Proof. Let R be a maximal orthogonal set of roots. By symmetry, we may reduce to the case where all the roots in R are positive. If R contains the root $\varepsilon _i \pm \varepsilon _j$ , then R must also contain the root $\varepsilon _i \mp \varepsilon _j$ because otherwise $R \cup \{\varepsilon _i \mp \varepsilon _j\}$ would be a set of orthogonal roots that was larger than R. It follows that R consists of $n/2$ pairs of roots such that each pair has the same support, and roots from distinct pairs have disjoint supports. This completes the proof of (i).

Part (ii) follows from (i) and the fact that if $1 \leq i < j \leq n$ , then the roots $\varepsilon _i \pm \varepsilon _j$ are positive.

Proof. Recall from Section 2 that the group $W(D_n)$ can be regarded as the group of signed permutations of n objects in which there is an even number of sign changes. Such a group acts transitively on the set described in Lemma 3.1 (i).

Now suppose W has type $E_7$ , and let $\alpha $ be a root of W. Then by Section 2, the stabilizer $W_\alpha $ is a Weyl group of type $D_6$ whose root system is the set $\Phi _\alpha $ of roots that are orthogonal to $\alpha $ . Since $W_\alpha $ acts transitively on $\Phi _\alpha $ , it follows that there is a well-defined bijection $[R]\mapsto [R\cup \{\alpha \}]$ from the set of $W_\alpha $ -orbits on $\mathcal {M}(W_\alpha )$ to the set of W-orbits on $\mathcal {M}(W)$ , where the orbit $[R]$ of every 6-tuple $R\in \mathcal {M}(W_\alpha )$ is sent to the orbit $[R\cup \{\alpha \}]$ of the 7-tuple $R\cup \{\alpha \}$ . It then follows that W acts transitively on $\mathcal {M}(W)$ , as desired.

Finally, if W has type $E_8$ , then for each root $\alpha $ of W, the stablizer $W_\alpha $ is of type $E_7$ . A similar argument to the one above shows that W acts transitively on $\mathcal {M}(W)$ because $W_\alpha $ acts transitively on $\mathcal {M}(W_\alpha )$ .

Proof. We first prove that (i) implies (iii). Assume that $\gamma $ is a root. Any root subsystem containing $Q\cup \{\gamma \}$ also contains $s_{\beta _i}(\beta _i) = -\beta _i$ for each i, as well as all roots of the form

$$ \begin{align*}s_{\beta_1}^{\epsilon_1} s_{\beta_2}^{\epsilon_2} s_{\beta_3}^{\epsilon_3} s_{\beta_4}^{\epsilon_4}(\gamma) ,\end{align*} $$

where we have $\epsilon _i \in \{0, 1\}$ for all i. The 16 roots listed above can also be expressed as

$$ \begin{align*}(\pm \beta_1 \pm \beta_2 \pm \beta_3 \pm \beta_4)/2 .\end{align*} $$

We have constructed all 24 roots in the set $\Psi $ listed in the statement, and this is the cardinality of a root system of type $D_4$ . To prove (iii), it now suffices to show that $\Psi $ is a root system of type $D_4$ . Because the elements of Q are orthogonal vectors of the same length, we may choose Euclidean coordinates $\beta _1 = \varepsilon _1 - \varepsilon _2$ , $\beta _2 = \varepsilon _1 + \varepsilon _2$ , $\beta _3 = \varepsilon _3 - \varepsilon _4$ and $\beta _4 = \varepsilon _3 + \varepsilon _4$ . With respect to these coordinates, we have

$$ \begin{align*}\Psi = \{\pm \varepsilon_i \pm \varepsilon_j : 1 \leq i < j \leq 4\} ,\end{align*} $$

which indeed forms a root system of type $D_4$ , as desired.

It is immediate that (iii) implies (ii).

In the usual notation for the simple roots of type $D_4$ , the orthogonal roots $\alpha _1$ , $\alpha _3$ , $\alpha _4$ , and $\alpha _1 + 2\alpha _2 + \alpha _3 + \alpha _4$ sum to $2 \alpha $ , where $\alpha $ is the root $\alpha _1 + \alpha _2 + \alpha _3 + \alpha _4$ . Lemma 3.2 applied to a root system of type $D_4$ then implies that the sum of every orthogonal quadruple of roots in a root system of type $D_4$ is equal to $2 \alpha '$ for some root $\alpha '$ . It follows that (ii) implies (i), which completes the proof.

Proof of Lemma 3.5.

