.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 $b\cdot \vec v$ 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.)

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    $\begingroup$ Maybe I'm missing something - what does enrichment over schemes have to do with the rest of the question? $\endgroup$ – S. Carnahan Sep 4 '10 at 4:09
  • $\begingroup$ Not all irreps Vec->Vec are Schur functors -- you could take the complex conjugate of your matrices, for example. $\endgroup$ – Allen Knutson Sep 4 '10 at 19:39

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