Wilson's book "The finite simple groups" gives (in section 2.7) a description of the double cover of the alternating groups. First, one constructs a double cover $2S_n$ of the symmetric group $S_n$. By definition, a permutation $\pi$ has two lifts to $2S_n$, denoted by $+\pi$ and $-\pi$. Elements of $2S_n$ will be denoted by bracket notation as opposed to parentheses, i.e. $\pm[a_1,\ldots,a_k]$ are the lifts of the $k$-cycle $(a_1,\ldots,a_k)\in S_n$. He then defines multiplication in $2S_n$ by:
- $[i,j]^{\pm\pi} = -[i^\pi,j^\pi]$, where $\pi$ is an odd permutation
- A $k$-cycle can be written as $[a_1,\ldots,a_k] = [a_1,a_2][a_1,a_3],\cdots[a_1,a_k]$
- A general element of $2S_n$ is a product of disjoint cycles
Then he says that you must be careful not to permute the cycles, or start a cycle at a different point.
His description is a bit confusing to me, mostly because he doesn't completely specify how to put elements of $2S_n$ into a ``canonical form''. I want to be certain that I'm understanding it correctly. Here are some questions:
(a) How do minus signs work when you multiply? For example, is $(-\pi)(-\sigma) = \pi\sigma$? (I would guess this is the case, as negation should presumably be viewed as multiplication by the nontrivial central element).
(b) What exactly is the difference between $[a_1,\ldots,a_k]$ and $[a_{k-i},\ldots,a_k,a_1,\ldots,a_{i-1}]$? (When are they the same?)
(c) Do disjoint cycles still commute in $2S_n$? (i.e., is $[1,2][3,4] = [3,4][1,2]$)? (In example 2.11, he shows that $[1,2][3,4] = -[3,4][1,2]$, but this doesn't answer the question)
(d) Are there other references for a similar "concrete" description of $2S_n$ that is more precise?
