I am following a course on introduction to manifolds by myself and I got stuck by a notation I don't understand.
It defines the permutation group $S_n$, and then the signature of a permutation as the map, $$\begin{matrix}S_n&\longrightarrow&\{-1,1\}\\\sigma&\longrightarrow&(-1)^\sigma\end{matrix}$$ as far as I understand $\sigma$ is a bijection from $\{1,2,\cdots,n\}$ to itself, so what does $(-1)^\sigma$ mean?
Then we use this to define a exterior r-form as a r-form $A$ verifying, $$A\circ\sigma=(-1)^\sigma A\text{ }\forall \sigma \in S_n$$ they denote the set of exterior r-forms as $\Omega^r(M)$ but right after they name it $\Lambda^r(\mathfrak{X}(M))$, what does $\Lambda^r$ mean in this context?
It must be something like $(-1)^{\text{parity of the permutation}}$, where the parity of the permutation can be defined (there are several possible definitions) as the parity of the number of transpositions into which the permutation is decomposed.
One shows that the number of factors of any two such decompositions has the same parity.
Another possible definition: it is the parity of the number of pairs $(i,j)$ such that $i<j$ and $\sigma(i)>\sigma(j)$.
The signature can also be calculated from the decomposition of $\sigma$ as a product of disjoint cycles: indeed the signature is multiplicative and the signature of a cycle of length $n$ is $(-1)^{n-1}$.