For the affine and hyperbolic root system the symmetric part of the centralizers of root subgroups in the corresponding Steinberg groups are calculated. In the affine case the corresponding root subsystems can be computed in term of the centralizers in the spherical root systems, while in the hyperbolic case there emerges a “zoo” of examples, many of them non-hyperbolic. This also delivers many examples of naturally occuring root subsystems of infinite rank.

§ INTRODUCTION

Let $A$ be a symmetrizable GCM, $\Phi$ the associated root system, $\Pi$ its fundamental roots, $\Phi_\mathrm{re}$ the set of real roots.

The Kac—Moody Steinberg group $\St(\Phi, R)$ over a commutative ring $R$ is defined by generators $x_\alpha(\xi)$, $\alpha\in\Phi_\mathrm{re}$, $\xi\in R$, and relations

\begin{align}
& x_\alpha(\xi)x_\alpha(\zeta) = x_\alpha(\xi+\zeta), \\
& [x_\alpha(\xi), x_\beta(\zeta)] = \prod_{\qquad\mathclap{i\alpha+j\beta\in C(\alpha,\beta)}} x_{i\alpha+j\beta}(N_{\alpha\beta ij}\xi^i\zeta^j),\quad \text{if the product on the RHS is finite,} \label{eq:comm-general} \\
& w_\alpha(\varepsilon)x_\beta(\xi)w_\alpha(-\varepsilon) = x_{s_\alpha\beta}(\eta_{\alpha\beta}\varepsilon^{-\beta(h_\alpha)}\xi), \quad \varepsilon\in R^*, \label{eq:weyl-conj}
\end{align}

where $C(\alpha, \beta) = (\mathbb{Z}_{>0}\alpha+\mathbb{Z}_{>0}\beta)\cap\Phi_\mathrm{re}$, $N_{\alpha\beta ij}$ are certain integers, $\eta_{\alpha\beta}=\pm1$ and $w_\alpha(\varepsilon)=x_\alpha(\varepsilon)x_{-\alpha}(-\varepsilon^{-1})x_\alpha(\varepsilon)$, while $s_\alpha$ is the simple reflection associated to $\alpha$.

Assume $\alpha\in\Phi_\mathrm{re}$, we are interested in the structure of $Y=C(x_\alpha(\xi))$, the centralizer of the elementary root unipotent, in $G(\Phi, R)$ or in $\St(\Phi, R)$. An explicit description of $Y$ for finite type root systems was used, for example, by M. Stein in his study of central extensions of Chevalley groups over rings [10].

From the definition of $\St(\Phi, R)$ one sees that (in generic case/characteristic $0$) $x_\beta(\zeta)\in Y$ if and only if $C(\alpha, \beta)=\varnothing$, while $h_\beta(\varepsilon)\in Y$ if and only if $\beta(h_\alpha)=0$.

\[ Z(\alpha) = \{ \beta\in\Phi \mid C(\alpha, \beta)=\varnothing \}. \]

This set splits into the union of two parts:

\begin{align*}
& Z_s(\alpha) = Z(\alpha) \cap -Z(\alpha), \\
& Z_u(\alpha) = Z(\alpha) \setminus Z_s(\alpha).
\end{align*}

We are interested in the structure of $Z_s(\alpha)$ and the corresponding subgroup of $\St(\Phi, R)$, which is not the whole (Levi part of the) centralizer, but is reasonably close.

We mostly focus our attention on the case of affine and hyperbolic root systems, because in this case there is a uniform criterion for real roots, which we will use implicitly.

If $A$ is a GCM of finite, affine or hyperbolic type, then

\[ \Phi_\mathrm{re} = \left\{ \alpha=\sum_i k_i\alpha_i \ \middle|\ |\alpha|^2>0,\ k_i\cdot|\alpha_i|^2/|\alpha|^2\in\mathbb{Z} \right\}. \]

The paper is organized as follows. In <ref> we recall and establish some basic properties of $Z_s(\alpha)$. In <ref> the centralizers are calculated for the root systems of affine types. The rest of the paper is devoted to the study of indefinite type root systems, with the main focus on the hyperbolic ones. The latter split into three principal cases:

* root systems of rank $3$ (<ref>), which is the easiest part;

* root systems of rank $\geqslant4$ with finite rank centralizers (<ref>), where some additional computational effort delivers a complete answer;

* root systems of rank $\geqslant4$ with the centralizers of (apparently) infinite rank (<ref>), where we only collect some experimental observations.

An example of an infinite rank subsystem inside a rank $3$ root system was presented in [7] by providing an infinite family of real roots which serve as a “weak basis” for a certain subsystem (for a definition of weak independence see <ref>). The coefficients of these roots are given by polynomials. Here we find many examples of (apparently) infinite-rank subsystems cut out by a simple condition on its elements. Inside it one can find an infinite weakly independent family of roots which is very likely the basis of this subsystem. However, apart from the simplest cases this family is not given by a single polynomial formula. The most striking example is the root system with the following Dynkin diagram:

\[
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\dv{0}{(0,0)}{1}
\dv{1}{(45:1)}{2}
\dv{2}{($(45:1)+(-45:1)$)}{3}
\dv{3}{(-45:1)}{4}
\deA{a0}{a1}\deA{a0}{a2}\deA{a0}{a3}\deA{a1}{a2}\deA{a2}{a3}
\end{tikzpicture}
\]

Here the basis of $Z_s(\alpha_4)$ seems to be covered by infinitely many polynomial families parametrized by a product of two primes congruent to $\pm1$ modulo $12$.

§ PROPERTIES OF $Z_S(\ALPHA)$

For $\alpha,\beta\in\Phi_\mathrm{re}$ the following statements are equivalent:

* $C(\alpha, \beta) = C(\alpha, -\beta) = \varnothing$;

* $\langle \beta, \alpha^\vee \rangle = 0$ and $\alpha+\beta\notin \Phi_\mathrm{re}$.

The proof is very similar to the proof of <cit.>.

Consider the $\alpha$-string through $\beta$:

\[ \beta-p\alpha,\ldots,\beta,\ldots,\beta+q\alpha. \]

The first and the last roots of the root string are real, so if $C(\alpha, \beta)=\varnothing$, then $q=0$. Similarly, $p=0$ if $C(\alpha, -\beta) = \varnothing$. But $\langle \beta, \alpha^\vee \rangle = p-q = 0$.

Now suppose that $\langle \beta, \alpha^\vee \rangle = 0$ and $\alpha+\beta\notin\Phi_\mathrm{re}$. Assume that $\gamma = m\alpha+n\beta\in\Phi$ for some $m,n>0$. Note that $w_\alpha w_\beta(\gamma) = -\gamma$, so $\gamma$ is real.

We can choose $\gamma$ so that $m$ is minimal. Consider the $\alpha$-string through $\gamma$:

\[ \gamma-p\alpha,\ldots,\gamma,\ldots,\gamma+q\alpha. \]

Since $m$ is minimal, $\gamma-\alpha\notin\Phi$ unless $m=1$. But then $\gamma-\alpha=n\beta$, which is not a root unless $n=1$, while $\alpha+\beta\notin \Phi_\mathrm{re}$ by assumption. Thus we conclude that $\gamma$ is the first root of this root string and $p=0$. But $p-q=\langle \gamma, \alpha^\vee \rangle = 2m$, so $q<0$, a contradiction.

Since $\alpha+\beta\notin\Phi_\mathrm{re}$ implies $w_\beta(\alpha+\beta)=\alpha-\beta\notin\Phi_\mathrm{re}$ when $\langle \beta, \alpha^\vee \rangle=0$, it follows that $C(\alpha, -\beta)$ is empty as well.

As a corollary, one can describe $Z_s(\alpha)_\mathrm{re}$ as

\[ Z_s(\alpha)_\mathrm{re} = \{ \beta\in\Phi_\mathrm{re} \mid \langle\alpha,\beta^\vee\rangle=0 \text{ and } \alpha+\beta\notin\Phi_\mathrm{re} \}. \]

$Z_s(\alpha)$ is closed, i.e. if $\beta,\gamma\in Z_s(\alpha)$ and $\beta+\gamma\in\Phi$, then $\beta+\gamma\in Z_s(\alpha)$.

Let $\beta,\gamma\in Z_s(\alpha)$ be two roots such that $\beta+\gamma\in\Phi$. Suppose that $\beta+\gamma\notin Z_s(\alpha)$. Then, since $\langle \beta+\gamma, \alpha^\vee \rangle = \langle \beta, \alpha^\vee \rangle + \langle \gamma, \alpha^\vee \rangle = 0$, $\alpha+\beta+\gamma$ is a root. Consider the $\beta$-string through $\alpha+\beta+\gamma$:

\[ \alpha+\beta+\gamma,\ldots,\alpha+\beta+\gamma + q\beta. \]

$\alpha+\beta+\gamma$ is the first root of this string because $\alpha+\gamma\notin\Phi$. Then

\[ 0-q = \langle \alpha+\beta+\gamma, \beta^\vee \rangle = 0+2+\langle \gamma, \beta^\vee \rangle, \quad \text{so} \quad \langle \gamma, \beta^\vee \rangle\leqslant-2. \]

Similarly, considering the $\gamma$-string through $\alpha+\beta+\gamma$, one gets $\langle \beta, \gamma^\vee \rangle\leqslant-2$. But in a root system of type $\begin{psmallmatrix}2 & -b \\ -a & 2\end{psmallmatrix}$ with $a,b\geqslant2$ the sum of the two fundamental roots is never a real root, a contradiction.

$Z_s(\alpha)$ is closed under the action of the corresponding Weyl group.

Since $\alpha+\gamma\notin\Phi$, also $s_\beta(\alpha+\gamma)\notin\Phi$. But $\langle \beta,\alpha^\vee \rangle = 0$, so $s_\beta(\alpha+\gamma)=\alpha+s_\beta(\gamma)$, hence $s_\beta(\gamma)\in Z_s(\alpha)$.

This allows to describe the subgroup of $\St(\Phi,R)$ generated by all $X_\beta$ for $\beta\in Z_s(\alpha)$ as the image of $\St(Z_s(\alpha),R)$ under the homomorphism induced by the inclusion $Z_s(\alpha)\hookrightarrow\Phi$.

§ CENTRALIZERS IN AFFINE ROOT SYSTEMS

In the affine case the calculation of centralizers can be briefly described as “affinization of the centralizer in a finite subsystem” (the centralizers in the finite case are listed in <cit.>). Namely, using the standard notation for affine root systems <cit.>, one divides them into three classes as follows: if $\Phi=\Psi^{(r)}$, where $\Psi$ is of finite type and $r=1,2,3$, then the set of real roots is described as follows:

If $r=1$,

\[ \Phi_\mathrm{re} = \{ \alpha+n\delta \mid \alpha\in\Psi,\ n\in\mathbb{Z} \}; \]

If $r=2$ or $3$, $\Phi\neq\rA_{2\ell}^{(2)}$,

\[ \Phi_\mathrm{re} = \{ \alpha+n\delta \mid \alpha\in\Psi^<,\ n\in\mathbb{Z} \} \cup \{ \alpha+nr\delta \mid \alpha\in\Psi^>,\ n\in\mathbb{Z} \}; \]

If $\Phi = \rA_{2\ell}^{(2)}$,

\begin{multline*}
\Phi_\mathrm{re} = \{ \nicefrac{1}{2}\cdot(\alpha+(2n-1)\delta) \mid \alpha\in\Psi^>,\ n\in\mathbb{Z} \} \cup {} \\ \cup \{ \alpha+n\delta \mid \alpha\in\Psi^<,\ n\in\mathbb{Z} \} \cup \{ \alpha+2n\delta \mid \alpha\in\Psi^>,\ n\in\mathbb{Z} \}.
\end{multline*}

Here $\Psi^{<}$ and $\Psi^{>}$ denote the subsets of all short and long roots of $\Psi$ respectively, and $\delta$ is (in the basis of the fundamental roots) the unique entry-wise positive primitive vector in the kernel of the Cartan matrix of $\Phi$ (listed in <cit.>). In particular, $\delta$ is an isotropic root and is orthogonal to all real roots of $\Phi$.