The ‘if’ implication holds since the four roots in the given form sum to twice the positive root $\varepsilon _i+\varepsilon _k$ . To prove the ‘only if’ implication, let $Q=\{\beta _1,\beta _2,\beta _3,\beta _4\}$ be a coplanar quadruple. Recall from Section 2 that two roots in type $D_n$ are orthogonal if and only if they have the same or disjoint support. It follows that if no two roots in Q have the same support, then the supports of $\beta _1,\beta _2,\beta _3,\beta _4$ contain a total of eight distinct coordinates $\varepsilon _i$ , in which case the sum $\gamma =\beta _1+\beta _2+\beta _3+\beta _4$ cannot be twice a root. We may therefore assume, without loss of generality, that $\{\beta _1,\beta _2\}=\{\varepsilon _i\pm \varepsilon _j\}$ for some $i<j$ . This implies $\beta _1+\beta _2=2\varepsilon _i$ . The condition that Q is an orthogonal set summing to twice a root then forces us to have $\{\beta _3,\beta _4\}=\{\varepsilon _k\pm \varepsilon _l\}$ for some elements $k,l$ distinct from i and j with $k<l$ .

Proof. To prove (i), it suffices by Lemma 3.2 to do so for a fixed R. Suppose first that W is of type $D_n$ , and choose

$$ \begin{align*}R = \{ \varepsilon_1 + \varepsilon_2, \ \varepsilon_1 - \varepsilon_2,\ \varepsilon_3 + \varepsilon_4, \ \varepsilon_3 - \varepsilon_4, \ \ldots, \ \varepsilon_{n-1} + \varepsilon_n, \ \varepsilon_{n-1} - \varepsilon_n \}.\end{align*} $$

The root $\alpha $ must be of the form $\pm \varepsilon _i \pm \varepsilon _j$ , where i and j come from different parts of the partition $\{\{1, 2\}, \{3, 4\}, \ldots , \{n-1, n\}\}$ . It follows that the support of $\alpha $ has one element in common with the support of each of precisely four elements of R making up two collinear pairs, and that $\alpha $ is orthogonal to all the other elements of R. Furthermore, the roots $\beta _1$ , $\beta _2$ , $\beta _3$ and $\beta _4$ that are not orthogonal to $\alpha $ are of the form $\pm \varepsilon _h \pm \varepsilon _i$ and $\pm \varepsilon _j \pm \varepsilon _k$ , where $|\{h, i, j, k\}| = 4$ . It follows that $2\alpha $ can be written in the form $\pm \beta _1\pm \beta _2\pm \beta _3\pm \beta _4$ for suitable choices of signs.

Next, suppose that W has type $E_8$ , and choose

$$ \begin{align*}R = \{ 2(\varepsilon_1 + \varepsilon_2), \ 2(\varepsilon_1 - \varepsilon_2), \ 2(\varepsilon_3 + \varepsilon_4), \ 2(\varepsilon_3 - \varepsilon_4), \ 2(\varepsilon_5 + \varepsilon_6), \ 2(\varepsilon_5 - \varepsilon_6), \ 2(\varepsilon_7 + \varepsilon_8), \ 2(\varepsilon_7 - \varepsilon_8) \}.\end{align*} $$

If $\alpha $ has the form $2(\pm \varepsilon _i \pm \varepsilon _j)$ , then the proof is completed using the same argument as in type $D_8$ . The other possibility is that we have $\alpha = \sum _{i = 1}^8 \pm \varepsilon _i$ , where the signs are chosen so that there is an even number of minus signs. In this case, $\alpha $ is orthogonal to precisely one of the roots $\{2(\varepsilon _j - \varepsilon _{j+1}), \ 2(\varepsilon _j + \varepsilon _{j+1})\}$ , according as $\varepsilon _j$ and $\varepsilon _{j+1}$ occur in $\alpha $ with the same or with opposite coefficients. It follows that $\alpha $ is orthogonal to precisely four elements of R and that $2 \alpha $ can be expressed in the required form.

Now suppose that W has type $E_7$ . By Section 2.3, we may identify the root system of W with the set of roots orthogonal to the highest root $\theta $ in the root system of type $E_8$ , so that $R \cup \{\theta \}$ is a maximal set of orthogonal roots in type $E_8$ . By the above paragraph, $\alpha $ is orthogonal to four of the roots in $R \cup \{\theta \}$ , but one of these roots is $\theta $ itself. It follows that $\alpha $ is orthogonal to three elements of R and that $2 \alpha $ can be expressed as a signed sum of the other four elements of R, as required. This completes the proof of (i).

