Weyl group ,root poset ,rationality graph ,type \(D\) ,Coxeter group

1 Introduction↩︎

Let \(G\) be a connected complex reductive algebraic group with Weyl group \(W\). In the study of rational decompositions \[g=N(g)B(g)\overline{u}N(g)^{-1}, \qquad N:G\dashrightarrow N_-,\quad B:G\dashrightarrow B_+, \label{eq:intro-normal-form}\tag{1}\] Voloshyn associated to every \(u\in W\) a descending sequence \[\nu_0(u)=u(\Pi_+)\cap \Pi_+, \qquad \nu_k(u)=u(\mathop{\mathrm{Adj}}\nu_{k-1}(u))\cap \Pi_+, \qquad k\geq 1, \label{eq:intro-nu}\tag{2}\] and called \(u\) rational when the stable value \(\nu(u)\) is empty [1]. Equivalently, the oriented graph with vertex set \(\nu_0(u)\) and arrows \[\alpha\longrightarrow \beta \quad\Longleftrightarrow\quad u^{-1}(\alpha)\leq \beta \label{eq:intro-arrow}\tag{3}\] has no directed cycle. The rational Weyl group elements form the vertex set of another graph, denoted \(\Gamma(W)\), in which two vertices are adjacent when they differ by left multiplication by a simple reflection. Voloshyn proved that \(\Gamma(W)\) is connected and non-trivial only in types \(A_r\), \(D_r\) for odd \(r\), and \(E_6\); he also conjectured that in type \(D_r\), \(r\) odd, the number of rational elements is \[\#\{u\in W(D_r):u\text{ rational}\}=2^r-1 . \label{eq:intro-count-conj}\tag{4}\] The computed values \[31,127,511,2047,8191,32767 \label{eq:intro-computed-values}\tag{5}\] for \(r=5,7,9,11,13,15\) support 4 . A second expectation in the same work is that the type-\(D\) rationality graph has exactly two vertices of valency one; two such vertices had already been constructed explicitly in [1]. The Coxeter-theoretic conventions used below are standard. We use the usual length function, simple reflections, Bruhat length criterion, and root-poset terminology following [2], together with the combinatorial Coxeter-group viewpoint of [3]. Our root-system conventions agree with Bourbaki’s presentation [4], and the signed-permutation realization of classical Weyl groups is the standard one described, for example, in Carter’s account of groups of Lie type [5]. The purpose of this paper is to prove a classification theorem which simultaneously proves 4 , gives the precise graph \(\Gamma(D_r)\), and verifies the valency-one expectation. We use the standard signed-permutation model of \(W(D_r)\). Thus \(W(D_r)\) is the subgroup of the signed symmetric group on an orthonormal basis \(e_1,\ldots,e_r\) consisting of signed permutations with an even number of sign changes. The positive and simple roots are \[\begin{align} \Pi_+ &=\left\{e_i-e_j:1\leq i<j\leq r\right\} \cup \left\{e_i+e_j:1\leq i<j\leq r\right\}, \tag{6} \\ \Delta &=\left\{\alpha_i=e_i-e_{i+1}:1\leq i\leq r-1\right\} \cup \left\{\alpha_r=e_{r-1}+e_r\right\}. \tag{7} \end{align}\] For \(r\) odd the longest element is \[w_0(e_i)=-e_i\quad(1\leq i\leq r-1), \qquad w_0(e_r)=e_r. \label{eq:w0-intro}\tag{8}\]

For a subset \(I=\{i_1<\cdots<i_k\}\subseteq\{1,\ldots,r-1\}\), \(I\neq\varnothing\), define a cycle \(p_I\in S_r\) by \[p_I(i_a)=i_{a+1}\;(1\leq a<k), \qquad p_I(i_k)=r, \qquad p_I(r)=i_1, \qquad p_I(j)=j\;(j\notin I\cup\{r\}). \label{eq:pI-intro}\tag{9}\] Equivalently, \(p_I\) is the cycle \((i_1\;i_2\;\cdots\;i_k\;r)\) in the convention that \(p_I(a)\) is the image of \(a\). We define two signed permutations \(c_I,d_I\in W(D_r)\) by \[\begin{align} c_I(e_j)&=-e_{p_I(j)}\quad(1\leq j\leq r-1), & c_I(e_r)&=e_{i_1}, \tag{10} \\ d_I(e_j)&=-e_{p_I(j)}\quad(j\in\{1,\ldots,r-1\}\setminus\{i_k\}), & d_I(e_{i_k})&=e_r, & d_I(e_r)&=-e_{i_1}. \tag{11} \end{align}\] The main theorem is the following.

The element \(c_{\{1\}}\) is exactly the element denoted \(C\) in [1]; its inverse is \(d_{\{1\}}\). Thus Theorem 1 is a simultaneous strengthening of Voloshyn’s type-\(D\) counting conjecture and his valency-one expectation. The proof is not a computation over the full signed-permutation group. Its core is the following two-step mechanism.

First, for each \(I\) the graph \(\Gamma(c_I)\) has a two-level form. If \(n_I(a)\) denotes the next element of \(I\cup\{r\}\) after \(a\in I\), then \[\nu_0(c_I)=A_I\sqcup B_I, \quad A_I=\left\{e_b-e_{n_I(a)}:a\in I,\\ a<b<n_I(a)\right\}, \quad B_I=\left\{e_{i_1}-e_q:i_1<q\leq r\right\}. \label{eq:intro-two-level}\tag{12}\] Every arrow out of \(A_I\) lands in \(B_I\), and no arrow leaves \(B_I\). Hence \(c_I\) is rational. The elements \(d_I\) follow from the diagram automorphism interchanging the two spin nodes.

