rt.representation theory - Analogy between canonical basis of U(n_-) and Schur functors, each under restriction

.1. For any category $mathcal C$, possibly enriched over schemes, define $Rep({mathcal C})$ to be the functor category ${mathcal C} to {bf Vec}$ with direct sum inherited from $bf Vec$. (If $mathcal C$ is enriched over schemes, like $bf Vec$ is since it's enriched over itself and vector spaces are schemes, then I probably only want functors that preserve this.) So it's easy to define irrep, etc. of $mathcal C$.

Fun fact: consider the irreps of $bf Vec$, called the Schur functors. If we restrict them to reps of the single-object category (i.e. monoid) $End({mathbb C}^n)$, some irreps restrict to $0$, and the ones that don't go to $0$ stay irreducible and give all the irreps of $End({mathbb C}^n)$ exactly once! In standard indexing, the Schur functors correspond to partitions, the partitions with more than $n$ rows restrict to $0$, and the partitions with $leq n$ rows give the irreps of $End({mathbb C}^n)$.

.2. I am told that one of the nice properties of Lusztig's canonical basis $B =$ { $b$ } of $U({mathfrak n}_-)$ is that on any irrep $V$ of $G$ with high weight vector $vec v$, the nonzero $bcdot vec v$ give a basis of $V$.

Is there a common framework for these two facts, perhaps involving categorifying the second one?

(I don't have anything riding on the answer... it's just a question I'd thought of a number of years ago and was reminded of by another mO question.)

• Next rt.representation theory - Global dimenson of quivers with relations
• Previous rt.representation theory - Embedding into Permutation Representation