One representation of "braid groups" is the Burau representation first propounded by Werner Burau in the 1930s. Later work has shown that this representation is "unfaithful" for n>=5, where n is the number of braids. That makes sense to me (a layman), insofar as the greater number of items, the greater the complexity, and the less "tractable" various formulations become.
But another representation of braid groups produced seemingly opposite results. In papers first published by Edward Formanek, and expanded by Inna Sysoeva, it was possible for Formanek to classify "irreducible complex representations B of Artin braid groups" to a dimension n-1. More to the point, Sysoeva showed that it was possible to classify these representations to a higher dimension, n, for n>=9. Here, the larger the n, the more manageable the braid groups are, and the more "tractable" the problem is.
I find these two results "paradoxical" to say the least. Is it, in fact the case, that while both results are true, one or both are counterintuitive? Or am I confusing seemingly similar concepts when in fact there is no relationship between them?
Or is it true that "nice properties" move in opposite directions in these two cases? To take a very simple example, 1< 1000 in the integer realm, but when you invert them to 1 and 1/1000, the first is greater than the second? (That is, you want to move toward more digits in the integer case, and toward fewer digits in the fractional case, if you want to maximize the number.)
