notabloids centertableaux

# Real roots in the root system $\mathsf{T}_{2,p,q}$

Karin Baur, Jian-Rong Li, and Andrei Smolensky  Karin Baur, School of
Mathematics, University of Leeds, Leeds, LS2 9JT, Currently on leave from the
University of Graz, Graz, Austria<EMAIL_ADDRESS>Jian-Rong Li, Faculty of
Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna,
Austria<EMAIL_ADDRESS>Andrei Smolensky, Department of Mathematics and
Mechanics, Saint Petersburg State University, Saint Petersburg, Russia.
<EMAIL_ADDRESS>
###### Abstract.

Motivated by the recent advances in the categorification of the cluster
structure on the coordinate rings of Grassmannians of $k$-subspaces in
$n$-space, we investigate a particular construction of root systems of type
$\mathsf{T}_{2,p,q}$, including the type $\mathsf{E}_{n}$. This construction
generalizes Manin’s “hyperbolic construction” of $\mathsf{E}_{8}$ and reveals
a lot of otherwise hidden regularities in this family of root systems.

##  1\. Introduction

The real roots of root systems of finite, affine and hyperbolic type can be
characterized in terms of the coefficients of their decomposition into the
linear combination of simple roots [Kac, Proposition 5.10]. For the root
system with a simply-laced diagram this description boils down to the
following: the real roots are the elements of the root lattice having the same
norm as the simple roots. However, for non-hyperbolic root systems this
condition is only necessary, but not sufficient. There is at present no
general description of real roots available for non-hyperbolic root systems.

We investigate the root system of type
$\mathsf{J}_{k,n}=\mathsf{T}_{2,k,n-k-2}$, $k\leqslant n$, which has the
following diagram:

$\alpha_{1}$$\alpha_{2}$$\alpha_{k}$$\alpha_{k+1}$$\beta$$\alpha_{n-1}$

Here we denote $\beta=\alpha_{n}$. The root system $\mathsf{J}_{3,n}$ is
usually called the $\mathsf{E}_{n}$ root system, while
$\mathsf{J}_{1,n}=\mathsf{A}_{n}$ and $\mathsf{J}_{2,n}=\mathsf{D}_{n}$. In
general, the root system $\mathsf{J}_{k,n}$ is non-finite, non-affine, and
non-hyperbolic. With this paper, we give a characterisation of real roots for
a large class of root systems.

Such root systems appear naturally in the study of generalized Del Pezzo
varieties, that is, roughly speaking, the blow-ups of $\mathbb{P}^{m}$ at some
finite set of points, see [Coble, DO]. In particular, for the case $k=3$ and
its relation to the Picard lattice of Del Pezzo surfaces see [Manin, Section
25].

Another motivation for the present paper is the study of the rigid
indecomposable modules in Grassmannian cluster categories
$\mathrm{CM}(B_{k,n})$, see [JKS, BBG] and cluster variables in Grassmannian
cluster algebras $\mathbb{C}[{\rm Gr}_{k,n}]$ [Scott]. Scott proved that there
is a cluster algebra structure on the coordinate ring $\mathbb{C}[{\rm
Gr}_{k,n}]$ of the Grassmannian varieties ${\rm Gr}_{k,n}$. Jensen, King and
Su in [JKS] showed that the category ${\rm CM}(B_{k,n})$ of Cohen-Macaulay
modules over a quotient $B_{k,n}$ of a preprojective algebra of affine type
$A$ provides an additive categorification of $\mathbb{C}[{\rm Gr}_{k,n}]$ and
they showed that there is a cluster character on ${\rm CM}(B_{k,n})$ which
sends rigid indecomposable modules to cluster variables in $\mathbb{C}[{\rm
Gr}_{k,n}]$. In their paper, the authors associated the root system
$\mathsf{J}_{k,n}$ to $\mathrm{CM}(B_{k,n})$. They pointed out that rigid
indecomposable modules in ${\rm CM}(B_{k,n})$ seem to correspond to (real or
imaginary) roots of $\mathsf{J}_{k,n}$. Thus studying the roots of
$\mathsf{J}_{k,n}$ will help to study the rigid indecomposable modules in
Grassmannian cluster categories $\mathrm{CM}(B_{k,n})$ [JKS] and cluster
variables in Grassmannian cluster algebras $\mathbb{C}[{\rm Gr}_{k,n}]$
[Scott].

In this paper, we give a characterization of the real positive roots in the
root system $\mathsf{J}_{k,n}$. A real positive root
$\gamma\in\mathsf{J}_{k,n}$ is said to have degree $d$ if when $\gamma$ is
written as a linear combination of simple roots, the coefficient of $\beta$ in
$\gamma$ is $d$. Degree $0$ positive roots are just positive roots of the
natural root subsystem of type $\mathsf{A}_{n-1}$ given by the nodes
$\alpha_{1},\dots,\alpha_{n-1}$. They are all of the form
$\alpha_{i}+\cdots+\alpha_{j-1}$ for some $1\leqslant i<j\leqslant n$.

In an arbitrary root system there is a procedure to check whether a positive
element of the root lattice is a real root: for a positive real root there
exists a sequence of simple reflections which at each step lowers the height
(and eventually leads to a simple root), see [Kac, Proposition 5.1(e)].
However, there is no systematic way to find this sequence other than by trial
and error.

Our main result is that if one realizes the root lattice $\mathbb{Z}\Delta$ as
a sublattice of $\mathbb{Z}^{n}$ (see Section 2), then in terms of the ambient
lattice the above procedure can be done much faster and easier, as follows.
theoremmaintheoremrestate $x=(x_{1},\ldots,x_{n})^{\top}\in\mathbb{Z}\Delta$
is a positive real root of degree $\geqslant 1$ if and only if

  1. (1)

$0\leqslant x_{i}\leqslant\deg(x)$ for all $i=1,\ldots,n$,

  2. (2)

$q(x)=2$,

  3. (3)

repeated application of $x\longmapsto s_{\beta}(\operatorname{dec}(x))$
preserves property (1) until it changes the sign of all entries of $x$.

Here $q(x)$ is a quadratic form (2.1) on $\mathbb{Z}^{n}$, $s_{\beta}$ is the
simple reflection associated with $\beta$,

$s_{\beta}\left((x_{1},\ldots,x_{n})^{\top}\right)=(x_{1}+r,\ldots,x_{k}+r,x_{k+1},\ldots,x_{n})^{\top},$

$r=x_{k+1}+\ldots+x_{n}-2\deg(x)$, $\deg(x)=\frac{1}{k}\sum_{i=1}^{n}x_{i}$,
and $\operatorname{dec}(x)\in\mathbb{Z}\Delta$ is the element obtained from
permuting the entries of $x_{i}$ to have them in decreasing order, i.e. if
$\operatorname{dec}(x)=(x_{1}^{\prime},\dots,x_{n}^{\prime})$ then
$x_{1}^{\prime}\geqslant x_{2}^{\prime}\geqslant\ldots\geqslant
x_{n}^{\prime}$.

The procedure in Section 1 allows a very efficient enumeration of real roots.
This enumeration reveals many regularities which are otherwise harder to see.
Among other things, it provides another view on Manin’s “hyperbolic
construction” of $\mathsf{E}_{8}$, which can be seen as the inclusion
$\mathsf{E}_{8}\subset\mathsf{J}_{4,9}$. This also highlights the connection
between the affine roots of $\mathsf{E}_{9}$ inside $\mathsf{J}_{4,10}$ and
the exceptional curves on del Pezzo surfaces.

The paper is organized as follows. In Section 2 we construct the root lattice
and the action of the Weyl group on it, and give a characterization of real
root. In Section 3 we note various relations between the root systems
$\mathsf{J}_{k,n}$ for distinct $k,n$. Section 4 is devoted to the enumeration
of real roots and to some particular families of real roots. In 4.2 we discuss
the finite types, i.e. the types $\mathsf{A}_{n}$, $\mathsf{D}_{n}$,
$\mathsf{E}_{6}$, $\mathsf{E}_{7}$ and $\mathsf{E}_{8}$ and give a simple
description of the fundamental weights. Section 4.3 provides a simpler
description of isotropic roots in root systems of affine types
$\mathsf{J}_{3,9}=\mathsf{E}_{8}^{(1)}$ and
$\mathsf{J}_{4,8}=\mathsf{E}_{7}^{(1)}$. Also, in Section 4.4 we introduce the
notion of “almost real roots”. These are not roots but closely resemble the
real roots. Section 5 compares the description given in the present paper with
Manin’s “hyperbolic construction” of $\mathsf{E}_{8}$. Section 6 describes in
greater details the connection to the cluster structures on the coordinate
rings of Grassmannians mentioned above.

### Acknowledgments

K. B. was supported by a Royal Society Wolfson Fellowship. She is currently on
leave from the University of Graz. She would like to thank the Isaac Newton
Institute for Mathematical Sciences, Cambridge, for support and hospitality
during the programme CAR where work on this paper was undertaken. J.-R.L. was
supported by the Austrian Science Fund (FWF): M 2633-N32 Meitner Program and P
34602 individual project. A.S. was supported by Russian Science Foundation
(RSF) (project No. 17-11-01261).

##  2\. Real roots in $\mathsf{J}_{k,n}$ root system

###  2.1. Root lattice

Jensen, King, and Su gave a description of the root system $\mathsf{J}_{k,n}$
[JKS, Section 2]. This description of the root system arises naturally as the
lattice that grades the Grassmannian cluster algebra ${\rm Gr}_{k,n}$. They
observed that, for the right quadratic form $q(x)$ there seems to be a
relationship between cluster variables and positive degree roots. We recall
their results in the following.

