I am reading Marc Levine's paper 'Motivic Euler characteristics and Witt-valued characteristic classes'. In that paper he considers the $BN_T(SL_n)$, namely the classifying space of the group of matrices generated by permutation matrices and torus (det=1).
Every oriented vector bundle of rank n on $X$ corresponds to a map $X\to BSL_n$. I wonder for which vector bundles do this map factor through $BN_T(SL_n)$?