Shane Cernele and Zinovy Reichstein, with an appendix by Athena Nguyen: Essential dimension and error-correcting codes

shane.cernele@gmail.com, reichst@math.ubc.ca, athena@math.ubc.ca

Submission: 2015, Sep 10, Replacement of 535.)

One of the important open problems in the theory of central simple algebras is to compute the essential dimension of GL_n/ mu_m, 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= (GL_{n_1} x ... x GL_{n_r})/C, where C is a central subgroup. Equivalently, we are interested in the essential dimension of a generic r-tuple (A_1, \dots, A_r) of central simple algebras such that deg(A_i) = n_i and the Brauer classes of A_1, \dots, A_r 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 n_1, ... , n_r 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). Surprisingly, for many groups of the above form the essential dimension becomes easier to estimate when r > 2; 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 GL_{n_1} x ... x GL_{n_r})/C. This description is used throughout the paper.

2010 Mathematics Subject Classification: 20G15, 16K20, 16K50, 94B05

Keywords and Phrases: essential dimension, central simple algebra, error-correcting code.

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