Essential dimension and error-correcting codes

Abstract

One of the important open problems in the theory of central simple algebras is to compute the essential dimension of $\operatorname{GL}_n/\mu_m$, 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 \[ G=(\operatorname{GL}_{n_1} \times \dots \times \operatorname{GL}_{n_r})/C \, , \] where $C$ is a central subgroup of $\operatorname{GL}_{n_1} \times \dots \times \operatorname{GL}_{n_r}$. Equivalently, we are interested in the essential dimension of a generic $r$-tuple $(A_1, \dots, A_r)$ of central simple algebras such that $\operatorname{deg}(A_i) = n_i$ 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 $\operatorname{Code}(C)$ which we naturally associate to $C$. We focus on the case where $n_1, \dots, n_r$ are powers of the same prime. The upper and lower bounds on $\operatorname{ed}(G)$ we obtain are expressed in terms of coding-theoretic parameters of $\operatorname{Code}(C)$, such as its weight distribution. Surprisingly, for many groups of the above form the essential dimension becomes easier to estimate when $r \geq 3$; 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 $(\operatorname{GL}_{n_1} \times \dots \times \operatorname{GL}_{n_r})/C$. This description and its corollaries are used throughout the paper.