Let $n$ and $k<n$ be two natural numbers and let $e_{1},\ldots,e_{n}$ be the
standard basis of $\mathbb{R}^{n}$. Let

$\alpha_{i}=e_{i+1}-e_{i}\quad\text{for}\quad
i=1,\ldots,n-1,\qquad\beta=e_{1}+\ldots+e_{k},$

and consider the lattice

$\mathbb{Z}\Delta=\langle\beta,\alpha_{1},\ldots,\alpha_{n-1}\rangle_{\mathbb{Z}}=\left\\{(x_{1},\ldots,x_{n})^{\top}\in\mathbb{Z}^{n}\
\middle|\ k\ \text{divides}\ x_{1}+\ldots+x_{n}\right\\},$

called the _root lattice_ and equipped with the quadratic form

$\displaystyle
q(x)=\sum_{i=1}^{n}x_{i}^{2}+\frac{2-k}{k^{2}}\bigg{(}\sum_{i=1}^{n}x_{i}\bigg{)}^{2}.$
(2.1)

and an inner product given by its polarization
$(x,y)=\frac{1}{2}\big{(}q(x+y)-q(x)-q(y)\big{)}$.

Any element $\alpha\in\mathbb{Z}\Delta$ can be written as
$\alpha=m_{\beta}\beta+m_{1}\alpha_{1}+\ldots+m_{n-1}\alpha_{n-1}$, and the
coefficient $m_{\beta}$ is called the _degree_ of $\alpha$, denoted by
$\deg(\alpha)$. If $x=(x_{1},\ldots,x_{n})^{\top}\in\mathbb{Z}\Delta$, then

$\deg(x)=(x_{1}+\ldots+x_{n})/k\qquad\text{and}\qquad
q(x)=x_{1}^{2}+\ldots+x_{n}^{2}+(2-k)\deg(x)^{2}.$

A direct calculation shows that the inner products are
$(\alpha_{i},\alpha_{i})=2$ for $i=1,\dots,n-1$ and $(\beta,\beta)=2$ and that

$(\alpha_{i},\alpha_{j})=\begin{cases}-1,&\text{if }|i-j|=1,\\\
0,&\text{otherwise,}\end{cases}\qquad(\beta,\alpha_{i})=\begin{cases}-1,&\text{if
}i=k,\\\ 0,&\text{otherwise},\end{cases}$

so that the Gram matrix $A$ of this inner product with respect to the basis
$\beta,\alpha_{1},\ldots,\alpha_{n-1}$ is the generalized Cartan matrix of the
root system of type $\mathsf{J}_{k,n}$.

The matrix of the basis change from
$\\{\beta,\alpha_{1},\dots,\alpha_{n-1}\\}$ to $\\{e_{1},\dots,e_{n}\\}$ is

$C=\begin{pmatrix}1&-1\\\ 1&1&-1\\\ \vdots&&1&-1\\\ 1&&&1&\ddots\\\
0&&&&\ddots&-1\\\ \vdots&&&&&1&-1\\\ 0&&&&&&1\end{pmatrix}.$

Therefore if

$\displaystyle\gamma=m_{\beta}\beta+m_{1}\alpha_{1}+\ldots+m_{n-1}\alpha_{n-1},$
then $\displaystyle\gamma=x_{1}e_{1}+\ldots+x_{n}e_{n},$ where
$\displaystyle(x_{1},\ldots,x_{n})^{\top}=C\cdot(m_{\beta},m_{1},\ldots,m_{n-1})^{\top}.$

The inverse for $C$ is calculated as $C^{-1}=VDU$, where $V$ is the lower
triangular matrix that differs from the identity matrix only in its first
column, which equals

$(1,1-k,2-k,\ldots,-1,0,\ldots,0)^{\top},$

$D=\operatorname{diag}(\frac{1}{k},1\ldots,1)$, and $U$ is an upper triangular
matrix having $1$ in all of its entries on and above the diagonal. Now

$Ux=(x_{1}+\ldots+x_{n},x_{2}+\ldots+x_{n},\ldots,x_{n})^{\top},$

so the first entry equals $kd$, where $d$ is the degree of $x$. Thus the first
entry of $DUx$ equals $d$, and the same holds for $VDUx$.

For the other entries of $VDUx$, note that $x_{i}+\ldots+x_{n}$ can be
rewritten as $kd-x_{1}-\ldots-x_{i-1}$. Thus for $i=2,\ldots,k$ the $i$-th
entry of $VDUx$ equals

$(kd-x_{1}-\ldots-x_{i-1})+(i-k-1)d=(i-1)d-x_{1}-\ldots-x_{i-1}.$

In particular, for $i=k$ it is

$(k-1)d-x_{1}-\ldots-x_{k-1}=x_{k}+\ldots+x_{n}-d.$

For $i>k$ the $i$-th entry is the same as the $i$-th entry of $Ux$, that is,
$x_{i}+\ldots+x_{n}$.

Many standard notions can be directly expressed in terms of the $e_{i}$ basis.
For example, if $\gamma=d\beta+\sum m_{i}\alpha_{i}$, the scalar product
$(\beta,\gamma)=2d-m_{k}$ can be computed as $2d-(x_{k+1}+\ldots+x_{n})$.

######  Example 2.1.

We demonstrate how the correspondence between the two bases described above
works in the second simplest case, that is in the case $k=2$ and
$\mathsf{J}_{k,n}=D_{n}$. In $\mathsf{D}_{n}$ the roots of degree $1$ are of
one of the following three forms (written in the $e_{i}$’s on the left and in
terms of the simple roots on the right):

$\displaystyle
e_{1}+e_{j}\quad\leftrightsquigarrow\quad\begin{matrix}0&1&1&\ldots&1&0&\ldots&0\\\
&1&&&\mathrlap{\lcurvearrownw j-1}\end{matrix}\quad\quad\text{with}\
2\leqslant j\leqslant n-1,$ $\displaystyle
e_{2}+e_{j}\quad\leftrightsquigarrow\quad\begin{matrix}1&1&1&\ldots&1&0&\ldots&0\\\
&1&&&\mathrlap{\lcurvearrownw j-1}\end{matrix}\quad\quad\text{with}\
3\leqslant j\leqslant n-1,$ $\displaystyle
e_{i}+e_{j}\quad\leftrightsquigarrow\quad\setcounter{MaxMatrixCols}{11}\begin{matrix}1&2&2&\ldots&2&1&\ldots&1&0&\ldots&0\\\
&1&&&\mathllap{i-1\rcurvearrowne}&&&\mathrlap{\lcurvearrownw
j-1}\end{matrix}\quad\text{with}\ 3\leqslant i<j\leqslant n-1.$

So in this case, the description of the positive roots in terms of the
$e_{i}$’s coincides (up to the ordering of the simple roots) with the standard
realization [Bou1, Ch. VI, §4, no. 8]: the positive roots of $\mathsf{D}_{n}$
are

$\displaystyle e_{i}\pm e_{j}\quad\text{with}\quad 1\leqslant i<j\leqslant n.$
and the simple roots are $\displaystyle\alpha_{i}=e_{i+1}-e_{i},\quad
1\leqslant i\leqslant n-1,\qquad\beta=e_{1}+e_{2}.$

Note that in [Bou1, Ch. VI, §4, no. 8] the numbering of the simple roots is
reversed, and $\beta$ is attached to $\alpha_{n-2}$.

###  2.2. Weyl group action

For $i=1,\ldots,n-1$ denote by $s_{i}$ the involution on $\mathbb{Z}^{n}$
induced by the transposition $(i,i+1)$ on the basis $e_{1},\ldots,e_{n}$, and
its restriction on $\mathbb{Z}\Delta$. Denote also by $s_{\beta}$ the linear
map

$x=(x_{1},\ldots,x_{n})^{\top}\longmapsto(x_{1}+r,\ldots,x_{k}+r,x_{k+1},\ldots,x_{n})^{\top},$

where $r=x_{k+1}+\ldots+x_{n}-2\deg(x)$. The map $s_{\beta}$ acts on
$\mathbb{Z}\Delta$, indeed, if $x\in\mathbb{Z}\Delta$, then

$(x_{1}+r)+\ldots+(x_{k}+r)+x_{k+1}+\ldots+x_{n}=\sum x_{i}+kr$

is also divisible by $k$, and $\deg(s_{\beta}x)=\deg(x)+r$.

######  Lemma 2.2.

The above formulas define an action of the Weyl group $W(\mathsf{J}_{k,n})$ on
$\mathbb{Z}\Delta$.

###### Proof.

To show that the action of $s_{\beta},s_{1},\ldots,s_{n}$ define an action of
the Weyl group, it is enough to show that they satisfy the defining relations
of $W(\mathsf{J}_{k,n})$ in its standard presentation as a Coxeter group, that
is

$\displaystyle s_{\beta}^{2}=s_{1}^{2}=\ldots=s_{n-1}^{2}=\operatorname{id},$
$\displaystyle
s_{i}s_{j}s_{i}=s_{j}s_{i}s_{j}\quad\text{for}\quad|i-j|=1,\quad i,j\leqslant
n-1,$ $\displaystyle s_{i}s_{j}=s_{j}s_{i}\quad\text{for}\quad|i-j|>1,\quad
i,j\leqslant n-1,$ $\displaystyle
s_{\beta}s_{k}s_{\beta}=s_{k}s_{\beta}s_{k},$ $\displaystyle
s_{\beta}s_{i}=s_{i}s_{\beta}\quad\text{for}\quad i\neq k.$

Note that the relations not involving $s_{\beta}$ are satisfied because
$s_{1},\ldots,s_{n-1}$ are defined as the fundamental transpositions, which
are known to be the standard Coxeter generators of the permutation group
$S_{n}$ [Wilson, Section 2.8.1].