Second, rationality is rigid under one-step descent. Voloshyn’s descent lemma says that every rational \(u\neq w_0\) admits a simple reflection \(s_a\) such that \(s_a u\) is rational and \(\ell(s_a u)=\ell(u)+1\) [1]. We prove that whenever \(v\in\mathcal{F}_r\) and \(s_a v\) is rational, then \(s_a v\in\mathcal{F}_r\). All excluded cases are killed by explicit loops or two-cycles in the root-poset graph 3 . Induction on \(\ell(w_0)-\ell(u)\) then gives the reverse inclusion.

2 Root-poset and rationality preliminaries↩︎

Throughout the paper \(r\geq 5\) is odd, and the type-\(D_r\) root system is realized as in 6 –7 . We write \[\rho\leq \eta \quad\Longleftrightarrow\quad \eta-\rho\in \sum_{a=1}^r \mathbb{Z}_{\geq 0}\alpha_a \label{eq:root-order}\tag{13}\] for the usual root-poset order on \(\Pi_+\). If \[x=(x_1,\ldots,x_r)\in \mathbb{R}^r, \qquad x=\sum_{a=1}^r m_a(x)\alpha_a, \label{eq:coeff-def}\tag{14}\] then the coordinates \(m_a(x)\) are \[\begin{align} m_a(x)&=x_1+\cdots+x_a, &&1\leq a\leq r-2, \tag{15} \\ m_{r-1}(x)&=\frac{x_1+\cdots+x_{r-1}-x_r}{2}, & m_r(x)&=\frac{x_1+\cdots+x_{r-1}+x_r}{2}. \tag{16} \end{align}\] Consequently \[\rho\leq \eta \quad\Longleftrightarrow\quad m_a(\eta-\rho)\geq 0\quad(1\leq a\leq r). \label{eq:root-order-coeff}\tag{17}\]

For a subset \(S\subseteq \Pi_+\) we use \[\mathop{\mathrm{Adj}}(S)=\left\{\gamma\in \Pi_+:\gamma\leq \beta\text{ for some }\beta\in S\right\}. \label{eq:adj-def}\tag{24}\] The following definitions are those of Voloshyn [1]; they are included to fix the notation used in the proof.

We use the following two facts proved in [1].

The length assertion in ?? is the standard Coxeter criterion \[\ell(s_a u)=\ell(u)+1 \quad\Longleftrightarrow\quad u^{-1}(\alpha_a)>0, \label{eq:length-criterion}\tag{25}\] which we shall use without further comment; see, for example, [2], [3], [4], or [5].

3 The signed cyclic family↩︎

Let \[[r-1]=\{1,2,\ldots,r-1\}. \label{eq:rminusone}\tag{26}\] For every non-empty \(I=\{i_1<\cdots<i_k\}\subseteq [r-1]\), let \(p_I\) be the cycle in 9 . We shall use the following notation throughout: \[n_I(i_a)=i_{a+1}\quad(1\leq a<k), \qquad n_I(i_k)=r. \label{eq:next-map}\tag{27}\] Thus \(n_I(a)\) is the next marked point of \(I\cup\{r\}\) after \(a\in I\). The elements \(c_I,d_I\) are as in 10 –11 . It is useful to set \[c_\varnothing=d_\varnothing=w_0. \label{eq:empty-convention}\tag{28}\] When \(I\neq\varnothing\) this convention is not used to define the signs; it only abbreviates the graph formulas.

Let \(\tau\) be the diagram automorphism of the based root system defined by \[\tau(e_j)=e_j\quad(1\leq j\leq r-1), \qquad \tau(e_r)=-e_r. \label{eq:tau-def}\tag{29}\] Then \[\tau(\alpha_a)=\alpha_a\quad(1\leq a\leq r-2), \qquad \tau(\alpha_{r-1})=\alpha_r, \qquad \tau(\alpha_r)=\alpha_{r-1}. \label{eq:tau-simple}\tag{30}\] Thus \(\tau\) preserves \(\Pi_+\) and the root order. It normalizes \(W(D_r)\) inside the full automorphism group of the root system, and \[d_I=\tau c_I\tau^{-1}, \qquad \tau s_{r-1}\tau^{-1}=s_r, \qquad \tau s_a\tau^{-1}=s_a\quad(1\leq a\leq r-2). \label{eq:tau-conjugation}\tag{31}\] Consequently \(u\) is rational if and only if \(\tau u\tau^{-1}\) is rational, because \[\nu_0(\tau u\tau^{-1})=\tau(\nu_0(u)), \qquad (\tau u\tau^{-1})^{-1}(\tau\alpha)\leq \tau\beta \Longleftrightarrow u^{-1}(\alpha)\leq \beta. \label{eq:tau-rationality}\tag{32}\] Thus it is enough to prove most assertions for \(c_I\); the corresponding assertions for \(d_I\) follow by applying \(\tau\).

4 Acyclicity of the family↩︎

We now prove that every element of \(\mathcal{F}_r\) is rational. The case \(w_0\) is immediate from \[w_0(\Pi_+)\subseteq \Pi_- , \qquad \nu_0(w_0)=\varnothing. \label{eq:w0-nu-empty}\tag{33}\] The main point is an explicit description of \(\nu_0(c_I)\).

Fix \(I=\{i_1<\cdots<i_k\}\neq\varnothing\). For \(a\in I\) put \(n=n_I(a)\). Since there is no element of \(I\) strictly between \(a\) and \(n\), one has \[p_I(a)=n, \qquad p_I(b)=b\quad(a<b<n). \label{eq:pI-gap}\tag{34}\] Define two sets of positive roots \[\begin{align} A_I&=\left\{e_b-e_{n_I(a)}:a\in I,\; a<b<n_I(a)\right\}, \tag{35}\\ B_I&=\left\{e_{i_1}-e_q:i_1<q\leq r\right\}. \tag{36} \end{align}\] The roots in \(A_I\) are the roots coming from ordinary inversions of \(p_I\) on the negative part \(\{1,\ldots,r-1\}\); the roots in \(B_I\) are the roots created by the unique positive sign at \(e_r\).

