2014/06/11 by Shane Cernele, Zinovy Reichstein, Cernele, Shane +3 · 2 citations
Computer Science · Mathematics · #16K20 #16K50 #20G15 #Advanced Algebra and Geometry #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1406.2953
openalex publication_date 2014/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
One of the important open problems in the theory of central simple algebras is to compute the essential dimension of GLn/μm, i.e., the essential dimension of a generic division algebra of degree n and exponent dividing m. In this paper we study the essential dimension of groups of the form G=(GLn1 × … × GLnr)/C , where C is a central subgroup of GLn1 × … × GLnr. Equivalently, we are interested in the essential dimension of a generic r-tuple (A1, …, Ar) of central simple algebras such that deg(Ai) = ni and the Brauer classes of A1, …, Ar satisfy a system of homogeneous linear equations in the Brauer group. The equations depend on the choice of C via the error-correcting code Code(C) which we naturally associate to C. We focus on the case where n1, …, nr are powers of the same prime. The upper and lower bounds on ed(G) we obtain are expressed in terms of coding-theoretic parameters of Code(C), such as its weight distribution. Surprisingly, for many groups of the above form the essential dimension becomes easier to estimate when r ≥ 3; in some cases we even compute the exact value. The Appendix by Athena Nguyen contains an explicit description of the Galois cohomology of groups of the form (GLn1 × … × GLnr)/C. This description and its corollaries are used throughout the paper.