Now set $x=(x_{1},\ldots,x_{n})\in\mathbb{Z}\Delta$. Then

$x\xmapsto{\ \ \mathclap{s_{\beta}}\ \
}(x_{1}+r,\ldots,x_{k}+r,x_{k+1},\ldots,x_{n})^{\top}\xmapsto{\ \
\mathclap{s_{\beta}}\ \
}(x_{1}+r+r^{\prime},\ldots,x_{k}+r+r^{\prime},x_{k+1},\ldots,x_{n})^{\top},$

where $r=x_{k+1}+\ldots+x_{n}-2\deg(x)$ and
$r^{\prime}=x_{k+1}+\ldots+x_{n}-2\deg(s_{\beta}x)$. But
$\deg(s_{\beta}x)=\deg(x)+r$, hence $r^{\prime}=r-2r$, so
$s_{\beta}^{2}(x)=x$.

The commutation relation for $s_{\beta}$ and $s_{i}$, $i\neq k$ are obvious
from the definition.

To prove the braiding relation for $s_{\beta}$ and $s_{k}$ consider

$\displaystyle x$ $\displaystyle\xmapsto{\ \ \mathclap{s_{k}}\ \
}(x_{1},\ldots,x_{k-1},x_{k+1},x_{k},x_{k+2},\ldots,x_{n})^{\top}$
$\displaystyle\xmapsto{\ \ \mathclap{s_{\beta}}\ \
}(x_{1}+r,\ldots,x_{k-1}+r,x_{k+1}+r,x_{k},x_{k+2},\ldots,x_{n})^{\top}$
$\displaystyle\xmapsto{\ \ \mathclap{s_{k}}\ \
}(x_{1}+r,\ldots,x_{k-1}+r,x_{k},x_{k+1}+r,x_{k+2},\ldots,x_{n})^{\top},$

where $r=x_{k}+x_{k+2}+\ldots+x_{n}-\deg(x)$, and

$\displaystyle x$ $\displaystyle\xmapsto{\ \ \mathclap{s_{\beta}}\ \
}(x_{1}+t,\ldots,x_{k}+t,x_{k+1},\ldots,x_{n})^{\top}$
$\displaystyle\xmapsto{\ \ \mathclap{s_{k}}\ \
}(x_{1}+t,\ldots,x_{k-1}+t,x_{k+1},x_{k}+t,x_{k+2},\ldots,x_{n})^{\top}$
$\displaystyle\xmapsto{\ \ \mathclap{s_{\beta}}\ \
}(x_{1}+t+t^{\prime},\ldots,x_{k-1}+t+t^{\prime},x_{k+1}+t^{\prime},x_{k}+t,x_{k+2},\ldots,x_{n})^{\top},$

where $t=x_{k+1}+\ldots+x_{n}-2\deg(x)$ and $t^{\prime}=x_{k}+t+x_{k+2}+\ldots
x_{n}-2\deg(s_{k}s_{\beta}x)$. But $\deg(s_{k}s_{\beta}x)=\deg(x)+t$, so
$t^{\prime}=x_{k}-x_{k+1}$. Thus $r=t+t^{\prime}$, and also
$x_{k+1}+t^{\prime}=x_{k}$ and $x_{k}+t=x_{k+1}+r$, which means that
$s_{k}s_{\beta}s_{k}(x)=s_{\beta}s_{k}s_{\beta}(x)$. ∎

######  Lemma 2.3.

$W(\mathsf{J}_{k,n})$ acts on $\mathbb{Z}\Delta$ by isometries, that is,
$q(wx)=q(x)$ for any $w\in W(\mathsf{J}_{k,n})$.

###### Proof.

The quadratic form $q$ is invariant under the action of $s_{i}$, because $q$
is defined in terms of symmetric polynomials. Concerning the action of
$s_{\beta}$, denote $r=x_{k+1}+\ldots+x_{n}-2d$, where $d=\deg(x)$, and
consider

$\displaystyle q(s_{\beta}x)$
$\displaystyle=\sum_{i=1}^{k}(x_{i}+r)^{2}+\sum_{\mathclap{i=k+1}}^{n}x_{i}^{2}+\frac{2-k}{k^{2}}\bigg{(}kr+\sum_{i=1}^{n}x_{i}\bigg{)}^{2}$
$\displaystyle=\sum_{i=1}^{n}x_{i}^{2}+2r\bigg{(}\sum_{i=1}^{k}x_{i}\bigg{)}+kr^{2}+(2-k)(d^{2}+2dr+r^{2})$
$\displaystyle=q(x)+2r\bigg{(}\sum_{i=1}^{k}x_{i}+2d-kd+r\bigg{)}$
$\displaystyle=q(x)+\bigg{(}r+2d-\sum_{\mathclap{i=k+1}}^{n}x_{i}\bigg{)}=q(x).\qed$

######  Remark 2.4.

This action is faithful and coincides with the standard action of
$W(\mathsf{J}_{k,n})$ on the root lattice.

###### Proof.

Straightforward check for the action of the generators on the simple roots. ∎

###  2.3. Real roots and degree change

The set of real roots $\Delta_{\mathrm{re}}$ of the root system $\Delta$ is
defined as the union of the Weyl group orbits of its simple roots. Since
$\mathsf{J}_{k,n}$ is simply-laced, all simple roots lie in the same orbit,
and so $\Delta_{\mathrm{re}}=W(\mathsf{J}_{k,n})\beta$. Recall that in the
basis $e_{1},\ldots,e_{n}$ one has $\beta=(1,\ldots,1,0,\ldots,0)^{\top}$.

The set of positive real roots is denoted by $\Delta_{\mathrm{re}}^{+}$.

######  Remark 2.5.

Real roots of degree $0$ form a subsystem of type $\mathsf{A}_{n-1}$ and are
of the form $e_{i}-e_{j}$, $i\neq j$. The root $e_{i}-e_{j}$ is positive if
$i>j$.

######  Lemma 2.6.

If $x=(x_{1},\ldots,x_{n})^{\top}\in\Delta_{\mathrm{re}}^{+}$ and
$\deg(x)=d\geqslant 1$, then $0\leqslant x_{i}\leqslant d$ for all
$i=1,\ldots,n$.

###### Proof.

Induction by $\deg(x)$.

Consider first the case $\deg(x)=1$. This means that $x_{1}+\ldots+x_{n}=k$.
On the other hand, for a real root $x$ one has $q(x)=2$. But

$q(x)=\sum x_{i}^{2}+\frac{2-k}{k^{2}}k^{2}=\sum x_{i}^{2}+2-k,$

hence $x_{1}^{2}+\ldots+x_{n}^{2}=k$. It follows that all $x_{i}$ are either
$0$ or $1$ (otherwise $\sum x_{i}^{2}>\sum x_{i}$).

Now if $x\in\Delta_{\mathrm{re}}$ has $\deg(x)>1$, then there is
$y\in\Delta_{\mathrm{re}}^{+}$ such that $\deg(y)<\deg(x)$ and $x=w(y)$ for
some $w\in W(A)$. The assumption $\deg(y)<\deg(x)$ holds because $w$ can be
chosen to be a sequence of reflections which only increase the height, see
[Moody, Proposition 1].

Since $w=\sigma_{1}s_{\beta}\sigma_{2}\dots s_{\beta}\sigma_{m}$ for some
$\sigma_{1},\ldots,\sigma_{m}\in W(\mathsf{A}_{n-1})\cong S_{n}$, one can
replace $y$ by $=\sigma_{2}s_{\beta}\dots\sigma_{m}(y)$ and $x$ by
$\sigma_{1}^{-1}(x)$, so that $x=s_{\beta}(y)$.

Now $y=s_{\beta}(x)=(x_{1}+r,\ldots,x_{k}+r,x_{k+1},\ldots,x_{n})^{\top}$ with
$r<0$, and $\deg(y)=\deg(s_{\beta}x)=\deg(x)+r$. By the induction hypothesis,
$0\leqslant x_{i}+r\leqslant\deg(y)$ for $i=1,\ldots,k$, and $0\leqslant
x_{i}\leqslant\deg(y)$ for $i>k$, so $0\leqslant x_{i}\leqslant\deg(x)$ for
all $i$. ∎

######  Remark 2.7.

For $x\in\mathbb{Z}\Delta$, which is not a real root, the inequalities
$0\leqslant x_{i}\leqslant\deg(x)$ are only guaranteed to be preserved by
$s_{\beta}$ in case this action increases the degree. That is, if $0\leqslant
x_{i}\leqslant\deg(x)$, $\deg(x)>0$ and $0<\deg(s_{\beta}(x))<\deg(x)$, then
$s_{\beta}(x)$ can have negative entries or entries greater than its degree.
See Section 4.4 for examples.

We will now establish some conditions on $x\in\mathbb{Z}\Delta$ which
guarantee that $s_{\beta}$ lowers the degree of $x$.

######  Lemma 2.8.

If $x=(x_{1},\ldots,x_{n})^{\top}\in\mathbb{R}^{n}$ and $d\in\mathbb{R}$
satisfy

$\displaystyle d\geqslant x_{1}\geqslant\ldots\geqslant x_{n}\geqslant 0,$
$\displaystyle x_{1}+\ldots+x_{n}=kd,$ $\displaystyle q(x)=\sum
x_{i}^{2}+(2-k)d^{2}>0,$

then $x_{k+1}+\ldots+x_{n}<2d$.

###### Proof.

The statement of the lemma can be reformulated as follows. Consider the
polyhedron $P$ given by the inequalities

$P=\left\\{x\in\mathbb{R}^{n}\ \middle|\ \begin{aligned} &d\geqslant
x_{1}\geqslant\ldots\geqslant x_{n}\geqslant 0,\\\
&x_{k+1}+\ldots+x_{n}\geqslant 2d,\\\
&x_{1}+\ldots+x_{n}=kd\end{aligned}\right\\}.$

One has to show that $P$ lies in a closed ball of radius $d\sqrt{k-2}$
centered at the origin.

