to 1 (the trivial map) or to −1 (the sign map). One must also show that the sign map is well-defined, but assuming that, this gives the first homology of Sn. The second homology (concretely, the Schur multiplier) is: This was computed in , and corresponds to the double cover of the symmetric group, 2 · Sn. Note that the exceptional low-dimensional homology of the alternating group ( corresponding to non-trivial abelianization, and due to the exceptional 3-fold cover) does not change the homology of the symmetric group; the alternating group phenomena do yield symmetric group phenomena – the map extends to and the triple covers of A6 and A7 extend to triple covers of S6 and S7 – but these are not homological – the map does not change the abelianization of S4, and the triple covers do not correspond to homology either. The homology "stabilizes" in the sense of stable homotopy theory: there is an inclusion map , and for fixed k, the induced map on homology is an isomorphism for sufficiently high n. This is analogous to the homology of families Lie groups stabilizing. The homology of the infinite symmetric group is computed in , with the cohomology algebra forming a Hopf algebra. Representation theory The representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to problems of quantum mechanics for a number of identical particles. The symmetric group Sn has order n!. Its conjugacy classes are labeled by partitions of n. Therefore, according to the representation theory of a finite group, the number of inequivalent irreducible representations, over the complex numbers, is equal to the number of partitions of n. Unlike the general situation for finite groups, there is in fact a natural way to parametrize irreducible representation by the same set that parametrizes conjugacy classes, namely by partitions of n or equivalently Young diagrams of size n. Each such irreducible representation can be realized over the integers (every permutation acting by a matrix with integer coefficients); it can be explicitly constructed by computing the Young symmetrizers acting on a space generated by the Young tableaux of shape given by the Young diagram. Over other fields the situation can become much more complicated. If the field