### Seminar in Algebra, Lie Theory, and Geometry

Abstract: Let $$\mathfrak{g}_n$$'s be classical Lie algebras of the same type, where the rank of $$\mathfrak{g}_n$$ equals $$n$$. Let $$Q=\bigsqcup Q_n$$, where $$Q_n$$ is a subset of the set of simple finite-dimensional $$\mathfrak{g}_n$$-modules. We denote by $$Q_n\big|\mathfrak{g}_{n-1}$$ the set of all irreducible components of $$v|\mathfrak{g}_{n-1}$$ for every element $$v\in Q_n$$. We call $$Q$$ a precoherent local system (p.l.s. for short) if $$Q_n\big|\mathfrak{g}_{n-1}$$ is contained in $$Q_{n-1}$$. By definition, $$Q$$ is a coherent local system (c.l.s. for short), if $$Q_n|\mathfrak{g}_{n-1} = Q_{n-1}$$. We say that two p.l.s. $$Q$$ and $$R$$ are equivalent if there exists $$n$$ such that $$Q_k = R_k$$ for all $$k \geq n$$.