Note that by scaling everything down $d$ times it is sufficient to prove this
statement for $d=1$. Note also that the maximal distance from $0$ over all
points of $P$ is attained at one of its vertices.

Denote by $A$ the following $(n+2){\times}n$ matrix, by $b$ the following
column vector in $\mathbb{R}^{n+2}$ and by $c$ the following row vector in
$\mathbb{R}^{n}$:

$A=\begin{pmatrix}1\\\ -1&1\\\ &-1&1\\\ &&\ddots&\ddots\\\ &&&-1&1\\\
&&&&-1\\\ 0\ \mathrlap{\ldots}&0&-1&\ldots&-1\end{pmatrix},\quad
b=\begin{pmatrix}1\\\ 0\\\ \\\ \vdots\\\ \\\ 0\\\ -2\end{pmatrix},\quad
c=(1,\ldots,1),$

so that $P=\\{x\in\mathbb{R}^{n}\mid Ax\leqslant b,\ c\cdot x=k\\}$.

Denote also by $A(i,j,l)$ the square matrix of the form
$\begin{pmatrix}\mathsf{A}_{I,*}\\\ c\end{pmatrix}$, where $\mathsf{A}_{I,*}$
is the submatrix of $A$ consisting of rows with indices in
$I=\\{1,\ldots,n\\}\setminus\\{i,j,l\\}$. Finally, denote by
$b(i,j,l)=\begin{pmatrix}b_{I}\\\ k\end{pmatrix}$.

Then the vertices of $P$ are those points of $P$ which satisfy the equation

$A(i,j,l)x=b(i,j,l),$

where $1\leqslant i<j<l\leqslant n+2$ are such that $A(i,j,l)$ is non-
singular.

First note that $i\leqslant k$. Indeed, if $i>k$, then $A(i,j,l)$ contains the
first $i$ rows of $A$. The equation coming from $A_{1,*}$ implies $x_{1}=1$,
while the next $i-1$ equations mean that $x_{1}=x_{2}=\ldots=x_{i}$, so
$x_{1}+\ldots+x_{n}\geqslant x_{1}+\ldots+x_{i}=i>k$.

Note also that $i,j,l$ cannot all be simultaneously $\leqslant k+1$, because
there is a linear dependence between the rows from $k+2$ to $n+2$, namely,

$\mathsf{A}_{k+2,*}+2\mathsf{A}_{k+3,*}+3\mathsf{A}_{k+4,*}+\ldots+(n-k)\mathsf{A}_{n+1,*}=\mathsf{A}_{n+2,*}.$

Assume first that $l=n+2$. Then $j\geqslant k+1$, because otherwise
$x_{k+1}=\ldots=x_{n}=0$ and thus $x_{k+1}+\ldots+x_{n}<2$. The solution is of
the form

$x=(\underbrace{1,\ldots,1}_{i-1},\underbrace{y,\ldots,y}_{j-i},0,\ldots,0),\quad
y=\frac{k+1-i}{j-i}.$

Since $j\geqslant k+1$, the inequalities of the form $x_{s}\geqslant x_{s+1}$
are also satisfied. Now $x_{k+1}+\ldots+x_{n}=y\cdot(j-k-1)$.

Let $s=k+1-i$, so that $y=\frac{s}{j-i}$. If $x$ is a vertex, then

$x_{k+1}+\ldots+x_{n}=y\cdot(j-k-1)\geqslant 2,$

or, equivalently,

$\frac{s}{j-i}(j-i-s)\geqslant 2.$

The squared distance from the origin to $x$ equals

$x_{1}^{2}+\ldots+x_{n}^{2}=(i-1)+(j-i)y^{2}=(i-1)+\frac{(k+1-i)^{2}}{j-i}=\\\
=k-s+\frac{s^{2}}{j-i}=k+\frac{s}{j-i}(s+i-j)\leqslant k-2.$

Now assume that $l<n+2$. Then the solution is of the form

$x=(\underbrace{1,\ldots,1}_{i-1},\underbrace{y_{1},\ldots,y_{1}}_{j-i},\underbrace{y_{2},\ldots,y_{2}}_{l-j},0,\ldots,0).$

The values of $y_{1}$ and $y_{2}$ are subject to the following two equations.
The first one is

$x_{1}+\ldots+x_{n}=k,\quad\text{i.e.}\quad(i-1)+y_{1}\cdot(j-i)+y_{2}\cdot(l-j)=k.$

The form that the equation $x_{k+1}+\ldots+x_{n}=2$ takes depends on $j$.

If $j\leqslant k+1$, the second equation is $y_{2}\cdot(l-k-1)=2$. In this
case

$y_{2}=\frac{2}{l-k-1},\quad
y_{1}=\frac{(k+1-i)(l-k-1)+2(j-l)}{(l-k-1)(j-i)}.$

Write

$\displaystyle s=k+1-i,\quad t=k+1-j,\quad r=l-k-1,$ so that $\displaystyle
y_{1}=\frac{sr-2(t+r)}{r(s-t)},\quad y_{2}=\frac{2}{r}.$

The inequality $y_{2}\leqslant y_{1}$ can be reformulated as $2(s+r)\leqslant
sr$, while the inequality $y_{1}\leqslant 1$ means $rt\leqslant 2(t+r)$. Now

$x_{1}^{2}+\ldots+x_{n}^{2}=k-s+(s-t)\frac{(sr-2(t+r))^{2}}{r^{2}(s-t)^{2}}+(t+r)\frac{4}{r^{2}}.$

This sum being not greater than $k-2$ can be expressed as

$2-s+\frac{(sr-2(t+r))^{2}}{r^{2}(s-t)}+\frac{4(t+r)}{r^{2}}\leqslant 0,$

or, multiplying by $r^{2}(s-t)$,

$(2-s)(s-t)r^{2}+(sr-2(t+r))^{2}+4(t+r)(s-t)\leqslant 0.$

But the left-hand side can be rewritten as

$(sr-2(s+r))\cdot(rt-2(t+r)),$

which is non-positive.

If $j>k+1$, the second equation becomes $y_{1}\cdot(j-k-1)+y_{2}\cdot(l-j)=2$.
This system of equations is equivalent to

$\displaystyle\begin{pmatrix}j-i&l-j\\\
i-k-1&0\end{pmatrix}\begin{pmatrix}y_{1}\\\
y_{2}\end{pmatrix}=\begin{pmatrix}k+1-i\\\ i-k+1\end{pmatrix},$ thus
$\displaystyle y_{1}=\frac{i-k+1}{i-k-1},\quad
y_{2}=\frac{(k+1-i)(i-k-1)+(i-j)(i-k+1)}{(i-k-1)(l-j)}.$

Again, write

$\displaystyle s=k+1-i,\quad t=j-k-1,\quad r=l-k-1,$ so that $\displaystyle
y_{1}=\frac{s-2}{s},\quad
y_{2}=\frac{s^{2}+(s+t)(2-s)}{s(r-t)}=\frac{2(s+t)-st}{s(r-t)}.$

The inequalities $0\leqslant y_{2}\leqslant y_{1}$ are equivalent to
$st\leqslant 2(s+t)$ and $sr\geqslant 2(s+r)$. Now

$x_{1}^{2}+\ldots+x_{n}^{2}=k-s+(s+t)\frac{(s-2)^{2}}{s^{2}}+(r-t)\frac{(2(s+t)-st)^{2}}{s^{2}(r-t)^{2}},$

and this sum being not greater than $k-2$ is equivalent to

$(2-s)(r-t)s^{2}+(s+t)(s-2)^{2}(r-t)+(2(s+t)-st)^{2}\leqslant 0.$

The left-hand side can be rewritten as

$(st-2(s+t))\cdot(sr-2(s+r)),$

which is non-positive. ∎

Denote by $\operatorname{dec}(x)\in\mathbb{Z}\Delta$ the permutation of
entries of $x=(x_{1},\ldots,x_{n})^{\top}$ such that $x_{1}\geqslant
x_{2}\geqslant\ldots\geqslant x_{n}$.

######  Corollary 2.9.

If $x\in\Delta_{\mathrm{re}}^{+}$ is such that $x=\operatorname{dec}(x)$, then
$\deg(s_{\beta}x)<\deg(x)$.

###### Proof.

The entries of the real root $x=(x_{1},\ldots,x_{n})^{\top}$ satisfy the
assumptions of the Lemma 2.8, so $\deg(s_{\beta}x)=\deg(x)+x_{k+1}+\ldots
x_{n}-2\deg(x)<\deg(x)$. ∎

###### Proof.

For any real root $x$ with $\deg(x)\geqslant 1$, all three properties are
satisfied, because they are satisfied for
$\beta=(1,\ldots,1,0,\ldots,0)^{\top}$ and are preserved by the operation
$x\longmapsto s_{\beta}(\operatorname{dec}(x))$ by Lemmas 2.6, 2.3 and 2.9.

Now assume that $x\in\mathbb{Z}\Delta$ satisfies these three properties, and
consider the sequence

$\big{(}x^{(i)}\big{)}_{i\in\mathbb{Z}_{\geqslant 0}}\quad\text{with}\quad
x^{(0)}=x,\quad x^{(i+1)}=s_{\beta}\big{(}\operatorname{dec}(x^{(i)})\big{)}.$

By Lemma 2.8 $\deg(x^{(i+1)})<\deg(x^{(i)})$. Denote by $m$ the smallest index
such that $x^{(m)}$ has non-negative entries but $x^{(m+1)}$ only has non-
positive entries. Then $\operatorname{dec}(x^{(m)})$ is of the form
$(y_{1},\ldots,y_{k},0,\ldots,0)$ for some non-negative $y_{i}$, because
$s_{\beta}$ only affects the first $k$ entries and must change the signs of
all entries by property (3).