Consider the case $r=1$. It is easy to see that $\beta+n\delta\in Z_s(\alpha)$ if and only if $\beta\in Z_s(\alpha)$. Indeed, since the expression of a root in $\Psi$ in the form $\beta+n\delta$ is unique, $\alpha+\beta+n\delta\in\Phi_\mathrm{re}$ if and only if $\alpha+\beta\in\Psi$. On the other hand, by the definition of $\delta$ one has $(\beta+n\delta,\alpha) = (\beta,\alpha)$.

A very similar calculation works for other cases. The computation of the centralizers is summarized in <ref>.

to 0.9c@    c@    l@    X
$\Phi$     $\alpha$     $Z_s(\alpha)$     particular cases
$\rA_\ell^{(1)}$     long     $\rA_{\ell-2}^{(1)}$     $\varnothing$ for $\ell=2$
$\rB_\ell^{(1)}$     long     $\rB_{\ell-2}^{(1)}\oplus\rA_1^{(1)}$     $\rC_2^{(1)}\oplus\rA_1^{(1)}$ for $\ell=4$, $\rA_1^{(1)}\oplus\rA_1^{(1)}$ for $\ell=3$

short     $\rD_{\ell-1}^{(1)}$     $\rA_3^{(1)}$ for $\ell=4$, $\rA_1^{(1)}\oplus\rA_1^{(1)}$ for $\ell=3$
$\rC_\ell^{(1)}$     long     $\rC_{\ell-1}^{(1)}$

short     $\rC_{\ell-2}^{(1)}$     $\rA_1^{(1)}$ for $\ell=3$, $\varnothing$ for $\ell=2$
$\rD_\ell^{(1)}$     long     $\rD_{\ell-2}^{(1)}\oplus\rA_1^{(1)}$     $\rA_3^{(1)}\oplus\rA_1^{(1)}$ for $\ell=5$, $3\rA_1^{(1)}$ for $\ell=4$
$\rG_2^{(1)}$     any     $\rA_1^{(1)}$
$\rF_4^{(1)}$     long     $\rC_3^{(1)}$

short     $\rA_3^{(1)}$
$\rE_6^{(1)}$     long     $\rA_5^{(1)}$
$\rE_7^{(1)}$     long     $\rD_6^{(1)}$
$\rE_8^{(1)}$     long     $\rE_7^{(1)}$
$\rA_{2\ell}^{(2)}$     long     $\rA_{2\ell-2}^{(2)}$     $\varnothing$ for $\ell=1$

medium     $\rA_{2\ell-4}\oplus\rA_1^{(1)}$     $\rA_1^{(1)}$ for $\ell=2$

short     $\rA_{2\ell-3}^{(2)}$
$\rA_{2\ell-1}^{(2)}$     long     $\rA_{2\ell-3}^{(2)}$

short     $\rA_{2\ell-5}^{(2)}\oplus\rA_1^{(1)}$     $2\rA_1^{(1)}$ for $\ell=3$
$\rD_{\ell+1}^{(2)}$     long     $\rD_{\ell-1}^{(2)}\oplus\rA_1^{(1)}$     $2\rA_1^{(1)}$ for $\ell=3$, $\rA_1^{(1)}$ for $\ell=2$

short     $\rB_{\ell-1}^{(1)}$     $\rC_2^{(1)}$ for $\ell=3$, $\rA_1^{(1)}$ for $\ell=2$
$\rE_6^{(2)}$     long     $\rA_7^{(2)}$

short     $\rB_3^{(1)}$
$\rD_4^{(3)}$     any     $\rA_1^{(1)}$

Centralizers for the root systems of affine types

§ CENTRALIZERS IN HYPERBOLIC ROOT SYSTEMS

Following [7], we say that a tuple $\mathcal{D}=(A,\Pi,\Pi^\vee,V,V^\vee,\langle \cdot,\cdot\rangle)$ is a set of root data if $A=(A_{ij})_{i,j\in J}$ is a GCM, $V$ and $V^\vee$ are vector spaces over $\mathbb{R}$, $\langle\cdot,\cdot\rangle\colon V\times V^\vee\to\mathbb{R}$ is a non-degenerate pairing, $\Pi=\{\alpha_i\}_{i\in J}\subset V$ and $\Pi^\vee=\{\alpha_i^\vee\}_{i\in J}\subset V^\vee$ are such that $\langle \alpha_i,\alpha_j^\vee\rangle=A_{ij}$ and $Q=\sum_J\mathbb{Z}\alpha_i$, $Q^\vee=\sum_J\mathbb{Z}\alpha_i^\vee$ are free abelian groups with bases $\{\gamma_i\}_{i\in I}\subset Q$, $\{\gamma_i^\vee\}_{i\in I}\subset Q^\vee$, satisfying

\[ \Pi\subset\bigoplus_I \mathbb{Z}_{\geqslant0}\gamma_i,\quad \Pi^\vee\subset\bigoplus_I \mathbb{Z}_{\geqslant0}\gamma_i^\vee. \]

The fundamental roots $\alpha_i$ are not assumed to be linearly independent, but satisfy the weak independence property:

For all $k\in J$

\[
\mathbb{Z}\alpha_k \cap \sum_{j\neq k}\mathbb{Z}_{\geqslant0}\alpha_j = \{0\},\quad
\mathbb{Z}\alpha_k^\vee \cap \sum_{j\neq k}\mathbb{Z}_{\geqslant0}\alpha_j^\vee = \{0\}.
\]

In <cit.> the following procedure for finding a (weak) basis $\Upsilon$ of a subsystem $\Omega$ is indicated:

* Fix a well-ordering $\beta_1,\beta_2,\ldots$ on $\Omega_\mathrm{re}^+$, which respects the height.

* $\beta_1\in\Upsilon$.

* For $n>1$ set $\beta_n\in\Upsilon$ if and only if $\beta_n\notin\sum_{i=1}^{n-1}\mathbb{Z}_{\geqslant0}\beta_i$.

Checking the weak independence is done by means of integer linear programming. Namely, we use the following variant of integer Farkas' lemma.

Given $A\in M(m,n,\mathbb{Z})$ and $b\in\mathbb{Z}^m$,

\[ \forall x\in\mathbb{Z}_{\geqslant0}^n\ \ \forall r\in\mathbb{Z}_{>0}\ Ax\neq br \Longleftrightarrow \exists y\in\mathbb{Z}^m\ \ \text{such that}\ \ y^\top A\geqslant0,\ y^\top\cdot b<0. \]

Here inequalities of the form “$\geqslant0$”, “$<0$”, etc. are applied entry-wise. Note that setting $A$ to be the matrix with columns formed by the coefficient of the first $n$ roots $\gamma_1,\ldots,\gamma_n$ in $\Upsilon$ and $b$ the coefficients column of $\beta$ the left hand side of the equivalence expresses the weak independence of $\beta$ with respect to $\gamma_1,\ldots,\gamma_n$.

Note also that to check the weak independence for a set $\Gamma$ of positive roots it is enough to only check that $m\gamma$, $\gamma\in\Gamma$, $m\neq0$, cannot be expressed as a nonnegative integer linear combination of positive roots $\beta\in\Gamma$ of smaller height than $\gamma$.

So this iterative procedure will reach every element of a basis in finite time. However, a priori there is no definite termination point, when one can be sure that all the elements of a basis have been found.

We propose the following procedure to certify the computation of a basis, depending on the rank of $\Phi$. This procedure covers most of the cases, but leaves some for additional manual calculation. Each such case which can be worked out is elaborated below. The notation is as follows: we display the Dynkin diagram of $\Phi$, the matrix of the bilinear form $B$ in the basis of the fundamental roots (which is the symmetrization of the Cartan matrix of $\Phi$), and the root $\alpha$ (meaning that we calculate $Z_s(\alpha)$).

The calculations for the hyperbolic root systems are summarized in the Appendices. The order of appearance of the root systems coincides with the order in the classification in [2]. The numbering of the root systems is different, for we are only interested in the $142$ symmetrizable systems. The numbering of the fundamental roots is mostly the same, with a few exceptions (those dealt in <ref>). The particular choice of $\alpha$ is such that the number of the fundamental roots of $\Phi$ lying in $Z_s(\alpha)$ is maximized and also such that the symmetry of the diagram is most preserved. We choose only one fundamental root in each Weyl group orbit, and we also unite roots mapped to each other by a diagram automorphism.

§.§ Root systems of rank $3$

There is no rank $2$ root system in dimension $1$.

Suppose that the rank $2$ root system with the fundamental roots $\alpha$ and $\beta$ and GCM $\begin{psmallmatrix}
2 & -b \\ -a & 2
\end{psmallmatrix}$ is realized in dimension $1$. That is, $V=\langle v\rangle$ and $V^\vee=\langle v^\vee\rangle$, so that for some non-zero real numbers $\lambda, \lambda^\vee, \mu, \mu^\vee$ one has

\[ \alpha = \lambda v,\quad \alpha^\vee = \lambda^\vee v^\vee,\quad \beta = \mu v,\quad \beta^\vee = \mu^\vee v^\vee. \]

Then one can calculate the values of the pairing $\langle \cdot,\cdot\rangle$ on the pairs of roots:

\begin{align*}
& 2 = \langle \alpha, \alpha^\vee \rangle = \lambda\lambda^\vee\langle v,v^\vee\rangle,
&& -b = \langle \alpha,\beta^\vee \rangle = \lambda\mu^\vee\langle v,v^\vee\rangle, \\
& 2 = \langle \beta, \beta^\vee \rangle = \mu\mu^\vee\langle v,v^\vee\rangle,
&& -a = \langle \beta,\alpha^\vee \rangle = \mu\lambda^\vee\langle v,v^\vee\rangle.
\end{align*}

Denote $\nu=\langle v,v^\vee\rangle$. Note that $\nu\neq0$ by the non-degeneracy of the pairing. Then

\[ \lambda^\vee = \frac{2}{\lambda\nu}, \qquad \mu^\vee = \frac{2}{\mu\nu}, \qquad \frac{-b}{\nu} = \lambda\mu^\vee = \frac{2\lambda}{\mu\nu}, \qquad \frac{-a}{\nu} = \mu\lambda^\vee = \frac{2\mu}{\lambda\nu}. \]

Hence $2\lambda=-b\mu$ and $2\mu=-a\lambda$, and so $2\lambda = \frac{ab\lambda}{2}$, thus $ab=4$.

If $(a,b)=(2,2)$, then $\lambda=-\mu$. If $(a,b)=(1,4)$, then $\lambda=-2\mu$. In both cases the weak independence property fails, a contadiction.

There is no rank $3$ root system in dimension $2$.

Suppose that the rank $3$ root system $\Phi$ with the fundamental roots $\alpha,\beta,\gamma$ is realized in dimension $2$. Then, in particular, each of the $3$ subsystems

\[ (\mathbb{Z}\alpha + \mathbb{Z}\beta)\cap\Phi, \qquad (\mathbb{Z}\alpha + \mathbb{Z}\gamma)\cap\Phi, \qquad (\mathbb{Z}\beta + \mathbb{Z}\gamma)\cap\Phi \]

is a rank $2$ root system in dimension $2$.

Assume first that all of these subsystems are infinite. Then by the structure of affine and hyperbolic rank $2$ root systems [3] each one posesses a non-empty cone of imaginary roots, lying between the two fundamental roots. This cones form one of the configurations shown at <ref>. Since these root systems share the bilinear form, both options are impossible (in one case, the positive part of the cone is disconnected, in the other one of the fundamental roots is not real).

Possible planar configurations of $\alpha$, $\beta$ and $\gamma$ (up to reordering) and the imaginary cones of the corresponding rank $2$ subsystems.

If one of these subsystems if finite, then the bilinear form on the ambient $2$-dimensional space is positive-definite, thus $\Phi$ is finite, but in dimension $2$ there are only five non-empty finite root systems ($\rA_1$, $\rA_1\times\rA_1$, $\rA_2$, $\rC_2$, $\rG_2$), all of rank $\leqslant2$.

As a corollary, once one finds two weakly independent roots in $Z_s(\alpha)$, one can stop, since this subsystem is realized in the $2$-dimensional space $\langle \alpha^\vee \rangle^\perp$. In fact, it is enough to find any two roots, for otherwise one gets an $\rA_1$ root system or no roots at all.

All centralizers in hyperbolic root systems of rank $3$ are listed in <ref>. Below we discuss how to certify the basis of $Z_s(\alpha)$ when it is either $\rA_1$ or empty.

