A combinatorial approach to the index
of seaweed subalgebras of Kac–Moody algebras
April 02, 2025
Abstract
In 2000, Dergachev and Kirillov introduced subalgebras of "seaweed type" in \(\mathfrak{gl}_n\) (or \(\mathfrak{sl}_n\)) and computed their index using certain graphs. Then seaweed subalgebras \(\mathfrak q\subset\mathfrak g\) were defined by Panyushev for any reductive \(\mathfrak g\). A few years later Joseph generalised this notion to the setting of (untwisted) affine Kac–Moody algebras \({\widehat{\mathfrak g}}\). Furthermore, he proved that the index of such a seaweed can be computed by the same formula that had been known for \(\mathfrak g\). In this paper, we construct graphs that help to understand the index of a seaweed \(\mathfrak q\subset{\widehat{\mathfrak g}}\), where \({\widehat{\mathfrak g}}\) is of affine type A or C.
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1 Introduction↩︎
Since the pioneering work of Dergachev and Kirillov [1], seaweed subalgberas of reductive Lie algebras have attracted a great deal of attention [2]–[9]. The seaweeds form a wide class of Lie algebras, which includes all parabolics and their Levi subalgebras. Unlike an arbitrary Lie algebra, it is reasonable to study the coadjoint action of a seaweed. An important numerical characterisation of the coadjoint representation is the index. In this paper, we study the index of a finite-dimensional seaweed in the Kac–Moody setting from a combinatorial point of view.
The index of a Lie algebra \({\mathfrak q}\), \({\mathsf{ind\,}}{\mathfrak q}\), is the minimal dimension of a stabiliser for the coadjoint representation of \({\mathfrak q}\). It can be regarded as a generalisation of the notion of rank. That is, \({\mathsf{ind\,}}{\mathfrak q}\) equals the rank of \({\mathfrak q}\), if \({\mathfrak q}\) is reductive. If \(\mathfrak q\) is non-reductive, then a calculation of \({\mathsf{ind\,}}\mathfrak q\) can be difficult. Nevertheless, there are classes of Lie algebras, for which the index can be determined. The seaweeds comprise one of these classes.
For \({\mathfrak{gl}}_n\), the seaweed subalgebras have been introduced in [1] and their index has been computed using certain graphs, which are said to be type-A meander graphs. A general definition suited for arbitrary reductive Lie algebras \({\mathfrak g}\) appears in [2]. Namely, if \({\mathfrak p}_1,{\mathfrak p}_2\subset {\mathfrak g}\) are parabolic subalgebras such that \({\mathfrak p}_1+{\mathfrak p}_2={\mathfrak g}\), then \({\mathfrak q}={\mathfrak p}_1\cap{\mathfrak p}_2\) is called a seaweed in \({\mathfrak g}\). (For this reason, some people began to use later the term “biparabolic subalgebra" for such \({\mathfrak q}\).)
An inductive (reduction) procedure for computing the index of the seaweeds in the classical Lie algebras is introduced in [2]. It helps to answer some subtle questions on the coadjoint action [4], [7]. A general algebraic formula for the index of the seaweeds had been proposed by Tauvel and Yu [3] and then proved by Joseph [5]. For these reasons we call it the Tauvel–Yu–Joseph ( TYJ for short) formula. Then it was shown in [6] that the same formula applies in the setting of (untwisted) affine Kac–Moody algebras \({\widehat{\mathfrak g}}\).
Let now \(\mathfrak p_1\subset{\widehat{\mathfrak g}}\) be a proper standard parabolic subalgebra and \(\mathfrak p_2\subset{\widehat{\mathfrak g}}\) the opposite of a proper standard parabolic. Then \({\widehat{\mathfrak q}}=\mathfrak p_1\cap\mathfrak p_2\) is a finite-dimensional Lie algebra, see Section 2 for an explanation. We say that \({\widehat{\mathfrak q}}\) is a seaweed in \({\widehat{\mathfrak g}}\). If \({\widehat{\mathfrak g}}\) is of affine type A, then \({\widehat{\mathfrak q}}\) has an easy description as a semi-direct product, see Proposition 2.
By construction, \({\widehat{\mathfrak g}}\) is a \(2\)-dimensional extension of the loop algebra \(\mathfrak g[t,t^{-1}]=\mathfrak g\otimes\Bbbk[t,t^{-1}]\), see 1 . The central part of this extension, \(\Bbbk C\), is contained in \({\widehat{\mathfrak q}}\), but not in \([{\widehat{\mathfrak q}},{\widehat{\mathfrak q}}]\) [6]. Therefore there is a subalgebra \(\bar{\mathfrak q}\subset{\widehat{\mathfrak q}}\) such that \(\bar{\mathfrak q}\cong{\widehat{\mathfrak q}}/\Bbbk C\) and \({\widehat{\mathfrak q}}=\bar{\mathfrak q}\oplus\Bbbk C\). Clearly \({\mathsf{ind\,}}{\widehat{\mathfrak q}}=1+{\mathsf{ind\,}}\bar{\mathfrak q}\). For \(\mathfrak g\) equal to \(\mathfrak{sl}_n\) or \(\mathfrak{sp}_{2n}\), we interpret \({\mathsf{ind\,}}\bar{\mathfrak q}\) in terms of the type-A and type-C affine meander graphs \(\Gamma\) and \(\Gamma^{\sf C}\). Our proofs are based on the TYJ formula.
The graphs \(\Gamma\) and \(\Gamma^{\sf C}\) have \(n\) and \(2n\) vertices, respectively, which are placed on a circle. Some edges are drawn outside of the circle and the other ones inside. The outside edges are defined by the first parabolic \(\mathfrak p_1\) and the inside ones by \(\mathfrak p_2\). A particular attention must be paid to the position of the centre \(o\) of the circle, see Figure 1. Then one counts the numbers of segments and cycles in the graph and checks, whether \(o\) lies inside of a cycle. This information is used in a combinatorial formula for the index, see Theorems 6 and 11.
Our combinatorial approach simplifies calculations of \({\mathsf{ind\,}}\bar{\mathfrak q}\). For example, if \({\widehat{\mathfrak q}}\) is the intersection of two maximal standard parabolics in \(\widehat{\mathfrak{sl}_n}\), then \[\bar{\mathfrak q}=(\mathfrak s(\mathfrak{gl}_d\oplus\mathfrak{gl}_{n-d})\oplus\Bbbk d)\ltimes (2\Bbbk^d{\otimes}(\Bbbk^{n-d})^*),\] where \(2\Bbbk^d{\otimes}(\Bbbk^{n-d})^*=\Bbbk^d{\otimes}(\Bbbk^{n-d})^*\oplus \Bbbk^d{\otimes}(\Bbbk^{n-d})^*\) is an Abelian ideal of \(\bar{\mathfrak q}\) and \({\mathrm{ad}}(d)\) acts on its direct summands with eigenvalues \(0\) and \(-1\), see Example 3. We have \[{\mathsf{ind\,}}\bar{\mathfrak q}=\gcd(n,2d)-\iota,\] where \(\iota=0\) if \(\gcd(n,2d)\) divides \(d\) and \(\iota=2\) otherwise, see Example 9. The formula for \(\iota\) shows that describing \(\iota\) is a non-trivial task, which would be more difficult without a graph.
In Section 5, we explain that the TYJ formula can be used for justifications of the graph interpretations for \({\mathsf{ind\,}}\mathfrak q\), where \(\mathfrak q\subset\mathfrak g\) is a usual, not affine, seaweed. The original type A approach of [1] is generalised in [8], [9] to all classical types. Although the TYJ formula is used in [9], proofs in [1], [8], [9] rely heavily on the inductive procedures, in particular, on the one developed in [2].
Throughout the paper, the ground field \(\Bbbk\) is of characteristic zero.
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Acknowledgement. I am grateful to Mamuka Jibladze for inspiring conversations.
2 Seaweed subalgebras of Kac–Moody algebras↩︎
Let \({\mathfrak g}\) be a simple finite-dimensional non-Abelian Lie algebra. We fix a Borel subalgebra \(\mathfrak b\subset \mathfrak g\) and a Cartan subalgebra \(\mathfrak t\subset\mathfrak b\). Set \(r={\mathsf{rk\,}}\mathfrak g=\dim\mathfrak t\). Let \(\Pi=\{\alpha_1,\ldots,\alpha_r\}\) be the set of simple roots associated with \((\mathfrak b,\mathfrak t)\) and \(\lambda\in\mathfrak t^*\) the highest root.
Let \({\widehat{\mathfrak g}}\) be an affine Kac–Moody algebra with a set of simple roots \(\widehat{\Pi}=\Pi\sqcup\{\alpha_0\}\). Then \[\label{KM-sum} {\widehat{\mathfrak g}}=\mathfrak g[t,t^{-1}]\oplus \Bbbk C\oplus\Bbbk d,\tag{1}\] where \(C\) is a central element and \([d,\xi t^k]=k\xi t^k\). Other commutator relations in \({\widehat{\mathfrak g}}\) are described in [10]. By the construction, \(\mathfrak h=\mathfrak t\oplus \Bbbk C\oplus\Bbbk d\) is a Cartan subalgebra of \({\widehat{\mathfrak g}}\).
