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In finite-dimensional complex representations of finite groups, I would like to understand what I can learn by looking at a single matrix coefficient. In particular, I would like to look at "diagonal" coefficients of the form $\varphi_v(g)=\langle g\cdot v,v \rangle$ for a nonzero vector $v$.

If $\rho:G\rightarrow\text{GL}(V)$ is an irreducible (unitary) representation of the finite group $G$, then, for every nonzero vector $v\in V$, the subspace $\text{span}\{G\cdot v\}$ is $G$-invariant, and thus equals $V$. Thus, all matrix coefficients of $\rho$ are determined by the single matrix coefficient $\varphi_v(g)=\langle g\cdot v,v \rangle$.

So, in the irrducible case, a single diagonal matrix coefficient actually determines the entire representation.

What about the non-irreducible case? Let's say I have a representation $\rho:G\rightarrow \text{GL}(V)$ (not assumed to be irreducible), and I have a nonzero vector $v\in V$, which gives me a matrix coefficient $\varphi_v(g)=\langle \rho(g)v,v\rangle$, and this matrix coefficient happens to be equal exactly to a certain matrix coefficient of an irreducible representation of $G$?

Does it mean that this irreducible representation occurs as a subrepresentation of $\rho$?

And another related question: As I said, a diagonal matrix coefficient $\varphi_v(g)=\langle g\cdot v,v \rangle$ of an irreducible representation determines the representation. However, unlike characters, there are many diagonal matrix coefficients (they depend on the basis). It would be helpful to have some kind of characterization of the matrix coefficients of a certain irreducible (or not) representation, as functions. For example, I may like to solve the following problem:

If I have a finite group $G$, an irreducible representation $\rho:G\rightarrow\text{GL}(V)$, and a linear operator $A:V\rightarrow V$, how can I check if there is a diagonal matrix coefficient of $\rho$ which is an eigenvector of $A$?

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  • $\begingroup$ As for my second question, a clarification: There are infinitely many diagonal matrix coefficients, one for every choice of nonzero vector $v$, so I can't check them one by one to see if they are eigenvectors of $A$. So, at the very least, I would like a finite procedure to determine the answer. $\endgroup$ – Abed Mar 22 '14 at 19:27

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