§.§.§ Centralizers of rank $<2$

Let us now consider the case when there are no two distinct positive roots $\beta\in Z_s(\alpha)$.

Denote $\beta=x_1\alpha_1+x_2\alpha_2+x_3\alpha_3$. Since such $\beta$ is orthogonal to $\alpha$, the coefficients $x_i$ must satisfy the equation of the form

\[ c_1x_1+c_2x_2+c_3x_3 = 0 \]

for some integers $c_i$ (they form a column equal to $B\alpha$). Thus one can eliminate one of the variables, say, $x_1$, from the ternary quadratic form $(\beta,\beta)\in\mathbb{Z}[x_1,x_2,x_3]$ to obtain a binary quadratic form $f$ in variables $x_2,x_3$. In most cases this quadratic form factors into the product of two degree $1$ terms. If so, one can find all non-negative integer solutions to the equations $f(y,z)=|\alpha_i|^2$ for each $i$ (there are only finitely many solutions) and then check whether they deliver a root $\beta$ that is strictly orthogonal to $\alpha$ or, indeed, define a real root at all.

The three remaining cases when no root $\beta\in Z_s(\alpha)$ can be found and the quadratic form $f$ does not factorize are dealt with below.

\[
\begin{tikzpicture}[baseline={(a1.base)}]
\dv{0}{(120:.75)}{1}\dv{1}{(0:.75)}{2}\dv{2}{(-120:.75)}{3}
\deCtwo{a1}{a0}\deA{a0}{a2}\deCtwo{a1}{a2}
\end{tikzpicture}
\qquad
B = \begin{pmatrix}
2 & -1 & -1 \\ -1 & 1 & -1 \\ -1 & -1 & 2
\end{pmatrix}
\qquad
\alpha = \alpha_2
\qquad
\]

Compute $0=(\beta,\alpha) = -x+y-z$ and $1=(\beta,\beta) = 2x^2-2xy-2xz+y^2-2yz+2z^2$, the latter simplifies (modulo the former) to $6x^2-6xy+y^2=1$. This implies $y\equiv 1 \pmod{2}$. If $\beta$ is a real root orthogonal to $\alpha$, then $|\beta+\alpha|^2 = |\beta|^2+|\alpha|^2 = 2$, so $\beta+\alpha$ being a real root is equvalent to

\[ 2x\divby2,\quad y+1\divby2,\quad 2z\divby2, \]

which are all satisfied. Hence no root $\beta$ is strictly orthogonal to $\alpha$.

\[
\begin{tikzpicture}[baseline={(a1.base)}]
\dv{0}{(120:.75)}{1}\dv{1}{(0:.75)}{2}\dv{2}{(-120:.75)}{3}
\deCtwo{a0}{a1}\deA{a0}{a2}\deCtwo{a2}{a1}
\end{tikzpicture}
\qquad
B = \begin{pmatrix}
2 & -2 & -1 \\ -2 & 4 & -2 \\ -1 & -2 & 2
\end{pmatrix}
\qquad
\alpha = \alpha_1
\qquad
|\beta|^2 = 2
\]

Again, such $\beta\in Z_s(\alpha)$ must satisfy $2x-2y-z=0$ and hence $6x^2-24xy+20y^2=2$, which implies $2x^2\equiv2\pmod{4}$, hence $x$ is odd. Now $|\beta+\alpha|^2=4$, and $\beta+\alpha$ is a real root if

\[ 2(x+1)\divby4,\quad 4y\divby4,\quad 2z\divby4, \]

and these conditions are always satisfied.

\[
\begin{tikzpicture}[baseline={(a1.base)}]
\dv{0}{(120:.75)}{1}\dv{1}{(0:.75)}{2}\dv{2}{(-120:.75)}{3}
\deCtwo{a1}{a0}\deAoo{a0}{a2}\deCtwo{a1}{a2}
\end{tikzpicture}
\qquad
B = \begin{pmatrix}
2 & -1 & -2 \\ -1 & 1 & -1 \\ -2 & -1 & 2
\end{pmatrix}
\qquad
\alpha = \alpha_2
\qquad
|\beta|^2 = 1
\]

$(\beta,\alpha) = -x+y-z$, so $(\beta,\beta) = 8x^2-8xy+y^2=1$, thus $y$ is odd. Since $|\beta+\alpha|^2=2$ and

\[ 2x\divby2,\quad y+1\divby2,\quad 2z\divby2, \]

$\beta+\alpha$ is a real root.

§.§ Root systems of rank $\geqslant4$

For $v\in V^\vee\setminus\{0\}$ define the hyperplane and two half-spaces

\[
H_v=v^\perp=\{x\in V\mid \langle x, v\rangle=0\},\quad
H_v^\pm=\{x\in V\mid \langle x, v\rangle\in\pm\mathbb{R}_{>0}\}
\]

Define the fundamental chamber for $\Phi$ as

\[ F = \{x\in V\mid \langle x, \alpha_j^\vee\rangle >0 \text{ for all } j\in J\}. \]

For a set $\mathcal{H}$ of hyperplanes define the equivalence relation $\sim_\mathcal{H}$ as

\[ x\sim_\mathcal{H}y\quad \Leftrightarrow \quad \text{for every $H\in\mathcal{H}$ either $x,y\in H$ or $x,y\in H^+$ or $x,y\in H^-$}. \]

For a subsystem $\Omega$ of $\Phi$ set $\mathcal{H}(\Omega) = \{ H_{\alpha^\vee} \mid \alpha\in\Omega \}$. Define the fundamental chamber for $\Omega$ as

\[ F(\Omega) = \{x\in V\mid \langle x, \alpha^\vee\rangle >0 \text{ for all } \alpha\in\Omega^+\}. \]

The equivalence relation $\sim_{\mathcal{H}(\Omega)}$ is coarser than $\sim_{\mathcal{H}(\Phi)}$, and hence $F(\Omega)\supset F(\Phi)$. Now $F(\Phi)$ can be defined in term of its fundamental roots as

\[ F(\Phi) = \{x\in V\mid \langle x, \alpha_j^\vee\rangle >0 \text{ for all } j\in J\}. \]

Suppose that $\Gamma\subset\Omega^+$ is a set of roots. They determine a convex cone

\[ C(\Gamma) = \{x\in V\mid \langle x, \alpha^\vee\rangle >0 \text{ for all } \alpha\in\Gamma\}. \]

If $\Gamma$ is a basis for $\Omega$, then $C(\Gamma)=F(\Omega)$, i.e. there is no $\beta\in\Omega$ such that $H_{\beta^\vee}$ cuts $C(\Gamma)$.

Given the chain of closed convex polyhedral cones inclusions $\overline{F(\Phi)}\subseteq \overline{F(\Omega)}\subseteq \overline{C(\Gamma)}$, one can represent each of $F=\overline{F(\Phi)}$ and $C=\overline{C(\Gamma)}$ as the conical hull of a set of vectors:

\[
F = \mathbb{R}_{\geqslant0}r_1+\ldots+\mathbb{R}_{\geqslant0}r_n,\quad
C = \mathbb{R}_{\geqslant0}s_1+\ldots+\mathbb{R}_{\geqslant0}s_k.
\]

With the bounding hyperplanes of the convex polyhedral cone known, the spanning vectors $r_i$ can be calculated via the Double Description method [8, 4].

Now for each face of $F$ one can determine its supporting hyperplane $H$ and find $\alpha\in\Phi$ such that $H=H_{\alpha^\vee}$. Then, if $\alpha\notin\Omega$, the reflection $\sigma$ of $F$ with respect to $H$ leaves the images $\sigma(r_1),\ldots,\sigma(r_n)$ inside $\overline{F(\Omega)}$. One can repeat this procedure for each of the cones $\sigma(F)$ and for each of its faces. If one reaches $s_i$ without ever crossing hyperplanes from $\mathcal{H}(\Omega)$, then one proves that $s_i\in\overline{F(\Omega)}$. If $s_i\in\overline{C(\Gamma)}$ for all $i$, then $C\subseteq\overline{F(\Omega)}$, and $\Gamma$ is the basis of $\Omega$.

However, this procedure never works as stated for $\Omega=Z_s(\alpha)$ and infinite $\Phi$, because in this case $\overline{F(\Omega)}$ contains the line $\mathbb{R}\alpha$, while the Tits cone $\cup_{w\in W}w\overline{F(\Phi)}$ is pointed.

Note that for any $\beta\in\Omega$ one has $\alpha\in H_{\beta^\vee}$, so the hyperplanes from $\mathcal{H}(\Omega)$ cutting $C$ can already be seen on $H_{\alpha^\vee}$. Thus one can look at projections $s_i'$ and $r_i'$ of $s_i$ and $r_i$ onto $H_{\alpha^\vee}$, as well as the projections of all $\sigma(r_i)$, see <ref>.

The convex cone $F$, spanned by three rays $r_1$, $r_2$, $r_3$, and its reflection $\sigma F$ with respect to $H_{\beta^\vee}$. Once all of $s_i'$ are in the expanding list of $r_i'$, it is proved that $\Gamma$ is a basis for $\Omega$.

The bases and the diagrams for all symmetrizable root systems of rank $\geqslant4$ are listed in <ref>, with the exceptions discussed in <ref>.

The procedure described above allows to certify the basis for most hyperbolic root systems, but fails to work in two extreme cases: no roots in $\Omega$ or too many roots in $\Upsilon$.

§.§.§ Empty centralizers

\[
\begin{tikzpicture}[baseline={($(current bounding box.center)+(0,-4pt)$)}]
\dv{1}{(0,1)}{1}\dv{2}{(1,1)}{2}\dv{3}{(1,0)}{3}\dv{4}{(0,0)}{4}
\deA{a1}{a2}\deA{a1}{a3}\deA{a1}{a4}\deA{a2}{a3}\deA{a2}{a4}\deA{a3}{a4}
\end{tikzpicture}
\qquad
B = \begin{pmatrix}
\end{pmatrix}
\qquad
\alpha=\alpha_1
\]

The root $\beta=x\alpha_1+y\alpha_2+z\alpha_3+w\alpha_4$ is orthogonal to $\alpha_1$ if $2x-y-z-w=0$. Substituting the expression for $w$ into the quadratic form $(\beta,\beta)$ one gets that

\[ 2 = |\beta|^2 = 6(x^2-2xy-2xz+y^2+yz+z^2), \]

which is impossible for integer $x,y,z$, thus $Z_s(\alpha_1) = \varnothing$.

The above argument works for any complete graph on $\geqslant2$ points (including $\rA_2$ and $\rA_2^{(1)}$).

\[
\begin{tikzpicture}[baseline={(a4.base)}]
\dv{1}{(120:1)}{1}\dv{2}{(-120:1)}{2}\dv{3}{(1,0)}{3}\dv{4}{(0,0)}{4}
\deCtwo{a4}{a1}\deCtwo{a4}{a2}\deCtwo{a4}{a3}
\end{tikzpicture}
\qquad
B = \begin{pmatrix}
2 & 0 & 0 & -2 \\ 0 & 2 & 0 & -2 \\ 0 & 0 & 2 & -2 \\ -2 & -2 & -2 & 4
\end{pmatrix}
\qquad
\alpha=\alpha_4
\]

The root $\beta=x\alpha_1+y\alpha_2+z\alpha_3+w\alpha_4$ is orthogonal to $\alpha_4$ if $2w=x+y+z$. Then

\[ 2 = |\beta|^2 = x(x-2y-2z)+(y-z)^2. \]

This equation has no integer solutions, because it has no solutions modulo $4$. Hence $Z_s(\alpha_4) = \varnothing$.