For a root \(\alpha\) of \((\mathfrak g,\mathfrak t)\), let \(e_\alpha\in\mathfrak g\) be a root vector. Set \(\widehat{\mathfrak b}=(\mathfrak b+\mathfrak h)\oplus t{\mathfrak g}[t]\) and \(e_{-\alpha_0}=e_{\lambda} t^{-1}\in{\mathfrak g}[t^{-1}]\). To a subset \(S\subset \widehat{\Pi}\), one associates a standard parabolic \(\mathfrak p=\mathfrak p(S)\subset{\widehat{\mathfrak g}}\), which is generated by \(\widehat{\mathfrak b}\) and \(\{e_{-\beta}\mid \beta\in S\}\). If \(S=\widehat{\Pi}\), then clearly \(\mathfrak p={\widehat{\mathfrak g}}\).
For convenience of the reader, we give a description of \(\mathfrak p(S)\) with a proper subset \(S\). Let \(\mathfrak p(S\cap \Pi)\subset\mathfrak g\) be the standard parabolic associated with \(S\cap\Pi\) and \(\mathfrak n(S\cap \Pi)\) the nilpotent radical of \(\mathfrak p(S\cap \Pi)\). Then let \(\mathfrak z\subset \mathfrak n(S\cap \Pi)\) be the centre of \(\mathfrak n(S\cap \Pi)\).
Lemma 1. Suppose that \(S\ne \widehat{\Pi}\).
(i) If \(\alpha_0\not\in S\), then \(\mathfrak p(S)=t\mathfrak g[t]+\mathfrak h+\mathfrak p(S\cap \Pi)\).
(ii) If \(\alpha_0\in S\), then \(\mathfrak p(S)=t\mathfrak g[t]+\mathfrak h+(\mathfrak p(S\cap \Pi)\ltimes \mathfrak z t^{-1})\), where \(\mathfrak z
t^{-1}\) is an Abelian ideal of \(\mathfrak p(S\cap \Pi)\ltimes \mathfrak z t^{-1}\).
Proof. The positive part \(t\mathfrak g[t]\) of \({\widehat{\mathfrak g}}\) is contained in \(\widehat{\mathfrak b}\) and hence in any standard parabolic subalgebra. Since \(\alpha_0(d)=1\) and \([d,\mathfrak g]=0\), we have \(\mathfrak p(S)\cap \mathfrak g=\mathfrak p(S\cap\Pi)\). Part (i) is now clear.
Suppose \(\alpha_0\in S\). Then \(\Pi\not\subset S\). Standard facts from the theory of simple Lie algebras are that \(\lambda=\sum_{i=1}^r n_i \alpha_i\) with \(n_i\geqslant 1\) and that \(\beta= \sum_{i=1}^r c_i \alpha_i\) with \(0\leqslant c_i\leqslant n_i\) for any positive root \(\beta\) of \((\mathfrak g,\mathfrak t)\). Therefore no combination \[-k\alpha_0+\left(\sum_{i=1}^{r} d_i \alpha_i\right)-\sum_{\alpha_i\in S\cap\Pi} k_i \alpha_i\] with \(k\geqslant 2\) and \(d_i, k_i\geqslant 0\) is an \(\mathfrak h\)-weight of \({\widehat{\mathfrak g}}\). Hence \(\mathfrak p(S)\cap t^{-1}\mathfrak g[t^{-1}]=:\mathfrak p_{-1}\) is contained in \(\mathfrak g t^{-1}\). Since \(\mathfrak p(S)\) is a Lie subalgebra, we have \([\mathfrak p(S\cap \Pi),\mathfrak p_{-1}]\subset\mathfrak p_{-1}\) and \([\mathfrak p_{-1},\mathfrak p_{-1}]=0\).
We have \(\mathfrak p_{-1}=Vt^{-1}\), where \(V\) is the sum of \(\mathfrak g_\beta\) with \(\beta\in U:=\lambda +\langle S\cap\Pi\rangle_{{\mathbb{Z}}}\). Hence \(V\) is contained in the nilpotent radical \(\mathfrak n(S\cap\Pi)\). For any \(\beta\in U\) and any \(\alpha_i\not\in S\), the sum \(\beta+\alpha_i\) is not a root of \(\mathfrak g\). Hence \([e_{\alpha_i},V]=0\). This is true for all elements of \(\mathfrak n(S\cap \Pi)\). Hence \(V\subset\mathfrak z\).
Let \(\mathfrak l(S\cap\Pi)\subset\mathfrak p(S\cap\Pi)\) be the standard Levi subalgebra. Note that \(V\) is a simple \(\mathfrak l(S\cap\Pi)\)-module with the highest weight \(\lambda\). Clearly \([\mathfrak l(S\cap\Pi),\mathfrak z]\subset\mathfrak z\).
Assume that \(V_1\subset\mathfrak z\) is a non-zero simple \(\mathfrak l(S\cap\Pi)\)-module with a highest weight vector \(e_\gamma\), where \(\gamma\ne\lambda\). Since \(\gamma\ne\lambda\), where is \(\alpha_i\in\Pi\) such that \([e_{\alpha_i},e_\gamma]\ne 0\). If \(\alpha_i\in S\), then this contradicts the assumption that \(\gamma\) is a highest weight for \(\mathfrak l(S\cap\Pi)\). If \(\alpha_i\not\in S\), then \(e_{\alpha_i}\in\mathfrak n(S\cap\Pi)\) and \([e_{\alpha_i},V_1]=0\), because \(V_1\subset\mathfrak z\). Thus \(V=\mathfrak z\). ◻
Let \(\omega\) be an involution of \({\widehat{\mathfrak g}}\) such that \(\omega|_{\mathfrak h}=-{\rm id}_{\mathfrak h}\) and \(\omega(\mathfrak g t^k)=\mathfrak g t^{-k}\) for all \(k\in\mathbb{Z}\). Then the opposite parabolic \(\mathfrak p^-\) is defined by \(\mathfrak p^-=\omega(\mathfrak p)\). Having two standard parabolics \(\mathfrak p=\mathfrak p(S)\) and \(\mathfrak r=\mathfrak p(S')\), we define a seaweed subalgebra \({\widehat{\mathfrak q}}=\mathfrak p\cap\mathfrak r^-=:{\widehat{\mathfrak q}}(S,S')\). The description of Lemma 1 implies that \({\widehat{\mathfrak q}}\) is finite-dimensional if and only if both \(\mathfrak p,\mathfrak r\) are smaller than \({\widehat{\mathfrak g}}\), i.e., if \(S\ne\widehat{\Pi}\) and \(S'\ne\widehat{\Pi}\). In the following, we always assume that both subsets \(S\) and \(S'\) are proper. Clearly \(C\in {\widehat{\mathfrak q}}\), but \(C\not\in[{\widehat{\mathfrak q}},{\widehat{\mathfrak q}}]\) by [6]. This implies that \({\mathsf{ind\,}}\bar{\mathfrak q}={\mathsf{ind\,}}{\widehat{\mathfrak q}}-1\) for \(\bar{\mathfrak q}={\widehat{\mathfrak q}}/\Bbbk C\).
A formula for the index of \(\bar{\mathfrak q}\) is given in [6]. It is an analogue of the formula for the index of a seaweed in \(\mathfrak g\) suggested by Tauvel and Yu [3] and proven initially in [5].
Let \({\mathcal{K}}({\mathfrak l}(S))=:{\mathcal{K}}(S)\subset\mathfrak h^*\) be the cascade of strongly orthogonal roots (= Kostant’s cascade) in the Levi subalgebra \({\mathfrak l}(S)\subset\mathfrak p(S)\), see [3], [11] for the details. In particular, \({\mathcal{K}}({\mathfrak g})={\mathcal{K}}(\Pi)\subset\mathfrak t^*\) is the cascade in \({\mathfrak g}\). Set \(\hat{{\mathfrak t}}=\mathfrak t\oplus\Bbbk d\). Let \(\mathfrak t_{{\mathbb{R}}}\subset\mathfrak t\) be the standard real form, i.e., \({\mathrm{ad}}(h)\) has real eigenvalues for each \(h\in\mathfrak t_{{\mathbb{R}}}\). Then set \(\hat{{\mathfrak t}}_{{\mathbb{R}}}={\mathfrak t}_{{\mathbb{R}}}\oplus{\mathbb{R}}d\). Since each root of \({\widehat{\mathfrak g}}\) is zero on \(C\), we may safely assume that \({\mathcal{K}}(S)\subset\hat{{\mathfrak t}}^*_{{\mathbb{R}}}\). Then let \(E_S\) be the \({\mathbb{R}}\)-linear span of \({\mathcal{K}}(S)\) in \(\hat{{\mathfrak t}}^*_{{\mathbb{R}}}\). We have \(\dim E_S=\# {\mathcal{K}}(S)\), since the elements of a cascade are linearly independent. The TYJ formula reads now: \[\label{eq:tau-yu} {\mathsf{ind\,}}\bar{{\mathfrak q}}(S,S')=|\widehat{\Pi}| + \dim E_S +\dim E_{S'} -2\dim (E_S+E_{S'}) ,\tag{2}\] see [3], [5], [6], and also [9]. If \(S=S_1\sqcup S_2\), where the subsets \(S_1\) and \(S_2\) are orthogonal, then clearly \({\mathcal{K}}(S)={\mathcal{K}}(S_1)\sqcup{\mathcal{K}}(S_2)\).
