Rank matrices for type $D$ Bruhat order

The answer to my questions is that I was wrong. I'll switch to the more standard notation of having $D_n$ act on $\{ -n, -n+1, \ldots, -2, -1, 1, 2, \ldots, n-1, n \}$ in order to match the reference of Proctor.

Here is the simplest example to demonstrate my error. Take $n \geq 4$ and look at $D_n \cdot [2]$. Then my belief above was that we should have $(2,1) > (1,-2)$, since $(2,1) > (1,-2)$ and $(2,-1) > (-1,-2)$ are both valid inequalities in type $B$. But this is wrong; any type $D$ transposition which lowers $(2,1)$ takes it below $(1,-2)$. In terms of weights, we are taking about $e_2+e_1$ and $-e_2+e_1$. They differ by $2 e_2 = (e_2+e_1) + (e_2-e_1)$, which is a sum of positive roots. However, subtracting either $e_2+e_1$ or $e_2-e_1$ from $e_2+e_1$ gives $0$ and $2 e_1$ respectively, neither of which are in the $D_n$ orbit of $e_2+e_1$.

The paper of Proctor which Sam Hopkins pointed me to gives the correct criterion in Theorem 5D: Let $I=\{ i_1 < i_2 < \cdots < i_k \}$ and $J=\{ j_1 < j_2 < \cdots < j_k \}$ be two $k$-element subsets of $\{ \pm 1, \pm 2, \ldots, \pm n \}$, both of which have the property of not containing both $a$ and $-a$. Then $I \leq J$ if and only if two conditions hold:

• $i_r \leq j_r$ and

• If there is a set of indices $p, p+1, \ldots, q$ such that $\{ |i_p|, |i_{p+1}|, \ldots, |i_q| \}$ and $\{ |j_p|, |j_{p+1}|, \ldots, |j_q| \}$ are each permutations of $\{ 1,2,\ldots, q-p+1 \}$, then the number of negative elements among $(i_p, i_{p+1}, \ldots, i_q)$ and $(j_p, j_{p+1}, \ldots, j_q)$ must have the same parity.

At the moment, I don't see how to make this second condition sound like a rank matrix condition.

Here are some more thoughts about how the Type D Bruhat order is more complicated than the Type A and Type B/C orders. These ideas might even suggest that giving a "rank matrix"-like description of the partial order is "impossible" in Type D.

There is a certain property of posets called "clivage" (by Lascoux and Schützenberger) or "dissective" (by Reading). Lascoux and Schützenberger (cited below) showed that the Type A and Type B Bruhat orders are dissective. Meanwhile, Geck and Kim (cited below) showed that Type D Bruhat order is not dissective (they address the exceptional types as well). Also, it is known that a finite poset $P$ is dissective if and only if the MacNeille completion of $P$ is distributive (see Theorem 7 of the paper of Reading cited below).

(As an aside, the MacNeille completion of the Bruhat order of Type A is the distributive lattice of Monotone Triangles (a.k.a. Alternating Sign Matrices) with componentwise order. See the paper of Brualdi and Schroeder cited below for more on this lattice.)

So, Type D Bruhat order lacks a nice property ("dissective") which Type A and B have. But what does this have to do with the possibility of a "rank matrix"-like description of the partial order in Type D? Well, intuitively at least, if you have a partial order defined by componentwise order on some arrays of numbers satisfying certain inequalities, then to formally extend your partial order to be a lattice, what you'll end up doing is taking $\min$'s and $\max$'s of the entries until you get enough new arrays so that everyone has a meet and join. And if so, the result will be distributive because $\min$'s and $\max$'s distribute over one another. But we know Type D Bruhat order does not have a distributive lattice as its completion, so it "can't" have a partial order given by comparing arrays of numbers in this way.

Brualdi, Richard A.; Schroeder, Michael W., Alternating sign matrices and their Bruhat order, Discrete Math. 340, No. 8, 1996-2019 (2017). ZBL1366.15024.

Geck, Meinolf; Kim, Sungsoon, Bases for the Bruhat-Chevalley order on all finite Coxeter groups, J. Algebra 197, No. 1, 278-310 (1997). ZBL0977.20033.

Lascoux, Alain; Schützenberger, Marcel-Paul, Lattices and bases of Coxeter groups, Electron. J. Comb. 3, No. 2, Research paper R27, 35 p. (1996); printed version J. Comb. 3, No. 2, 633-667 (1996). ZBL0885.05111..

Reading, Nathan, Order dimension, strong Bruhat order and lattice properties for posets, Order 19, No. 1, 73-100 (2002). ZBL1007.05097.