gr.group theory - Presentation for the double cover of A_n

Yeah, Schur did this a long time ago. Let $tilde Sigma_n to Sigma_n$ be a double cover (there are two) -- lets denote them $tilde Sigma_n = Sigma_n^epsilon$ where $epsilon in {+1, -1}$.

Schur uses the notation $[a_1 a_2 cdots a_k]$ for a specific lift of the cycle $(a_1 a_2 cdots a_k) in Sigma_n$ to $Sigma_n^epsilon$ -- might as well call these $k$-cycles. Then his presentation goes like this:

$$[a_1 a_2 cdots a_k] = [a_1 a_2 cdots a_i][a_i a_{i+1} cdots a_k] forall 1 < i< k$$

and all $k$-cycles, $k>1$.

$$[a_1 a_2 cdots a_k]^{[b_1 b_2 cdots b_j]} = (-1)^{j-1}[phi(a_1) phi(a_2) cdots phi(a_k)]$$

where $phi$ is the cycle $(b_1 b_2 cdots b_j)$

$$[a_1 a_2 cdots a_k]^k = epsilon$$

for all $k$-cycles -- ie this is always $+1$ or $-1$ depending on which extension of $Sigma_n$ you're interested in. And:

$$[a_1 a_2 cdots a_k][b_1 b_2 cdots b_j] = (-1)^{(k-1)(j-1)}[b_1b_2 cdots b_j][a_1 a_2 cdots a_k]$$

provided the cycles $(a_1 a_2 cdots a_k)$ and $(b_1 b_2 cdots b_j)$ are disjoint.

The map $tilde Sigma_n to Sigma_n$ sends $[a_1 cdots a_k]$ to $(a_1 cdots a_k)$. So this gives you a corresponding presentation of the double of $A_n$ -- take your favourite presentation of $A_n$, lift the relators and see what happens using the above relations.

A small extra tidbit -- think of $Sigma_n$ as being the group of orientation-preserving isometries of $mathbb R^{n}$ that preserves a regular (n-1)-simplex. Then if you lift this group to $Spin(n)$, the extension you want is the one where $[a_1 a_2 cdots a_k]^k = -1$.

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