The proof gives the stronger nilpotence bound \[\nu_2(c_I)=\varnothing, \qquad \nu_2(d_I)=\varnothing \qquad(I\neq\varnothing). \label{eq:nu2-empty}\tag{49}\] For example, when \(I=\{1\}\), one obtains \[\begin{align} \nu_0(c_{\{1\}}) &=\left\{e_1-e_q:2\leq q\leq r\right\} \cup \left\{e_b-e_r:2\leq b\leq r-1\right\}, \tag{50}\\ \nu_1(c_{\{1\}}) &=\left\{e_b-e_r:2\leq b\leq r-1\right\}, \qquad \nu_2(c_{\{1\}})=\varnothing, \tag{51} \end{align}\] which recovers the element constructed in [1].

5 The complete internal arrow calculus↩︎

The proof of rationality above used only the absence of arrows from \(B_I\) and of arrows from \(A_I\) to \(A_I\). For later reference, and to make the stabilization statement completely explicit, we record the exact arrows. This also gives a useful independent check on the formulas for \(\nu_1\) and on the length computations in Section 10.

Fix \(I=\{i_1<\cdots<i_k\}\neq\varnothing\). For \(a\in I\) and \(a<b<n_I(a)\), write \[A(a,b)=e_b-e_{n_I(a)}\in A_I, \qquad A(a,b)^-=c_I^{-1}A(a,b)=e_a-e_b. \label{eq:Aab-notation}\tag{52}\] For \(q\) with \(i_1<q\leq r\), write \[B(q)=e_{i_1}-e_q\in B_I, \qquad B(q)^-=c_I^{-1}B(q)=e_{t(q)}+e_r, \label{eq:Bq-notation}\tag{53}\] where \(t(q)\) is the unique element of \([r-1]\) satisfying \(p_I(t(q))=q\). Thus the arrow relation is \[X\longrightarrow Y \quad\Longleftrightarrow\quad X^-\leq Y, \qquad X,Y\in A_I\sqcup B_I. \label{eq:arrow-Xminus}\tag{54}\]

The exact arrow formula admits a matrix description. Order the vertices of \(A_I\) lexicographically by the pair \((a,b)\) and the vertices of \(B_I\) increasingly by \(q\). The adjacency matrix of \(\Gamma_{c_I}\) has block form \[M_I= \begin{pmatrix} 0 & R_I\\ 0 & 0 \end{pmatrix}, \qquad (R_I)_{(a,b),q}= \begin{cases} 1,& b\leq q,\\ 0,& b>q, \end{cases} \label{eq:MI-block}\tag{57}\] where \((a,b)\) ranges over \(a\in I\), \(a<b<n_I(a)\) and \(q\) ranges over \(i_1<q\leq r\). In particular \[M_I^2=0. \label{eq:MI-square-zero}\tag{58}\] This is a stronger nilpotence property than acyclicity: the adjacency algebra generated by \(M_I\) is already killed in degree two. The number of arrows is also explicit. Summing ?? gives \[\begin{align} \#\operatorname{Arr}(\Gamma_{c_I}) &=\sum_{a\in I}\sum_{b=a+1}^{n_I(a)-1}\#\{q:i_1<q\leq r,b\leq q\} \notag\\ &=\sum_{a\in I}\sum_{b=a+1}^{n_I(a)-1}(r-b+1). \label{eq:number-arrows-cI} \end{align}\tag{59}\] The same number is obtained for \(\Gamma_{d_I}\).

The exact internal arrow calculus also explains why the family is rigid under left multiplication. A left multiplication by \(s_a\) acts on the target side. InCoxeter-theoretic terms, the inversion set of \(s_a v\) differs from that of \(v\) by the single root \(v^{-1}(\alpha_a)\); in the signed permutation model it swaps the corresponding target labels. The coordinate analysis below determines exactly when this operation preserves rationality.

6 One-step rigidity↩︎

The reverse inclusion in Theorem 1 is proved by induction. The induction step requires the following exact description of which simple left multiplications of a family element can remain rational.

6.1 Simple reflections acting on the family↩︎

Let \(I\subseteq [r-1]\), and use the convention 28 . For \(1\leq a\leq r-2\), left multiplication by \(s_a\) interchanges the target labels \(a\) and \(a+1\). Thus, in terms of the cycle \(p_I\), it swaps the labels \(a,a+1\) in the image of \(p_I\). The following identities are immediate from 9 –10 ; they are recorded because they are the combinatorial skeleton of \(\Gamma(D_r)\).

The remaining simple reflections are not merely outside the family; they are not rational. We first prove this for the \(c\)-half.

6.2 Non-admissible ordinary reflections↩︎

Applying \(\tau\) gives the identical statement for the \(d\)-half.

6.3 Non-admissible spin reflections↩︎

For the \(c\)-half, the admissible spin reflection is \(s_{r-1}\); the other spin reflection \(s_r\) is excluded except at \(w_0\), where it gives \(d_{\{r-1\}}\).

The diagram automorphism gives the dual spin exclusion.

Combining the action and exclusion lemmas gives the promised one-step rigidity.

7 A certificate catalogue for the excluded moves↩︎

The induction proof in Section 8 needs only the one-step rigidity statement. Nevertheless, it is useful to isolate the local obstruction certificates because they show that the exclusion is not a global enumeration phenomenon. Each forbidden neighbouring element fails rationality for a visibly small reason: either a loop, or in one boundary case a two-cycle.

