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Journal of the Ramanujan Mathematical Society

Volume 35, Issue 4, December 2020  pp. 357–372.

Essential dimension of double covers of symmetric and alternating groups

Authors:  Zinovy Reichstein and Abhishek Kumar Shukla
Author institution:Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

Summary:  I. Schur studied double covers S{±}{n} and A{n} of symmetric groups S{n} and alternating groups A{n}, respectively. Representations of these groups are closely related to projective representations of S{n} and S{n}; there is also a close relationship between these groups and spinor groups. We study the essential dimension ed(S{±}{n}) and ed(A{n}). We show that over a base field of characteristic ≠ 2, ed(S{±}{n}) and ed(A{n}) grow exponentially with n, similar to ed(Spin{n}). On the other hand, in characteristic 2, they grow sublinearly, similar to ed(S{n}) and ed(A{n}). We give an application of our main result to the theory of trace forms.

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