It follows from (i) that we have $\alpha \in \Psi $ , because condition (i) implies condition (iii) in Proposition 3.3. The element $\gamma =(\beta _1+\beta _2+\beta _3+\beta _4)/2$ is a root because Q is a coplanar quadruple, and the set $s_\alpha (Q)$ is a coplanar quadruple because its elements sum to twice $s_\alpha (\gamma )$ , which is a root because $\gamma $ is. We also have $s_\alpha (Q)\subseteq \Psi $ because both $\alpha $ and Q are in $\Psi $ . This implies that $\Psi $ must be the $D_4$ -subsystem associated to $s_\alpha (Q)$ , proving (ii).

Proof. By Section 2.3, the roots orthogonal to a given root in type $D_4$ form a subsystem of type $3A_1$ . Therefore, each positive root lies in a unique quadruple of mutually orthogonal positive roots, which proves (i).

Let $\{\alpha _1,\alpha _2,\alpha _3,\alpha _4\}$ be the induced simple roots of $\Psi $ , with $\alpha _2$ corresponding to the branch node in the Dynkin diagram. Then the three quadruples from (i) are given by

The roots in $\Psi ^+_3$ add up to twice the root $\alpha = \alpha _1 + \alpha _2 + \alpha _3 + \alpha _4$ . The root $\alpha $ is strictly lower in the $\le $ order than one of the roots in $\Psi ^+_3$ ; therefore, $\Psi ^+_3$ is an alignment by Definition 3.9. The roots in $\Psi ^+_1$ and $\Psi ^+_2$ both add up to $2 \theta $ , where $\theta = \alpha _1 + 2\alpha _2 + \alpha _3 + \alpha _4$ . The root $\theta $ is strictly higher in the $\le $ order than each element of $\Psi ^+_1$ and $\Psi ^+_2$ . Note that we have $s_\theta (\Psi ^+_1) = -\Psi ^+_2$ and $s_\theta (\Psi ^+_2) = -\Psi ^+_1$ . Furthermore, the roots $\alpha _2$ and $\alpha _1 + \alpha _2 + \alpha _3 + \alpha _4$ are the unique $\leq $ -minimal and unique $\leq $ -maximal elements of $\Psi ^+_1 \ \cup \ \Psi ^+_2$ , respectively. This implies that $\Psi ^+_1$ is a crossing and that $\Psi ^+_2$ is a nesting, and it also follows that none of $\Psi ^+_1, \Psi ^+_2$ and $\Psi ^+_3$ is both a crossing and a nesting. The quadruple Q must be one of $\Psi ^+_1, \Psi ^+_2$ and $\Psi ^+_3$ , and (ii) follows.

Part (iii) follows from (ii) since each of $\Psi ^+_C, \Psi ^+_N$ and $\Psi ^+_A$ uniquely determines $\Psi ^+$ as its associated $D_4$ -subsystem by Proposition 3.3 (iii).

Part (iv) holds as each of the quadruples in $\{\Psi ^+_C, \Psi ^+_N, \Psi ^+_A\}$ is a basis for the span of $\Psi ^+$ , so that any root that is orthogonal to every element of one quadruple is also orthogonal to every element of each of the other quadruples.

Finally, the claims in (v) and (vi) can all be verified by inspection or direct computation based on the description of $\Psi ^+_C=\Psi ^+_1, \Psi ^+_N=\Psi ^+_2$ and $\Psi ^+_A=\Psi ^+_3$ . For the equation $\gamma _C=\gamma _N+\gamma _A$ and the assertion about $s_\alpha $ in (vi), one can alternatively prove them using the usual realizations of the root system and Weyl group of type $D_4$ , where the simple roots are $\alpha _1 = \varepsilon _1 - \varepsilon _2$ , $\alpha _2 = \varepsilon _2 - \varepsilon _3$ , $\alpha _3 = \varepsilon _3 - \varepsilon _4$ and $\alpha _4 = \varepsilon _3 + \varepsilon _4$ and the group W acts as signed permutations. Under this realization, we have

