Essential dimension of double covers of symmetric and alternating groups

Abstract: I. Schur studied double covers

w i d e t i l d e {S y m}_{n}^{p m}

and

w i d e t i l d e {A l t}_{n}

of symmetric groups

S y m_{n}

and alternating groups

A l t_{n}

, respectively. Representations of these groups are closely related to projective representations of

S y m_{n}

and

A l t_{n}

; there is also a close relationship between these groups and spinor groups. We study the essential dimension

e d (w i d e t i l d e {S y m}_{n}^{p m})

and

e d (w i d e t i l d e {A l t}_{n})

. We show that over a base field of characteristic

e q 2

,

e d (w i d e t i l d e {S y m}_{n}^{p m})

and

e d (w i d e t i l d e {A l t}_{n})

grow exponentially with

n

, similar to

e d (S p i n_{n})

. On the other case, in characteristic

2

, they grow sublinearly, similar to

e d (S y m_{n})

and

e d (A l t_{n})

. We give an application of our result in good characteristic to the theory of trace forms.

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