We record the data on the cascades in \({\mathfrak{gl}}_n\) and \(\mathfrak{sp}_{2n}\). Let \(\varepsilon_i\in{\mathfrak{gl}}_n^*\) with \(1\leqslant i\leqslant n\) be the standard linear functions such that \(\varepsilon_i(A)=a_{ii}\) for a matrix \(A=(a_{ij})\). We regard each \(\varepsilon_i\) also as a linear function on the standard Cartan subalgebra of \({\mathfrak{gl}}_n\). Similarly for \(\mathfrak g=\mathfrak{sp}_{2n}\), let \(\{\varepsilon_1,\ldots,\varepsilon_n\}\subset\mathfrak t^*\) be the commonly used basis. We have \[\label{eq:K-Pi-gl} {\mathcal{K}}({{\mathfrak{gl}}_n})=\{\varepsilon_i-\varepsilon_{n+1-i}\mid i=1,\dots, \left\lfloor \frac{n}{2}\right\rfloor\} \,\text{ and } \;\# {\mathcal{K}}({{\mathfrak{gl}}_n})=\left\lfloor \frac{n}{2}\right\rfloor;\tag{3}\] as well as \[\label{eq:K-Pi-sp} {\mathcal{K}}({\mathfrak{sp}_{2n}})=\{2\varepsilon_i \mid i=1,\dots, n\} \,\text{ and } \;\# {\mathcal{K}}({\mathfrak{sp}_{2n}})=n.\tag{4}\]
2.1 Seaweed subalgebras in type \(\widetilde{{\sf A}}_r\)↩︎
Suppose that \({\mathfrak g}=\mathfrak{sl}_{r+1}\). Let \({\widehat{\mathfrak q}}(S,S')\subset{\widehat{\mathfrak g}}\) be a standard finite-dimensional affine seaweed. The extended Dynkin diagram of type \(\widetilde{{\sf A}}_r\) is a cycle. Therefore we can change the enumeration of simple roots cyclicly. Since \(S'\) is a proper subset of \(\widehat{\Pi}\), we may assume that \(\alpha_0\not\in S'\). Let \(\mathfrak r^-\subset\mathfrak g\) be the parabolic subalgebra generated by \(\mathfrak b^-=\omega(\mathfrak b)\) and \(\{e_{\beta} \mid \beta\in S'\}\).
Proposition 2. Suppose \(S'\subset\Pi\). Then a finite-dimensional standard seaweed subalgebra \({\widehat{\mathfrak q}}={\widehat{\mathfrak q}}(S,S')\subset{\widehat{\mathfrak g}}\) is isomorphic either to \(\mathfrak q \oplus\Bbbk d\oplus\Bbbk C\) or to a semi-direct product \((\mathfrak q \oplus\Bbbk d\oplus\Bbbk C)\ltimes \mathfrak z t^{-1}\), where \(\mathfrak q=\mathfrak p(S\cap\Pi)\cap \mathfrak r^-\) is a seaweed in \({\mathfrak g}\) and \(\mathfrak z\) is the centre of \(\mathfrak n(S\cap\Pi)\).
Proof. By Lemma 1, \(\mathfrak p(S')^-=t^{-1}{\mathfrak g}[t^{-1}]+\mathfrak h+\mathfrak r^-\) and either \(\mathfrak p(S)=t\mathfrak g[t]+\mathfrak h+\mathfrak p(S\cap \Pi)\) or \(\mathfrak p(S)=t\mathfrak g[t]+\mathfrak h+(\mathfrak p(S\cap \Pi)\ltimes \mathfrak z t^{-1})\). Since \({\widehat{\mathfrak q}}=\mathfrak p(S)\cap\mathfrak p(S')^-\), the result follows. ◻
Example 3. Suppose that \(S=\widehat{\Pi}\setminus\{\alpha_d\}\) with \(1\leqslant d\leqslant r\) and \(S'=\Pi\). Then \[\mathfrak p(S\cap\Pi)=\mathfrak s(\mathfrak{gl}_d\oplus\mathfrak{gl}_{n-d})\ltimes (\Bbbk^d{\otimes}(\Bbbk^{n-d})^*)\] with \(\mathfrak z=\mathfrak n(S\cap\Pi)=\Bbbk^d{\otimes}(\Bbbk^{n-d})^*\). By Proposition 2, we have \[\bar{\mathfrak q}(S,\Pi)=(\mathfrak s(\mathfrak{gl}_d\oplus\mathfrak{gl}_{n-d})\oplus\Bbbk d)\ltimes (2\Bbbk^d{\otimes}(\Bbbk^{n-d})^*),\] where \(2\Bbbk^d{\otimes}(\Bbbk^{n-d})^*=\mathfrak z\oplus\mathfrak z t^{-1}\), \([d,\mathfrak z]=0\), and \({\mathrm{ad}}(d)\) acts on \(\mathfrak z t^{-1}\) as \(-{\rm id}\).
2.2 Construction of a graph in type \(\widetilde{{\sf A}}_r\)↩︎
Suppose that \(S=\{\alpha_i \mid i\not\in I\}\), \(S'=\{\alpha_i \mid i\not\in I'\}\), where \(I,I'\subset\{0,1,\ldots,r\}\). We put \(n=r+1\) nodes on a circle, as vertices of a regular \(n\)-gon, labelling them consequently clockwise with the cosets \(0+n{\mathbb{Z}}\), \(1+n{\mathbb{Z}},\ldots, r+n{\mathbb{Z}}\). For \(a\in{\mathbb{Z}}\), set \(\overline{a}=a+n{\mathbb{Z}}\). Let \(o\) be the centre of the circle.
Suppose that \(I=\{i_1,\ldots,i_a\}\), where \(0\leqslant i_1<i_2<\ldots < i_a\leqslant r\). Then the node \(\overline{i_1+1}\) is connected by an arc to the node \(\overline{i_2}\) on the outside of the circle. Say that this arc goes from \(\overline{i_1+1}\) to \(\overline{i_2}\). We continue on the outside joining the vertex \(\overline{i_1+2}\) with \(\overline{i_2-1}\), the vertex \(\overline{i_1+3}\) with \(\overline{i_2-2}\) and so on. If \(i_2-i_1\) is odd, then there is a middle node with no adjacent outside arc. For an arc of the graph going from \(A\) to \(B\), we call the arc of the circle going from \(A\) to \(B\) clockwise its shadow. It is essential that \(o\) lies outside the area bounded by an arc and its shadow.
Then we do the same for the pairs \((i_2,i_3), \ldots, (i_{a-1},i_a), (i_a,i_1)\). If \(a=1\), then we have just one pair \((i_1,i_1)\). Different outside arcs are disjoint. The same procedure applies to \(I'\), but the outside arcs are replaced here with inside arcs. The rule concerning \(o\) remains the same. Figure 1 explains this. Let \(\Gamma=\Gamma(S,S')\) be the resulting graph.
Without loss of generality, we assume that the arcs of \(\Gamma\) meet only in vertices. The nodes of \(\Gamma\) can be labeled by representatives of the cosets \(a+n{\mathbb{Z}}\).
Figure 1: Rules concerning \(o\).
In Figure 2, we present three graphs \(\Gamma\). For the first one, \(n=10\), \(I=\{9\}\), and \(I'=\{4,8\}\). For the second one, \(n=9\), \(I=\{3,8\}\), and \(I'=\{2,6\}\). Finally for the third one, \(n=8\), \(I=\{1,5\}\), \(I'=\{3,7\}\).
Figure 2: Examples of graphs \(\Gamma\).
Each node of \(\Gamma\) is a vertex of at most \(2\) arcs. Therefore a connected component of \(\Gamma\) is either a segment or a cycle. There are no loops in \(\Gamma\). But note that \(\Gamma\) does not have to be a simple graph. It may contain a cycle with two vertices \(i,j\) and two arcs \((i,j)_{\sf outs}\), \((i,j)_{\sf ins}\), where the first one is an outside and the second an inside arc. In that case, “an arc \((i,j)\) of \(\Gamma\)" refers to any of the two.
3 Combinatorial interpretation of the index in type \(\widetilde{{\sf A}}_r\)↩︎
In this section, \(\mathfrak g=\mathfrak{sl}_{n}\). We choose \(\{1,\ldots,n\} \subset{\mathbb{Z}}\) as a set of representatives for \({\mathbb{Z}}/n{\mathbb{Z}}\). Now the vertices of \(\Gamma\) are numbered from \(1\) to \(n\). To the arc of the circle joining a vertex \(i\) with \(i{+}1\) clockwise we assign the simple root \(\alpha_i\). The arc joining \(n\) and \(1\) is labeled with \(\alpha_0\).