§.§.§ Infinite rank centralizers

In case the Dynkin diagram of $\Phi$ is of the form “simply-laced cycle + something”, the procedure continues finding new fundamental roots for $Z_s(\alpha)$ indefinitely. This phenomena also occurs for root systems with multiply-laced Dynkin diagrams obtained from those described above by the operation of “diagram folding”. Below we describe several cases when $Z_s(\alpha)$ has infinite rank.

\[
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\dv{0}{(0,0)}{1}
\dv{1}{(30:1)}{2}
\dv{2}{(-30:1)}{3}
\dv{3}{(-1,0)}{4}
\deA{a0}{a1}\deA{a0}{a2}\deA{a0}{a3}\deA{a1}{a2}
\end{tikzpicture}
\]

The basis for $Z_s(\alpha_4)$ is provided by the family of roots

\[ \beta_n = \aenroot{\&\& n(3n+2) \\ 3n(n+1) \& 6n(n+1) \\ \&\& (n+1)(3n+1) \\}{(r-2-1)--(r-2-2)--(r-1-3)--(r-3-3)--(r-2-2)},\quad n\in\mathbb{Z}_{\geqslant0} \]

and those obtained from these by swapping the coefficients of $\alpha_2$ and $\alpha_3$ (we call them $\beta'_n$). In particular, $\beta_0=\alpha_3$ and $\beta'_0=\alpha_2$. Set

\begin{gather*}
y = (-n,n,n-1,1),\qquad y' = (-n,n-1,n,1),\\
\intertext{so that for $m<n$}
y\cdot\beta_m = n-m-1 \geqslant 0, \qquad y\cdot\beta'_m = m+n \geqslant 0, \qquad y\cdot\beta_n = -1 < 0, \\
y'\cdot\beta_m = m+n \geqslant 0, \qquad y'\cdot\beta'_m = n-m-1 \geqslant 0, \qquad y'\cdot\beta'_m = -1 < 0.
\end{gather*}

Hence by Integer Farkas lemma $\beta_n$ and $\beta'_n$ cannot be expressed in terms of $\beta_m,\beta'_m$, $m<n$. Since $\hei(\beta_n)=\hei(\beta'_n)$, we see that $\beta_1,\beta'_1,\ldots,\beta_n,\beta'_n$ are weakly independent.

\[
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\dv{4}{(-1,0)}{5}
\dv{0}{(0,0)}{1}
\dv{1}{(45:1)}{2}
\dv{2}{($(45:1)+(-45:1)$)}{3}
\dv{3}{(-45:1)}{4}
\deA{a0}{a1}
\deA{a1}{a2}
\deA{a2}{a3}
\deA{a3}{a0}
\deA{a4}{a0}
\end{tikzpicture}
\]

The basis for $Z_s(\alpha_5)$ is given by $\alpha_2,\alpha_3,\alpha_4,2\alpha_1+\alpha_2+\alpha_4+\alpha_5$ together with the family of roots

\[
\aenroot{\& \& n(2n+1) \\ 2n(n+1) \& 4n(n+1) \& \& 2n(n+1) \\ \& \& (n+1)(2n+1) \\}{(r-2-1)--(r-2-2)--(r-1-3)--(r-2-4)--(r-3-3)--(r-2-2)},\quad n\in\mathbb{Z}_{\geqslant1}
\]

and those obtained frome these by swapping the coefficients for $\alpha_2$ and $\alpha_4$.

\[
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\dv{5}{(-1,0)}{6}
\dv{0}{(0,0)}{1}
\dv{1}{(30:1)}{2}
\dv{2}{($(30:1)+(0:1)$)}{3}
\dv{3}{($(-30:1)+(0:1)$)}{4}
\dv{4}{(-30:1)}{5}
\deA{a0}{a1}
\deA{a1}{a2}
\deA{a2}{a3}
\deA{a3}{a4}
\deA{a4}{a0}
\deA{a5}{a0}
\end{tikzpicture}
\]

In this case experimental evidence suggests that the basis of $Z_s(\alpha_6)$ is not given by polynomials. It starts with $\alpha_2,\ldots,\alpha_5$ together with the following roots (and their symmetric images):

\[
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\dv{6}{(-1,0)}{7}
\dv{0}{(0,0)}{1}
\dv{1}{(45:1)}{2}
\dv{2}{($(45:1)+(0:1)$)}{3}
\dv{3}{($(45:1)+(0:1)+(-45:1)$)}{4}
\dv{4}{($(-45:1)+(0:1)$)}{5}
\dv{5}{(-45:1)}{6}
\deA{a0}{a1}
\deA{a1}{a2}
\deA{a2}{a3}
\deA{a3}{a4}
\deA{a4}{a5}
\deA{a5}{a0}
\deA{a6}{a0}
\end{tikzpicture}
\]

The basis of $Z_s(\alpha_7)$ starts with $\alpha_2,\ldots,\alpha_6$ and the following roots (together with their symmetric images):

$\rootAEn{216, 100, 104, 108, 112, 117, 108}$

$\rootAEn{270, 126, 130, 135, 140, 145, 135}$,
$\rootAEn{330, 155, 160, 165, 170, 176, 165}$,

The roots listed above are covered (up to symmetry) by two disjoint polynomial families

\begin{align*}
& \aenroot{\&\& 6n^2-5n+1 \& 6n^2-4n \\ 3n(2n-1) \& 6n(2n-1) \&\&\& 3n(2n-1) \\ \&\& 6n^2-n \& 6n^2-2n \\}{(r-2-1)--(r-2-2)--(r-1-3)--(r-1-4)--(r-2-5)--(r-3-4)--(r-3-3)--(r-2-2)},\quad n\in\mathbb{Z}_{\geqslant1}, \\
& \aenroot{\&\& 6n^2+n \& 6n^2+2n \\ 3n(2n+1) \& 6n(2n+1) \&\&\& 3n(2n+1) \\ \&\& 6n^2+5n+1 \& 6n^2+4n \\}{(r-2-1)--(r-2-2)--(r-1-3)--(r-1-4)--(r-2-5)--(r-3-4)--(r-3-3)--(r-2-2)},\quad n\in\mathbb{Z}_{\geqslant1}.
\end{align*}

Note that they can be combined into a single “almost polynomial” family as even and odd members of the sequence:

\[
\aenroot{\& \& \frac{1}{2}n(3n+1) \& \makebox[\widthof{666}][c]{?} \\ \frac{3}{2}n(n+1) \& 3n(n+1) \& \& \& \frac{3}{2}n(n+1) \\ \& \& \frac{1}{2}(3n^2+5n+2) \& \makebox[\widthof{666}][c]{?} \\}{(r-2-1)--(r-2-2)--(r-1-3)--(r-1-4)--(r-2-5)--(r-3-4)--(r-3-3)--(r-2-2)},\quad n\in\mathbb{Z}_{\geqslant1}
\]

\[
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\dv{0}{(0,0)}{1}
\dv{1}{(45:1)}{2}
\dv{2}{($(45:1)+(-45:1)$)}{3}
\dv{3}{(-45:1)}{4}
\deA{a0}{a1}\deA{a0}{a2}\deA{a0}{a3}\deA{a1}{a2}\deA{a2}{a3}
\end{tikzpicture}
\]

$\rootK{4,1,8,6}$     $\rootK{8,13,28,18}$     $\rootK{28,37,92,60}$     $\rootK{28,61,104,66}$     $\rootK{60,73,192,126}$     $\rootK{52,97,188,120}$     $\rootK{104,121,328,216}$     $\ldots$

$\rootK{20,1,28,24}$     $\rootK{20,13,52,36}$     $\rootK{48,37,132,90}$     $\rootK{120,61,288,204}$     $\rootK{88,73,248,168}$     $\rootK{168,97,420,294}$     $\rootK{140,121,400,270}$

$\rootK{48,1,60,54}$     $\rootK{120,13,192,156}$     $\rootK{368,37,580,474}$     $\rootK{500,61,820,660}$     $\rootK{748,73,1172,960}$     $\vdots$     $\vdots$

$\rootK{88,1,104,96}$     $\rootK{168,13,252,210}$     $\rootK{448,37,680,564}$     $\vdots$      $\vdots$

$\rootK{140,1,160,150}$     $\rootK{388,13,512,450}$     $\vdots$

$\rootK{204,1,228,216}$     $\rootK{472,13,608,540}$

$\rootK{280,1,308,294}$     $\rootK{812,13,988,900}$

$\rootK{368,1,400,384}$     $\rootK{932,13,1120,1026}$

$\rootK{468,1,504,486}$     $\vdots$

The basis of $Z_s(\alpha_4)$ in the root system №45. Each root comes with its symmetric image (the coefficients for $\alpha_1$ and $\alpha_3$ are swapped).

The beginning of the basis of $Z_s(\alpha_4)$ is listed in <ref>. From the experimental evidence it seems that some subsets of the basis of $Z_s(\alpha_4)$ are (up to symmetry swapping $\alpha_1$ and $\alpha_3$) parametrized by polynomials. The roots with $c_2=1$ are of the form

\[
\aenroot{\& \makebox[\widthof{$6n^2$}][c]{1} \\ 2n(3n-1) \& \& 2n(3n+1) \\ \& 6n^2 \\}{(r-2-1)--(r-1-2)--(r-2-3)--(r-3-2)--(r-2-1)--(r-2-3)},\quad n\in\mathbb{Z}_{\geqslant0}.
\]

Roots with $c_2=13$ are covered by

\begin{align*}
& \aenroot{\& \makebox[\widthof{666}][c]{13} \\ 78n^2-122n+52 \& \& 78n^2-70n+20 \\ \& 78n^2-96n+36 \\}{(r-2-1)--(r-1-2)--(r-2-3)--(r-3-2)--(r-2-1)--(r-2-3)},\quad n\in\mathbb{Z}_{\geqslant0}, \\
& \aenroot{\& \makebox[\widthof{666}][c]{13} \\ 78n^2-86n+28 \& \& 78n^2-34n+8 \\ \& 78n^2-60n+18 \\}{(r-2-1)--(r-1-2)--(r-2-3)--(r-3-2)--(r-2-1)--(r-2-3)},\quad n\in\mathbb{Z}_{\geqslant0}
\end{align*}

Roots with $c_2=37$ are covered by

\begin{align*}
& \aenroot{\& \makebox[\widthof{666}][c]{37} \\ 222n^2-326n+132 \& \& 22n^2-178n+48 \\ \& 222n^2-252n+90 \\}{(r-2-1)--(r-1-2)--(r-2-3)--(r-3-2)--(r-2-1)--(r-2-3)},\quad n\in\mathbb{Z}_{\geqslant0}, \\
& \aenroot{\& \makebox[\widthof{666}][c]{37} \\ 222n^2-266n+92 \& \& 222n^2-118n+28 \\ \& 222n^2-192n+60 \\}{(r-2-1)--(r-1-2)--(r-2-3)--(r-3-2)--(r-2-1)--(r-2-3)},\quad n\in\mathbb{Z}_{\geqslant0}
\end{align*}

The found possible values of $c_2$ are

\[ 1,13,37,61,73,97,121,181,193,241,253,277,337,349,397,421,433,481,517,673,781,\ldots \]

The order of their first appearances is not coherent with height. All of them except for $1$ are either primes $p$, $p\equiv1\pmod{12}$, or products of two primes $pq$, $p\equiv q\equiv\pm1\pmod{12}$. It is not clear whether the products with $p=q$ appear in this sequence except fo $p=q=11$.

[1]
Y. Billig and A. Pianzola, Root strings with two consecutive real roots,
Tohoku Math. J.(2) 47 (1995), no. 3, 391–403.

[2]
L. Carbone, S. Chung, L. Cobbs, R. McRae, D. Nandi, Y. Naqvi, and D. Penta,
Classification of hyperbolic Dynkin diagrams, root lengths and Weyl
group orbits, J. Phys. A 43 (2010), no. 15, 155–209.

[3]
L. Carbone, M. Kownacki, S. Murray, and S. Srinivasan, Root subsystems of
rank 2 hyperbolic root systems, arXiv preprint arXiv:1506.05405 (2015).

[4]
K. Fukuda and A. Prodon, Double description method revisited,
Franco-Japanese and Franco-Chinese Conference on Combinatorics and
Computer Science, Springer, 1995, pp. 91–111.

[5]
J. Hurrelbrink, J. Morita, and U. Rehmann, On the homological $\pi_0$ of
Kac-Moody groups over fields, Cont. Math 126 (1992), 71–77.

[6]
V. Kac, Infinite-Dimensional Lie Algebras, 3 ed., Cambridge
University Press, 1990.

[7]
R. Moody and A. Pianzola, On infinite root systems, Trans. Amer. Math.
Soc. 315 (1989), no. 2, 661–696.

[8]
T. Motzkin, H. Raiffa, G. Thompson, and R. Thrall, The double description
method, Contributions to the Theory of Games 2 (1953), no. 28,

[9]
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Proc. Natl. Acad. Sci. USA 80 (1983), no. 6, 1778–1782.

