Tensor-product multiplicities for finite-dimensional simple modules

Assume the Axiom of Choice. Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$, positive system ${\mathrm{\Phi }}^{+}$ with base ${\alpha }_1,\dots ,{\alpha }_r$, positive cone $Q_{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, Weyl vector $\rho$, weight lattice $P$ and set of dominant integral weights ${\mathrm{\Lambda }}^{+}=\{\lambda \in P:⟨\lambda ,{\alpha }_i^{\vee }⟩\ge 0\text{ for all }i\}$ (Finite Weyl root system, lattice and chamber conventions, Integral, dominant, and strictly dominant weights).

For $\lambda \in {\mathrm{\Lambda }}^{+}$ let $L(\lambda )$ denote the finite-dimensional simple $\mathfrak{g}$-module of highest weight $\lambda$ (Highest-weight classification), and let $L(\lambda )\otimes L(\mu )$ carry the tensor-product action $x\cdot (v\otimes w)=xv\otimes w+v\otimes xw$ (Direct-sum, dual, Hom, and tensor representations). By Weyl's complete reducibility theorem (Weyl's complete reducibility theorem) and the classification of finite-dimensional simple modules there is a decomposition $L(\lambda )\otimes L(\mu )\cong {⨁}_{\nu \in {\mathrm{\Lambda }}^{+}}L(\nu {)}^{\oplus c_{\lambda \mu }^{\nu }}$ with uniquely determined integers $c_{\lambda \mu }^{\nu }\ge 0$. The direct sum is finite because $L(\lambda )\otimes L(\mu )$ is finite-dimensional, so its completely reducible decomposition has only finitely many nonzero simple summands. Each constituent highest weight is a weight and lies below $\lambda +\mu$ in the partial order of weights: every tensor-product weight is a sum of a weight of $L(\lambda )$, which lies in $\lambda -Q_{+}$, and a weight of $L(\mu )$, which lies in $\mu -Q_{+}$ (Highest weight modules lie below the top weight).

The integers $c_{\lambda \mu }^{\nu }=[L(\lambda )\otimes L(\mu ):L(\nu )]$ are the tensor-product multiplicities of $\mathfrak{g}$. More generally, for a finite-dimensional completely reducible $\mathfrak{g}$-module $V$ we write $[V:L(\nu )]$ for the number of summands isomorphic to $L(\nu )$ in any decomposition of $V$ into simple modules; this number does not depend on the chosen decomposition. By Schur's lemma (Schur’s lemma for irreducible Lie-algebra representations) the multiplicity has the equivalent hom-space description $c_{\lambda \mu }^{\nu }=\dim {\mathrm{Hom}}_{\mathfrak{g}}(L(\nu ),L(\lambda )\otimes L(\mu ))\mathrm{.}$