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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