[10]
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commutative rings, Amer. J. Math. 93 (1971), no. 4, 965–1004.

§ CENTRALIZERS IN RANK $3$

every picture/.append style=show background rectangle,inner frame sep=1ex,background rectangle/.style=use as bounding box

№     $\Phi$      $\alpha$    $Z_s(\alpha)$     basis

1(0, 0)12(1.5, 0)2

(1, 2, 0)(3, 2, 4)

$\varnothing$ [-6pt]0pt28pt

$\varnothing$ [-6pt]0pt28pt

1(0, 0)12(1.5, 0)2

(2, 1, 0)(0, 1, 2)

1(0, 0)12(1.5, 0)2

(1, 2, 0)(3, 2, 4)

1(0, 0)12(1.5, 0)2

(2, 3, 0)(0, 3, 2)

1(0, 0)12(1.5, 0)2

(3, 2, 0)(9, 2, 12)

1(0, 0)12(1.5, 0)2

(2, 1, 0)(0, 1, 2)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)12(1.5, 0)2

(0, 0, 1)(1, 2, 0)

1(0, 0)12(1.5, 0)2

(0, 1, 2)(2, 3, 0)

1(0, 0)12(1.5, 0)2

(1, 0, 0)(1, 6, 6)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)12(1.5, 0)2

(0, 0, 1)(3, 2, 0)

1(0, 0)12(1.5, 0)2

(2, 1, 0)(2, 3, 2)

1(0, 0)12(1.5, 0)2

(1, 0, 0)(0, 2, 1)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)12(1.5, 0)2

(0, 0, 1)(1, 2, 0)

1(0, 0)12(1.5, 0)2

(2, 3, 0)(2, 9, 6)

1(0, 0)12(1.5, 0)2

(1, 0, 0)(0, 2, 1)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)12(1.5, 0)2

(0, 0, 1)(3, 2, 0)

1(0, 0)12(1.5, 0)2

(2, 1, 0)(0, 1, 2)

1(0, 0)12(1.5, 0)2

(1, 0, 0)(1, 2, 2)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)12(1.5, 0)2

(0, 0, 1)(1, 2, 0)

1(0, 0)12(1.5, 0)2

(0, 1, 2)(2, 3, 0)

1(0, 0)12(1.5, 0)2

(1, 0, 0)(0, 2, 3)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)12(1.5, 0)2

(0, 0, 1)(1, 2, 0)

1(0, 0)12(1.5, 0)2

(2, 3, 0)(0, 3, 2)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)12(1.5, 0)2

(0, 0, 1)(3, 2, 0)

1(0, 0)12(1.5, 0)2

(2, 1, 0)(0, 1, 2)

$\varnothing$ [-6pt]0pt28pt

$\varnothing$ [-6pt]0pt28pt

$\varnothing$ [-6pt]0pt28pt

$\varnothing$ [-6pt]0pt28pt

$\varnothing$ [-6pt]0pt28pt

$\varnothing$ [-6pt]0pt28pt

$\varnothing$ [-6pt]0pt28pt

1(0, 0)12(1.5, 0)2

(1, 0, 2)(7, 6, 2)

1(0, 0)1

(2, 1, 0)

$\varnothing$ [-6pt]0pt28pt

1(0, 0)1

(1, 2, 0)

$\varnothing$ [-6pt]0pt28pt

$\varnothing$ [-6pt]0pt28pt

1(0, 0)12(1.5, 0)2

(2, 1, 0)(0, 1, 2)

$\varnothing$ [-6pt]0pt28pt

1(0, 0)1

(1, 0, 2)

1(0, 0)12(1.5, 0)2

(2, 0, 3)(0, 2, 3)

1(0, 0)1

(3, 0, 2)

1(0, 0)12(1.5, 0)2

(2, 0, 1)(0, 2, 1)

$\varnothing$ [-6pt]0pt28pt

$\varnothing$ [-6pt]0pt28pt

$\varnothing$ [-6pt]0pt28pt

$\varnothing$ [-6pt]0pt28pt

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)1

(0, 0, 1)

1(0, 0)1

(1, 0, 0)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)1

(0, 0, 1)

1(0, 0)1

(1, 0, 0)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)1

(0, 0, 1)

1(0, 0)1

(1, 0, 0)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)1

(0, 0, 1)

$\varnothing$ [-6pt]0pt28pt

1(0, 0)12(1.5, 0)2

(1, 0, 0)(0, 2, 1)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)1

(0, 0, 1)

1(0, 0)1

(0, 1, 2)

1(0, 0)12(1.5, 0)2

(1, 0, 0)(1, 4, 4)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)12(1.5, 0)2

(0, 0, 1)(1, 2, 0)

$\varnothing$ [-6pt]0pt28pt

1(0, 0)1

(1, 0, 0)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)12(1.5, 0)2

(0, 0, 1)(8, 8, 1)

1(0, 0)1

(2, 1, 0)

1(0, 0)1

(1, 0, 0)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)1

(0, 0, 1)

$\varnothing$ [-6pt]0pt28pt

1(0, 0)12(1.5, 0)2

(1, 0, 0)(0, 2, 1)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)1

(0, 0, 1)

1(0, 0)1

(0, 1, 2)

1(0, 0)12(1.5, 0)2

(1, 0, 0)(1, 2, 2)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)1

(0, 0, 1)

$\varnothing$ [-6pt]0pt28pt

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)1

(0, 0, 1)

1(0, 0)1

(0, 1, 2)

1(0, 0)12(1.5, 0)2

(1, 0, 0)(0, 2, 3)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)1

(0, 0, 1)

1(0, 0)1

(0, 3, 2)

1(0, 0)12(1.5, 0)2

(1, 0, 0)(0, 2, 1)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)1

(0, 0, 1)

$\varnothing$ [-6pt]0pt28pt

1(0, 0)1

(1, 0, 0)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)1

(0, 0, 1)

$\varnothing$ [-6pt]0pt28pt

1(0, 0)1

(1, 0, 0)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)1

(0, 0, 1)

1(0, 0)1

(0, 3, 2)

1(0, 0)12(1.5, 0)2

(1, 0, 0)(0, 2, 1)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)1

(0, 0, 1)

1(0, 0)1

(0, 1, 2)

1(0, 0)12(1.5, 0)2

(1, 0, 0)(0, 2, 3)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)12(1.5, 0)2

(0, 0, 1)(1, 2, 0)

1(0, 0)1

(2, 3, 0)

1(0, 0)1

(1, 0, 0)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)12(1.5, 0)2

(0, 0, 1)(3, 2, 0)

1(0, 0)1

(2, 1, 0)

1(0, 0)1

(1, 0, 0)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)1

(0, 0, 1)

$\varnothing$ [-6pt]0pt28pt

1(0, 0)1

(1, 0, 0)

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)1

(0, 0, 1)

$\varnothing$ [-6pt]0pt28pt

1(-1.25, 0)12(0, 0)23(1.25, 0)3

1(0, 0)1

(0, 0, 1)

$\varnothing$ [-6pt]0pt28pt

§ CENTRALIZERS IN RANK $\GEQSLANT 4$

every picture/.append style=scale=0.75
dynkin-vertex/.append style=font=

every picture/.append style=show background rectangle,inner frame sep=1ex,background rectangle/.style=use as bounding box
№     $\Phi$     $\alpha$     $Z_s(\alpha)$     basis

[baseline=(current bounding box.center)]

see <ref>

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_2$, $\alpha_3$(12,8,5,6)(12,5,8,6)(36,21,16,18)(36,16,21,18)(72,40,33,36)(72,33,40,36)(120,65,56,60)(120,56,65,60)(180,96,85,90)(180,85,96,90)

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_2$, $\alpha_3$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_2$, $\alpha_3$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_2$, $\alpha_3$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$(0,0,2,1)

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$,(1,2,0,0)

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_4$,(0,2,1,2)(2,4,2,1)(2,6,3,2)

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_3$,(0,2,1,2)(1,4,0,4)

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$,(1,2,0,0)

$\alpha_1$, $\alpha_3$,(4,8,1,4)(1,8,4,4)(9,24,4,12)(4,24,9,12)(16,48,9,24)(9,48,16,24)(25,80,16,40)(16,80,25,40)(36,120,25,60)(25,120,36,60)(49,168,36,84)(36,168,49,84)(64,224,49,112)

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_3$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_2$, $\alpha_3$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_2$, $\alpha_3$,

-4pt[0pt][0pt]$\varnothing$ [-12pt]0pt36pt

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]
1(-1.5, 0)12(0,0)23(1.5, 0)3

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]
1(-1.5, 0)12(0,0)23(1.5, 0)3

$\alpha_1$, $\alpha_2$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]
1(-1.5, 0)12(0,0)23(1.5, 0)3

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]
1(-1.5, 0)12(0,0)23(1.5, 0)3

$\alpha_1$, $\alpha_2$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]
1(-1.5, 0)12(0,0)23(1.5, 0)3

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]
1(-1.5, 0)12(0,0)23(1.5, 0)3

$\alpha_1$, $\alpha_2$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]
1(-1.5, 0)12(0,0)23(1.5, 0)3

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]
1(-1.5, 0)12(0,0)23(1.5, 0)3

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]
1(-1.5, 0)12(0,0)23(1.5, 0)3

$\alpha_1$, $\alpha_2$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]
1(-1.5, 0)12(0,0)23(1.5, 0)3

$\alpha_1$, $\alpha_2$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]
1(-1.5, 0)12(0,0)23(1.5, 0)3

$\alpha_1$, $\alpha_2$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]
1(-1.5, 0)12(0,0)23(1.5, 0)3

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]
1(-1.5, 0)12(0,0)23(1.5, 0)3

$\alpha_1$, $\alpha_2$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]
1(-1.5, 0)12(0,0)23(1.5, 0)3

$\alpha_1$, $\alpha_2$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]
1(-1.5, 0)12(0,0)23(1.5, 0)3

$\alpha_1$, $\alpha_2$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]
1(-1.5, 0)12(0,0)23(1.5, 0)3

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$,

[baseline=(current bounding box.center)]

$\alpha_2$, $\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_5$,

[baseline=(current bounding box.center)]

$\alpha_2$, $\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_5$,

[baseline=(current bounding box.center)]

$\alpha_2$, $\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

$\alpha_4$, $\alpha_5$,

[baseline=(current bounding box.center)]

$\alpha_2$, $\alpha_3$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_4$, $\alpha_5$,

$\alpha_2$, $\alpha_3$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_3$,

[baseline=(current bounding box.center)]
1(120:1.5)1 2($(-60:1.5)+(1.5,0)$)2 3(-120:3)3 4(-120:1.5)4 5(-60:1.5)5 6(0,0)6 7(60:1.5)7 8(-60:3)8 9($(-120:1.5)+(-1.5,0)$)9
a1a624 a1a7
[to path=[rounded corners=.3cm] – ++(60:1.5) – ++(0:1) – ($(a8)+(0:1.5)+(60:1)$) – ($(a8)+(0:1.5)$) – ()]a1a844
a1a944 a2a524 a2a744 a2a8
[to path=[rounded corners=.3cm] – ++(-60:1.5) – ++(-120:1) – ($(a9)+(-120:1.5)+(-60:1)$) – ($(a9)+(-120:1.5)$) – ()]a2a944
[to path=[rounded corners=.3cm] – ++(180:1.5) – ++(120:1) – ($(a7)+(120:1.5)+(180:1)$) – ($(a7)+(120:1.5)$) – ()]a3a744
a3a844 a3a9 a4a5 a4a6 a4a942 a5a6 a5a842 a6a742

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_3$,

[baseline=(current bounding box.center)]
[to path=[rounded corners=.4cm] – ++(0:1.5) – (0:3) – ($(a6)+(0:1.5)$) – ()]a4a6

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$, $\alpha_5$,

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$, $\alpha_5$,

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$, $\alpha_5$,

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_5$,

$\alpha_1$, $\alpha_2$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$, $\alpha_5$,

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_5$,

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$, $\alpha_5$,

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_5$,

[baseline=(current bounding box.center)]
1(-120:1.5)1 2(3,0)2 3($(1.5,0)+(-60:1.5)$)3 4(0,0)4 5(1.5,0)5 6(-60:1.5)6 7(-1.5,0)7
[to path=[rounded corners=.4cm] – (120:1.5) – ++(3,0) – ()]a7a244