Denote $d=\deg(x^{(m)})$, so that by property (3) $y_{i}\leqslant d$. Now
since $(y_{1}+\ldots+y_{k})/k=d$, it follows that $y_{i}=d$ and
$\operatorname{dec}(x^{(m)})=(d,\ldots,d,0,\ldots,0)^{\top}$. But then

$q\big{(}\operatorname{dec}(x^{(m)})\big{)}=kd^{2}+(2-k)d^{2}=2d^{2},$

so by property (2) $d=1$ and $\operatorname{dec}(x^{(m)})=\beta$ (and
$x^{(m+1)}=-\beta$). This implies that $x\in W\beta$ and hence it is a real
root. ∎

######  Remark 2.10.

It follows from the proof of the above theorem that property (3) can be
replaced by the following: repeated application of $x\longmapsto
s_{\beta}(\operatorname{dec}(x))$ leads to $-\beta$, and it does so in at most
$\deg(x)$ steps (in particular, for a real root $x$ of degree $1$ one has
$\operatorname{dec}(x)=\beta$ and $s_{\beta}(\operatorname{dec}(x))=-\beta$).

######  Remark 2.11.

The process described above also works for the elements of the root lattice
close to the real roots. In particular, it allows to distinguish non-roots
admitting a sequence of height-lowering simple reflections, see Section 4.4.
The latter are related to the indecomposable modules appearing in the
categorification of Grassmannian cluster algebras, see Section 6.

##  3\. Symmetries and embeddings

There is a natural correspondence between $\mathsf{J}_{k,n}$ and
$\mathsf{J}_{n-k,n}$ as their graphs are isomorphic, i.e. they have the same
root system, with a different ordering of the simple roots.

######  Remark 3.1.

If $x=(x_{1},\ldots,x_{n})^{\top}$ is an element of the root lattice
$\mathsf{J}_{k,n}$, then the corresponding element of the root lattice
$\mathsf{J}_{n-k,n}$ is $x^{\prime}=(d-x_{n},\ldots,d-x_{1})^{\top}$, where
$d=\deg(x)$.

###### Proof.

This correspondence is linear, maps simple roots to simple roots in symmetric
positions and preserves the quadratic form:

$\displaystyle q_{n-k,k}(x^{\prime})$
$\displaystyle=\sum(d-x_{i})^{2}+\big{(}2-(n-k)\big{)}d^{2}$
$\displaystyle=nd^{2}-2d\cdot\sum x_{i}+\sum x_{i}^{2}+(2-n+k)d^{2}$
$\displaystyle=\sum x_{i}^{2}-2kd^{2}+(2+k)d^{2}=q_{k,n}(x).\qed$

The root system $\mathsf{J}_{k,n}$ can be considered as a subsystem of both
$\mathsf{J}_{k,n+1}$ and $\mathsf{J}_{k+1,n+1}$ in the natural way, meaning
that the branch node of the tree is mapped to the branch point of the larger
graph. In terms of the $\alpha_{i},\beta$, we can consider any positive root
for a larger system containing it. The next remark explains how the subsystem
arises in terms of the $x_{i}^{\prime}s$.

######  Remark 3.2.

If $x=(x_{1},\ldots,x_{n})^{\top}$ is an element of the root lattice
$\mathsf{J}_{k,n}$ of degree $d$, then the corresponding elements of the root
latices $\mathsf{J}_{k,n+1}$ and $\mathsf{J}_{k+1,n+1}$ are

$(x_{1},\ldots,x_{n},0)^{\top}\quad\text{and}\quad(d,x_{1},\ldots,x_{n})^{\top},$

respectively. Note that both have degree $d$ in their respective root lattice.

###### Proof.

Both correspondences are linear, map simple roots to the respective simple
roots and preserve the quadratic form:

$d^{2}+\sum x_{i}^{2}+\big{(}2-(k+1)\big{)}d^{2}=\sum
x_{i}^{2}+(2-k)d^{2}.\qed$

In particular, iterating the above, this provides a description of the
infinite rank root system $\mathsf{J}_{\infty,\infty}$ as a set of
$\mathbb{Z}$-indexed sequences $(x_{i})_{i\in\mathbb{Z}}$ of integers such
that there exists $M\in\mathbb{N}$ such that

  1. (1)

$(x_{-M},\ldots,x_{M})^{\top}$ is an element of $\mathsf{J}_{M,2M+1}$ of
degree $d$,

  2. (2)

$x_{M+1}=x_{M+2}=\ldots=0$,

  3. (3)

$x_{-M-1}=x_{-M-2}=\ldots=d$.

Note that the particular choice of such $M$ does not change the degree $d$.
Indeed, if $x_{-M}+\ldots+x_{M}=Md$, then $x_{-M-m}+\ldots+x_{M+m}=(M+m)d$.

This naturally extend to the description of the inifinite rank root lattice
$\mathbb{Z}\mathsf{J}_{\infty,\infty}$. It is equipped with the inner product
$q$ defined as the value of $q$ on $\mathbb{Z}\mathsf{J}_{M,2M+1}$ for a
suitable $M$.

We also define $\operatorname{dec}(x)$ as follows: for
$(x_{i})_{i\in\mathbb{Z}}\in\mathbb{Z}\mathsf{J}_{\infty,\infty}$ with
$x_{i}\geqslant 0$ and of degree $d$ denote
$x^{\prime}=\operatorname{dec}\left((x_{-M},\ldots,x_{M})^{\top}\right)\in\mathbb{Z}\mathsf{J}_{M,2M+1}$
and set $\operatorname{dec}(x)$ to be the image of $x^{\prime}$ in
$\mathbb{Z}\mathsf{J}_{\infty,\infty}$. Note that if $x$ is a real root of
$\mathsf{J}_{\infty,\infty}$, then so is $\operatorname{dec}(x)$.

For every $x\in\mathbb{Z}\mathsf{J}_{\infty,\infty}$ there exist the smallest
$k,n$ such that $x$ comes from an element of $\mathbb{Z}\mathsf{J}_{k,n}$ by
means of Remark 3.2. Namely, $k$ is the smallest natural number such that
$x_{-k-1}=x_{-k-2}=\ldots=\deg(x)$, while $n$ is the smallest such that
$x_{n-k}=x_{n-k+1}=\ldots=0$.

##  4\. Enumeration of roots

###  4.1. Real roots

In order to enumerate roots, we can use the action of the $\mathsf{A}_{n-1}$
type subsystem in $\mathsf{J}_{k,n}$ as explained in the following remark.

######  Remark 4.1.

The Weyl group $W(\mathsf{A}_{n-1})\cong S_{n}$ of the root subsystem
$\langle\alpha_{1},\ldots,\alpha_{n-1}\rangle$ of type $\mathsf{A}_{n-1}$ acts
on the root lattice by permutations on the entries of $x$ while keeping the
degree. Thus the enumeration of roots reduces to the enumeration of the orbits
of this action on the roots of each degree. This will be our strategy in this
section. In each such orbit we choose one representative which is ordered.
This representative has the smallest height over its orbit, and since the
support of a root is connected, this root belongs to a particular natural
subsystem of the smallest rank.

All orbits of real roots of degrees up to $5$ (assuming $k$ and $n$ are large
enough) are listed in Table 1, in the coordinates $(x_{i})_{i}$.