(3.1) $$ \begin{align} \gamma_C = (\varepsilon_1^2 - \varepsilon_3^2)(\varepsilon_2^2 - \varepsilon_4^2),\; \gamma_N=(\varepsilon_1^2 - \varepsilon_4^2)(\varepsilon_2^2 - \varepsilon_3^2),\;\text{and}\; \gamma_A=(\varepsilon_1^2 - \varepsilon_2^2)(\varepsilon_3^2 - \varepsilon_4^2), \end{align} $$

and the equation $\gamma _C=\gamma _N+\gamma _A$ follows as the terms expressed in coordinates satisfy the Ptolemy relation

$$\begin{align*}(A-C)(B-D) = (A-D)(B-C) + (A-B)(C-D).\\[-37pt] \end{align*}$$

Proof. Let $\Psi $ be the type $D_4$ subsystem associated to Q, and let $\{\alpha _1, \alpha _2, \alpha _3, \alpha _4\}$ be the simple system of $\Psi $ induced by $\Phi ^+$ , with $\alpha _2$ corresponding to the branch node in the Dynkin diagram. Then as in the proof of Theorem 3.10, the set $\Psi ^+$ decomposes into the union of the crossing $\Psi ^+_C$ , nesting $\Psi ^+_N$ and alignment $\Psi ^+_A$ given below:

If $Q=\Psi ^+_A$ , then we have $\beta _4 - \beta _1 - \beta _2 - \beta _3 = 2\alpha _2$ , and $\beta _4 + \beta _1 - \beta _2 - \beta _3 = 2\alpha _2 + 2\alpha _i$ for some $i \in \{1, 3, 4\}$ .

If $Q=\Psi ^+_N$ , then we have $\beta _1 < \beta _j$ for all $j \in \{2, 3, 4\}$ . We also have $\beta _4 - \beta _1 - \beta _2 - \beta _3 = -2\alpha _i - 2\alpha _2$ for some $i \in \{1, 3, 4\}$ , and $\beta _4 + \beta _1 - \beta _2 - \beta _3 = -2\alpha _i$ for some $i \in \{1, 3, 4\}$ .

If $Q=\Psi ^+_C$ , then we have $\beta _4> \beta _j$ for all $j \in \{1, 2, 3\}$ . We also have $\beta _4 - \beta _1 - \beta _2 - \beta _3 = -2\alpha _2$ and $\beta _4 + \beta _1 - \beta _2 - \beta _3 = 2\alpha _i$ for some $i \in \{1, 3, 4\}$ . Furthermore, none of the $\beta _i$ is a simple root of $\Phi $ because none of the $\beta _i$ is even a simple root of $\Psi $ .

All the claims in the proposition follow from the above observations. Note that the conditions in (ii), (iii), and (iv) are exclusive and exhaustive because of part (i).

Proof. By [Reference Carter11, Lemma 11 (iii)] and its proof, if $W = W(E_8)$ , then W acts transitively on ordered triples of orthogonal roots, and the set of roots orthogonal to three given mutually orthogonal roots is a root system of type $A_1 + D_4$ . Since the action of orthogonal triples of roots is transitive, it suffices to prove (i) for a fixed choice of $\beta _1$ , $\beta _2$ and $\beta _3$ . If we choose $\beta _1 = \alpha _3 = 2(\varepsilon _2 - \varepsilon _1)$ , $\beta _2 = \alpha _5 = 2(\varepsilon _4 - \varepsilon _3)$ and $\beta _3 = \alpha _7 = 2(\varepsilon _6 - \varepsilon _5)$ , then it follows from the explicit description of the root system in Section 2.3 that the only positive root that forms a coplanar quadruple with $\beta _1,\beta _2$ and $\beta _3$ is the root $\beta _4 = 2(\varepsilon _8 - \varepsilon _7)$ . This proves (i).

The uniqueness property of (i) proves that the set of coplanar quadruples corresponds to the $A_1$ summand of the $A_1 + D_4$ subsystem. This implies that if $\beta $ is any positive root that is orthogonal to all of $\beta _1$ , $\beta _2$ and $\beta _3$ , then either $\beta _4 = \beta $ , or $\beta _4$ is orthogonal to $\beta $ . The maximality of R in the statement of (ii) implies that $\beta _4$ cannot be orthogonal to all elements of R. It follows that $\beta _4 \in R$ , which proves (ii).

Proof. Part (i) is immediate from Lemma 3.17. To prove part (ii), we need to show that any two distinct quadruples from the Steiner quadruple system are either disjoint or overlap in precisely two elements. This can be proved by an exhaustive check, or by arguing as follows.

The quadruples in the left column of the table consist of the element 8 together with three points forming a line in the Fano plane (see Section 6.2). Any two such quadruples intersect in two elements: the element 8 and the unique point on the intersection of the two lines of the Fano plane. The general case follows by combining this observation with the fact that each quadruple in the right column is the complement of the corresponding quadruple in the left column.

