so the b_n root poset...

(1,1,0,0)
=== (1,0,1,0)
(0,1,1,0) == (1,0,0,1)
=== (0,1,0,1) == (1,0,0,0)
(0,0,1,1) == (0,1,0,0) == (1,0,0,-1)
=== (0,0,1,0) == (0,1,0,-1) == (1,0,-1,0)
(0,0,0,1) == (0,0,1,-1) == (0,1,-1,0) == (1,-1,0,0)

is there a nice way to identify this with some sort of "triangular self-adjoint matrixes" (or something...)? ... imitating the a_n case, as baez suggested...

??hmm, so suppose that we compare the "matrix picture" of an a-series bruhat cell to the "matrix picture" of its zariski tangent space ... ???or something...