degree $1$ |  $\underbracket{1\ldots 1}_{k}$ (A1$+$,A2$-$,B1$+$,B2$-$,C1$\pm$,D,E)
---|---
degree $2$ |  $\underbracket{2\ldots 2}_{k-3}\underbracket{1\ldots 1}_{6}$ (A1$-$,A2$+$,B1$-$,B2$+$,C0$\pm$,C2$\pm$,D,E)
degree $3$ |  $\underbracket{3\ldots 3}_{k-3}\underbracket{2}_{1}\underbracket{1\ldots 1}_{7}$ (A0$\pm$,A3$\pm$,D)
|  $\underbracket{3\ldots 3}_{k-4}\underbracket{2\ldots
2}_{4}\underbracket{1\ldots 1}_{4}$ (C1$\pm$,E)
|  $\underbracket{3\ldots 3}_{k-5}\underbracket{2\ldots
2}_{7}\underbracket{1}_{1}$ (B0$\pm$,B3$\pm$,D’)
degree $4$ |  $\underbracket{4\ldots 4}_{k-3}\underbracket{3}_{1}\underbracket{1\ldots 1}_{9}$ (D) |  $\underbracket{4\ldots 4}_{k-3}\underbracket{222}_{3}\underbracket{1\ldots 1}_{6}$ (A1$+$,A2$-$)
|  $\underbracket{4\ldots 4}_{k-4}\underbracket{33}_{2}\underbracket{222}_{3}\underbracket{1\ldots 1}_{4}$ |  $\underbracket{4\ldots 4}_{k-4}\underbracket{3}_{1}\underbracket{2\ldots 2}_{6}\underbracket{1}_{1}$ (C0$\pm$,C2$\pm$)
|  $\underbracket{4\ldots 4}_{k-5}\underbracket{3\ldots 3}_{5}\underbracket{1\ldots 1}_{5}$ (E) | $\underbracket{4\ldots 4}_{k-5}\underbracket{3\ldots 3}_{4}\underbracket{222}_{3}\underbracket{11}_{2}$
|  $\underbracket{4\ldots 4}_{k-6}\underbracket{3\ldots 3}_{6}\underbracket{222}_{3}$ (B1$+$,B2$-$) |  $\underbracket{4\ldots 4}_{k-7}\underbracket{3\ldots 3}_{9}\underbracket{1}_{1}$ (D’)
degree $5$ |  $\underbracket{5\ldots 5}_{k-3}\underbracket{4}_{1}\underbracket{1\ldots 1}_{11}$ (D) | $\underbracket{5\ldots 5}_{k-3}\underbracket{3}_{1}\underbracket{222}_{3}\underbracket{1\ldots 1}_{6}$
|  $\underbracket{5\ldots 5}_{k-3}\underbracket{2\ldots 2}_{6}\underbracket{111}_{3}$ (A1$-$,A2$+$) | $\underbracket{5\ldots 5}_{k-4}\underbracket{44}_{2}\underbracket{2\ldots 2}_{4}\underbracket{1\ldots 1}_{4}$
|  $\underbracket{5\ldots 5}_{k-4}\underbracket{4}_{1}\underbracket{333}_{3}\underbracket{2}_{1}\underbracket{1\ldots 1}_{5}$ | $\underbracket{5\ldots 5}_{k-4}\underbracket{4}_{1}\underbracket{33}_{2}\underbracket{2\ldots 2}_{4}\underbracket{11}_{2}$
|  $\underbracket{5\ldots 5}_{k-4}\underbracket{3\ldots 3}_{5}\underbracket{2}_{1}\underbracket{111}_{3}$ |  $\underbracket{5\ldots 5}_{k-4}\underbracket{3\ldots 3}_{4}\underbracket{2\ldots 2}_{4}$ (C1$\pm$)
|  $\underbracket{5\ldots 5}_{k-5}\underbracket{444}_{3}\underbracket{33}_{2}\underbracket{22}_{2}\underbracket{111}_{3}$ | $\underbracket{5\ldots 5}_{k-5}\underbracket{444}_{3}\underbracket{3}_{1}\underbracket{2\ldots 2}_{5}$
|  $\underbracket{5\ldots 5}_{k-5}\underbracket{44}_{2}\underbracket{3\ldots 3}_{4}\underbracket{22}_{2}\underbracket{1}_{1}$ |  $\underbracket{5\ldots 5}_{k-6}\underbracket{4\ldots 4}_{6}\underbracket{1\ldots 1}_{6}$ (E)
|  $\underbracket{5\ldots 5}_{k-6}\underbracket{4\ldots 4}_{5}\underbracket{3}_{1}\underbracket{222}_{3}\underbracket{1}_{1}$ | $\underbracket{5\ldots 5}_{k-6}\underbracket{4\ldots 4}_{4}\underbracket{3\ldots 3}_{4}\underbracket{11}_{2}$
|  $\underbracket{5\ldots 5}_{k-6}\underbracket{444}_{3}\underbracket{3\ldots 3}_{6}$ (B1$-$,B2$+$) | $\underbracket{5\ldots 5}_{k-7}\underbracket{4\ldots 4}_{6}\underbracket{333}_{3}\underbracket{2}_{1}$
|  $\underbracket{5\ldots 5}_{k-9}\underbracket{4\ldots 4}_{11}\underbracket{1}_{1}$ (D’) |
Table 1. $W(\mathsf{A}_{n-1})$-orbits of real roots of degrees $1$ to $5$,
presented in the $x_{i}$. $0$’s are omitted. Some orbits are marked by the
families of roots, see below.

The completeness of this table is justified by the following lemma.

######  Lemma 4.2.

If $x\in\mathbb{Z}\mathsf{J}_{\infty,\infty}$ has degree $d\geqslant 1$, all
its entries are non-negative, $q(x)>0$ and $x=\operatorname{dec}(x)$, then it
is a root lattice element of the natural $\mathsf{J}_{2d-1,4d-2}$ subsystem.

###### Proof.

There are minimal $k,n$ such that $x$ is a root lattice element of a natural
$\mathsf{J}_{k,n}$ subsystem. Write $x$ in terms of the simple roots for this
subsystem. Denote by $m_{i}$ the coefficient of $\alpha_{i}$ in $x$, so that
$x=m_{\beta}\beta+m_{1}\alpha_{1}+\ldots+m_{n-1}\alpha_{n-1}$. Then
$m_{k}=(VDUx)_{k+1}=x_{k+1}+\ldots+x_{n}$, which by Lemma 2.8 is at most
$2d-1$. It follows that $m_{1},\ldots,m_{k}$ is a strictly increasing
sequence, while $m_{k},\ldots,m_{n-1}$ is strictly decreasing. Thus
$k,n-k\leqslant 2d-1$, so $n\leqslant 4d-2$. ∎

The above lemma gives a tool to enumerate the real roots of degree $\leqslant
d$ as follows:

  1. (1)

enumerate all length $4d-2$ decreasing sequences of numbers from
$\\{0,1,\ldots,d\\}$ such that the sum of entries is divisible by $2d-1$;

  2. (2)

for each such sequence check whether $q$ evaluates to $2$;

  3. (3)

if it does, perform the procedure of Section 1 to establish whether this
sequence correspond to a real root.

######  Remark 4.3.

Since $x=\operatorname{dec}(x)$, the sequences $m_{1},\ldots,m_{k}$ and
$m_{k},\ldots,m_{n-1}$ are convex in the sense that $2m_{i}\leqslant
m_{i-1}+m_{i+1}$ for $i=2,\ldots,k-1$ and $k+1,\ldots,n-2$.

######  Remark 4.4.

Experimental evidence suggests that in fact such $x$ is an element of a
natural $\mathsf{J}_{k,n}$ subsystem for some $n\leqslant 2d+2$ and some $k$.
The roots $\gamma_{d}$ and $\delta_{d}$ (see below) display the extreme cases
with $n=2d+2$ and $k=3$ and $k=d+1$ respectively.

The rest of Section 4 is devoted to the description of particular series of
roots. We will show that in terms of the $e_{i}$’s even the structure of
finite type root systems is more transparent.

We start with marking two distinguished families of roots, one in each degree
$d\geqslant 2$:

$\displaystyle\gamma_{d}$ $\displaystyle=\underbracket{d\ldots
d}_{k-3}\underbracket{d-1}_{1}\underbracket{1\ldots 1}_{2d+1},$ (D)
$\displaystyle\delta_{d}$ $\displaystyle=\underbracket{d\ldots
d}_{k-d-1}\underbracket{d-1\ldots d-1}_{d+1}\underbracket{1\ldots 1}_{d+1},$
(E)

corresponding respectively to

$\gamma_{d}=\begin{matrix}1&d&2d-1&2d-2&\cdots&2&1\\\
&&d\end{matrix}\quad\text{and}\quad\delta_{d}=\begin{matrix}1&2&\cdots&d&d+1&d&\cdots&2&1\\\
&&&&d\end{matrix}.$

The family of roots dual to $\gamma_{d}$ (see Remark 3.1) are marked by (D’)
in Table 1.

Among the roots of these series are $\gamma_{2}=\delta_{2}$, the maximal root
of the natural $\mathsf{E}_{6}$ subsystem, and
$\delta_{3}=\beta+\delta_{4,8}\in\mathsf{J}_{4,8}$, see Section 4.3. Their
inners products are

$(\gamma_{d},\gamma_{d^{\prime}})=2-|d-d^{\prime}|,\qquad(\delta_{d},\delta_{d^{\prime}})=2-|d-d^{\prime}|,\qquad(\gamma_{d},\delta_{d})=d(3-d).$

To see that they are indeed real roots note that the sum of last $n-k$ entries
of $\gamma_{d}$ equals $2d-1$, thus $(\gamma_{d},\beta)=1$ and

$s_{\beta}(\gamma_{d})=\underbracket{d-1\ldots
d-1}_{k-3}\underbracket{d-2}_{1}\underbracket{00}_{2}\underbracket{1\ldots
1}_{2d-1}\in W(\mathsf{A}_{n-1})\gamma_{d-1}.$

Similarly, the sum of the last $n-k$ entries of $\delta_{d}$ equals $d+1$, so
$(\delta_{d},\beta)=d-1$ and

$s_{\beta}(\delta_{d})=\underbracket{1\ldots 1}_{k-d-1}\underbracket{0\ldots
0}_{d+1}\underbracket{1\ldots 1}_{d+1}\in W(\mathsf{A}_{n-1})\beta.$

###  4.2. Root systems of finite types

Let us now consider the case of finite root systems
$\mathsf{J}_{3,n}=\mathsf{E}_{n}$, $n=6,7,8$. It can be seen in Table 1 that
in $\mathsf{E}_{6}$ and $\mathsf{E}_{7}$ there are no degree $3$ roots (as in
these cases, $\sum x_{i}<3k$). In $\mathsf{E}_{6}$ there is a single degree
$2$ root $(1,1,1,1,1,1)^{\top}$, which is the maximal root $\gamma_{2}$. In
$\mathsf{E}_{7}$ there are $7$ such roots, all conjugate under
$W(\mathsf{A}_{6})$ to the image of $\gamma_{2}$ in $\mathsf{E}_{7}$ and of
the form $\operatorname{dec}(x)=(1,1,1,1,1,1,0)^{\top}$. In $\mathsf{E}_{8}$
there is a single $W(\mathsf{A}_{7})$-orbit of degree $3$ roots, of the form
$\operatorname{dec}(x)=\gamma_{3}=(2,1,1,1,1,1,1,1)^{\top}$.

In Fig. 1 the positive roots of $\mathsf{E}_{6}$ are displayed by means of the
weight diagram of its adjoint representation. The weight diagram of a
representation is a graph with vertices corresponding to the weights of the
representation (with multiplicities). An edge labeled $i$ joins the weights
$\lambda$ and $\mu$ if $\lambda-\mu=\pm\alpha_{i}$. The weights of the adjoint
representation are the roots of the root system together with zero weights
corresponding to the simple roots. To determine the root $\alpha$
corresponding to a given vertex one can find a path joining this vertex to a
zero weight and going from left to right. Then $\alpha=\sum\alpha_{i}$, where
sum is taken over all labels $i$ occuring in this path. For more details
concerning weight diagrams see [PSV].