Recall that \(\varepsilon_j\in\mathfrak{gl}_n^*\) with \(1\leqslant j\leqslant n\) are the standard linear functions. We regard them also as functions on \({\mathfrak t}^*\) and on \(\mathfrak h\) by setting \(\varepsilon_i(d)=\varepsilon_i(C)=0\) for each \(i\). Of course, \(\sum_{i=1}^n \varepsilon_i\) is zero on the Cartan subalgebra \(\mathfrak t\subset\mathfrak{sl}_n\). This circumstance makes several arguments more involved.
Set \(\delta=\sum_{i=0}^r \alpha_i \in\mathfrak h^*\), where \(r=n-1\). Then \(\delta\) is an imaginary root such that \(\delta|_{\mathfrak t}=0\), \(\delta(d)=1\), and \(\delta(C)=0\), while \(\alpha_0=\varepsilon_n-\varepsilon_1+\delta\) and \(\alpha_i=\varepsilon_{i}-\varepsilon_{i+1}\) for \(i>0\). In this terms, the label of an arc \((i,i+1)\) is \(\varepsilon_{i}-\varepsilon_{i+1}\) for \(1\leqslant i\leqslant r\).
There is a bijections between the outside arcs of \(\Gamma\) and \({\mathcal{K}}(S)\). Namely, an arc \((i,j)\) going from a vertex \(i\) to \(j\) corresponds to \[\sum_{u\in[i,j-1]}\alpha_u, \; \text{ where } \;[i,j-1]=\{i,i+1,\ldots,r,0,1,\ldots,j-1\} \;\text{ if } j<i.\] The sum is equal to \(\varepsilon_i-\varepsilon_j\) if \(j>i\) and to \(\varepsilon_i-\varepsilon_j+\delta\), if \(j<i\), cf. 3 . The similar bijection exists between \({\mathcal{K}}(S')\) and the inside arcs. Uniformly, we assign to an arc of \(\Gamma\) the sum of simple roots belonging to its shadow.
Let \((i,j)\) be an arc of \(\Gamma\), which may go from \(i\) to \(j\) or from \(j\) to \(i\). Let \(\beta(i,j)\in {\mathcal{K}}(S)\cup {\mathcal{K}}(S')\) be the element assigned to it. Then \[\beta(i,j)\in\{\varepsilon_i-\varepsilon_j, \varepsilon_j-\varepsilon_i, \varepsilon_i-\varepsilon_j+\delta, \varepsilon_j-\varepsilon_i+\delta \}.\]
Lemma 4. Let \(B\subset \Gamma\) be a connected subgraph consisting of the pairwise distinct vertices \(j_1,\ldots,j_m\) and the edges \((j_i,j_{i+1})\) with \(1\leqslant i<m\). Then the set \[\bar K(B):=\{ \beta(j_i,j_{i+1})|_{\mathfrak t} \mid 1\leqslant i<m\} \subset\mathfrak t^*\] is linearly independent and \(|\bar K(B)|=m-1\).
Proof. Suppose that \(X=\sum_{i=1}^{m-1} c_i \bar\beta(j_i,j_{i+1})=0\) for some \(c_i\in\Bbbk\). We have \[X=\pm c_1 \varepsilon_{j_1}+ \left(\sum_{i=2}^{m-1} (\pm c_{i-1}\pm c_i) \varepsilon_{j_i}\right) \pm c_{m-1} \varepsilon_{j_m}.\] Each difference \(\varepsilon_i-\varepsilon_j\) is zero on the centre \(\Bbbk I_n\subset\mathfrak{gl}_n\). Hence \(X\), regarded an an element of \(\mathfrak{gl}_n^*\), is zero on \(\Bbbk I_n\). Thereby \(X\) is zero on \(\mathfrak{gl}_n\) and \(c_1=c_{m-1}=0\). Furthermore, \(c_{i-1}=\pm c_{i}\) for \(2\leqslant i \leqslant m-1\). Thus, \(c_i=0\) for each \(i\). ◻
For a connected component \(B\) of \(\Gamma\), let \({\mathcal{V}}(B)\) be the set of vertices of \(B\) and \({\mathcal{E}}(B)\) the set of edges. Set \(K(B)=\{ \beta(i,j) \mid (i,i)\in {\mathcal{E}}(B)\}\subset\mathfrak h^*\) and \[\bar K(B)= \{ \beta(i,j)|_{\mathfrak t} \mid (i,i)\in {\mathcal{E}}(B)\}\subset\mathfrak t^*.\]
Lemma 5. Let \(B\) be a cycle in \(\Gamma\) that does not contain \(o\) in its interior. Let \(j_1,\ldots,j_m\) be the consecutive vertices of \(B\). For each \(\boldsymbol{e}\in{\mathcal{E}}(B)\) joining a vertex \(j_i\) with \(j_{i+1}\) or \(j_m\) with \(j_1\), let \(c_{\boldsymbol{e}}\in\{1,-1\}\) be a coefficient defined by the following rule. If the shadow of \(\boldsymbol{e}\) goes from \(j_i\) to \(j_{i+1}\) (or from \(j_m\) to \(j_1\)) clockwise, then \(c_{\boldsymbol{e}}=1\), otherwise \(c_{\boldsymbol{e}}=-1\). Then \(Y(B):=\sum\limits_{\boldsymbol{e}\in{\mathcal{E}}(B)} c_{\boldsymbol{e}} \beta(\boldsymbol{e})=0\).
Proof. Consider an arrow \(\overrightarrow{og}\) starting at \(o\) and ending at a point \(g\) of \(B\). Let \(g\) move along \(B\) making one round trip. The arrow is moving accordingly, rotating around \(o\). Since \(o\) does not lie in the interior of \(B\), the total angle of the rotation is zero. We extend \(\overrightarrow{og}\) to a half-line \(L_{\overrightarrow{og}}\) and let \(\ell(g)\) be the intersection point of \(L_{\overrightarrow{og}}\) and the circle. Then the total distance, counted with a sign, traveled by \(\ell(g)\) on the circle is zero. The equality \(Y(B)=0\) is an analogue of this fact.
Assume that \(g=j_1\) at first and that it goes to \(j_2\), \(j_3\) and so on until \(j_m\) and then back to \(j_1\). When \(g\) moves from \(u\) to \(v\) along an edge \((u,v)\in {\mathcal{E}}(B)\), the point \(\ell(g)\) moves from \(u\) to \(v\) along the shadow of \((u,v)\). For each arc \((\overline{i},\overline{i+1})\) of the circle, \(\ell(g)\) travels from \(\overline{i}\) to \(\overline{i+1}\) clockwise as many times as from \(\overline{i+1}\) to \(\overline{i}\) anti-clockwise, because the total rotation angle is zero. Hence each \(\alpha_j\in\widehat{\Pi}\) appears in \(Y(B)\) with the zero coefficient and \(Y(B)=0\). ◻
Theorem 6. Let \({\widehat{\mathfrak q}}={\widehat{\mathfrak q}}(S,S')\) be a standard finite-dimensional seaweed subalgebra of \({\widehat{\mathfrak g}}\) for \(\mathfrak g=\mathfrak{sl}_n\). Set \(\bar{\mathfrak q}={\widehat{\mathfrak q}}/\Bbbk C\). Then \[{\mathsf{ind\,}}\bar{\mathfrak q}= 2\#\{\text{the cycles of}\;\Gamma\} + \#\{\text{the segments of \Gamma}\}-\iota\] for \(\Gamma=\Gamma(S,S')\), where \(\iota=0\) if \(\Gamma\) has no cycles with \(o\) in the interior and \(\iota=2\) otherwise.
Proof. We use 2 . The relation between the arcs of \(\Gamma\) and the elements of \({\mathcal{K}}(S)\) and \({\mathcal{K}}(S')\) shows that \(\dim E_S +\dim E_{S'}\) is equal to the number of arcs in \(\Gamma\), say \(a\). Since each node of \(\Gamma\) is a vertex of at most \(2\) arcs, we have \(a=n-\#\{\text{the segments of \Gamma}\}\).
In order to compute \(\dim(E_S+E_{S'})\), we describe relations among the elements of \({\mathcal{K}}(S,S'):={\mathcal{K}}(S)\cup-{\mathcal{K}}(S')\). The minus sign in this formula is used because \({\mathcal{K}}(S)\cap{\mathcal{K}}(S')\) does not have to be empty.
For each connected component \(B\) of \(\Gamma\), we have \(\bar K(B)\subset \langle \varepsilon_i \mid i \in {\mathcal{V}}(B)\rangle_{{\mathbb{R}}}\). If \(B_1\) and \(B_2\) are distinct components, then \({\mathcal{V}}(B_1)\cap{\mathcal{V}}(B_2)=\varnothing\). Since each difference \(\varepsilon_i-\varepsilon_j\) is zero on \(\Bbbk I_n\), a minimal relation among elements of \({\mathcal{K}}(S,S')\) is (up to signs) a non-trivial linear dependence among \(\bar\beta(\boldsymbol{e})=\beta(\boldsymbol{e})|_{\mathfrak t}\) with \(\boldsymbol{e}\in {\mathcal{E}}(B)\) for some \(B\). By Lemma 4, there are no such dependences if \(B\) is a segment and at most one if \(B\) is a cycle.