Proof. If W has type $D_{2k}$ , then R determines a perfect matching of the set $\{1, 2, \ldots , 2k\}$ with k blocks by Lemma 3.1, and the coplanar quadruples in R correspond bijectively to pairs of these blocks by Remark 3.6. It follows that M does not depend on R and equals $\binom {k}{2}$ .

In type $E_8$ , the result follows from Lemma 3.18 (i).

If W has type $E_7$ , then as in the proof of Proposition 3.8, we may again identify the root system of W with the set of roots orthogonal to the the highest root $\theta $ in the root system of type $E_8$ . The set $R \cup \{\theta \}$ is then a maximal orthogonal set of roots in type $E_8$ . The coplanar quadruples of R are in bijection with the quadruples of $S(3,4,8)$ that exclude a fixed element, and there are $7$ such quadruples, so we are done.

Proof. If W has type $D_n$ , then the assertions follow from Lemma 3.5.

Suppose that W has type $E_8$ . In this case, part (i) follows from Lemma 3.18 (ii). If $|Q_1 \cap Q_2| = 2$ , then $|Q_1 \cup Q_2| = 6$ , and there are precisely two elements $\alpha , \beta \in R$ that are orthogonal to every root in $Q_1 \cup Q_2$ . Let $\Psi $ be the set of roots in $\Phi $ that are orthogonal to $\alpha $ and $\beta $ . Then $\Psi $ forms a root system of type $D_6$ by Section 2.3, proving the first assertion of (ii), and the sets $Q_1, Q_2, Q_1\cap Q_2$ and $Q_1\cup Q_2$ all lie in $\Psi $ . The other assertions of (ii) now follow by applying the type $D_n$ case of the result to $\Psi $ .

Finally, suppose that W has type $E_7$ . We identify the root system with the set of roots orthogonal to the the highest root $\theta $ in the root system of type $E_8$ as usual. Then $R \cup \{\theta \}$ is a maximal orthogonal set of roots in type $E_8$ . The assertions in this case follow by applying the argument of the previous paragraph with $\alpha = \theta $ .

Proof of Lemma 4.8.

Throughout the proof, we will identify both R and the coplanar quadruples within R with their corresponding matchings (as in Lemma 3.1 and Remark 3.6). Recall that we will often write a 2-block $\{a,b\}$ in a matching as $ab$ .

Suppose that $\alpha _i$ is a simple root, so that the reflection $s_{\alpha _i}$ acts as the transposition $(i, i+1)$ . We will assume that $\alpha _i \not \in R$ , so that we have $R=\{ai,(i+1)b,ef\}$ and the coplanar quadruple moved by $s_{\alpha _i}$ is $Q=\{ai,(i+1)b\}$ . Let $Q'$ be any other coplanar quadruple in R. Then $Q'$ is of the form $\{xy,ef\}$ with $x\in \{a,b\}$ and $y\in \{i,i+1\}$ , and we have $s_{\alpha _i}(Q')=\{\{x,s_{\alpha _i}(y)\},ef\}$ . Since the numbers i and $i+1$ are only distance 1 apart, the four elements $x, s_{\alpha _i}(y), e,f$ appearing in $Q'$ have the same relative order as the numbers $x,y,e,f$ ; therefore, $Q'$ and $s_{\alpha _i}(Q')$ have the same type. It follows that Q is the only quadruple in R that is changed to a quadruple of another type by $s_{\alpha _i}$ . Note that Q will be an alignment if $a < i < i+1 < b$ ; Q will be a crossing if $a < b < i$ or $i+1 < a < b$ or $b < i < i+1 < a$ ; and Q will be a nesting if $b < a < i$ or $i+1 < b < a$ . We have $\lambda (s_{\alpha _i}(R)) = \lambda (R)+1$ in the first three of these six cases and $\lambda (s_{\alpha _i}(R))=\lambda (R)-1$ in the last three cases. The first of the six cases corresponds to the situation in (1), and the second and third cases correspond to the situation in (2). Part (i) follows.

Suppose that Q and $Q'$ are as in the statement of (ii), with Q being the alignment $\{a_1a_2,b_1b_2\}$ for some $a_1 < a_2 < b_1 < b_2$ . If $b_1 = a_2 + 1$ , then the simple root $s_{a_2}$ moves an A to a C by moving Q, and we have $R'=s_{\alpha _2}(R)$ ; therefore, $R'$ has type $A^{p-1}C^{q+1}N^{r}$ by (i). If $b_1>a_2+1$ , then Q must be one of the following five quadruples:

$$\begin{align*}\{12,56\},\quad \{12,45\},\quad \{12, 46 \},\quad \{23, 56\},\quad\text{and}\quad \{13,56\}. \end{align*}$$

Direct computation shows that d equals $5,1,3,1$ and $3$ in these cases, respectively. It follows that d is a positive odd number.