Figure 1. Weight diagram of the adjoint representation of $\mathsf{E}_{6}$.
Only the part corresponding to the positive roots and to zero weights is
shown. The rightmost vertices correspond to the zero weights numbered by the
simple roots, and all other vertices correspond to positive roots of
$\mathsf{E}_{6}$. In any square (parallelogram) the labels on opposide sides
coincide and thus some of them are omitted. The vertices of degree $1$ roots
are labeled by $abc$, meaning that the corresponding roots is
$e_{a}+e_{b}+e_{c}$. The leftmost vertex corresponds to
$\gamma_{2}=e_{1}+e_{2}+\dots+e_{6}$.

Another instance where the $e_{i}$ basis reveals more symmetry is the
expression for the fundamental weights. Recall that by definition fundamental
weights (of a simply-laced root system) form the basis dual to the basis of
the fundamental simple roots. The expansion of the fundamental weights in
terms of the simple roots can be obtained by taking the columns of the matrix
$A^{-1}$, the inverse of the Cartan matrix. Thus the expressions in terms of
the $e_{i}$ basis can be calculated as the columns of $CA^{-1}$.

In case $k=2$ (so that $\mathsf{J}_{k,n}=\mathsf{D}_{n}$) this gives (after
the renumbering of the simple roots, see Example 2.1) the standard description
[Bou1, Ch. VI, §4, no. 8(VI)]

$\displaystyle\varpi_{\beta}=\frac{1}{2}(1,\ldots,1)^{\top},$
$\displaystyle\varpi_{1}=\frac{1}{2}(-1,1,\ldots,1)^{\top},$
$\displaystyle\varpi_{i}=e_{i+1}+\ldots+e_{n}\quad\text{for}\quad 2\leqslant
i\leqslant n-1.$

The fundamental weights for $\mathsf{E}_{6},\mathsf{E}_{7}$ and
$\mathsf{E}_{8}$ are listed in Tables 2, 3 and 4.

Similarly, the sum of all positive roots, which equals twice the sum of the
fundamental weights, is

$\displaystyle\mathsf{D}_{n}$
$\displaystyle\colon\quad(0,1,\ldots,n-1)^{\top},$
$\displaystyle\mathsf{E}_{6}$ $\displaystyle\colon\quad(3,4,5,6,7,8)^{\top},$
$\displaystyle\mathsf{E}_{7}$
$\displaystyle\colon\quad(15,17,19,21,23,25,27)^{\top},$
$\displaystyle\mathsf{E}_{8}$
$\displaystyle\colon\quad(22,23,24,25,26,27,28,29)^{\top}.$

$\varpi_{\beta}$ |  $\begin{matrix}1&2&3&2&1\\\ &&2\end{matrix}$ | $(1,1,1,1,1,1)^{\top}$
---|---|---
$\varpi_{1}$ |  $\dfrac{1}{3}\left(\begin{matrix}4&5&6&4&2\\\ &&3\end{matrix}\right)$ | $\frac{1}{3}(-1,2,2,2,2,2)^{\top}$
$\varpi_{2}$ |  $\dfrac{1}{3}\left(\begin{matrix}5&10&12&8&4\\\ &&6\end{matrix}\right)$ | $\frac{1}{3}(1,1,4,4,4,4)^{\top}$
$\varpi_{3}$ |  $\begin{matrix}2&4&6&4&2\\\ &&3\end{matrix}$ | $(1,1,1,2,2,2)^{\top}$
$\varpi_{4}$ |  $\dfrac{1}{3}\left(\begin{matrix}4&8&12&10&5\\\ &&6\end{matrix}\right)$ | $\frac{1}{3}(2,2,2,2,5,5)^{\top}$
$\varpi_{5}$ |  $\dfrac{1}{3}\left(\begin{matrix}2&4&6&5&4\\\ &&3\end{matrix}\right)$ | $\frac{1}{3}(1,1,1,1,1,4)^{\top}$
Table 2. Fundamental weights of $\mathsf{E}_{6}$. $\varpi_{\beta}$ |  $\dfrac{1}{2}\left(\begin{matrix}4&8&12&9&6&3\\\ &&7\end{matrix}\right)$ | $\frac{1}{2}(3,3,3,3,3,3,3)^{\top}$
---|---|---
$\varpi_{1}$ |  $\begin{matrix}2&3&4&3&2&1\\\ &&2\end{matrix}$ | $(0,1,1,1,1,1,1)^{\top}$
$\varpi_{2}$ |  $\begin{matrix}3&6&8&6&4&2\\\ &&4\end{matrix}$ | $(1,1,2,2,2,2,2)^{\top}$
$\varpi_{3}$ |  $\begin{matrix}4&8&12&9&6&3\\\ &&6\end{matrix}$ | $(2,2,2,3,3,3,3)^{\top}$
$\varpi_{4}$ |  $\dfrac{1}{2}\left(\begin{matrix}6&12&18&15&10&5\\\ &&9\end{matrix}\right)$ | $\frac{1}{2}(3,3,3,3,5,5,5)^{\top}$
$\varpi_{5}$ |  $\begin{matrix}2&4&6&5&4&2\\\ &&3\end{matrix}$ | $(1,1,1,1,1,2,2)^{\top}$
$\varpi_{6}$ |  $\dfrac{1}{2}\left(\begin{matrix}2&4&6&5&4&3\\\ &&3\end{matrix}\right)$ | $\frac{1}{2}(1,1,1,1,1,1,3)^{\top}$
Table 3. Fundamental weights of $\mathsf{E}_{7}$. $\varpi_{\beta}$ |  $\begin{matrix}5&10&15&12&9&6&3\\\ &&8\end{matrix}$ | $(3,3,3,3,3,3,3,3)^{\top}$
---|---|---
$\varpi_{1}$ |  $\begin{matrix}4&7&10&8&6&4&2\\\ &&5\end{matrix}$ | $(1,2,2,2,2,2,2,2)^{\top}$
$\varpi_{2}$ |  $\begin{matrix}7&14&20&16&12&8&4\\\ &&10\end{matrix}$ | $(3,3,4,4,4,4,4,4)^{\top}$
$\varpi_{3}$ |  $\begin{matrix}10&20&30&24&18&12&6\\\ &&15\end{matrix}$ | $(5,5,5,6,6,6,6,6)^{\top}$
$\varpi_{4}$ |  $\begin{matrix}8&16&24&20&15&10&5\\\ &&12\end{matrix}$ | $(4,4,4,4,5,5,5,5)^{\top}$
$\varpi_{5}$ |  $\begin{matrix}6&12&18&15&12&8&4\\\ &&9\end{matrix}$ | $(3,3,3,3,3,4,4,4)^{\top}$
$\varpi_{6}$ |  $\begin{matrix}4&8&12&10&8&6&3\\\ &&6\end{matrix}$ | $(2,2,2,2,2,2,3,3)^{\top}$
$\varpi_{7}$ |  $\begin{matrix}2&4&6&5&4&3&2\\\ &&3\end{matrix}$ | $(1,1,1,1,1,1,1,2)^{\top}$
Table 4. Fundamental weights of $\mathsf{E}_{8}$.

###  4.3. Affine roots and roots coming from affine subsystems

Among the root systems of type $\mathsf{J}_{k,n}$ there are two affine type
root systems, namely, for $(k,n)=(3,9)$ or $(4,8)$ (there is also
$\mathsf{J}_{6,9}\cong\mathsf{J}_{3,9}$). In a simply-laced affine root system
$\Phi$ roots come in families of the form $\alpha+m\delta$, where $\alpha$ is
a root of the canonical finite type subsystem $\mathring{\Phi}$ and
$\delta=\delta_{k,n}$ is the smallest element-wise positive vector such that
$A\delta=0$ for $A$ the Cartan matrix of $\Phi$, and $m\in\mathbb{Z}$.

Denote by $J$ the $n{\times}n$-matrix consisting of $1$’s, so that the Gram
matrix of the inner product with respect to the basis $e_{1},\ldots,e_{n}$
equals $I-\frac{k-2}{k^{2}}J$. On the other hand, it must be equal to
$C^{-\top}AC^{-1}$, so $A\delta=0$ implies
$C^{\top}C\delta=\frac{k-2}{k^{2}}C^{\top}JC\delta$, which, in turn, means
that $\delta^{\prime}=C\delta$ satisfies
$\delta^{\prime}=\frac{k-2}{k^{2}}J\delta^{\prime}$. If
$\delta^{\prime}=(x_{1},\ldots,x_{n})^{\top}$, then $x_{1}=\ldots=x_{n}=x$ and
$x=\frac{k-2}{k^{2}}nx$. There are only three positive integer solutions to
$n=\frac{k^{2}}{k-2}$, namely, $(k,n)=(3,9)$, $(4,8)$ or $(6,9)$. Recall that
the element $x$ must also satisfy the restriction that $\sum x_{i}=nx$ is
divisible by $k$. In cases $(k,n)=(3,9)$ and $(4,8)$ the minimal such $x$
equals $1$, and for $(k,n)=(6,9)$ one has $x=2$. Thus

$\displaystyle\delta^{\prime}_{3,9}=(1,1,1,1,1,1,1,1,1)^{\top},$
$\displaystyle\delta^{\prime}_{4,8}=(1,1,1,1,1,1,1,1)^{\top},$
$\displaystyle\delta^{\prime}_{6,9}=(2,2,2,2,2,2,2,2,2)^{\top},$

which correspond, respectively, to

$\delta_{k,n}=\begin{matrix}2&4&6&5&4&3&2&1\\\
&&3\end{matrix},\quad\begin{matrix}1&2&3&4&3&2&1\\\
&&&2\end{matrix},\quad\begin{matrix}1&2&3&4&5&6&4&2\\\ &&&&&3\end{matrix}.$

One can consider $\delta^{\prime}_{k,n}$ as an element of a larger root system
by means of Remark 3.2, and below we will use such identifications implicitly.
One also can obtain $\delta^{\prime}_{6,9}$ from $\delta^{\prime}_{3,9}$ by
Remark 3.1.

