Presentation for the double cover of A_n

Question

The wikipedia page Covering groups of the alternating and symmetric groups gives explicit presentations for the double covers of the symmetric group S_n (n ≥ 4). Can someone provide a similar presentation, or better yet an explicit combinatorial description, of the double cover of the alternating group A_n? (I really only care about the case of large n, in case it matters.)

Answer 1

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 $[a_1 a_2 \cdots a_k]^k = -1$.