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My motivation was some research I'm doing on relatives of $U_q(\mathfrak{g}_2)$, where I needed to understand all 1-dimensional representations of B_3/N_3B_3/T_3. This was easy enough to work out directly, but it made me realize I'd really like to know about all small representations of B_n/N_n B_n/T_n for small n. Essentially this is because "low weight spaces" in ribbon planar algebras (i.e. ribbon categories with a fixed favorite object) always have the structure of a representation of B_n/N_n B_n/T_n (where you're looking at low-weight vectors in the n-box space, i.e. invariants of the nth tensor power of your fixed object).
My motivation was some research I'm doing on relatives of $U_q(\mathfrak{g}_2)$, where I needed to understand all 1-dimensional representations of B_3/N_3. This was easy enough to work out directly, but it made me realize I'd really like to know about all small representations of B_n/N_n for small n. Essentially this is because "low weight spaces" in ribbon planar algebras (i.e. ribbon categories with a fixed favorite object) always have the structure of a representation of B_n/N_n (where you're looking at low-weight vectors in the n-box space, i.e. invariants of the nth tensor power of your fixed object).