For an element \(u\in W(D_r)\), call a positive root \(\gamma\) a loop certificate if \[u(\gamma)\in \Pi_+, \qquad u^{-1}(u(\gamma))=\gamma\leq u(\gamma). \label{eq:loop-certificate-def}\tag{81}\] Equivalently, the vertex \(u(\gamma)\in\nu_0(u)\) has a loop in \(\Gamma_u\). Similarly, call an ordered pair \((\gamma,\delta)\) of positive roots a two-cycle certificate if \[u(\gamma),u(\delta)\in\Pi_+, \qquad \gamma\leq u(\delta), \qquad \delta\leq u(\gamma). \label{eq:twocycle-certificate-def}\tag{82}\] Then \(u(\gamma)\to u(\delta)\to u(\gamma)\) in \(\Gamma_u\).

For ordinary non-admissible reflections, the obstruction is always a loop and there are only two local patterns. Let \(u=s_a c_I\) with \(1\leq a\leq r-2\) and \(a+1\notin I\). If \(I=\varnothing\) or \(a\notin I\), then the loop root is the simple root \(\alpha_a\). If \(a\in I\), put \(m=n_I(a)\); then \(m>a+1\) and the loop root is \(e_a-e_m\). The computations are summarized by \[\begin{array}{c|c|c|c} \text{case}&\gamma&u(\gamma)&\gamma\leq u(\gamma)\\ \hline I=\varnothing&\alpha_a&\alpha_a&\alpha_a\leq\alpha_a\\ I\neq\varnothing,\,a\notin I&\alpha_a&\alpha_a&\alpha_a\leq\alpha_a\\ I\neq\varnothing,\,a\in I,\,a+1\notin I&e_a-e_{a+1}&e_a-e_{n_I(a)}&e_a-e_{a+1}\leq e_a-e_{n_I(a)}. \end{array} \label{eq:ordinary-certificate-table}\tag{83}\] The last inequality is the interval containment \[[a,a+1]\subseteq [a,n_I(a)] \label{eq:last-ordinary-interval}\tag{84}\] inside the root poset for roots \(e_x-e_y\).

For the wrong spin reflection \(u=s_r c_I\), \(I\neq\varnothing\), there are three patterns. If \(r-1\notin I\), the source root is \(e_{i_k}+e_r\) and the positive image is \(e_{i_1}+e_{r-1}\). If \(r-1\in I\) but \(I\neq\{r-1\}\), with \(b=\max(I\setminus\{r-1\})\), the source root is \(e_b+e_r\) and the positive image is \(e_{i_1}+e_r\). If \(I=\{r-1\}\), the spin roots form a two-cycle. In formula form, \[\begin{array}{c|c|c|c} \text{case}&\gamma&u(\gamma)&\text{certificate}\\ \hline r-1\notin I&e_{i_k}+e_r&e_{i_1}+e_{r-1}&e_{i_k}+e_r\leq e_{i_1}+e_{r-1}\\ r-1\in I,\;I\neq\{r-1\}&e_b+e_r&e_{i_1}+e_r&e_b+e_r\leq e_{i_1}+e_r\\ I=\{r-1\}&(\alpha_{r-1},\alpha_r)&(\alpha_r,\alpha_{r-1})&\alpha_{r-1}\leftrightarrow\alpha_r. \end{array} \label{eq:spin-certificate-table}\tag{85}\] Here the first inequality is ?? , the second is ?? , and the last entry abbreviates the two arrows in 79 .

Applying \(\tau\) gives the complete obstruction catalogue for the \(d\)-half. Thus every forbidden move from a vertex of \(\mathcal{F}_r\) is detected by one of the following positive roots: \[\begin{array}{l|l} \text{forbidden element} & \text{loop vertex \(u(\gamma)\) or two-cycle pair}\\ \hline \substack{s_a c_I,\;1\leq a\leq r-2\\ a+1\notin I} &\alpha_a\text{ or }e_a-e_{n_I(a)}\\ \substack{s_r c_I\\ I\neq\varnothing} &e_{i_1}+e_{r-1},\;e_{i_1}+e_r,\text{ or }(\alpha_{r-1},\alpha_r)\\ \substack{s_a d_I,\;1\leq a\leq r-2\\ a+1\notin I} &\tau(\alpha_a)\text{ or }\tau(e_a-e_{n_I(a)})\\ \substack{s_{r-1}d_I\\ I\neq\varnothing} &\tau(e_{i_1}+e_{r-1}),\;\tau(e_{i_1}+e_r),\text{ or }(\alpha_r,\alpha_{r-1}). \end{array} \label{eq:complete-certificate-table}\tag{86}\] The notation in 86 is deliberately redundant. In the ordinary \(d\)-case, \(\tau\) fixes \(\alpha_a\) and fixes \(e_a-e_{n_I(a)}\) unless \(n_I(a)=r\), in which case \(\tau(e_a-e_r)=e_a+e_r\); in the spin case it interchanges the two spin nodes.

8 Classification and graph structure↩︎

We can now prove the classification theorem. Let \[N=\ell(w_0)=\#\Pi_+=r(r-1). \label{eq:N-length}\tag{87}\] Recall that \(w_0\) is the unique element of length \(N\).

9 Recognition of rational elements as signed permutations↩︎

The classification can be reformulated as an intrinsic test on signed one-line notation. This is useful because the definition of rationality refers to the whole root-poset graph, whereas the theorem reduces the decision problem to the cycle structure and signs of a signed permutation.