Let \(B\) be a cycle with the pairwise distinct vertices \(j_1,\ldots,j_m\) and the edges \((j_i,j_{i+1})\) with \(1\leqslant i\leqslant m\), where \((j_m,j_{m+1})\) is an arc connecting \(j_m\) with \(j_1\). Then \[\label{B} \sum\limits_{(u,v)\in{\mathcal{E}}(B)} c_{(u,v)} \bar\beta(u,v) =0\tag{5}\] for \(c_{(u,v)}\in \{1,-1\}\). Thus, \[\dim((E_S+E_{S'})|_{\mathfrak t}) = \dim E_S + \dim E_{S'} - \#\{\text{the cycles of}\;\Gamma\}.\]
It remains to compare \(\dim(E_S+E_{S'})\) and \(\dim((E_S+E_{S'})|_{\mathfrak t})\). These dimensions are equal if \(\delta\not\in E_S+E_{S'}\) and differ by \(1\) otherwise. For each cycle \(B\) of \(\Gamma\), set \(X(B)=\sum\limits_{\boldsymbol{e}\in{\mathcal{E}}(B)} c_{\boldsymbol{e}} \beta(\boldsymbol{e})\), where the coefficients \(c_{\boldsymbol{e}}\) are the same as in 5 . Then \(X(B)\in {\mathbb{R}}\delta\). Let \(F(B)\) be the face of \(\Gamma\) bounded by \(B\). Now we show that \(X(B)\ne 0\) if and only if \(o\in F(B)\).
Let \(C\) be the half-line starting at \(o\) and going through the middle of the circle arc \((n,1)\). Without loss of generality, assume that \(C\cap\Gamma\) is a finite set and that \(C\) intersects each arc of \(\Gamma\) at most once and transversally. Then \(C\) intersects the shadow of an arc \((i,j)\) of \(\Gamma\) if and only if it intersects \((i,j)\). Note that \(\beta(i,j)=\delta\pm(\varepsilon_i -\varepsilon_j)\) if and only if \(C\) intersects the shadow of \((i,j)\).
Suppose first that \(o\in F(B)\). Since \(C\) intersects \(B\) odd number of times, the number of \((u,v)\in{\mathcal{E}}(B)\) such that \(C\cap (u,v)\ne\varnothing\) is odd. Hence \(X(B) = k \delta\) for \(k\in 1+2{\mathbb{Z}}\).
Suppose now that \(o\not\in F(B)\). Then Lemma 5 implies that \(X(B)=0\). This can be seen in another way, if we define negative and positive crossings of \(C\) and \(B\). Roughly speaking, a crossing is positive, if \(C\) enters \(F(B)\) and negative otherwise. Depending on this, \(\beta(i,j)=\delta\pm(\varepsilon_i -\varepsilon_j)\) appears in \(X(B)\) either with coefficient \(1\) or \(-1\). Since the numbers of negative and positive crossings coincide, the coefficient of \(\delta\) in \(X(B)\) is zero.
Recall that a minimal relation among the elements of \({\mathcal{K}}(S,S')|_{\mathfrak t}\) is of the form 5 for some cycle \(B\). Thus, \(\delta\in E_S+E_{S'}\) if and only if \(\delta\in \langle K(B)\rangle_{{\mathbb{R}}}\) for a cycle \(B\). Now we have \(\dim(E_S+E_{S'})=\dim((E_S+E_{S'})|_{\mathfrak t}) + \frac{1}{2}\iota\). Rewriting 2 brings \[\begin{align} & {\mathsf{ind\,}}\bar{\mathfrak q}=n+\dim E_S +\dim E_{S'}-2\left(\dim((E_S+E_{S'})|_{\mathfrak t}) + \frac{1}{2}\iota\right)= \\ & = n+a-2\left(\dim E_S +\dim E_{S'} - \#\{\text{the cycles of}\;\Gamma\}+ \frac{1}{2}\iota\right)= \\ & = n - a + 2\#\{\text{the cycles of}\;\Gamma\} -\iota = \#\{\text{the segments of \Gamma}\} + 2\#\{\text{the cycles of}\;\Gamma\} -\iota. \end{align}\] This finishes the proof. ◻
Remark 7. Our proof of Theorem 6 shows that there is another recipe for computing \(\iota\). We call an arc \((i,j)\) of \(\Gamma\) affine if its shadow contains \(\alpha_0\), i.e., if \(\beta(i,j)=\delta\pm(\varepsilon_i-\varepsilon_j)\). Then \(o\in F(B)\) if and only if \({\mathcal{E}}(B)\) contains an odd number of affine arcs.
Example 8. Consider the graphs presented in Figure 2 and corresponding seaweeds. The first graph is a cycle and \(o\) lies in its interior. This \(\Gamma\) has \(3\) affine arcs, namely \((0,9)\), \((0,3)\), and \((9,4)\). Hence \(\iota=2\) and \({\mathsf{ind\,}}\bar{\mathfrak q}=2-2=0\).
The second graph has one segment and one cycle \(B\) with \(o\not\in F(B)\). Here \(B\) has two affine arcs, \((7,2)\) and \((8,1)\). We have \(\iota=0\) and \({\mathsf{ind\,}}\bar{\mathfrak q}=1+2=3\).
The third graph consists of two cycles. Both of them have \(o\) in the interior. Hence \({\mathsf{ind\,}}\bar{\mathfrak q}=4-2=2\).
Example 9. As an illustration, we consider the case, where \(|I|=|I'|=1\). Suppose \(I=I'\). Then \(\bar{\mathfrak q}=\mathfrak{sl}_n\oplus \Bbbk d\). The graph \(\Gamma\) has \(\left\lfloor\frac{n}{2}\right\rfloor\) cycles and \(n-2\!\left\lfloor\frac{n}{2}\right\rfloor\) segments. The centre \(o\) is not contained in \(F(B)\) for any of the cycles. By Theorem 6, \({\mathsf{ind\,}}\bar{\mathfrak q}=2\!\left\lfloor\frac{n}{2}\right\rfloor + n-2\!\left\lfloor\frac{n}{2}\right\rfloor=n=1+{\mathsf{ind\,}}\mathfrak{sl}_n\).
Suppose now that \(I=\{0\}\) and \(I'=\{d\}\) with \(1\leqslant d \leqslant n/2\). The corresponding seaweed subalgebra is described in Example 3. Consider two reflections \(s,s'\) that are symmetries of the \(n\)-gon, where \(s(1)=n\) and \(s'(d)=d+1\). By the construction, two nodes of \(\Gamma\) lie in the same connected component if and only if they lie in one and the same orbit of the group \({\tt H}=\left<s,s'\right>\). The composition \(s'{\circ}s\) is the rotation \(i\mapsto i+2d \;(\!\!\!\!\mod n)\). The order of \(s'{\circ}s\) is \(\frac{n}{\gcd(n,2d)}\). It is easier to describe the connected component of \(\Gamma\) for even and odd \(n\) separately.
Suppose \(n=2k\). Then \(\Gamma\) has no segments and each cycle in \(\Gamma\) has \(|{\tt H}|=\frac{2k}{\gcd(k,d)}\) nodes. The number of cycles is therefore \(\gcd(k,d)\). We obtain, \(2{\cdot}\#\{\text{the cycles of}\;\Gamma\}=2\gcd(k,d)=\gcd(2k,2d)=\gcd(n,2d)\).
Consider now \(n=2k+1\). Here \(\Gamma\) has one segment. The segment contains \(\frac{1}{2}|{\tt H}|=\frac{n}{\gcd(n,2d)}\) nodes. Each cycle in \(\Gamma\) has \(|{\tt H}|\) nodes. Hence there are \[\left(n-\frac{n}{\gcd(n,2d)}\right)\frac{\gcd(n,2d)}{2n}=\frac{1}{2}(\gcd(n,2d)-1)\] cycles in \(\Gamma\). Thus, \(2{\cdot}\#\{\text{the cycles of}\;\Gamma\} + \#\{\text{the segments of \Gamma}\}=\gcd(n,2d)\).
Now we determine the value of \(\iota\) following the idea of Remark 7. There are \(d\) affine arcs in \(\Gamma\). All of them are inside arcs. They join \(d\) with \(d{+}1\), the node \(d{-}1\) with \(d{+}2\), and so on until the arc \((1,2d)\). Note that \(2d\leqslant n\). For a cycle \(B\), the number of affine arcs in \(B\) is equal to \(|{\mathcal{V}}(B)\cap\{1,2,\ldots,d\}|\).
If \(n\) is even and \(d\) is odd, then there is at least one cycle with an odd number of affine arcs. Thus here \(\iota=2\). Suppose that \(n\) is odd. Set \(a=\gcd(n,2d)=\gcd(n,d)\). Then \(a|d\). For each cycle \(B\), we have \(|{\mathcal{V}}(B)\cap\{1,2,\ldots,a\}|=2\) and hence \(|{\mathcal{V}}(B)\cap\{1,2,\ldots,d\}|=2\frac{sec:d}{a}\) is even. Here \(\iota=0\).
