Roughly, this question asks how the Bruhat (strong) order in type $D$ can be understood like the Bruhat orders in types A and B=C. I'll review how types A and B work before asking my question. As a side note, I tried to ask this question earlier today, then deleted it with the intention of fixing some errors and reasking it, but I can't find the deleted question.

**Notation:** I write $[n]$ for $\{1,2,\ldots, n \}$.

**Type A** The Bruhat order of type $A_{n-1}$ is a partial order on the group of permutation of $[n]$. It can be described in the two following manners:

("Tableaux criterion") Put a partial order on the set of $k$ element subsets of $[n]$ as follows: For $I$, $J$ two such subsets, sort $I = \{ i_1 < i_2 < \cdots < i_k \}$ and $J = \{ j_1 < j_2 < \cdots < j_k \}$, and define $I \leq J$ if $i_1 \leq j_1$, $i_2 \leq j_2$, ..., $i_k \leq j_k$. Then, for permutations $u$ and $v$, we have $u \leq v$ if and only if $u[k] \leq v[k]$ for $1 \leq k \leq n-1$.

("Rank matrix criterion") We have $u \leq v$ if and only if, for all $1 \leq k, \ell \leq n-1$, we have $\#(u[k] \cap [\ell]) \leq \#(v[k] \cap [\ell])$.

The numbers $r_{k \ell} = \#(u[k] \cap [\ell])$ are called the rank matrix of $u$. It is often useful to formally define $r_{0k}=r_{k0} = 0$ and $r_{kn}=r_{nk}=k$. With those boundary definitions, rank matrices are characterized by $0 \leq r_{(k+1)\ell} - r_{k \ell},\ r_{k(\ell+1)} - r_{k \ell} \leq 1$ and, if $r_{(k+1)\ell} = r_{k(\ell+1)} = r_{k\ell}+1$ then $r_{(k+1)(\ell+1)} = r_{k\ell}+2$.

**Type B** Let $\sigma : [2n] \to [2n]$ be the involution $\sigma(k) = 2n+1-k$. Then the Coxeter group of type $B$ is the centralizer of $\sigma$ in $S_{2n}$. Bruhat order in type $B$ is the induced order from $S_{2n}$, and thus can be described as by either the Tableaux Criterion or the Rank Matrix Criterion. In either case, we may cut roughly in half the number of cases which need to be checked, because $u[k]$ determines $u[2n-k]$, so we only need check the conditions for $1 \leq k \leq n$.

**Type D** The Coxeter group of type $D_n$ is the index two subgroup of $B_n$ consisting of permutations for which $\#(u [n] \cap [n]) \equiv n \bmod 2$. The Bruhat order is no longer induced from $B_n$. It seems to follow from other things I will say below that there is a quick definition of it as an induced order though, so that will be my first question:

Let $\tau$ be the permutation $(n \ n+1)$ in $B_n$; this permutation is not in $D_n$ but normalizes $D_n$. Embed $D_n$ into $B_n \times B_n$ by $u \mapsto (u, \tau u \tau^{-1})$. Is the Bruhat order on $D_n$ simply the $B_n \times B_n$ Bruhat order restricted to the image?

In any case, what I can find in sources is that something like the tableaux criterion holds. Namely, in any Coxeter group, there is a collection of subgroups called the maximal parabolics, and there are partial orders on the quotients by the maximal parabolics such that $u \leq v$ if and only if their cosets in by each maximal parabolic $P$ obey $u P \leq v P$. To make this sound more like the tableaux criterion, note that the maximal parabolics in type $D_n$ are the stabilizers of $[1]$, $[2]$, ..., $[n-2]$, $[n]$ and $[n]':=\tau([n])$. So we can identify cosets of maximal parabolics with $D_n$ orbits of these sets. So we can detect whether $u \leq v$ by comparing $uX$ and $vX$ for some poset relation, with $X$ in the list $[1]$, $[2]$, ..., $[n-2]$, $[n]$ and $[n]':=\tau([n])$. But I haven't found a source which spells out how to order the $D_n X$'s.

What, explicitly, is the order on the $D_n X$'s?

I tried to work this out myself, and I believe I got the following: Let $Y$ and $Z \in D_n X$ for $X$ as above. Then the poset relation is that we have both $Y \leq Z$ and $\tau(Y) \leq \tau(Z)$, in our order on subsets of $[2n]$.

Is this right? Is there a source for this?

It seems to me I can also encode this in a rank matrix style. Namely, for $X$ and $Y$ one of $[1]$, $[2]$, ..., $[n-2]$, $[n]$, $[n]'$, $[n+2]$, ..., $[2n-2]$, $[2n-1]$ and $u$ in the $D_n$ Coxeter group, let $r_{XY}(u) = \#(uX \cap Y)$.

Is it true that $u \leq v$ if and only if $r_{XY}(u) \leq r_{XY}(v)$ for all such $X$, $Y$? Is there a source for this?

Finally, one could ask for a simple local characterization of the $r_{XY}$'s, similar to the characterization I gave above for type $A$ rank matrices.

Is such a characterization known?

minusculeweights (see Theorem 4.3 of arxiv.org/abs/1108.5245), but Type D has many non-minuscule weights so this probably does not help you. $\endgroup$ – Sam Hopkins Nov 30 '19 at 3:00