Let \(u\in W(D_r)\) be written as \[u(e_j)=\varepsilon_j e_{\pi(j)}, \qquad \varepsilon_j\in\{\pm1\}, \qquad \pi\in S_r. \label{eq:signed-one-line}\tag{94}\] For a non-empty subset \(I=\{i_1<\cdots<i_k\}\subseteq [r-1]\), the two cyclic families have the same unsigned permutation \[\pi=p_I=(i_1\;i_2\;\cdots\;i_k\;r) \label{eq:pI-cycle-notation}\tag{95}\] where the cycle is read in the source-to-target direction \(j\mapsto p_I(j)\). Thus the non-trivial orbit of \(\pi\) is exactly \(I\cup\{r\}\) and its entries appear in increasing order before returning from \(r\) to \(i_1\).

The proposition gives a linear-time recognition procedure once the cycle of \(\pi\) containing \(r\) has been extracted. Define \[O_r(\pi)=\{r,\pi(r),\pi^2(r),\ldots\} \label{eq:orbit-r}\tag{96}\] for the \(\pi\)-orbit of \(r\). If \(O_r(\pi)=\{r\}\), then rationality is possible only for \(w_0\). If \(O_r(\pi)\neq\{r\}\), write \[O_r(\pi)\setminus\{r\}=\{i_1<\cdots<i_k\}. \label{eq:orbit-I-sorted}\tag{97}\] Then \(\pi=p_I\) if and only if \[\pi(i_j)=i_{j+1}\quad(1\leq j<k), \qquad \pi(i_k)=r, \qquad \pi(r)=i_1, \label{eq:cycle-test}\tag{98}\] and \(\pi(j)=j\) for every \(j\notin I\cup\{r\}\). Equations ?? and ?? then decide between the two halves. Thus rationality can be tested by the chain of equivalences \[\begin{align} u\text{ rational} &\Longleftrightarrow u=w_0 \notag\\ &\quad\text{or }\exists I\neq\varnothing: \bigl(\pi=p_I\text{ and }\varepsilon=(-,\ldots,-,+)_r\bigr) \notag\\ &\quad\text{or }\exists I\neq\varnothing: \bigl(\pi=p_I\text{ and }\varepsilon_{\max I}=+1, \varepsilon_j=-1\;(j\neq\max I)\bigr). \label{eq:decision-equivalence} \end{align}\tag{99}\] Here \((-,\ldots,-,+)_r\) denotes the sign vector with the unique positive sign at the source \(r\). The only sorting needed in 97 may be avoided by checking that the orbit segment from \(\pi(r)\) to \(r\) is increasing: \[\pi(r)<\pi^2(r)<\cdots<\pi^{k}(r)<r, \qquad \pi^{k+1}(r)=r. \label{eq:increasing-cycle-test}\tag{100}\] Equivalently, in the cycle notation starting at the smallest non-\(r\) entry, the unique non-trivial cycle is \[(i_1\;i_2\;\cdots\;i_k\;r). \label{eq:increasing-cycle-again}\tag{101}\]

This recognition theorem is often the shortest route to applying the main result. Instead of constructing \(\Gamma_u\) for an arbitrary signed permutation \(u\), one checks whether the unsigned permutation is an increasing cycle through the spin coordinate \(r\), and then checks whether the sign vector is one of the two allowed vectors. The exclusion lemmas in Section 6 show that every deviation from these two sign patterns is detected locally by a loop or a two-cycle in the root-poset graph.

10 Explicit formulas for lengths and descents↩︎

Although the classification proof above does not require closed length formulas for \(c_I\) and \(d_I\), the formulas clarify the shape of the graph and provide a useful check on the induction. We include them because they make the subset parametrization completely transparent.

For \(I=\{i_1<\cdots<i_k\}\neq\varnothing\), define the gap lengths \[g_a=n_I(i_a)-i_a-1 \qquad(1\leq a\leq k), \label{eq:gap-lengths}\tag{102}\] where \(n_I(i_k)=r\). The gaps start at \(i_1\), not at \(1\), and therefore \[g_a=\#\{b:i_a<b<n_I(i_a)\}, \qquad \sum_{a=1}^k g_a=r-i_1-k. \label{eq:gap-sum}\tag{103}\] The set \(A_I\) has cardinality \[\#A_I=\sum_{a=1}^k g_a=r-i_1-k, \label{eq:AI-card}\tag{104}\] whereas \[\#B_I=r-i_1. \label{eq:BI-card}\tag{105}\] Hence \[\#\nu_0(c_I)=\#A_I+\#B_I=2r-2i_1-k. \label{eq:nu0-card}\tag{106}\] Since \(\ell(u)=\#\{\beta\in\Pi_+:u(\beta)\in\Pi_-\}\) and \(\#\Pi_+=r(r-1)\), this gives \[\ell(c_I)=r(r-1)-(2r-2i_1-k). \label{eq:length-cI}\tag{107}\] The same formula holds for \(d_I\) by \(\tau\)-symmetry: \[\ell(d_I)=\ell(c_I). \label{eq:length-dI}\tag{108}\] For \(I=\{1\}\), one obtains \[\ell(c_{\{1\}})=\ell(d_{\{1\}})=r(r-1)-(2r-3), \label{eq:length-C}\tag{109}\] which agrees with Voloshyn’s formula \[\frac{1}{2}\bigl(r(r-1)+(r-2)(r-3)\bigr) =r(r-1)-(2r-3) \label{eq:Voloshyn-length-agree}\tag{110}\] for \(r\) odd.