To prove the second assertion in (ii), we prove its contrapositive. If $R'$ does not have type $A^{p-1}C^{q+1}N^r$ , then Q must be one of five quadruples listed in the last paragraph. We claim that in each case, for every positive root $\alpha $ such that $s_\alpha (Q)=Q'$ , there exists a positive root $\alpha '<\alpha $ that moves another alignment in R other than Q to a crossing. Specifically, we may always take $\alpha '$ to be $\varepsilon _2-\varepsilon _3$ in the first three cases and $\varepsilon _4-\varepsilon _5$ in the last two cases. For example, the only possibilities for $\alpha $ if $Q=\{12,56\}$ are $\varepsilon _2\pm \varepsilon _5$ and $\varepsilon _1\pm \varepsilon _6$ , and for all these possibilities the root $\alpha '=\varepsilon _2-\varepsilon _3$ is smaller than $\alpha $ and moves an alignment in R other than Q to a crossing. This completes the proof of the desired contrapositive.

A similar argument proves (iii). This time, we have $Q= \{a_1a_2,b_1b_2\}$ for some $a_1 < b_1 < a_2 < b_2$ . If we have $b_1=a_1+1$ or $b_2=a_2+1$ , then the simple root $\alpha _{a_1}$ or $\alpha _{a_2}$ moves Q to a nesting $Q'$ , so $R'$ has type $A^pC^{q-1}N^{r+1}$ by (i). The only remaining possibility for Q is $\{14,36\}$ . If $\alpha $ is a positive root such that $s_\alpha (Q)=Q'$ , then $\alpha \in \{\varepsilon _1\pm \varepsilon _3,\varepsilon _4\pm \varepsilon _6\}$ . Direct computation shows that any simple root $\alpha '<\alpha $ moves some crossing in R other than Q to a nesting.

Finally, (iv) follows by combining (ii) and (iii) since $R'$ can be obtained from by first replacing Q in R with its corresponding crossing $Q"$ and then replacing $Q"$ in the result with $Q'$ .

Proof of Proposition 4.7.

If W has type $D_4$ then there are only three possibilities for R, and all the assertions follow by direct verification. By Lemma 4.8, we may therefore assume that the rank of W is at least $7$ .

For each coplanar quadruple $Q\subseteq R$ , we define $H_Q$ to be the set of all 6-subsets H of R such that there exists a $D_6$ -subsystem of $\Phi $ containing both $\Psi _Q$ and H, where $\Psi _Q$ is the $D_4$ -subsystem associated with Q. By Proposition 3.20, if $Q_1$ is any coplanar quadruple in R, then either $Q_1=Q$ , or $Q_1 \cap Q = \emptyset $ , or $\lvert Q_1\cap Q \rvert =2$ and there is a unique element of $H_\Psi $ that contains both Q and $Q_1$ .

We now prove (iii). By tracking the contributions towards the crossing number made by the three types of coplanar quadruples $Q_1\subseteq R$ just mentioned, we note that

(4.1) $$ \begin{align} C(R') - C(R) = -1 + \sum_{H \in H_Q} \big(C((H \backslash Q) \cup Q') - C(H) + 1 \big). \end{align} $$

Here, the term $-1$ comes from the case $Q_1=Q$ since the crossing Q in R is replaced by the non-crossing feature $Q'$ as we change R to $R'$ . In the second case where $Q_1\cap Q=\emptyset $ , the quadruple $Q_1$ lies in both R and $R'$ and thus does not contribute to the difference $C(R')-C(R)$ . Finally, every $Q_1$ with $\lvert Q_1\cap Q \rvert =2$ appears together with Q in a unique element H of $H_Q$ and contributes a term in the sum over $H_Q$ , where we have added 1 to the difference $C((H\backslash Q)\cup Q')-C(H)$ to account for the fact that the change from Q to $Q'$ in H has been recorded by the term $-1$ in the first case.

