The wikipedia page [Covering groups of the alternating and symmetric groups](http://en.wikipedia.org/wiki/Covering_groups_of_the_alternating_and_symmetric_groups) gives explicit presentations for the double covers of the symmetric group Sn (n ≥ 4). Can someone provide a similar presentation, or better yet an explicit combinatorial description, of the double cover of the alternating group An? (I really only care about the case of large n, in case it matters.)



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 $[a\_1 a\_2 \cdots a\_k]^k = -1$. 