It is cleaner to discuss length changes in terms of the defect \[\delta(I)=r(r-1)-\ell(c_I)=2r-2\min I-\#I \qquad(I\neq\varnothing), \label{eq:defect-I}\tag{111}\] with the convention \(\delta(\varnothing)=0\) for \(w_0\). If \(J=I\mathbin{\triangle}\{a\}\) is an admissible ordinary toggle, then \[\ell(c_J)-\ell(c_I)=\delta(I)-\delta(J) =2(\min J-\min I)+(\#J-\#I), \label{eq:length-toggle-general}\tag{112}\] where the same formula holds for \(d_I\). This identity records all subcases: toggling the minimum has the opposite length effect from toggling an interior point. For the spin toggle \(a=r-1\) one obtains \[\ell(c_{I\mathbin{\triangle}\{r-1\}})-\ell(c_I) =\begin{cases} -1, & I=\varnothing,\\[1mm] +1, & I\neq\varnothing\text{ and } r-1\notin I,\\[1mm] -1, & r-1\in I\text{ and }I\neq\{r-1\},\\[1mm] +1, & I=\{r-1\}, \end{cases} \label{eq:length-toggle-spin}\tag{113}\] where the first and last lines use \(c_\varnothing=w_0\). The corresponding formulas for \(d_I\) are identical.

The defect distribution has a closed form. Let \[F_r(q)=\sum_{u\text{ rational}} q^{r(r-1)-\ell(u)}. \label{eq:defect-polynomial-def}\tag{114}\] Then Theorem 1 and 111 give \[\begin{align} F_r(q) &=1+2\sum_{\varnothing\neq I\subseteq [r-1]}q^{2r-2\min I-\#I} \notag\\ &=1+2\sum_{m=1}^{r-1}\sum_{J\subseteq\{m+1,\ldots,r-1\}} q^{2r-2m-1-\#J} \notag\\ &=1+2\sum_{n=0}^{r-2}q^{n+1}(1+q)^n. \label{eq:defect-polynomial} \end{align}\tag{115}\] Evaluating at \(q=1\) recovers the count, \[F_r(1)=1+2\sum_{n=0}^{r-2}2^n=2^r-1. \label{eq:defect-polynomial-count}\tag{116}\] Thus the enumeration is not only cardinal: it is refined by Coxeter length.

For \(r=5\), the defect polynomial is \[\begin{align} F_5(q) &=1+2\bigl(q+q^2+2q^3+3q^4+4q^5+3q^6+q^7\bigr) \label{eq:F5-polynomial} \end{align}\tag{120}\] so the length distribution, beginning from \(N=20\), is \[\begin{array}{c|cccccccc} N-\ell&0&1&2&3&4&5&6&7\\ \hline \#\{u\text{ rational}:N-\ell(u)=t\}&1&2&2&4&6&8&6&2. \end{array} \label{eq:D5-defect-table}\tag{121}\] This is the refined form of the count \(31\).

11 The rationality graph as a subset graph↩︎

Theorem 1 gives a compact model for \(\Gamma(D_r)\) that may be useful in applications to rational normal forms and cluster-type structures. We spell it out as a purely combinatorial graph.

Let \(\mathscr{B}_{r-1}\) be the graph with vertex set all subsets of \([r-1]\) and edges \[I\longleftrightarrow I\mathbin{\triangle}\{a\} \quad\text{if and only if}\quad \begin{cases} 1\leq a\leq r-2\text{ and }a+1\in I,\\ \text{or }a=r-1. \end{cases} \label{eq:B-graph}\tag{122}\] Then \(\Gamma(D_r)\) is obtained from two copies of \(\mathscr{B}_{r-1}\) by identifying their empty-set vertices and deleting no other vertices: \[\Gamma(D_r)\cong \mathscr{B}_{r-1}^{(c)}\cup_{\varnothing}\mathscr{B}_{r-1}^{(d)}. \label{eq:Gamma-as-B}\tag{123}\] The edge labelled \(a\) in the \(c\)-copy corresponds to left multiplication by \(s_a\) for \(a<r-1\) and by \(s_{r-1}\) for \(a=r-1\); in the \(d\)-copy it corresponds to \(s_a\) for \(a<r-1\) and by \(s_r\) for \(a=r-1\).

The graph \(\mathscr{B}_{r-1}\) is connected. We give a deletion algorithm because it is later useful for distance estimates. Put \(n=r-1\). If \(I\neq\varnothing\) and \(m=\max I\), then the word \[\omega_m= \begin{cases} (n),&m=n,\\ (n,n-1,\ldots,m,m+1,\ldots,n),&m<n, \end{cases} \label{eq:omega-m}\tag{124}\] removes \(m\) and leaves all elements \(<m\) unchanged. Indeed, the first half of the word adds the chain \(m+1,m+2,\ldots,n\) one element at a time until toggling \(m\) is allowed; the second half deletes the added chain from bottom to top. In formulas, for \(m<n\), \[I \xleftrightarrow{n} I\cup\{n\} \xleftrightarrow{n-1} I\cup\{n-1,n\} \xleftrightarrow{n-2}\cdots \xleftrightarrow{m} (I\setminus\{m\})\cup\{m+1,\ldots,n\} \label{eq:delete-first-half}\tag{125}\] followed by \[(I\setminus\{m\})\cup\{m+1,\ldots,n\} \xleftrightarrow{m+1} (I\setminus\{m\})\cup\{m+2,\ldots,n\} \xleftrightarrow{m+2}\cdots \xleftrightarrow{n} I\setminus\{m\}. \label{eq:delete-second-half}\tag{126}\] Every toggle displayed in 125 –126 is allowed by 122 . Repeating the procedure for the successive maxima of \(I\) reaches \(\varnothing\). This gives a purely combinatorial proof, in type \(D_r\) odd, of the connectedness part of Voloshyn’s theorem.

