# Schur Multipliers

I'd like to know H^2(G,C^*) for a few (finite) groups I have in mind. For example, a finite abelian group. Is this information in a table somewhere? If not, could someone point me to a basic introduction with examples?

Also, given an exact sequence 1 -> H -> G -> K -> 1, I'd like to use the long exact sequence in cohomology to figure out H^2(G,C^*) provided I know it for H and K. Thus it would be nice to have vanishing theorems for H^1 and H^3. Any references for this would also be very helpful.

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## 2 Answers

For finite abelian groups, see Section V.6 in Brown's book on group cohomology. For the finite simple groups, the wikipedia page has pretty good information.

As far as a general reference, aside from general books on group cohomology (like Brown's book above, which along with Adem-Milgram's book is my favorite reference) I have found Karpilovsky's book "The Schur multiplier" useful.

I don't know what you mean by the "long exact sequence" in cohomology. What does exist is the Hochschild-Serre spectral sequence. The hardest part of analyzing this is computing the differentials. For computing $H^2$, I have found Huebschmann's papers "Automorphisms of group extensions and differentials in the Lyndon-Hochschild-Serre spectral sequence" and "Group extensions, crossed pairs and an eight term exact sequence" helpful.

There is a five-term exact sequence in group cohomology arising from a short exact sequence. However, this is only useful for computing $H^2$ of the cokernel from information about $H^2$ of the central term and $H^1$ of the kernel. See Brown's book for a discussion of this.

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The group $H^2(G,\mathbb{C}^\times)$ plays a rôle in orbifold conformal field theory and is usually known as the discrete torsion group. In fact, in this context one actually needs the explicit cocycle and for the case of a finite simple abelian group it is very easy to compute explicitly.

Let $\varepsilon: G \times G \to \mathbb{C}^\times$ be the cocycle. Without loss of generality one can normalise it so that $$\varepsilon(0,g)=\varepsilon(g,0) = 1$$ for all $g \in G$. With this normalisation the cocycle conditions become, in addition, the following: $$\varepsilon(g,g)=1 \quad \varepsilon(g,g')= \varepsilon(g',g)^{-1}$$ and $$\varepsilon(g_1+g_2,g) = \varepsilon(g_1,g)\varepsilon(g_2,g)$$ from where it follows that if $G$ has order $N$, then for all $g,g' \in G$, $$\varepsilon(g,g')^N = 1$$

Let $G = \mathbb{Z}/N_1 \times \cdots \times \mathbb{Z}/N_k$ be a finite simple abelian group and let $\alpha_i$ be a generator of $\mathbb{Z}/N_i$, so that we can write any element of $G$ as a sum $\sum_i n_i \alpha_i$ where $n_i = 0,1,\ldots,N_i-1$.

Then one finds that all cocycles are given in terms of $B_{ij} = -B_{ji}$ taking the possible values $0,1,\ldots,\mathrm{gcd}(N_i,N_j)-1$, by the formula $$\varepsilon(\sum_i n_i\alpha_i,\sum_j m_j\alpha_j) = \exp 2\pi\sqrt{-1}\sum_{i,j} \frac{B_{ij} n_im_j}{\mathrm{gcd}(N_i,N_j)}$$

It is the bilinear $B_{ij}/\mathrm{gcd}(N_i,N_j)$ which is called the discrete torsion. It should be emphasised that torsion here is by analogy with the torsion of a connection in differential geometry and not with torsion as in group theory.

If you google "discrete torsion" and "orbifold" you might find suitable references, just like this paper of Vafa and Witten.

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I think by "finite simple group" you really mean "finite abelian group"... – Andy Putman Jan 16 '10 at 23:26
Oops! That was a typo, thanks! – José Figueroa-O'Farrill Jan 17 '10 at 0:13