Dependencies

Set \(\Delta = \{ [i] : i \in \{ 1,\ldots , n\} \} \). The pair \((\Phi ,\Delta )\) where \(\Phi \) is the root pairing constructed in Theorem 10 is a root pairing of type \(A\).

Fix a finite, crystallographic, reduced and irreducible root pairing \(\Phi =(R,M,N,\mathcal{L},I,\alpha ,\alpha ^\vee ,s)\) and a base \(\Delta \subset I\). The pair \((\Phi ,\Delta )\) is said to be of:

1. type \(A\) if there is a sequence \((i_1,\ldots ,i_n)\), where \(n=|\Delta |\), of distinct elements of \(\Delta \) such that

   \[ \mathcal{L}(\alpha _{i_r},\alpha _{i_{r+1}}^\vee )=-1 \quad \text{for all }1\le r{\lt}n; \]

2. type \(B\) if there is a sequence \((i_1,\ldots ,i_n)\) of distinct elements of \(\Delta \) such that

   \[ \mathcal{L}(\alpha _{i_r},\alpha _{i_{r+1}}^\vee )=-1 \quad \text{for all }1\le r{\lt}n-1, \qquad \mathcal{L}(\alpha _{i_n},\alpha _{i_{n-1}}^\vee )=-2; \]

3. type \(C\) if there is a sequence \((i_1,\ldots ,i_n)\) of distinct elements of \(\Delta \) such that

   \[ \mathcal{L}(\alpha _{i_r},\alpha _{i_{r+1}}^\vee )=-1 \quad \text{for all }1\le r{\lt}n-1, \qquad \mathcal{L}(\alpha _{i_{n-1}},\alpha _{i_n}^\vee )=-2; \]

4. type \(D\) if there exist distinct elements \(d_1,d_2,t\in \Delta \) such that

   \[ \mathcal{L}(\alpha _{d_a},\alpha _t^\vee )=-1 \quad (a=1,2), \]

   and \(\Delta \setminus \{ d_1,d_2\} \) is of type \(A\);

5. type \(E\) if there exist distinct elements \(d_1,d_2,t\in \Delta \) such that

   \[ \mathcal{L}(\alpha _{d_1},\alpha _{d_2}^\vee )=-1, \qquad \mathcal{L}(\alpha _{d_2},\alpha _t^\vee )=-1, \]

   and \(\Delta \setminus \{ d_1,d_2\} \) is of type \(A\).

A base \(\Delta \) is said to be irreducible if there exist \(k\in \mathbb {Z}_{{\gt}0}\) and \(a_1,\ldots ,a_k\in \Delta \) such that:

1. \(\{ a_1,\ldots ,a_k\} =\Delta \), and

2. \(\mathcal{L}(\alpha _{a_r},\alpha _{a_{r+1}}^\vee )\ne 0\) for every \(1\le r{\lt}k\).

By definition, a root pairing admitting an irreducible base is called irreducible.

A root pairing is a tuple \((R,M,N,\mathcal{L},I,\alpha ,\alpha ^\vee ,s)\) where:

• \((R,M,N,\mathcal{L})\) is a perfect pairing,

• \(I\) is a set,

• The maps

  \begin{align*} \alpha :& \; I\to M, \quad i\mapsto \alpha _i, \\ \alpha ^\vee :& \; I\to N, \quad i\mapsto \alpha _i^\vee , \\ s:& \; I\to \operatorname {Perm}(I), \quad i\mapsto s_i, \end{align*}

  are such that

  1. \(\alpha \) and \(\alpha ^\vee \) are injective,

  2. for all \(i\in I\), \(\mathcal{L}(\alpha _i,\alpha _i^\vee )=2\),

  3. for all \(i,j\in I\),

     \begin{align*} \alpha _{s_i(j)} & =\alpha _j-\mathcal{L}(\alpha _j,\alpha _i^\vee )\alpha _i,\\ \alpha _{s_i(j)}^\vee & =\alpha _j^\vee -\mathcal{L}(\alpha _i,\alpha _j^\vee )\alpha _i^\vee . \end{align*}

Given a root pairing \(\Phi =(R,M,N,\mathcal{L},I,\alpha ,\alpha ^\vee ,s)\), a set \(\Delta \subset I\) is called a base of \(\Phi \) if:

1. the sets \(\{ \alpha _i:i\in \Delta \} \) and \(\{ \alpha _i^\vee :i\in \Delta \} \) are \(R\)-linearly independent, and

2. for every \(x\in I\), there exist \(k,m\in \mathbb {Z}_{{\gt}0}\), \(a_1,\ldots ,a_k,b_1,\ldots ,b_m\in \Delta \), and \(\varepsilon ,\delta \in \{ \pm 1\} \) such that

   \[ \alpha _x=\varepsilon \sum _{r=1}^k\alpha _{a_r} \quad \text{and}\quad \alpha _x^\vee =\delta \sum _{r=1}^m\alpha _{b_r}^\vee . \]

Fix a root pairing \(\Phi =(R,M,N,\mathcal{L},I,\alpha ,\alpha ^\vee ,s)\). We say that \(\Phi \) is:

• crystallographic if \(\mathcal{L}(\alpha _i,\alpha _j^\vee )\in \mathbb {Z}\) for all \(i,j\in I\),

• reduced if, for all \(i,j\in I\), the roots \(\alpha _i\) and \(\alpha _j\) are linearly dependent over \(R\) if and only if \(\alpha _i=\pm \alpha _j\),

• finite if \(I\) is finite.

The maps \(\alpha :I_n\times \{ \pm 1\} \to \mathbb {Z}^n\) and \(\alpha ^\vee :I_n\times \{ \pm 1\} \to (\mathbb {Z}^n)^*\) are injective.

For every \(J\in I_n\times \{ \pm 1\} \), \(\mathcal{L}(\alpha _J,\alpha _J^\vee )=2\).

For signed intervals \(J\) and \(K\),

Let \(J=\varepsilon [i,j]\) and \(K=\delta [k,l]\) be signed intervals. Then

For all \(J,K\in I_n\times \{ \pm 1\} \),

For all \(J,K\in I_n\times \{ \pm 1\} \),

For all \(J\in I_n\times \{ \pm 1\} \), we have \(s_J\in \operatorname {Perm}(I_n\times \{ \pm 1\} )\).

Fix \(n\in \mathbb {Z}_{{\gt}0}\), and let

Let \(\mathcal{L}\) be the canonical pairing, let \(I=I_n\times \{ \pm 1\} \) be the set of signed intervals, let \(\alpha \) and \(\alpha ^\vee \) be as in (??) and (??), and let \(s\) be as in (??). Then

is a finite, crystallographic and reduced root pairing.