The degree formula 92 can be rewritten as \[\deg_{\mathscr{B}_{r-1}}(I)=1+\#(I\setminus\{1\}) \quad(I\neq\varnothing), \qquad \deg_{\mathscr{B}_{r-1}}(\varnothing)=1. \label{eq:B-degree}\tag{127}\] After gluing two copies at \(\varnothing\), the common vertex has degree \(2\). Thus the graph has \[\sum_{v\in \Gamma(D_r)}\deg(v) =2\sum_{I\subseteq [r-1]}\deg_{\mathscr{B}_{r-1}}(I) \label{eq:sum-degrees-start}\tag{128}\] with the empty degree counted twice before gluing. A direct summation gives \[\begin{align} \#E(\mathscr{B}_{r-1}) &=\frac{1}{2}\left(1+\sum_{\varnothing\neq I\subseteq [r-1]}(1+\#(I\setminus\{1\}))\right) \notag\\ &=\frac{1}{2}\left(1+(2^{r-1}-1)+(r-2)2^{r-2}\right) \notag\\ &=2^{r-2}+(r-2)2^{r-3}. \label{eq:edges-B} \end{align}\tag{129}\] Therefore \[\#E(\Gamma(D_r))=2^{r-1}+(r-2)2^{r-2}. \label{eq:edges-Gamma}\tag{130}\] For instance, \(\#E(\Gamma(D_5))=40\), agreeing with the degree distribution \[2\cdot 1+13\cdot 2+12\cdot 3+4\cdot 4=80. \label{eq:D5-degree-dist}\tag{131}\]

The local neighbourhoods can be read without reference to the ambient Weyl group. For \(I\neq\varnothing\), put \[P(I)=\{a\in [r-2]:a+1\in I\}, \qquad Q(I)=P(I)\cup\{r-1\}. \label{eq:PI-QI}\tag{132}\] Then the neighbours of \(c_I\) in \(\Gamma(D_r)\) are \[\mathcal{N}(c_I)=\{c_{I\mathbin{\triangle}\{a\}}:a\in Q(I)\}, \label{eq:neighbours-c}\tag{133}\] where \(c_\varnothing=w_0\), and similarly \[\mathcal{N}(d_I)=\{d_{I\mathbin{\triangle}\{a\}}:a\in Q(I)\}. \label{eq:neighbours-d}\tag{134}\] The neighbours of the glued vertex are \[\mathcal{N}(w_0)=\{c_{\{r-1\}},d_{\{r-1\}}\}. \label{eq:neighbours-w0}\tag{135}\] Consequently the graph distance from \(c_I\) to \(w_0\) is bounded by the total length of the deletion words in 124 . If \(I=\{i_1<\cdots<i_k\}\) and \(n=r-1\), the word \[\omega_{i_k}\omega_{i_{k-1}}\cdots\omega_{i_1} \label{eq:deleting-word}\tag{136}\] connects \(I\) to \(\varnothing\) in \(\mathscr{B}_{r-1}\). Hence \[\operatorname{dist}_{\Gamma(D_r)}(c_I,w_0) \leq \sum_{j=1}^k\bigl(2(r-1-i_j)+1\bigr), \label{eq:distance-bound}\tag{137}\] with the identical estimate for \(d_I\). In particular, because any path from a \(c\)-vertex to a \(d\)-vertex passes through \(w_0\), one has \[\operatorname{dist}_{\Gamma(D_r)}(c_I,d_J) =\operatorname{dist}_{\Gamma(D_r)}(c_I,w_0) +\operatorname{dist}_{\Gamma(D_r)}(w_0,d_J). \label{eq:c-d-distance-split}\tag{138}\] The exact diameter is not needed for the classification; the point is that connectedness, degrees, and cross-half distances follow from a finite state subset calculus rather than from a search in the Weyl group of order \(2^{r-1}r!\).

12 The complete \(D_5\) model↩︎

We record the smallest odd case in full. This is not used in the proof, but it provides a compact verification of the abstract subset graph and of the internal arrow formula. Put \(n=4\). In the abstract graph \(\mathscr{B}_4\), the edges with label \(4\) are present for every subset because the spin toggle is always allowed: \[\begin{align} E_4=\bigl\{ &\varnothing\!-\!4, 1\!-\!14, 2\!-\!24, 3\!-\!34, 12\!-\!124, 13\!-\!134, 23\!-\!234, 123\!-\!1234 \bigr\}. \label{eq:D5-E4} \end{align}\tag{139}\] Here, for instance, \(14\) means \(\{1,4\}\). The ordinary label-\(1\) edges are exactly those pairs in which \(2\) is present: \[E_1=\{2\!-\!12, 23\!-\!123, 24\!-\!124, 234\!-\!1234\}. \label{eq:D5-E1}\tag{140}\] Similarly, \[E_2=\{3\!-\!23, 13\!-\!123, 34\!-\!234, 134\!-\!1234\}, \label{eq:D5-E2}\tag{141}\] where the condition is the presence of \(3\), and \[E_3=\{4\!-\!34, 14\!-\!134, 24\!-\!234, 124\!-\!1234\}, \label{eq:D5-E3}\tag{142}\] where the condition is the presence of \(4\). Therefore \[\,\#E(\mathscr{B}_4)=\#E_4+\#E_1+\#E_2+\#E_3=8+4+4+4=20. \label{eq:D5-half-edges}\tag{143}\] The graph \(\Gamma(D_5)\) consists of a \(c\)-copy and a \(d\)-copy of this graph, glued at \(\varnothing=w_0\). In the \(c\)-copy the abstract label \(4\) is the simple reflection \(s_4=s_{r-1}\), while in the \(d\)-copy it is \(s_5=s_r\); the labels \(1,2,3\) are the same in both copies. Thus \[\#V(\Gamma(D_5))=1+2(2^4-1)=31, \qquad \#E(\Gamma(D_5))=2\cdot20=40. \label{eq:D5-vertices-edges}\tag{144}\]

The internal root-poset graphs for individual vertices also have the predicted two-level form. For \(I=\{1,3\}\), one has \(p_I=(1\;3\;5)\), and Proposition 2 gives \[\begin{align} A_{13}&=\{e_2-e_3, e_4-e_5\}, \tag{145}\\ B_{13}&=\{e_1-e_2, e_1-e_3, e_1-e_4, e_1-e_5\}. \tag{146} \end{align}\] By Proposition 4, the outgoing arrows are \[\begin{align} e_2-e_3&\longrightarrow e_1-e_2, e_1-e_3, e_1-e_4, e_1-e_5, \tag{147}\\ e_4-e_5&\longrightarrow e_1-e_4, e_1-e_5. \tag{148} \end{align}\] No arrow starts from any element of \(B_{13}\). Thus \(\nu_1(c_{13})=A_{13}\) and \(\nu_2(c_{13})=\varnothing\).