$\alpha_1$, $\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$, $\alpha_5$,

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_5$,

[baseline=(current bounding box.center)]
1(0,0)1 2($(1.5,0)+(-60:1.5)$)2 3(-60:1.5)3 4(3,0)4 5(-120:1.5)5 6(-1.5,0)6 7(1.5,0)7
[to path=[rounded corners=.4cm] – (120:1.5) – ++(3,0) – ()]a6a448

$\alpha_1$, $\alpha_3$, $\alpha_4$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]
1(120:1.5)1 2(-60:1.5)2 3(-1.5,0)3 4(0,0)4 5(-120:1.5)5 6(60:1.5)6
[to path=[rounded corners=.4cm] – (1.5,0) – ()]a2a6

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_3$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_3$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$, $\alpha_5$,

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$,

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_3$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$, $\alpha_5$,

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_5$,

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_3$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$, $\alpha_5$,

$\alpha_1$, $\alpha_5$,

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_3$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_3$, $\alpha_4$, $\alpha_5$,

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_5$,

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_3$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_4$, $\alpha_5$, $\alpha_6$,

$\alpha_1$, $\alpha_2$, $\alpha_3$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]
1(0,1.5)1 2($(-150:1.5)+(120:1.3)$)2 3($(-30:1.5)+(-120:1.3)$)3 4($(-150:1.5)+(-60:1.3)$)4 5($(-30:1.5)+(60:1.3)$)5 6(0,0)6 7(-30:1.5)7 8(-150:1.5)8 9(1.3,1.5)9 10(-1.3,1.5)10
[to path=[rounded corners=.4cm] – ++(120:1.25) – ++(60:1.25) – ($(a5)+(60:1.25)+(120:1.25)$) – ($(a5)+(60:1.25)$) – ()]a2a5
[to path=[rounded corners=.4cm] – ++(-120:1.25) – ++(-1.25,0) – ($(a10)+(-1.25,0)+(-120:1.25)$) – ($(a10)+(-1.25,0)$) – ()]a3a10
[to path=[rounded corners=.4cm] – ++(-60:1.25) – ++(1.25,0) – ($(a9)+(1.25,0)+(-60:1.25)$) – ($(a9)+(1.25,0)$) – ()]a4a9

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]
1(120:1.5)1 2(-120:1.5)2 3(0,0)3 4(-1.5,0)4 5(3,0)5 6(1.5,0)6 7($(-60:1.5)+(1.5,0)$)7 8($(60:1.5)+(1.5,0)$)8

$\alpha_1$, $\alpha_2$, $\alpha_3$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]
1(0,1.25)1 2(-30:1.25)2 3(-150:1.25)3 4($(-150:1.25)+(-120:1.5)$)4 5($(-30:1.25)+(1.5,0)$)5 6($(0,1.25)+(120:1.5)$)6 7(0,0)7 8($(-150:1.25)+(-1.5,0)$)8 9($(-30:1.25)+(-60:1.5)$)9 10($(0,1.25)+(60:1.5)$)10
[to path=[rounded corners=.5cm] – ++(-1.5,0) – ++(120:1.5) – ($(a10)+(120:1.5)+(-1.25,0)$) – ($(a10)+(120:1.5)$) – ()]a4a1044
[to path=[rounded corners=.5cm] – ++(-60:1.5) – ++(-120:1.5) – ($(a8)+(-120:1.5)+(-60:1.5)$) – ($(a8)+(-120:1.5)$) – ()]a5a844
[to path=[rounded corners=.5cm] – ++(60:1.5) – ++(1.5,0) – ($(a9)+(1.5,0)+(60:1.5)$) – ($(a9)+(1.5,0)$) – ()]a6a944

$\alpha_1$, $\alpha_2$, $\alpha_3$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_6$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_5$, $\alpha_6$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_3$, $\alpha_6$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_3$, $\alpha_6$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]
1(120:1.5)1 2(-1.5,0)2 3(3,0)3 4($(60:1.5)+(1.5,0)$)4 5(60:1.5)5 6(1.5,0)6 7(0,0)7

$\alpha_1$, $\alpha_2$, $\alpha_4$, $\alpha_5$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_4$, $\alpha_5$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]
1(120:1.5)1 2(-1.5,0)2 3(1.5,0)3 4(3,0)4 5($(1.5,0)+(60:1.5)$)5 6(-120:1.5)6 7($(1.5,0)+(-60:1.5)$)7 8(0,0)8

$\alpha_1$, $\alpha_2$, $\alpha_4$, $\alpha_5$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_3$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_3$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_1$, $\alpha_2$, $\alpha_6$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

$\alpha_4$, $\alpha_5$, $\alpha_6$,

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]

[baseline=(current bounding box.center)]


# Radio Resource Management and Path Planning in Intelligent Transportation
Systems via Reinforcement Learning for
Environmental Sustainability

S. Norouzi1, N. Azarasa2, M.R. Abedi1, N. Mokari1, S.E. Seyedabrishami2, H.
Saeedi31, E.A. Jorswieck4  1Department of Electrical and Computer Engineering,
Tarbiat Modares University, Tehran, Iran  2Department of Civil and
Environmental Engineering, Tarbiat Modares University, Tehran, Iran  3College
of Engineering, Doha University for Science and Technology, Doha, Qatar
4Institute of Communications Technology, TU Braunschweig, Lower Saxony,
Germany  Emails:{nader.mokari<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Efficient and dynamic path planning has become an important topic for urban
areas with larger density of connected vehicles (CV) which results in
reduction of travel time and directly contributes to environmental
sustainability through reducing energy consumption. CVs exploit the cellular
wireless vehicle-to-everything (C-V2X) communication technology to disseminate
the vehicle-to-infrastructure (V2I) messages to the Base-station (BS) to
improve situation awareness on urban roads. In this paper, we investigate
radio resource management (RRM) in such a framework to minimize the age of
information (AoI) so as to enhance path planning results. We use the fact that
V2I messages with lower AoI value result in less error in estimating the road
capacity and more accurate path planning. Through simulations, we compare road
travel times and volume over capacity (V/C) against different levels of AoI
and demonstrate the promising performance of the proposed framework.

###### Index Terms:

V2X, Path planning; Resource management; MADDPG, AoI.

††publicationid: pubid:  979-8-3503-4760-9/23/$31.00 ©2023 IEEE

##  I Introduction

Intelligent transportation systems suggest that connected vehicles (CVs) will
gradually take over the roads. Meanwhile, path planning remains an important
process to increase the driving efficiency. Path planning refers to the
selection of a path from a starting point to the terminal point within a
particular network under some standard criteria such as path minimization [1].
This is specially the case in traditional path planning algorithms where the
traffic volume is assumed static [2]. It means that traffic jams and accidents
are not taken into account.

A great deal of progress on path planning has been made in the literature [3],
mainly focusing on finding the shortest path/time [4, 5, 6]. To improve fuel
consumption efficiency, a path planning scheme using deep reinforcement
learning (DRL) is proposed in [7] in which task deadline and fuel consumption
of each vehicle in the platoon are considered. A novel three-architecture
approach for vehicle path planning (SEARCH) proposed in [8] which exploits
Unmanned Aerial Vehicles (UAVs), Vehicular Ad hoc Networks (VANETs), 5G based
cellular systems, and Software-Defined Networking (SDN) to address two main
issues for navigation management difficulties: lack of local knowledge
regarding navigation solutions as well as vehicles that are inflexible in
dealing with unforeseeable situations. Based on their findings, the proposed
architecture works effectively in terms of reducing the driving time to any
destination. Authors in [9], by considering the personalized requirements of
vehicle users, propose a digital-twin (DT) enabled path planning scheme to
facilitate traffic management using a Q-learning algorithm.

A key subject in intelligent transportation systems is the vehicle
communication. According to the existing literature, in [10], we examine Mode
4 of the 3GPP cellular V2X architecture [11]. A distributed algorithm is used
to schedule resources and manage interference between CVs. Since path planning
information is time-sensitive, it is particularly critical to ensure the
freshness of the information received by base station (BS) at intersections.
Traditional performance metrics, such as latency or throughput, are not
sufficient to measure the timeliness of updates. As such, a new criterion,
called the age of information (AoI), is introduced in [12] for this kind of
application. In [13] the authors design a radio resource management (RRM)
policy to optimize the AoI in vehicular networks that satisfy the demand for
real-time and reliable communications. In [14] authors minimize the age of
status updates sent by vehicles over a carrier-sense multiple access (CSMA)
network using the gradient descent scheme. The authors of [15] investigate the
problem of AoI-aware RRM for a platooning system. To solve the corresponding
optimization problem, they use the multi-agent reinforcement learning (MARL)
algorithm for distributed resource allocation.

As can be seen, no study has been conducted to investigate the impact of RRM
on path planning validation. However, this is essential to have accurate and
real-time information when planning the optimal path for each CV. Accordingly,
RRM provides traffic incident messages that comply with the above
requirements. In our work, we model capacity estimation error utilizing AoI
levels, followed by formulating an optimization problem to minimize the AoI of
vehicles to validate path planning.

The contributions of this work lies in the following key items:
$\bullet$ We formulate an optimization problem for each CV to minimize the
AoI.
$\bullet$ A model is developed to describe the spectrum access of multiple CVs
as a multi-agent problem and utilize the deep deterministic policy gradient
(MADDPG) [16] algorithm.
$\bullet$ We update urban capacities estimation errors according to different
AoI values in path planning simulation.
$\bullet$ Numerical experiments demonstrate that the average AoI quantity is
maintained within a range of 5-10 milliseconds and fast convergence occurs.
Accordingly, path planning performance has significantly improved in terms of
travel time (TT) and volume over capacity V/C.
The rest of the paper is organized as follows. In Section II, the system model
is described. We present multi-agent RL-based V2X RRM design in Section III.
Section IV provides the simulation results and concluding remarks follow in
Section V.

##  II SYSTEM MODEL AND PROBLEM FORMULATION

Figure 1: Radio resource management system model

###  II-A Network model

As depicted in Fig. 1, our study is based on the utilization of cellular V2X
technology for vehicular communication which consists of one BS and multiple
CVs. The BS is located in the center of the environment and is equipped with a
single antenna, while CVs are also equipped with a single antenna. The BS is
mainly responsible for managing the optimal path by collecting data from all
network nodes and building a global traffic information graph of all CVs on
the roads. For this purpose, the BS updates the road’s capacity with the
received V2I information from CVs.

###  II-B Radio system model formulation

We assume that $\mathcal{V}=\\{v_{1},v_{2},\ldots,v_{V}\\}$, with
$|\mathcal{V}|=V$ indicates the set of CVs, $V$ represents the total number of
CVs. $\mathcal{L}=\\{l_{1},l_{2},\dots,l_{L}\\}$, with $|\mathcal{L}|=L$
indicates the set of roads, where $L$ represents the total number of roads in
urban graph model. The time is discretized into equal scheduling slots of
length $\Delta t$ with index $t$. The system bandwidth is divided into
orthogonal sub-channels of size $W_{n}$, where indexed by $n\in\mathcal{N}$,
and $\mathcal{N}$ is the set of sub-channels. According to the proposed
scheme, CV $v$ in road $l$ exchanges highway traffic data with the BS in the
center of crossroad via V2I links. Orthogonal frequency division multiplexing
(OFDM) is used to handle frequency selective wireless channels. We assume that
the channel fading is independent across different sub-channels and remains
constant within one sub-channel. We model the V2I channel gain of CV to BS on
sub-channel $n$ during one $\Delta t$ $t$ as,

$\displaystyle h_{v}^{t}[n]=\alpha_{v}^{t}g_{v}^{t}[n],$ (1)

It is worth noting that $\alpha_{v}^{t}$ and $g_{v}^{t}[n]$ represent the
large-scale fading effect, which is the result of path loss and shadowing, and
the small-scale fading, respectively. In addition, we define a binary variable
$\rho_{v}^{t}[n]\in$ $\\{0,1\\}$ that indicates whether sub-channel $n$ is
allocated to CV $v$ at time slot $t$, $\rho_{v}^{t}[n]=1$, otherwise
$\rho_{v}^{t}[n]=0$. We can describe the instantaneous rates between CV $v$
and the BS according to the Shannon capacity formula as follows:

$\displaystyle\mathcal{C}_{v}^{t}[n]=W_{n}\log_{2}\left(1+\frac{\rho_{v}^{t}[n]p_{v}^{t}[n]h_{v}^{t}[n]}{I_{v}^{t}[n]+\sigma^{2}_{n}}\right),$
$\displaystyle I_{v}^{t}[n]=\sum_{v^{\prime}\in
V}\rho_{v^{\prime}}^{t}[n]p_{v^{\prime}}^{t}[n]h_{v^{\prime}}^{t}[n],\>v^{\prime}\neq
v,$ (2)

where $p_{v}^{t}[n]$ is the power level used by CV $v$ on sub-channel $n$ for
transmitting, $h_{v^{\prime}}^{t}[n]$ is the interfering channel to BS from CV
$v^{\prime}\in\mathcal{V}$, and $I_{v}^{t}[n]$ corresponds to the total
interference power. As previously described, the CV has to ensure timely
communication with the BS to exchange any dynamic incidents. As a result, we
note $A_{v}^{t}$ as the AoI of CV $v$ up to the beginning of the scheduling
slot t, it refers to the period of time that has passed since the last
successful V2I communication. It is worth noting that this communication
inform BS of dynamic traffic capacities. The AoI of CV $v$ evolves according
to the following recurrence relation:

$\displaystyle A_{v}^{t+1}=\begin{cases}\Delta t,&\text{ if
}\rho_{v}^{t}[n]\cdot\mathcal{C}_{v}^{t}[n]\geq\mathcal{C}_{v}^{\min}\\\
A_{v}^{t}+\Delta t,&\text{ otherwise }\end{cases},$ (3)

In this case, $\mathcal{C}_{v}^{\min}$ is the minimum capacity required for
V2I communication.