Similar arguments based on the facts that $N(Q')=N(Q)+1$ and $A(Q')=A(Q)$ show that

(4.2) $$ \begin{align} N(R') - N(R) = 1 + \sum_{H \in H_Q} \big(N((H \backslash Q) \cup Q') - N(H) - 1 \big) \end{align} $$

and

(4.3) $$ \begin{align} A(R') - A(R) = \sum_{H \in H_Q} \big(A((H \backslash Q) \cup Q') - A(H) \big). \end{align} $$

Since $\lambda =C+2N$ , it follows from Equations (4.1) and (4.2) that

(4.4) $$ \begin{align} \lambda(R') - \lambda(R) = 1 + \sum_{H \in H_Q} \big(\lambda((H \backslash Q) \cup Q') - \lambda(H) - 1 \big). \end{align} $$

By Lemma 4.8.(iii), each summand in the sum over $H_Q$ is zero in Equation (4.3) and is a nonnegative even number in Equation (4.4); therefore, we have $A(R')=A(R)$ , and the number $d=\lambda (R')-\lambda (R)$ is a positive odd number.

To prove the last assertion in (ii), suppose that $s_\alpha (Q)=Q'$ for some positive root $\alpha $ , but no positive root $\alpha ' < \alpha $ moves a C to an N in R. The same minimality condition then applies if R is replaced by an element of $H_Q$ , so every summand in the sums over $H_Q$ in Equations (4.1) and (4.2) is zero by Lemma 4.8 (iii). It follows that $R'$ has type $A^pC^{q-1}N^{r+1}$ , which proves Proposition 4.7 (iii).

The proof of Proposition 4.7 (ii) follows by a similar but shorter argument, and the proof of Proposition 4.7 (iv) follows by combining parts (ii) and (iii).

Finally, to prove Proposition 4.7 (i), assume that $\alpha _i$ is a simple root such that $\alpha _i \not \in R$ . We have already proved part (i) if W has type $D_4$ , and this implies that either $\alpha _i$ moves an A to a C or a C to an N. In the former case, the conclusions follow from part (ii), and in the latter case, they follow from part (iii), in each case because the simple root $\alpha _i$ is minimal in the order $\le $ .

Proof of Theorem 4.5.

We first prove (i). Proposition 4.7 (i) proves that $(X, \lambda )$ is a scaled W-set, so it suffices to show $(X,\lambda )$ satisfies the axioms (QP1) and (QP2). We do so by induction on n.

If $n=4$ or $n=6$ , then W has type D, and the axioms (QP1) and (QP2) can be proved by direct verification or as follows. Suppose $n=2k$ . By Remark 3.7, it suffices to show that $(X,\lambda )$ is a quasiparabolic set for the symmetric group $S_{n}$ . We may identify the set X with the set $X'$ of fixed-point free involutions in $S_{n}=W(A_{n-1})$ , with each collinear pair $\{\varepsilon _i\pm \varepsilon _j\}$ in a maximal set $R\in X$ corresponding to a factor $(i,j)$ in an involution $\iota \in X'$ . Under this identification, the actions of $S_n$ on $X'$ and X coincide with each other, so it suffices to show that $X'$ is a quasiparabolic set for $S_n$ under the level function $\lambda $ . Rains and Vazirani [Reference Rains and Vazirani43, Section 4] proved that $X'$ is a quasiparabolic set for $S_n$ under the level function h given by $h(\iota )=(\ell (\iota ) - k)/2$ , where $\ell $ denotes Coxeter length, so it further suffices to show that whenever an involution $\iota \in S_n$ corresponds to a maximal set R of positive orthogonal roots, we have $(\ell (\iota ) - k)/2 = \lambda (R)$ . This can be proved by an exhaustive check or by induction on $\lambda (R)$ by using the first three cases in the second paragraph of the proof of Lemma 4.8. For example, the involution

$$ \begin{align*}\iota = (13)(26)(45) = s_2 s_1 s_3 s_5 s_4 s_3 s_5 s_4 s_2 \in S_6\end{align*} $$

corresponds to the set $R=\{13,26,45\}$ of type $ACN$ . In this case, we have $(\ell (\iota ) - k)/2 = (9-3)/2$ and $\lambda (R)=3$ , as required.

Now assume $n\ge 7$ . Let $r=s_\alpha \in T$ be the reflection corresponding to a root $\alpha $ , and let $x \in X$ . If $rx \ne x$ , then $\pm \alpha $ are not in x, so r moves an A to a C, or a C to an N, or an A to an N, or vice versa by Remark 3.12. It follows from Proposition 4.7 (ii), (iii) and (iv) that $\lambda (rx)> \lambda (x)$ in all the first three cases and $\lambda (rx)<\lambda (x)$ in the last three cases; therefore, axiom (QP1) holds.

