical dimension are linked. These results let us relate essential dimension to canonical dimension of algebraic groups. In particular, using the recent computation of the essential dimension of algebraic tori by MacDonald, Meyer, Reichstein and the author, we establish a lower bound on the canonical dimension of algebraic tori. [ABSTRACT FROM PUBLISHER]

Let p be a prime integer, 1≤ s≤ r be integers and F be a field of characteristic different from p. We find upper and lower bounds for the essential p-dimension ed( $$ Al{{g}_{{{{p}^r},{{p}^s}}}} $$) of the class $$ Al{{g}_{{{{p}^r},{{p}^s}}}} $$ of central simple algebras of degree p and exponent dividing p. In particular, we show that ed( Alg)=ed( Alg)=8 and ed( $$ Al{{g}_{{{{p}^2},p}}} $$)= p+ p for p odd. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we find an upper bound for the essential dimension of finite cyclic groups over a field F of characteristic different from containing all the primitive -th roots of unity, where are distinct prime numbers. We then provide a lower bound for in terms of the canonical dimension of some central division algebra A. This lower bound coincides with the upper bound thus giving the value of if a conjecture for holds. [Copyright &y& Elsevier]

Abstract: We prove a new upper bound on the essential p-dimension of the projective linear group . [Copyright &y& Elsevier]

Abstract: An improved upper bound for the essential p-dimension of is computed using algebraic tori. It follows that if . [ABSTRACT FROM AUTHOR]

A fundamental problem in any communication system is: given a communication channel between a transmitter and a receiver, how many "independent" signals can be exchanged between them? Arbitrary communication channels that can be described by linear compact channel operators mapping between normed spaces are examined in this paper. The (well-known) notions of degrees of freedom (DOF) at level ε and essential dimension of such channels are developed in this general setting. We argue that the DOF at level ε and the essential dimension fundamentally limit the number of independent signals that can be exchanged between the transmitter and the receiver. We also generalize the concept of singular values of compact operators to be applicable to compact operators defined on arbitrary normed spaces which do not necessarily carry a Hilbert space structure. We show how these generalized singular values, which we call DOF singular values, can be used to calculate the DOF at level ε and the essential dimension of compact operators that describe communication channels. We describe physically realistic channels that require such general channel models. [ABSTRACT FROM AUTHOR]

Abstract: In a previous paper the author determined all possible cohomological invariants of -torsors in Galois cohomology with mod 2 coefficients, where J is a split central simple Jordan algebra of odd degree . In the present paper we extend these results to a larger class of groups indexed by , 1, 2 or 3, and positive integers that add to such that is odd. We also give precise values for the essential dimension of these groups. [Copyright &y& Elsevier]

Abstract: Let G be a finite group and an equivariant polynomial map between finite dimensional G-modules. We say that φ is faithful if G acts faithfully on . The covariant dimension of G is the minimum of the dimension of taken over all faithful φ. In [Hanspeter Kraft, Gerald W. Schwarz, Compression of finite group actions and covariant dimension, J. Algebra 313 (1) (2007) 268–291] we investigated covariant dimension and were able to determine it in many cases. Our techniques largely depended upon finding homogeneous faithful covariants. After publication of the paper, the junior author of this article pointed out several gaps in our proofs. Fortunately, this inspired us to find better techniques, involving multihomogeneous covariants, which have enabled us to extend and complete the results, simplify the proofs and fill the gaps of our previous work. [Copyright &y& Elsevier]

Abstract: The present paper deals with algebraic tori and essential dimension in three unrelated contexts. After some preliminaries on essential dimension, versal torsors and tori, we explicitly construct a versal torsor for , odd, defined over a field of transcendence degree over the base field. This recovers a result of Lorenz, Reichstein, Rowen and Saltman. We also discuss the so-called “tori method” which gives a geometric proof of a result of Ledet on the essential dimension of a cyclic p-group. In the last section we compute the essential dimension of the functor , the latter set being in bijection with the isomorphism classes of n-dimensional K-tori. [Copyright &y& Elsevier]

Abstract: Let G be a finite group and an equivariant morphism of finite-dimensional G-modules. We say that φ is faithful if G acts faithfully on . The covariant dimension of G is the minimum of the dimension of taken over all faithful φ. In this paper we investigate covariant dimension and are able to determine it for abelian groups and to obtain estimates for the symmetric and alternating groups. We also classify groups of covariant dimension less than or equal to 2. A byproduct of our investigations is the existence of a purely transcendental field of definition of degree for a generic field extension of degree . [Copyright &y& Elsevier]

Abstract: We give necessary and sufficient conditions for a finite group to have essential dimension 1 over an infinite ground field. [Copyright &y& Elsevier]

Abstract: We define and study a numerical invariant of an algebraic group action which we call the canonical dimension. We then apply the resulting theory to the problem of computing the minimal number of parameters required to define a generic hypersurface of degree d in . [Copyright &y& Elsevier]