###  II-C Path planning

For assigning the traffic demand between different origins and destinations
pairs of the network to available routes, one can take either user equilibrium
(UE) or system optimization (SO) approach. In UE condition, no driver can
unitedly reduce his/her travel time and according to Wardrop’s principles, all
the available routes between specific origin-destination pairs have the same
travel time. This condition naturally exists in the network because all drives
tend to optimize their travel time by changing routes. Under this condition
the total travel time of the network may not be minimum. On the other hand,
the objective function of SO problem is the total travel time spent in the
network and the traffic flow patterns that solves this problem, minimizes the
objective function. In our case study, the number of assigned CVs to each
route with both approaches are the same due to consistency of travel times in
each system iteration. For this reason, we take the UE assignment approach to
simulate the network and use the Dijkstra algorithm for finding the shortest
path based on travel time according to the bereau of public roads (BPR)
equation as below:

$\displaystyle
T_{l}(F_{l})=T_{l}(0)\left[1+\alpha_{u}\left(\frac{F_{l}}{Ca_{l}}\right)^{\beta_{u}}\right],$
(4)

where $l$ is an index of road, $T_{l}(0)$, $T_{l}$, $F_{l}$ and $Ca_{l}$ are
free flow time, travel time, traffic flow, and capacity of the road,
respectively. $\alpha_{u}$ and $\beta_{u}$ are two parameters of this equation
that are mostly regarded as $0.15$ and $4$[17]. At last, the Frank-Wolfe (FW)
algorithm is applied to solve an iterative first-order optimization. As
described earlier, different incidents can affect capacity of a road. In this
regard, we denote $Ca_{l}$ as the real capacity of road,

$\displaystyle Ca_{l}=\hat{Ca}_{l}+\Delta ca,$ $\displaystyle\Delta
ca=\left(-(A_{v})/A_{v,\mathfrak{(}R)}^{\text{max}}\right)\Delta m,$ (5)

where $\hat{Ca}_{l}$ and $\Delta ca$ denote the estimated capacity of each
road, and deviation from the actual value, respectively. The BS which is
responsible for path planning can estimate the roads capacities more
accurately with fresh data collected by V2I communications. $A_{v}$ is a
measure of the freshness of information. $A_{v,\mathfrak{(}R)}^{\text{max}}$
is the maximum AoI between CV $v$ and the BS, and $\Delta m$ is the maximum
amount of error in road’s capacities estimation. Thus, the optimization
problem for each CV $v$ can be expressed as follows:

$\displaystyle\min_{\boldsymbol{\rho},\boldsymbol{p}}$
$\displaystyle\left\\{\frac{1}{T}\sum_{t=1}^{T}A_{v}^{t}\right\\}$ (6) s.t.
$\displaystyle
C1:\mathcal{C}_{v}^{t}[n]\geq\mathcal{C}_{v}^{\min},\quad\forall
v\in\mathcal{V},\forall n\in\mathcal{N},$ (7) $\displaystyle
C2:\rho_{v}^{t}[n]\in\\{0,1\\},\quad\forall v\in\mathcal{V},\forall
n\in\mathcal{N},$ (8) $\displaystyle
C3:\sum_{n\in\mathcal{N}}\rho_{v}^{t}[n]\leq 1,\quad\forall
t\in\mathcal{N},\forall v\in\mathcal{V},$ (9) $\displaystyle
C4:p_{v}^{t}[n]\leq p_{v}^{\max},\quad\forall v\in\mathcal{V},\forall
n\in\mathcal{N}.$ (10)

The objective is to minimize the average AoI for every CV within every $T$
seconds, which leads to the correct estimation of the capacity and its optimal
use. Constraint C3 indicates that each CV is only permitted to access one sub-
channel per time slot. C4 mandates that CV $v$’s transmit power remain below
its maximum value $p_{v}^{\max}$. In the optimization problem, the sub-channel
selection parameter is a binary variable. The objective function is also non-
convex. Therefore, the optimization problem is an NP-hard combinatorial
optimization problem that is difficult to solve. To address the complexities
of the proposed optimization problem, we investigate the state-of-the-art
MADDPG method.

##  III MADDPG based solution

In a multi-agent environment, agents are intended to maximize their policy
functionalities, as outlined below:

$\displaystyle\max_{\pi}{J}^{v}\left(\pi^{v}\right),\>v\in\mathcal{V},\>\pi^{v}\in\Pi^{v},$
(11)

where
${J}^{v}\left(\pi^{v}\right)=\mathbb{E}\left[\sum_{t=0}^{\infty}\gamma^{t}\tilde{r}_{t+1}^{v}\mid
s_{0}^{v}\right]$ is our conditional expectation, $\pi^{v}$ describes the
policy of CV $v$, additionally, the set $\Pi^{v}$ includes all feasible
policies that are available for CV $v$. In the proposed model, a CV interacts
with the vehicular network environment takes actions in accordance with its
policies in order to address the optimization problem, 6. The agents strive to
improve their reward 11, which is linked to the objective function. At any
time $t$, the CV observes a state, $s^{t}$, and takes appropriate action,
$a^{t}$. As a result of the CV’s selected action, the environment transitions
into a new state $s^{t+1}$ and a reward is given to the CV. As a result of our
proposed system model, we define the state space, $\mathcal{S}$, the action
space, $\mathcal{A}$, and the reward function, $r^{t}$, as follows:
State space: Multiple components comprise the state observed by the CV at time
slot $t$: the instant channel information between CV $v$ and the BS,
$h_{v}^{t}[n]$ for all $n\in\mathcal{N}$, the previous interference from other
CVs to CV $v$, and the AoI of CV $v$, $A_{v}^{t}$. Accordingly, the state
space of CV $v$ is

$\displaystyle\mathbf{s}_{v}^{t}=[h_{v}^{t}[n],I_{v}^{t-1}[n],A_{v}^{t}],\>v\in\mathcal{V}.$
(12)

Action space: The action of each CV $v$ is defined as
$\mathbf{a}_{v}^{t}=\left\\{\rho_{v}^{t},p_{v}^{t}\right\\}$. Earlier, we
mentioned that $\rho_{v}^{t}[n]$ being 0 or 1 indicates if sub-channel $n$ has
been selected for a give $v$ and $t$, and $p_{v}^{t}$ represents the power at
time $t$ for CV $v$ on sub-carrier $n$. Using the deep deterministic policy
gradient (DDPG) method, the agent is able to select any power within the range
of 0 to $p_{v}^{\max}$. This is the advantage of policy gradient method where
we can apply continuous actions spaces and can converge to more accurate
results than conventional deep Q learning (DQNs) in which the power has to be
discretized.
Reward space: In this study, we use a learning problem that involves agents
receiving two reward, a global reward for their cooperation and an individual
reward for their performance. Two critics will be trained in the following
functions as part of this framework: global centralized critics use
observations and actions of all agents as inputs to estimate the global team
reward. The local critics, which are specific to each agent, estimate the
rewards based on the agent’s local observations and actions. The goal is to
optimize both global and local rewards simultaneously through the policy. We
first discuss MADDPG framework and then design our rewards.
We use $\pi$ to indicate the set for all CVs policies where CV $v$ policy is
denoted by
$\pi^{v}\left(\pi^{v}\in\boldsymbol{\pi}=\left\\{\pi^{1},\ldots,\pi^{V}\right\\}\right)$
with parameters $\tilde{\theta}^{v}$, $\tilde{Q}^{v}$ for CV critic
(Q-function) with parameter $\tilde{\phi}^{v}$ and $\tilde{Q}_{G}$ for global
Q-function with parameter $\tilde{\psi}$. In the first step, we form a neural
network, and in the second step, we discuss gradient policies. The number of
layers in neural networks for each agent action, critic, and global critic are
$L_{\pi}^{v},L_{q}^{v}$, and $L_{G}$, respectively. As a result of the above
information, we can develop our neural network in this manner
$\tilde{\Theta}_{i}=\left(W_{\pi}^{(1)},\ldots,W_{\pi}^{(L)}\right)$,
$\tilde{\Phi}_{q}=\left(W_{q}^{(1)},\ldots,W_{q}^{(L)}\right)$, and
$\tilde{\Psi}_{G}=\left(W_{G}^{(1)},\ldots,W_{G}^{(L)}\right)$ as actor, CV
critic and global critic. It can be concluded from the above that the gradient
policy has the following characteristics:

$\displaystyle\nabla_{\theta^{v}}{J}^{v}=\mathbb{E}\left[\left.\nabla_{{\tilde{\theta}^{v}}}\pi^{v}\left(a^{v}\mid
s^{v}\right)\nabla^{a^{v}}Q_{\pi}^{v}(\mathbf{s},\mathbf{a})\right|_{{a^{v}}={\pi^{v}}(s^{v})}\right],$
(13)

where $s=(s_{1},\dots,s_{N})$ and $a=(a_{1},\dots,a_{N})$ are the total state
and action vectors. In order to approximate Qfunction for CV $v$, all actions
and states in $Q_{\pi}^{v}(\mathbf{s},\mathbf{a})$ are integrated as inputs.
To develop the policy gradient, we utilize two critic networks, which are
reformulated as follows:

$\displaystyle\nabla_{\theta^{v}}{J}^{v}$
$\displaystyle=\underbrace{\mathbb{E}_{\mathbf{s},\mathbf{a}\sim\mathcal{D}}\left[\nabla_{\theta^{v}}\pi^{v}\left(a^{v}\mid
s^{v}\right)\nabla_{a^{v}}Q_{G}^{\psi}(\mathbf{s},\mathbf{a})\right]}_{\text{Global
Critic }}$ (14)
$\displaystyle+\underbrace{\mathbb{E}_{s^{v},a^{v}\sim\mathcal{D}}\left[\nabla_{\theta^{v}}\pi^{v}\left(a^{v}\mid
s^{v}\right)\nabla_{a^{v}}Q_{\phi^{v}}^{v}\left(s^{v},a^{v}\right)\right]}_{\text{CV
critic }},$

where $a^{v}=\pi^{v}(s^{v})$ is the action that agent $v$ chooses under policy
$\pi^{v}$ and $\mathcal{D}$ is the replay buffer. In (14), two terms are used:
the first shows the global critic, which receives the actions and states of
all CVs and estimates the global Q-function using the global rewards
$\tilde{r}_{G}$, and the second shows the critic of each CV, which receives
only its own actions and states. The loss function of the global critic is
updated by using

$\displaystyle\mathcal{L}(\psi)=\mathbb{E}_{\mathbf{s},\mathbf{a},\mathbf{r},\mathbf{s}}\left[\left(Q_{G}^{\psi}(\mathbf{s},\mathbf{a})-y_{G}\right)^{2}\right],$
(15)

where $y_{G}$ represents the target value of the estimation as follows:

