Kronecker product of two irreducible representations

Question

Suppose, I have two irreps $D^\mu(R)$ with dimension $n^\mu$ and $D^\nu(R)$ with dimension $n^\nu$ such that $n^\mu>n^\nu$ For $\forall R \in G$.Direct product of these two irreps will be reducible which means that it can be reduced to its irreducible bits. Suppose, One of those irreducible representation is $D^\sigma$ with dimension $n^\sigma$. My question- Is it possible to have an irrep with $n^\sigma<n^\mu/n^\nu$

Answer 1

Using Schur's Orthogonality, we have that for finite groups the number of times a representation $D^{\bar{\sigma}}$ is contained in $D^{\mu}\otimes D^{\nu}$ is the same under interchange of $\sigma,\mu$ and $\nu $ (Here, $D^{\bar{\sigma}}$ denotes the adjoint representation of $D^{\sigma}$).

Now, suppose the representation $\lambda$ is in $D^{\mu}\otimes D^{\nu}$. Since $D^{\lambda}\sim D^{\bar{\sigma}}$ for some $\sigma$, we have $D^{\bar{\sigma}}$ is in $D^{\mu}\otimes D^{\nu} \Rightarrow D^{\mu}$ is in $D^{\bar{\sigma}}\otimes D^{\nu} $. Counting dimensions, we get $n_{\mu}\le n_{\bar{\sigma}}n_{\nu}$. Since $\bar{\sigma}\sim \lambda$, we get $n_{\lambda}\ge \frac{n_{\mu}}{n_{\nu}}$ which is what we needed to show.