I'm working on the following exercise (Exercise 24.12) in Humphrey's book Introduction to Lie algebras and representation theory.
My question is how to solve the following exercise:
Exercise: Deduce from Steinberg's formula that the only possible $\lambda \in \Lambda^{+}$ for which $V(\lambda)$ can occur as a summand of $V(\lambda^{\prime}) \otimes V(\lambda^{\prime\prime})$ are those of the form $\mu + \lambda^{\prime\prime}$, where $\mu \in \Pi(\lambda^{\prime})$. .....
Notation issue: Here we are talking about representations of a semisimple Lie algebra $L$ with fixed maximal toral subalgebra $H$ (or say CSA $H$) and corresponding root system $\Phi$ with a fixed base (or say positive system) $\Delta$. $\mathcal{W}$ is the Weyl group of $(\Phi, \Delta)$. $\lambda$ is a dominant integral weight in $H^{\ast}$, $V(\lambda)$ is the irreducible highest weight representation of highest weight $\lambda$ (i.e. the irreducible quotient of the Verma module $Z(\lambda)$). Here $\Pi(\lambda) = \Pi(V(\lambda))$ is the set of all weights of $V(\lambda)$
Steinberg's formula: Let $\lambda^{\prime}, \lambda^{\prime\prime} \in \Lambda^{+}$. Then the number of times $V(\lambda)$, $\lambda \in \Lambda^{+}$, occurs in $V(\lambda^{\prime}) \otimes V(\lambda^{\prime\prime})$ is given by the formula $$ \sum_{\sigma, \tau \in \mathcal{W}} \mathrm{sn}(\sigma \tau) p(\lambda + 2\delta - \sigma(\lambda^{\prime} + \delta)-\tau(\lambda^{\prime\prime} + \delta)). $$ where $p(\lambda)$ is the number of of sets of nonnegative integers $\{k_{\alpha}, \alpha > 0 \}$ for which $-\lambda = \sum_{\alpha > 0} k_{\alpha} \alpha$ and $\mathrm{sn}(\sigma) := (-1)^{\ell(\sigma)}$, here $\ell$ is the length function on the Weyl group $\mathcal{W}$.
My attempts: As usual and as what I'm supposed to do, I shall write my attempts, but I just repeat the definition again and again but failed. This exercise carried on asking to decompose $V(1,3) \otimes V(4,4)$ in $\mathsf{A}_2$ with a previous exercise and I managed to do this. But I still cannot prove the part of this exercise quoted above. :(
Thank you all for commenting and answering! :)