For \(I=\{2,4\}\), the cycle is \(p_I=(2\;4\;5)\) and \[A_{24}=\{e_3-e_4\}, \qquad B_{24}=\{e_2-e_3,e_2-e_4,e_2-e_5\}. \label{eq:D5-A24-B24}\tag{149}\] The unique \(A\)-vertex points to all three \(B\)-vertices: \[e_3-e_4\longrightarrow e_2-e_3, \,e_2-e_4, \,e_2-e_5. \label{eq:D5-arrows-24}\tag{150}\] For the Coxeter-type subset \(I=\{1,2,3,4\}\), all gaps have length zero, so \[A_{1234}=\varnothing, \qquad B_{1234}=\{e_1-e_2,e_1-e_3,e_1-e_4,e_1-e_5\}, \qquad \nu_1(c_{1234})=\varnothing. \label{eq:D5-c1234-nu}\tag{151}\] These three examples represent the possible internal shapes in \(D_5\): two \(A\)-levels, one \(A\)-level, and no \(A\)-level. Applying \(\tau\) gives the corresponding \(d\)-vertices without changing the abstract two-level structure.

13 Consequences for rational normal forms↩︎

Voloshyn’s theorem associates a rational decomposition of the form 1 to every rational Weyl group element. Theorem 1 therefore gives the complete type-\(D\) list of Weyl-group indices for which Voloshyn’s construction produces rational normal forms. In the present notation this list is \[\overline{w_0}, \qquad \overline{c_I},\;\overline{d_I} \quad(\varnothing\neq I\subseteq [r-1]), \label{eq:normal-form-list}\tag{152}\] where bars denote representatives in \(N_G(H)\). Thus the number of such normal forms in type \(D_r\) odd is exactly \(2^r-1\).

The two-level proof of Proposition 3 also bounds the stabilization depth of Voloshyn’s recursive construction for the elements \(c_I,d_I\). Since 48 implies 49 , the recursion stabilizes after at most two non-trivial root-poset passes for every element in the two cyclic families: \[\nu_0(c_I)\supseteq \nu_1(c_I)=A_I\supseteq \nu_2(c_I)=\varnothing, \qquad \nu_0(d_I)\supseteq \nu_1(d_I)=\tau(A_I)\supseteq \nu_2(d_I)=\varnothing. \label{eq:stabilization-depth}\tag{153}\] This is stronger than mere rationality. It says that the possible obstruction graph has no path of length two, not only no directed cycle. Equivalently, for every \(I\neq\varnothing\), the adjacency relation inside \(\Gamma_{c_I}\) and \(\Gamma_{d_I}\) satisfies \[\alpha\to\beta\to\gamma \quad\Longrightarrow\quad \text{no such }(\alpha,\beta,\gamma)\text{ exists}. \label{eq:no-two-path}\tag{154}\] In applications where the recursive maps \(B_k,N_k\) of [1] are written explicitly, 153 is the combinatorial reason that no third correction step appears for type-\(D\) rational elements.

14 Conclusion↩︎

We have proved that, for odd \(r\geq 5\), the rational Weyl group elements of type \(D_r\) are exactly \[w_0, \qquad c_I,d_I\quad(\varnothing\neq I\subseteq [r-1]), \label{eq:conclusion-family}\tag{155}\] where \(c_I,d_I\) are the two signed cyclic elements defined in 10 –11 . This proves the closed formula \[\#\{u\in W(D_r):u\text{ rational}\}=2^r-1 \label{eq:conclusion-count}\tag{156}\] and determines the entire rationality graph. The graph is the union of two subset graphs glued at \(w_0\), and its two leaves are precisely \[c_{\{1\}}: e_1\mapsto -e_r,\;e_j\mapsto -e_j\;(2\leq j\leq r-1),\;e_r\mapsto e_1, \label{eq:leaf-c}\\[-2mm]\tag{157}\] \[d_{\{1\}}: e_1\mapsto e_r,\;e_j\mapsto -e_j\;(2\leq j\leq r-1),\;e_r\mapsto -e_1. \label{eq:leaf-d}\tag{158}\] The classification reduces the type-\(D\) rationality problem to a rigid and explicit subset calculus. It also suggests that the more irregular type-\(A\) enumerations in [1] cannot be explained by a single-cycle signed model of this kind; the type-\(D\) odd case is exceptional because the outer automorphism at the two spin nodes produces exactly two glued copies of the same subset graph.

Data availability↩︎

No data were used in this theoretical study.

Funding↩︎

No funding was received for this work.

Declaration of competing interest↩︎

The author placeholders declare no competing interests.

Data availability↩︎

No datasets were generated or analyzed in this work.

Declaration of Generative AI and AI-Assisted Technologies in the Writing Process↩︎

During the preparation of this work, the authors used DeepSeek to build a specialized agent for solving mathematical problems, which was employed to generate an initial proof of the main theorem. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the content of the published article.

References↩︎