To prove axiom (QP2), assume that we have $r \in T$ , $x \in X$ , $s \in S$ , $\lambda (rx)> \lambda (x)$ and $\lambda (srx) < \lambda (sx)$ . Then the definition of scaled W-sets forces $\lambda (rx) =\lambda (sx)= \lambda (x)+1$ , so each of r and s must be an $AC$ or $CN$ move by Proposition 4.7 (ii), (iii) and (iv). Let $Q_1$ and $Q_2$ be the coplanar quadruples of roots in R moved by r and s, respectively. Then $Q_1$ and $Q_2$ are disjoint, or coincide with each other, or intersect in two elements by Proposition 3.20. If $Q_1$ and $Q_2$ were disjoint, we would have $Q_2\subseteq r(R)$ , so that s would move $Q_2$ in $r(R)$ , and Proposition 4.7 (ii) and (iii) would imply that $ \lambda (srx)=\lambda (rx)+1>\lambda (rx)=\lambda (sx), $ contradicting the assumption that $\lambda (srx)<\lambda (sx)$ . If $Q_1=Q_2$ , then the fact that $\lambda (rx)=\lambda (sx)=\lambda (x)+1$ implies that s and r must both be $AC$ moves or both be $CN$ moves, according as $Q_1=Q_2$ is an alignment or crossing, by Proposition 4.7. It follows from Remark 3.12 that $rx=sx$ . Finally, if $\lvert Q_1\cap Q_2 \rvert =2$ , then Proposition 3.20 implies that there is a subsystem $\Sigma $ of type $D_6$ containing both $Q_1$ and $Q_2$ . Applying the inductive hypothesis to $\Sigma $ proves that $rx = sx$ , which completes the proof.

Proof. All the claims can be proved by examining the effects of various types of (simple) reflections on sums and levels of n-roots recorded in Theorem 3.10 and Proposition 4.7. We first prove (i). By the definition of $\le _Q$ , it is enough to consider the case where $r \in T$ is a reflection and $\gamma = r(\beta )$ satisfies $\lambda (\beta )<\lambda (\gamma )$ . By Proposition 4.7, this implies that r is an $AC$ , $CN$ or $AN$ move. The proof of (i) then follows from Theorem 3.10 (vi).

We now prove (ii). If $\alpha _i \in R$ , then $\sigma (s_i(R))=\sigma (R)-2\alpha _i$ because $s_{i}(\alpha _i) = -\alpha _i$ . If $\alpha _i$ is a $CN$ or $NC$ move, Theorem 3.10 (vi) implies that $\sigma (s_{\alpha _i}(R)) = \sigma (R)$ . It follows from Proposition 4.7 (i) that the only other possibility is that $\alpha _i$ moves an A to a C or vice versa. By Theorem 3.10 (vi), this can only happen if $\alpha _i$ is the unique simple root in the corresponding nesting (i.e., the root $\alpha _2$ in the explicitly constructed sets $\Psi _2^+=\Psi ^+_N$ in the proof of that theorem). Explicit computation shows that root sums in this case differ by $2\alpha _i$ in the precise manner described in the statement, which completes the proof of (ii).

The first part of (iii) follows immediately from (ii) and Proposition 4.7 (i). The assertion about nonnesting n-roots follows from the special case $r=0$ .

Proof. We recall that by Theorem 2.8, Remark 2.9 and Corollary 2.10 of [Reference Rains and Vazirani43], every orbit of a quasiparabolic set contains at most one W-minimal and at most one W-maximal element. If such a W-minimal or W-maximal element exists, then it can be identified as the unique element in the orbit with the minimal or maximal possible level, respectively.

The set X is finite, so it has at least one W-maximal and one W-minimal element. Since X consists of a single W-orbit by Lemma 3.2, it follows from the paragraph preceding this proposition that X has a unique W-maximal and a unique W-minimal element, and that they are the unique elements with the minimal and maximal possible level.

Let R be a maximal orthogonal set of positive roots. If R contains any coplanar quadruple Q that is a crossing or a nesting, then by Remark 3.12, we can find a reflection $s_\alpha $ that moves Q to an alignment or a crossing, respectively, and in both cases, Proposition 4.7 implies that $\lambda (s_\alpha (R))<\lambda (R)$ . Iterating this procedure proves the existence of an n-root of type $A^M$ , which achieves the lowest possible value of $\lambda $ . The uniqueness property of the previous paragraph then completes the proof of (i). Part (ii) can be proved similarly, by using the fact that any alignment or crossing in R would induce a level-increasing $AC$ or $CN$ move.