$\displaystyle y_{G}=\tilde{r}_{G}+\left.\gamma
Q_{G}^{\psi}\left(\mathbf{s}^{\prime},\mathbf{a}^{\prime}\right)\right|_{a^{\prime
v}=\pi^{\prime v}(s^{\prime v})}.$ (16)

The target policy is $\pi^{\prime}=\left\\{\pi^{\prime 1},\ldots,\pi^{\prime
V}\right\\}$. It is parameterized with
$\boldsymbol{\theta}^{\prime}=\left\\{\theta^{\prime 1},\ldots,\theta^{\prime
V}\right\\}$, and the CV loss function and its target update are as follows:

$\displaystyle\mathcal{L}^{v}\left(\phi^{v}\right)=\mathbb{E}_{\mathbf{s^{v}},\mathbf{a^{v}},\mathbf{r^{v}},\mathbf{s^{\prime
v}}}\left[\left(Q_{\phi^{v}}^{v}\left(s^{v},a^{v}\right)-y^{v}\right)^{2}\right],$
(17) $\displaystyle y^{v}=r^{v}+\left.\gamma Q_{\phi^{\prime
v}}^{v}\left(s^{\prime v},a^{\prime v}\right)\right|_{a^{\prime v}=\pi^{\prime
v}(s^{\prime v})}.$ (18)

On the basis of the results presented in [18], we replace the global critic
with the twin delayed (TD3) deterministic policy gradient in order to overcome
the problem of overestimation and suboptimal policies resulting from errors in
function approximation. In this regard, the gradient of policy is as follows:

$\displaystyle\nabla_{\theta^{v}}{J}^{v}$
$\displaystyle=\underbrace{\mathbb{E}_{\mathbf{s},\mathbf{a}\sim\mathcal{D}}\left[\nabla_{\theta^{v}}\pi^{v}\left(a^{v}\mid
s^{v}\right)\nabla_{a^{v}}Q_{G_{i}}^{\psi_{i}}(\mathbf{s},\mathbf{a})\right]}_{\text{TD3
Global Critic }}$ (19)
$\displaystyle+\underbrace{\mathbb{E}_{s^{v},a^{v}\sim\mathcal{D}}\left[\nabla_{\theta^{v}}\pi^{v}\left(a^{v}\mid
s^{v}\right)\nabla_{a^{v}}Q_{\phi^{v}}^{v}\left(s^{f},a^{f}\right)\right]}_{\text{CV
critic }},$

In (19), the twin global critics are updated as follows:

$\displaystyle\mathcal{L}\left(\psi_{i}\right)=\mathbb{E}_{\mathbf{s}.\mathbf{a},\mathbf{r},\mathbf{s^{\prime}}}\left[\left(Q_{G_{i}}^{\psi_{i}}(\mathbf{s},\mathbf{a})-y_{G}\right)^{2}\right],$
(20)

where $y_{G}$ is updated as follows:

$\displaystyle
y^{v}=r^{v}+\left.\gamma(1-\tilde{d})\min_{i=1,2}Q_{\phi_{i}^{\prime
v_{i}}}^{v_{i}}\left(s^{\prime v},a^{\prime v}\right)\right|_{a^{\prime
i}=\pi^{\prime i}(s^{\prime i})}.$ (21)

The agents’ critic networks are updated by 17 and 18 and there is a detailed
description of the MADDPG algorithm in 1. In TD3, the main idea is to wait for
the convergence of value estimates before updating the policy. We are now able
to design our rewards, each agent should keep updated connections with the BS
and maintain a minimal level of AoI. The global reward should assist the CVs
in choosing the proper sub-channel and power to minimize interference between
the CVs. To this end, we design individual agent and global reward as

$\displaystyle
r_{\ell}^{v}=-{\frac{A_{v}^{t}}{V}+0.05G\left(\mathcal{C}_{v}^{t}-\mathcal{C}_{v}^{\text{min
}}\right)},$ (22) $\displaystyle
r_{g}=\frac{1}{V}\sum_{v\in\mathcal{V}}r_{\ell}^{v},$ (23)

where $G(x)$ is the stepwise function used to satisfy $C3$.

1 Generate CVs and establish an environment

2 Input: For each agent, enter the number of states and actions

3 Initiate the main global critic networks $Q_{\phi_{1}}^{v}$ and
$Q_{\phi_{2}}^{v}$

4 Initiate the target global critic networks of $Q_{\phi_{1}}^{\prime v}$ and
$Q_{\phi_{2}}^{\prime v}$

5 Set up the policy and critic networks for each agent

6for  _each episode_ do

7 Provide an update on CV locations and respective channel gains

8 Reset the payload of the V2I message and maximum transmission time $T$ to
${100\mathrm{~{}ms}}$

9 for  _each timestep $t$_ do

10 for  _each agent $v$_ do

11 observe $s_{v}^{t}$ and Choose an action $a_{v}^{t}$

12 ${s_{t}=[s_{t}^{1},\cdots,s_{t}^{V}]}$ ,
${a_{t}=[a_{t}^{1},\cdots,a_{t}^{V}]}$

13 Receive global and local rewards, $r_{g}^{t}$ and $r_{l}^{t}$

14 Store $(s^{t},a^{t},r_{l}^{t},r_{g}^{t},s^{t+1})$ in replay buffer
$\mathcal{D}$

15 Sample minibatch of size $S$, $(s^{j},a^{j},r_{l}^{j},r_{g}^{j},s^{\prime
j})$, from the replay buffer $\mathcal{D}$

16 $y_{g}^{j}=r_{g}^{j}+\gamma\min_{i}Q_{\psi_{i}^{\prime}}^{g_{i}}(s^{\prime
j},a^{\prime j})$

17 Provide an update to global critics by minimizing the loss:

18
$\mathcal{L}(\psi_{i})=\frac{1}{S}\sum_{j}\left\\{{\left(Q_{\psi_{i}}^{g_{i}}(s^{j},a^{j})-y_{g}^{j}\right)}^{2}\right\\}$

19 Update target parameters:
$\psi_{i}^{\prime}\leftarrow\tau\psi_{i}+(1-\tau)\psi^{\prime}_{i}$

20 if  _episode mode d_ then

21 Train local critics and actors

22 for  _each agent $i$_ do

23 Provide an update to local critics by minimizing the loss:

24
$\mathcal{L}(\phi_{i})=\frac{1}{S}\sum_{j}\left\\{{\left(Q_{\phi_{i}}^{i}(s^{j}_{i},a^{j}_{i})-y_{i}^{j}\right)}^{2}\right\\}$

25 Provide an update to local critics:

26
$\nabla{J}\theta_{i}\approx\frac{1}{S}\sum_{j}\nabla\theta_{i}\pi_{i}\left(a_{i}\mid
s_{i}^{j}\right)\nabla_{a_{i}}Q_{\psi_{1}}^{g_{1}}(s^{j},a^{j})+\nabla_{\theta_{i}}\pi_{i}(a_{i}\mid
s_{j}^{j})\nabla_{a_{i}}Q_{\phi_{i}}^{i}(s_{i}^{j},a_{i}^{j})$

27 Provide an update to target networks parameters

28 $\theta_{i}^{\prime}\leftarrow\tau\theta_{i}+(1-\tau)\theta_{i}^{\prime}$ ,
$\phi_{i}^{\prime}\leftarrow\tau\phi_{i}+(1-\tau)\phi_{i}^{\prime}$

29

30

Algorithm 1 Two time frame modified MADDPG

##  IV SIMULATION RESULTS

TABLE I: Parameters for simulation

Parameter | Value
---|---
Frequency of carrier | $2\mathrm{GHz}$
Number of Resource blocks | 3
Each Resource block’s bandwidth | $180\mathrm{kHz}$
Number of CVs in RRM simulation | 20
CVs Speed | $10-14\mathrm{~{}m}/\mathrm{s}$
BS and CVs antenna heights | $25,1.5\mathrm{~{}m}$
BS and CVs antenna gains | $8,3\mathrm{dBi}$
BS and CVs receiver noise figure | $5,9\mathrm{~{}dB}$
CVs mobility model | Urban case of A.1.2 [11]
CVs maximum power | $30\mathrm{dBm}$
Noise power $\sigma^{2}$ | $-114\mathrm{dBm}$
Time constraint of V2I message dissemination, $T$, | $100\mathrm{~{}ms}$
V2I links path loss model | $128.1+37.6\log_{d}(10)$
Shadowing distribution | Log-normal
Standard deviation of shadowing for V2I links | $8\mathrm{~{}dB}$
Decorrelation distance for V2I links | $50\mathrm{~{}m}$
V2I path loss/shadowing update | Every $100\mathrm{~{}ms}$[11]
V2I fast fading update | Every $1\mathrm{~{}ms}$[11]
Fast fading | Rayleigh fading
Replay buffer size for experience | 100000
Size of the mini batch | 64
Hidden layer number and size of actor networks | $2/1024,512$
Hidden layer number and size of critic networks | $3/1024,512,256$
Learning rate of critic/actor networks | $0.001/0.0001$
Factor of discount | $0.99$
Target networks soft update parameter | $0.0005$
Number of CVs in urban simulation | $240$
Normal roads’ capacity | $30\mathrm{~{}veh/h}$
Roads’ length | $250,433\mathrm{~{}m}$
Free flow time | $18,31\mathrm{~{}s}$
Number of roads | $48$
$\alpha_{u}$ | $0.15$
$\beta_{u}$ | $4$

Figure 2: Average V/C with different value of $\Delta m$ Figure 3: Average
roads travel times with different value of $\Delta m$

To evaluate the effectiveness of our formulation, we analyzed a network of 16
nodes, 48 roads, and 240 origin-destination pairs. In this scenario, CV flow
of 240 CVs per hour is considered, and the capacity of each road is assumed to
be 30 CVs per hour. A summary of the simulation parameters can be found in I.
The simulation results for CVs driving in this scenario are depicted in Fig. 3
and 2. We plotted the roads’ average travel time and a variable of the roads’
quality of service named volume over capacity (V/C) ratio against average AoI
for four different values of $\Delta m$. V/C is a measure of capacity
sufficiency, that is, whether or not the physical geometry and path planning
system provides sufficient capacity for the network flow. In the case of low
traffic flow in the network, the V/C ratio tends to zero which shows an ideal
state of the network. In contrast, as the traffic flow in the network
increases, the V/C ratio tends to 1. In the case of an overcrowded state in
the network, this ratio passes 1 and the roads’ quality of service is very
poor.

Fig. 3 depicts the average roads travel times versus average AoI values. We
can figure out that the average travel time of a CV grows as the average AoI
increases. An increase in AoI causes more error in road capacities estimation
and, as a result, we face invalid path planning. Moreover, it is shown that in
lower values of $\Delta m$, the growth of average travel time is more
negligible compared to high values of $\Delta m$.

Figure 2 plots the values of the V/C ratio against average AoI values. These
plots start from a value in the range of zero and 1. As the average AoI
increases, the path planning framework fails to update the road capacities
accurately and the volume of roads grows, and congestion forms. As a result,
the volume of roads gradually grows to the capacity and the V/C ratio passes
$1$. As mentioned earlier, in lower values of $\Delta m$, the increase in the
V/C ratio is smaller compared to higher values of $\Delta m$.

Figure 4: Average reward over learning episodes. Figure 5: Average AoI over
learning episodes.

Fig. 4 shows the convergence of proposed MADDPG framework reward. In the
proposed framework, individual and global rewards are maximized simultaneously
for all agents, resulting in a stable performance. From Fig. 5, it is evident
that as the number of episodes increases, the AoI quantity decreases. The
utilized MADDPG framework maintains the average AoI values from 60 to below
10$\mathrm{~{}ms}$.

##  V CONCLUSION

As presented in this paper, a joint path planning scheme and radio resource
allocation was developed for vehicular systems in order to minimize AoI of CVs
while updating urban road capacities for effective path planning. In our work,
we were able to confirm that the presented RRM scheme is highly robust and
effective in terms of enticing CVs to improve the model of their systems by
minimizing their AoI, thereby reducing the error in estimating the capacity.
The future work will explore more complicated system models including other
technologies such as UAVs to monitor the urban environment for dynamic
incidents.

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