Finding the cohomology class of an irreducible representation of a normal subgroup - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T16:30:04Z http://mathoverflow.net/feeds/question/75941 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75941/finding-the-cohomology-class-of-an-irreducible-representation-of-a-normal-subgrou Finding the cohomology class of an irreducible representation of a normal subgroup Patrick Neumann 2011-09-20T12:53:42Z 2011-09-20T12:53:42Z <p>Dear Mathoverflow,</p> <p>I have a question concerning finite dimensional irrducible representations of normal subgroups and how to construct representations of the supergroup out of these representations with a computer (using GAP). The following text describes a construction, the question itself is on the bottom. Let $G$ be a finite group and let $N \triangleleft G$ be a normal subgroup (which is necessarily a finite index subgroup). Now let $\vartheta: N\to \mathbb{C}$ be an irreducible character. My goal is to construct all irreducible characters $\pi$ of $G$ whose restriction to $N$ is equal to a multiple of $\vartheta$. My references for this is the article "Representations Induced in an Invariant Subgroup" written by Alfred Clifford in 1937 and the book "Character theory of finite groups" by Martin Isaacs.</p> <p>The book of Isaacs (chapter 11) gives an explicit way for this construction: First we find the inertia subgroup of $\vartheta$ relative to $G$, that is the subgroup of $G$ defined by $\langle g\in G| \vartheta(h^g) = \vartheta(h)\ \forall h\in N\rangle$. Then we take an irreducible representation $\rho$ of $N$ affording the character $\vartheta$ and choose a projective representation $P$ such that the restriction of $P$ to $N$ is equal to $\rho$. This projective representation can be contructed for example by taking a coset transversal $T$ of $N$ in $G$ and then take for every $t\in T$ a matrix $P_t\in GL_n(\mathbb{C})$ such that</p> <p>$P_t^{-1}\rho(h)P_t = \rho(t^{-1}ht)\ \forall h\in N$</p> <p>and define the projective representation $P$ to be $P(th) = P_t\rho(h)$. Since $P$ is a projective representation, there is a well-defined 2-cocycle $\alpha\in Z^2(G/N, \mathbb{C}^* )$ defined via $\alpha(gN,hN) := P(g)P(h)P(gh)^{-1}$. This 2-cocycle has a cohomology class in the Schur-Multiplier</p> <p>$H^2(G/N, \mathbb{C}^* ) = Z^2(G/N, \mathbb{C}^* )/ B^2(G/N, \mathbb{C}^* )$</p> <p>and we take a projective representation $\beta$ of $G/N$ with a cocycle cohomologous to the inverse to $\alpha$, take the tensor product $P\otimes \beta$ and then take the induced representation of $P\otimes \beta$ to get an irreducible representation of $G$. </p> <p>Each of the steps above is realisable by a computer, I'm pretty sure. There is just one step I don't know how to do it. If we have a cocycle $\alpha$ which is a function from $(G/N)^2\to \mathbb{C}^*$ then we want to know which cohomology class it actually has in $H^2(G/N, \mathbb{C}^* )$. Since $H^2(G/N, \mathbb{C}^* )$ is abelian and finite in this case, there is an isomorphism between $H^2(G/N, \mathbb{C}^* )$ and its irreducible characters. So we can assign an irreducible character of $H^2(G/N, \mathbb{C}^* )$ to every 2-cocycle in $Z^2(G/N, \mathbb{C}^* )$. </p> <p>My question is: How can I assign the irreducible character to a 2-cocycle? I know that this character exists and I know how to construct the 2-cocycle, but I don't know how to assign a character to this 2-cocycle with an algorithm.</p> <p>Another question might be: Is there an easier way to find the cohomology class corresponding to $\vartheta$?</p> <p>A third way would be to have the 2-cocycle $\alpha$ and for every cohomlogy class we have a representative $\gamma$. Then we can check whether $\alpha\gamma^{-1} \in B^2(G/N,\mathbb{C}^* )$ or not. There I also don't know how to do this.</p> <p>Maybe someone here can help me. Thank You very much!</p>