The last case is \(n,d\in 2{\mathbb{Z}}\). If \(a=\gcd(n,2d)\) divides \(d\), then again \(|{\mathcal{V}}(B)\cap\{1,2,\ldots,d\}|=2\frac{sec:d}{a}\) for each cycle \(B\) and \(\iota=0\). Suppose that \(a\) does not divide \(d\). Since \(a|2d\), we have \(d=ca+\frac{a}{2}\) with \(c\in{\mathbb{Z}}\). Here \(|{\mathcal{V}}(B)\cap\{1,2,\ldots,d\}|=2c+1\) for each cycle \(B\) and \(\iota=2\).
The uniform answer is: \(\iota=0\) if \(\gcd(n,2d)\) divides \(d\) and \(\iota=2\) otherwise. We have \({\mathsf{ind\,}}\bar{\mathfrak q}=\gcd(n,2d)-\iota\).
4 Affine meander graphs in type \(\widetilde{{\sf C}}_r\)↩︎
In this section, \({\mathfrak g}=\mathfrak{sp}_{n}\) with \(n=2r\). To proper subsets \(S,S'\subset \widehat{\Pi}\) we associate a graph \(\Gamma^{\sf C}(S,S')\). Similar to type A, we put \(n\) nodes on a circle, as vertices of a regular \(n\)-gon, labelling them consequently clockwise with the numbers from \(1\) to \(n\). The arcs of the circle are labeled with the roots \(\alpha_i\in \widehat{\Pi}\). To the arc \((n,1)\) we attach \(\alpha_0\) and to \((r,r+1)\) the root \(\alpha_r\). Each of the arcs \((i,i+1)\) and \((2r-i,2r+1-i)\) is labeled with \(\alpha_i\) for \(1\leqslant i <r\). We illustrate this procedure in Figure 3. Let \(o\) be the centre of the circle.
Figure 3: Arcs and simple roots in type C.
4.1 Construction of the graph \(\Gamma^{\sf C}(S,S')\)↩︎
First we remove the arcs labelled with \(\alpha_i\) such that \(\alpha_i\not\in S\). The circle becomes a union of connected components. Let \(j_1,\ldots,j_m\) be the consecutive vertices in a connected component \(C_k\). For each \(1\leqslant i\leqslant m/2\), we draw an arc \((j_i,j_{m+1-i})\) of \(\Gamma^{\sf C}=\Gamma^{\sf C}(S,S')\), connecting the vertex \(j_i\) with \(j_{m+1-i}\). Each arc \((j_i,j_{m+1-i})\) lies outside of the circle. The shadow of \((j_i,j_{m+1-i})\) is a segment of \(C_k\) connecting \(j_i\) with \(j_{m+1-i}\). The construction applies to all connected components. The same procedure is repeated for \(S'\), but the outside arcs are replaced here with inside arcs. We require that \(o\) lies outside the area bounded by any arc of \(\Gamma^{\sf C}\) and its shadow. The arcs are drawn in such a way that the graph has no self-intersections.
4.2 Properties of the graph \(\Gamma^{\sf C}(S,S')\)↩︎
Each meander graph of type \(\widetilde{{\sf C}}_r\) is also a meander graph of type \(\widetilde{{\sf A}}_{2r-1}\). Each meander graph of type \(\widetilde{{\sf A}}_{2r-1}\) that is symmetric with respect to a reflection with no fixed vertices is a meander graph of type \(\widetilde{{\sf C}}_r\). We keep notation of Sections 2.2 and 3.
Let \(\sigma\!:{\mathbb{R}}^2\to{\mathbb{R}}^2\) be the reflection such that \(\sigma(1)=n\) for the vertices \(1\) and \(n\) on the circle. Without loss of generality, assume that \(\Gamma^{\sf C}\) is \(\sigma\)-stable. Let \(L\) be the axis of \(\sigma\) and \(\langle\sigma\rangle\subset{\rm O}({\mathbb{R}}^2)\) the subgroup of order \(2\) generated by \(\sigma\).
In the following lemma, we collect several observation on the structure of \(\Gamma^{\sf C}\).
Lemma 10. (i) Each connected component of \(\Gamma^{\sf C}\) is either a segment or a cycle.
(ii) Let \((i,j)\) be an arc of \(\Gamma^{\sf C}\). Then either \((i,j)\) is \(\sigma\)-stable and \(i+j=n+1\) or \((i,j)\cap L =\varnothing\).
(iii) Let \(B\) be a connected component of \(\Gamma^{\sf C}\). Then either \(B\) is \(\sigma\)-stable or \(B\cap L =\varnothing\).
(iv) Let \(B\) be a \(\sigma\)-stable connected component of \(\Gamma^{\sf C}\). If \(B\) is a segment, then exactly one arc
of \(B\) intersects \(L\). If \(B\) is a cycle, then exactly two different arcs of \(B\) intersect \(L\).
Proof. Statements (i) and (ii) are direct consequences of the construction rules.
Suppose \(B\cap L\ne\varnothing\). Then there is \((i,j)\in{\mathcal{E}}(B)\) such that \((i,j)\cap L\ne\varnothing\). By part (ii), the arc \((i,j)\) is \(\sigma\)-stable. Hence \(B\cap \sigma(B)\ne\varnothing\) and \(\sigma(B)=B\). This proves part (iii).
By (iii) and (ii), a \(\sigma\)-stable component \(B\) contains a \(\sigma\)-stable arc \((i,j)\). Assume that \(j>r\). Then \(i=2r+1-j\leqslant r\). Let \(i_1=i, i_2=j, i_3, \ldots, i_m\) be the maximal sequence of consecutive vertices of \(B\) such that \(i_k>r\) for \(2\leqslant k\leqslant m\). If \(i_m\) is a leaf (adjacent to only one arc), then \(B\) is a segment. Furthermore, \({\mathcal{V}}(B)=\{\sigma(i_m),\ldots,\sigma(i_3), i,j,i_3, \ldots, i_m\}\) and exactly one arc of \(B\) intersects \(L\).
Suppose that \(i_m\) is not a leaf. If \({\mathcal{V}}(B)=\{i,j\}\), then \(B\) has exactly two different arcs joining \(i\) and \(j\). Both of them intersect \(L\). If \(|{\mathcal{V}}(B)|>2\), then there is \((i_m,i_{m+1})\in{\mathcal{E}}(B)\), where \(i_{m+1}=\sigma(i_m)\leqslant r\) and \(i_{m+1}\ne i\). The vertices \(\{\sigma(i_m),\ldots,\sigma(i_3), i,j,i_3, \ldots, i_m\}\) belong to a cycle, which has to be equal to \(B\). We see that only the arcs \((i,j)\) and \((i_m,\sigma(i_m))\) intersect \(L\). Now part (iv) is settled. ◻
4.3 Interpretation of the index↩︎
If we remove from the circle the arcs labeled with \(\alpha_i\not\in S\), then there is a bijection between the \({\langle\sigma\rangle}\)-orbits on the connected components and the simple non-Abelian ideals of the Levi subalgebra \(\mathfrak l(S)\subset{\widehat{\mathfrak g}}\). Furthermore, there is a bijection between the \({\langle\sigma\rangle}\)-orbits on the outside arcs of \(\Gamma^{\sf C}\) and \({\mathcal{K}}(S)\). Similarly, there is a bijection between the \({\langle\sigma\rangle}\)-orbits on the inside arcs of \(\Gamma^{\sf C}\) and \({\mathcal{K}}(S')\). To each arc \((i,j)\) of \(\Gamma^{\sf C}\) we assign \(\beta(i,j)\in {\mathcal{K}}(S)\cup {\mathcal{K}}(S')\), which is the sum of simple roots belonging to the shadow of \((i,j)\). If \((u,v)=\sigma((i,j))\), then \(\beta(u,v)=\beta(i,j)\).
Recall that in type \({\sf C}_r\), we have a standard basis \(\{\varepsilon_1,\ldots,\varepsilon_r\}\subset\mathfrak t^*\). Assume that \(\varepsilon_i(d)=\varepsilon_i(C)=0\) for each \(i\). Then \(\alpha_i=\varepsilon_{i}-\varepsilon_{i+1}\) for \(1\leqslant i < r\) and \(\alpha_r=2\varepsilon_r\). Furthermore, \(\alpha_0=\delta-2\varepsilon_1\), where \(\delta=\alpha_0+\alpha_r+2\sum_{i=1}^{r-1}\alpha_i\) is an imaginary root with \(\delta(d)=1\). By the construction, \[\beta(i,n+1-i) \in \{ 2\varepsilon_i, \delta-2\varepsilon_i\} \;\text{ if } \;1\leqslant i\leqslant r\] and \(\beta(i,j)=\varepsilon_i-\varepsilon_j\) for \(1\leqslant i<j\leqslant r\), cf. 3 , 4 .
