so i think that the partial order on the b_n roots that gives the invariant distributions on the flag variety as its down-sets is as follows:


(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)

this is for the case b_4, but the pattern continues.

(the shape of the poset is suggestive, but i'm not sure what it means yet. i'm trying to connect it with vague memories of some stuff that arnold mentioned about bratteli diagrams and young diagrams and flag varieties and bruhat classes or stuff like that... but it definitely reminds me of certain bratteli diagrams and the representation theory of su(2) and so forth...)

let's try counting the down-sets as a function of n...

2,6,20,...

is this "2n choose n"?

is it obvious why it should be that?

let's try identifying the integrable invariant distributions here...
especially the maximal integrable ones...

2 ???

4,4 ???

8,11,6 ???

16,26,22,8 ???

well, i fed "4,11,26" into sloane's encyclopedia and it led me to the "triangle of eulerian numbers":

1 1 1 1 1 1 1 1 1
1 4 11 26 57 120 247 502
1 11 66 302 1191 4293 14608
1 26 302 2416 15619 88234
1 57 1191 15619 156190
1 120 4293 88234
1 247 14608
1 502
1

with the following comment:

T(n,k) = number of ways to write the Coxeter element s_{e1}s_{e1-e2}s_{e2-e3}s_{e3-e4}...s_{e_{n-1}-e_n} of the reflection group of type B_n, using s_{e_k} and as few reflections of the form s_{e_i+e_j}, where i = 1, 2, ..., n and j is not equal to i, as possible. - Pramook Khungurn (pramook(AT)mit.edu), Jul 07 2004

and perhaps some other potentially relevant and interesting comments.

on the other hand i don't quite completely get the relationship between my triangle above and theirs yet...

maybe try the next row of mine?

32,57,61,37,10 ???

1 + 1 + 2 + 4 + 8 + 16 = 32

1 + 1 + 2 + 4 + 8 + 15 + 26 = 57

1 + 1 + 2 + 4 + 7 + 11 + 15 + 20 = 61

1 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 37

1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 10


well, so anyway, what's going on here with the zariski tangent spaces of the b2 schubert varieties??

4=[0,1,2,3] 4=[0,1,2,3]
8=[0,1,2,3,3,4,5,6] 11=[0,1,2,2,3,3,4,4,5,6,7] 6=[0,1,2,3,4,5]
16 26 22 8
32 57 61 37 10

8 vertexes of cube.... "distances" from a flag...

...i'm getting a match-up in this case... the 8 distances here are [0,1,2,3,3,4,5,6]. matching the 8 dimensions of the invariant distributions extending the maximal integrable one...

??but that's not convincing me yet that the correspondence actually goes like that... because of the business about the singular 2d b2 schubert variety...

but now let's consider the 12 edges of the cube... the 12 distances that i'm getting at the moment are [0,1,2,2,3,3,4,4,5,5,6,7]... so how does that compare to the 11 dimensions?? hmm, just as i guessed: naively it looks like the 2 5's merged into just one... ??...