Theorem 11. Let \({\widehat{\mathfrak q}}={\widehat{\mathfrak q}}(S,S')\) be a standard finite-dimensional seaweed subalgebra of \({\widehat{\mathfrak g}}\) for \(\mathfrak g=\mathfrak{sp}_n\). Set \(\bar{\mathfrak q}={\widehat{\mathfrak q}}/\Bbbk C\). Then \[{\mathsf{ind\,}}\bar{\mathfrak q}= 1+ \#\{\text{the cycles of}\;\Gamma^{\sf C}\} + \frac{1}{2}\#\{\text{the non-\sigma-stable segments of \Gamma^{\sf C}}\}-\iota\] for \(\Gamma^{\sf C}=\Gamma^{\sf C}(S,S')\), where \(\iota=0\) if \(\Gamma^{\sf C}\) has no cycles with \(o\) in the interior and \(\iota=2\) otherwise.
Proof. We use 2 . The relation between the arcs of \(\Gamma^{\sf C}\) and the elements of \({\mathcal{K}}(S)\cup{\mathcal{K}}(S')\) shows that \(\dim E_S +\dim E_{S'}\) is equal to the number of \({\langle\sigma\rangle}\)-orbits on the arcs of \(\Gamma^{\sf C}\). Let this number be \(a\). Note that \(|\Pi|=r\) is the number of \({\langle\sigma\rangle}\)-orbits on the vertices of \(\Gamma^{\sf C}\). We interpret \(r-a\) in terms of the graph.
Let \(B\) be a connected component of \(\Gamma^{\sf C}\). Suppose first that \(B\) is a cycle. If \(B\) is not \(\sigma\)-stable, then \({\langle\sigma\rangle}\) has \(|{\mathcal{E}}(B)|=|{\mathcal{V}}(B)|\) orbits on \({\mathcal{E}}(B)\sqcup {\mathcal{E}}(\sigma(B))\) and on \({\mathcal{V}}(B)\sqcup{\mathcal{V}}(\sigma(B))\). If \(\sigma(B)=B\), then \({\langle\sigma\rangle}\) has \[2+\frac{1}{2}\left(|{\mathcal{E}}(B)|-2\right) = 1+\frac{1}{2}|{\mathcal{E}}(B)|\] orbits on \({\mathcal{E}}(B)\) by Lemma 10 (iv). The number of \({\langle\sigma\rangle}\)-orbits on \({\mathcal{V}}(B)\) is \(\frac{1}{2}|{\mathcal{V}}(B)|=\frac{1}{2}|{\mathcal{E}}(B)|\).
Suppose now that \(B\) is a segment. If \(B\) is not \(\sigma\)-stable, then \({\langle\sigma\rangle}\) has \(|{\mathcal{E}}(B)|=|{\mathcal{V}}(B)|-1\) orbits on \({\mathcal{E}}(B)\sqcup {\mathcal{E}}(\sigma(B))\) and \(|{\mathcal{V}}(B)|\) orbits on \({\mathcal{V}}(B)\sqcup{\mathcal{V}}(\sigma(B))\). If \(\sigma(B)=B\), then \({\langle\sigma\rangle}\) has \[1+\frac{1}{2}\left(|{\mathcal{E}}(B)|-1\right) = \frac{1}{2}\left(|{\mathcal{E}}(B)|+1\right)\] orbits on \({\mathcal{E}}(B)\) by Lemma 10 (iv). The number of \({\langle\sigma\rangle}\)-orbits on \({\mathcal{V}}(B)\) is the same \(\frac{1}{2}|{\mathcal{V}}(B)|=\frac{1}{2}(1+|{\mathcal{E}}(B)|)\). Summing up, \[\label{part1} r-a = \frac{1}{2}\#\{\text{the non-\sigma-stable segments of \Gamma^{\sf C}}\} - \#\{\text{the \sigma-stable cycles of}\;\Gamma^{\sf C}\}.\tag{6}\]
With obvious changes we repeat the strategy of the proof of Theorem 6. For each connected component \(B\) of \(\Gamma^{\sf C}\), we have \(\bar K(B)\subset \langle \varepsilon_i \mid i \in {\mathcal{V}}(B)\cup {\mathcal{V}}(\sigma(B)), 1\leqslant i\leqslant r\rangle_{{\mathbb{R}}}\). Hence a minimal relation among elements of \({\mathcal{K}}(S,S')|_{\mathfrak t}\) is (up to signs) a non-trivial linear dependence among \(\bar\beta(\boldsymbol{e})\) with \(\boldsymbol{e}\in {\mathcal{E}}(B)\) for some \(B\). If \(B\) is a segment, then there are no such dependences. If \(B\) is a cycle, then there is one dependence. Since \(\bar K(B)=\bar K(\sigma(B))\), the dimension of \((E_S+E_{S'})|_{\mathfrak t}\) is equal to \[\dim E_S + \dim E_{S'} - \#\{\text{the \sigma-stable cycles of}\;\Gamma^{\sf C}\} -\frac{1}{2} \#\{\text{the non-\sigma-stable cycles of}\;\Gamma^{\sf C}\}.\]It remains to compare \(\dim(E_S+E_{S'})\) and \(\dim((E_S+E_{S'})|_{\mathfrak t})\). These dimensions are equal if \(\delta\not\in E_S+E_{S'}\) and differ by \(1\) otherwise. Note that \(\delta\in E_S+E_{S'}\) if and only if there is a cycle \(B\) in \(\Gamma^{\sf C}\) such that \(\delta\in\langle K(B)\rangle_{{\mathbb{R}}}\). We say that an arc \((i,\sigma(i))\) with \(1\leqslant i\leqslant r\) of \(\Gamma^{\sf C}\) is affine, if \(\beta(i,j)=\delta-2\varepsilon_i\). Recall that \(\beta(i,j)=\pm(\varepsilon_i-\varepsilon_j)\), if \(j\ne \sigma(i)\) and \(i\leqslant r\). By Lemma 10, a \(\sigma\)-stable cycle \(B\) contains at most two affine arcs and a non-\(\sigma\)-stable cycle \(\tilde{B}\) contains no affine arcs. Here \(o\not\in F(\tilde{B})\) and \(\delta\not\in \langle K(\tilde{B})\rangle_{{\mathbb{R}}}\).
Suppose that \(o\in F(B)\). Then one of the \(\sigma\)-stable arcs of \(B\) is affine and another one is not. Here \(\delta\in \langle K(B)\rangle_{{\mathbb{R}}}\). Suppose that \(o\not\in F(B)\). Then either both \(\sigma\)-stable arcs of \(B\) are not affine and then clearly \(\delta\not\in \langle K(B)\rangle_{{\mathbb{R}}}\) or both of them are affine. We consider the latter case. Let \(i_1,i_2,\ldots,i_m,\sigma(i_m),\ldots,\sigma(i_2),\sigma(i_1)\) be the consecutive vertices of \(B\) with \(i_j\leqslant r\) for each \(j\). Then a relation among \(\bar\beta(\boldsymbol{e})\) with \(\boldsymbol{e}\in{\mathcal{E}}(B)\) looks as follows \[2\varepsilon_{i_1} + 2(\varepsilon_{i_2}-\varepsilon_{i_1})+\ldots +2(\varepsilon_{i_m}-\varepsilon_{i_{m-1}})-2\varepsilon_{i_m}=0.\] If we replace \(2\varepsilon_{i_1}\) with \(2\varepsilon_{i_1}-\delta\) and \(2\varepsilon_{i_m}\) with \(2\varepsilon_{i_m}-\delta\), the result is still zero. Thus, \(\delta\not\in \langle K(B)\rangle_{{\mathbb{R}}}\).
Summing up, \(\dim(E_S+E_{S'})=\dim((E_S+E_{S'})|_{\mathfrak t}) + \frac{1}{2}\iota\). We rewrite 2 using 6 and obtain \[\begin{align} & {\mathsf{ind\,}}\bar{\mathfrak q}=1+r+\dim E_S +\dim E_{S'}-2\!\left(\!\dim((E_S+E_{S'})|_{\mathfrak t}) + \frac{\iota}{2}\right)\!= 1{+}r-\!(\dim E_S +\dim E_{S'})\, + \\ & + 2\#\{\text{the \sigma-stable cycles of}\;\Gamma^{\sf C}\} + \#\{\text{the non-\sigma-stable cycles of}\;\Gamma^{\sf C}\}-\iota = \\ & = 1 + \frac{1}{2}\#\{\text{the non-\sigma-stable segments of \Gamma^{\sf C}}\} - \#\{\text{the \sigma-stable cycles of}\;\Gamma^{\sf C}\} + \\ & + 2\#\{\text{the \sigma-stable cycles of}\;\Gamma^{\sf C}\} + \#\{\text{the non-\sigma-stable cycles of}\;\Gamma^{\sf C}\}-\iota = \\ & = 1+ \frac{1}{2}\#\{\text{the non-\sigma-stable segments of \Gamma^{\sf C}}\} + \#\{\text{the cycles of}\;\Gamma^{\sf C}\} -\iota. \end{align}\] This finishes the proof. ◻
Example 12. We consider the case of maximal parabolics, where \(S=\widehat{\Pi}\setminus\{\alpha_i\}\) and \(S'=\widehat{\Pi}\setminus\{\alpha_j\}\). Let \(\Gamma^{\sf C}=\Gamma^{\sf C}(S,S')\) be the corresponding graph. Each arc of \(\Gamma^{\sf C}\) connects a vertex \(u\) with \(\sigma(u)\). Thereby \(\Gamma^{\sf C}\) has \(r\) cycles, where each cycle has two vertices and two \(\sigma\)-stable arcs. If \(i=j\), then there are no cycles with \(o\) in the interior and \({\mathsf{ind\,}}\bar{\mathfrak q}=r+1\). In this case, \(\bar{\mathfrak q}\) is isomorphic to \(\mathfrak{sp}_{2i}\oplus\mathfrak{sp}_{n-2i}\oplus\Bbbk\). If \(i\ne j\), there are cycles with \(o\) in the interior and \({\mathsf{ind\,}}\bar{\mathfrak q}=r-1\).
5 A glance at the finite-dimensional case↩︎
For \(\mathfrak g=\mathfrak{sl}_n\), a formula for the index of a seaweed subalgebra \(\mathfrak q\subset{\mathfrak g}\) in terms of the corresponding type-A meander graph is obtained in [1]. In [8], [9], a similar interpretation of \({\mathsf{ind\,}}{\mathfrak q}\) is given for \(\mathfrak{sp}_{2n}\) and \(\mathfrak{so}_n\). We explain how the TYJ formula, cf 2 , provides a proof for the combinatorial statements in all classical types,
If \(\mathfrak g\) is of type A or C, then one just simplifies the arguments of Theorems 6, 11. We have \(\alpha_0\not\in S\) and \(\alpha_0\not\in S'\). There are no affine arcs, the circle is replaced by a line and its centre disappears. In particular, in type A, each cycle in the graph gives rise to a relation among the elements of \({\mathcal{K}}(S)\cup-{\mathcal{K}}(S')\).
Example 13. Suppose that \(n=9\) and \(\mathfrak g=\mathfrak{sl}_9\). Choose \(S=\{\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_6,\alpha_8\}\) and \(S'=\{\alpha_1,\alpha_3,\alpha_4,\alpha_5,\alpha_7,\alpha_8\}\). Then \(\bar{{\mathfrak q}}=\mathfrak q(S,S')\oplus\Bbbk d\), where \(\mathfrak q(S,S')=:\mathfrak q\) is a seaweed in \(\mathfrak g\). Clearly \({\mathsf{ind\,}}\bar{\mathfrak q}=1+{\mathsf{ind\,}}\mathfrak q\). The type-A meander graph \(\Gamma_{\sf fin}\) of \({\mathfrak q}\) is shown in Figure 4, below the dotted line we indicate elements of \(\Pi\setminus S\) and above those of \(\Pi\setminus S'\).
Figure 4: A graph in a finite-dimensional setting.
We see that \[{\mathcal{K}}(S)=\{\varepsilon_1-\varepsilon_5,\varepsilon_2-\varepsilon_4,\alpha_6,\alpha_8\} \;\;\text{ and } \;\; {\mathcal{K}}(S')=\{\alpha_1,\varepsilon_3-\varepsilon_6,\alpha_4,\varepsilon_7-\varepsilon_9\}.\] There is a natural bijection between \({\mathcal{K}}(S)\) and the above arcs, as well as between \({\mathcal{K}}(S')\) and the below arcs. The unique cycle in the graph gives rise to a relation \[(\varepsilon_1-\varepsilon_5)-\alpha_4-(\varepsilon_2-\varepsilon_4)-\alpha_1=0\] among elements of \({\mathcal{K}}(S)\cup-{\mathcal{K}}(S')\). The TYJ formula asserts that \({\mathsf{ind\,}}\mathfrak q\) is equal to \[\begin{align} & (n-1)+\#\{\text{the arcs in \Gamma_{\sf fin}}\}-2(\#\{\text{the arcs in \Gamma_{\sf fin}}\}- \#\{\text{the cycles in \Gamma_{\sf fin}}\})= \\ & = \#\{\text{the segments of \Gamma_{\sf fin}}\} +2\{\text{the cycles in \Gamma_{\sf fin}}\} -1. \end{align}\] This is exactly the statement of [1]. In our example, \({\mathsf{ind\,}}\mathfrak q=2\).
5.1 The orthogonal case↩︎
Let \(\mathfrak q=\mathfrak q(S,S')\subset\mathfrak g\) be a standard seaweed defined by the subsets \(S,S'\subset\Pi\). A type-B meander graph \(\Gamma^{\sf B}_{\sf fin}=\Gamma^{\sf B}_{\sf fin}(S,S')\) is constructed by the same principle as in type C. For \(\mathfrak g=\mathfrak{so}_{2r+1}\), we put \(n=2r\) nodes on a horizontal line. The arcs of the line are labeled with the simple roots \(\alpha_1,\alpha_2,\ldots,\alpha_{r-1},\alpha_r,\alpha_{r-1},\ldots,\alpha_1\). The graph \(\Gamma^{\sf B}_{\sf fin}\) has edges above and below the horizontal line.
In order to construct the above edges, we remove the arcs labelled with \(\alpha_i\) such that \(\alpha_i\not\in S\) from the line. The interval \([1,n]\) becomes a union of connected components. Let \(j_1,\ldots,j_m\) be the consecutive vertices in a connected component \(C_k\). For each \(1\leqslant i\leqslant m/2\), we draw an edge \((j_i,j_{m+1-i})\). The shadow of \((j_i,j_{m+1-i})\) is a segment of \(C_k\) connecting \(j_i\) with \(j_{m+1-i}\) and \(\beta(j_i,j_{m+1-i})\in\mathfrak t^*\) is the sum of simple roots belonging to this segment. The same procedure applies to \(S'\) and produces the below edges. The resulting graph is stable with respect to the reflection \(\sigma\), which sends a vertex \(i\) to \(n+1-i\). Set \[{\mathcal{L}}(S)=\{\beta(\boldsymbol{e}) \mid \boldsymbol{e}\text{ is an above edge of } \Gamma^{\sf B}_{\sf fin}\} \;\;\text{ and } \;\; {\mathcal{L}}(S')=\{\beta(\boldsymbol{e}) \mid \boldsymbol{e}\text{ is a below edge of } \Gamma^{\sf B}_{\sf fin}\}.\] If \(\{\alpha+{r-1},\alpha_r\}\subset S\), then \({\mathcal{L}}(S)\ne{\mathcal{K}}(S)\), but in any case \(\langle{\mathcal{L}}(S)\rangle_{{\mathbb{R}}} =\langle{\mathcal{K}}(S)\rangle_{{\mathbb{R}}}\) and similarly \(\langle{\mathcal{L}}(S')\rangle_{{\mathbb{R}}} =\langle{\mathcal{K}}(S')\rangle_{{\mathbb{R}}}\). Using the TYJ formula and an argument similar to the proof of Theorem 11, we obtain \[{\mathsf{ind\,}}\mathfrak q= \#\{\text{the cycles of}\;\Gamma^{\sf B}_{\sf fin}\}+\frac{1}{2}\#\{\text{the non-\sigma-stable segments of \Gamma^{\sf B}_{\sf fin}}\}.\] This is the answer of [8].
A Dynkin diagram of type D\(_r\) with \(r\geqslant 4\) has a branching point and this makes a combinatorial interpretation of \({\mathsf{ind\,}}\mathfrak q\) much more complicated. Several type-D meander graphs have a crossing (edges meeting not at a vertex).
The argument of [9] relies on a reduction to either a parabolic or \(\mathfrak q_{\sf ec}:=\mathfrak q(S,S')\), where \(S=\Pi\setminus\{\alpha_r\}\) and \(S'=\Pi\setminus\{\alpha_{r-1}\}\). The index of a parabolic is computed with the help of the TYJ formula and that computation is quite involved, see [9]. The same lemma states that \({\mathsf{ind\,}}{\mathfrak q}_{\sf ec}=r-2\). The proof of the second statement uses the Raïs formula for the index of semi-direct products [12]. As we show now, it can be replaced by the TYJ formula.
Example 14. Suppose that \(\mathfrak g\) is of type \({\sf D}_r\). If \(S=\Pi\setminus\{\alpha_r\}\) and \(S'=\Pi\setminus\{\alpha_{r-1}\}\), then \(|{\mathcal{K}}(S)|=|{\mathcal{K}}(S')|=\left\lfloor \frac{r}{2}\right\rfloor\). Furthermore, \[{\mathcal{K}}(S)\cup{\mathcal{K}}(S') ={\mathcal{K}}(S) \sqcup \{\alpha_1+\alpha_2+\ldots+\alpha_{r-2}+\alpha_r\}.\] Hence \({\mathsf{ind\,}}{\mathfrak q}_{\sf ec}=r+2\left\lfloor \frac{r}{2}\right\rfloor-2(\left\lfloor \frac{r}{2}\right\rfloor+1)=r-2\).
The TYJ formula can replace also the reduction argument of [9]. This line of reasoning is quite involved and we do not give details here.