This means that if $k\geqslant 3$ and $n-k\geqslant 6$, there are the
following families of roots in $\mathsf{J}_{k,n}$, coming from the natural
$\mathsf{J}_{3,9}$ subsystem:

$\displaystyle\pm(e_{i}-e_{j})+m\delta_{3,9}\quad\text{for}\quad k-3\leqslant
j<i\leqslant k+5,$ (A0) $\displaystyle\pm\underbracket{1\ldots
1}_{k}+m\delta_{3,9}=\underbracket{3m\pm 1\ldots 3m\pm
1}_{k-3}\underbracket{m\pm 1\ldots m\pm 1}_{3}\underbracket{m\ldots m}_{6},$
(A1) $\displaystyle\pm\underbracket{2\ldots 2}_{k-3}\underbracket{1\ldots
1}_{6}+m\delta_{3,9}=\underbracket{3m\pm 2\ldots 3m\pm
2}_{k-3}\underbracket{m\pm 1\ldots m\pm 1}_{6}\underbracket{m\ldots m}_{3},$
(A2) $\displaystyle\pm\underbracket{3\ldots
3}_{k-3}\underbracket{2}_{1}\underbracket{1\ldots
1}_{7}+m\delta_{3,9}=\underbracket{3m\pm 3\ldots 3m\pm
3}_{k-3}\underbracket{m\pm 2}_{1}\underbracket{m\pm 1\ldots m\pm
1}_{7}\underbracket{m}_{1}.$ (A3)

In Table 1 the orbits that contain the roots from one of the series (A0–A3)
are marked by (Xi$\pm$), with the sign corresponding to the choice of the
signs in the formula.

Note that the roots of (A1$\pm$) and (A2$\mp$) families represent the same
$W(\mathsf{A}_{n-1})$-orbits, but with different numbering. Namely, the root
of (A1) family with a given value of $m=m_{1}$ and $+$ as the sign and the
root of (A2) family with $m=m_{1}+1$ and $-$ as the sign are obtained from one
another by a permutation of the last $9$ non-zero entries. The root of the
form (A1) with $m=m_{1}$ and $-$ as the sign is in the same
$W(\mathsf{A}_{n-1})$-orbit as the root of the form (A2) with $m=m_{1}-1$ and
$+$ for the sign.

The same holds for (A3$+$) and (A3$-$).

Similarly if $k\geqslant 6$ and $n-k\geqslant 3$, there are roots coming from
the natural $\mathsf{J}_{6,9}$ subsystem:

$\displaystyle\pm(e_{i}-e_{j})+m\delta_{6,9}\quad\text{for}\quad k-5\leqslant
j<i\leqslant k+3,$ (B0) $\displaystyle\pm\underbracket{1\ldots
1}_{k}+m\delta_{6,9}=\underbracket{3m\pm 1\ldots 3m\pm
1}_{k-6}\underbracket{2m\pm 1\ldots 2m\pm 1}_{6}\underbracket{2m\ldots
2m}_{3},$ (B1) $\displaystyle\pm\underbracket{2\ldots
2}_{k-3}\underbracket{1\ldots 1}_{6}+m\delta_{6,9}=\underbracket{3m\pm 2\ldots
3m\pm 2}_{k-6}\underbracket{2m\pm 2\ldots 2m\pm 2}_{3}\underbracket{2m\pm
1\ldots 2m\pm 1}_{6},$ (B2) $\displaystyle\pm\underbracket{3\ldots
3}_{k-5}\underbracket{2\ldots
2}_{7}\underbracket{1}_{1}+m\delta_{6,9}=\underbracket{3m\pm 3\ldots 3m\pm
3}_{k-6}\underbracket{2m\pm 3}_{1}\underbracket{2m\pm 2\ldots 2m\pm
2}_{7}\underbracket{2m\pm 1}_{1}.$ (B3)

Again, the roots of (B1$\pm$ and B2$\mp$) and (B3$+$ and B3$-$) represent the
same $W(\mathsf{A}_{n-1})$-orbits, but with different numbering.

If $k,n-k\geqslant 4$, there are roots coming from the natural
$\mathsf{J}_{4,8}$ subsystem:

$\displaystyle\pm(e_{i}-e_{j})+m\delta_{4,8}\quad\text{for}\quad k-4\leqslant
j<i\leqslant k+3,$ (C0) $\displaystyle\pm\underbracket{1\ldots
1}_{k}+m\delta_{4,8}=\underbracket{2m\pm 1\ldots 2m\pm
1}_{k-4}\underbracket{m\pm 1\ldots m\pm 1}_{4}\underbracket{m\ldots m}_{4},$
(C1) $\displaystyle\pm\underbracket{2\ldots 2}_{k-3}\underbracket{1\ldots
1}_{6}+m\delta_{4,8}=\underbracket{2m\pm 2\ldots 2m\pm
2}_{k-4}\underbracket{m\pm 2}_{1}\underbracket{m\pm 1\ldots m\pm
1}_{6}\underbracket{m}_{1}.$ (C2)

Here $W(\mathsf{A}_{n-1})$-orbits do not depend on the signs (after a suitable
renumbering), and, moreover, the orbits of (C0) and (C2) cover the same set of
roots.

###  4.4. Almost real roots

Every real root of degree $\geqslant 1$, when expressed in basis
$e_{1},\ldots,e_{n}$, say, $x=(x_{1},\ldots,x_{n})^{\top}$, satisfies the
following three properties by Lemmas 2.3 and 2.6:

  1. (1)

$x\in\mathbb{Z}\Delta$,

  2. (2)

$0\leqslant x_{1},\ldots,x_{n}\leqslant d$, where $d=(x_{1}+\ldots+x_{n})/k$,

  3. (3)

$q(x)=\sum x_{i}^{2}+\frac{2-k}{k^{2}}(\sum x_{i})^{2}=2$.

However, there are vectors $x$ satisfying all of the above, which are not real
roots. We call such elements of the root lattice _almost real roots_. They
exist in degrees $\geqslant 4$.

Almost real roots of degrees $4$ and $5$ (in $\mathsf{J}_{k,n}$ for large
enough $k,n$) are listed in Table 5. Note that the statement of Lemma 4.2 also
holds for almost real roots.

degree $4$ |  $\underbracket{4\ldots 4}_{k-4}\underbracket{333}_{3}\underbracket{1\ldots 1}_{7}$ $\underbracket{4\ldots 4}_{k-6}\underbracket{3\ldots 3}_{7}\underbracket{111}_{3}$
---|---
degree $5$ |  $\underbracket{5\ldots 5}_{k-3}\underbracket{33}_{2}\underbracket{1\ldots 1}_{9}$ $\underbracket{5\ldots 5}_{k-4}\underbracket{44}_{2}\underbracket{3}_{1}\underbracket{2}_{1}\underbracket{1\ldots 1}_{7}$ $\underbracket{5\ldots 5}_{k-5}\underbracket{4\ldots 4}_{4}\underbracket{22}_{2}\underbracket{1\ldots 1}_{5}$
|  $\underbracket{5\ldots 5}_{k-6}\underbracket{4\ldots
4}_{5}\underbracket{33}_{2}\underbracket{1\ldots 1}_{4}$
$\underbracket{5\ldots 5}_{k-7}\underbracket{4\ldots
4}_{7}\underbracket{3}_{1}\underbracket{2}_{1}\underbracket{11}_{2}$
$\underbracket{5\ldots 5}_{k-8}\underbracket{4\ldots
4}_{9}\underbracket{22}_{2}$
Table 5. $W(\mathsf{A}_{n-1})$-orbits of almost real roots of degrees $4$ and
$5$.

Calculating the minimal subsystems for each orbit ot almost real roots in
Table 5 we see that degree $4$ almost real roots are present in all root
systems of type $\mathsf{J}_{k,n}$ which contain $\mathsf{J}_{4,10}$ or
$\mathsf{J}_{6,10}$, and that every $\mathsf{J}_{k,n}$ root system containing
$\mathsf{J}_{3,11}$ or $\mathsf{J}_{8,11}$ has an almost real root of degree
$5$. This implies that in every non-finite, non-affine, non-hyperbolic root
system of type $\mathsf{J}_{k,n}$ there are almost real roots.

Let $x$ be a positive almost real root, and assume that
$x=\operatorname{dec}(x)$. Then repeating the operation
$x\mapsto\operatorname{dec}(s_{\beta}(x))$ lowers the height and eventually
leads to an element $x^{\prime}=(x_{1},\ldots,x_{n})$ of degree $d$ such that
either some of the entries $x_{i}$ are negative or greater than $d$. When
translated to the root basis, this means that the coefficient of some simple
root $\alpha_{i}$ is negative.

######  Example 4.5.

For $(k,n)=(4,10)$ set $x=(3,3,3,1,1,1,1,1,1,1)^{\top}$, so that in the root
basis

$x=\begin{matrix}1&2&3&6&5&4&3&2&1\\\ &&&4\end{matrix}.$

Then $\operatorname{dec}(s_{\beta}(x))=(1,1,1,1,1,1,1,1,1,-1)^{\top}$, which
it the roots basis equals

$\operatorname{dec}(s_{\beta}(x))=\begin{matrix}1&2&3&4&3&2&1&0&-1\\\
&&&2\end{matrix}.$

Similarly, for $(k,n)=(6,10)$ and $x=(3,3,3,3,3,3,3,1,1,1)^{\top}$ in terms of
the root basis one has

$x=\begin{matrix}1&2&3&4&5&6&3&2&1\\\ &&&&&4\end{matrix},$

while $\operatorname{dec}(s_{\beta}(x))=(3,1,1,1,1,1,1,1,1,1)^{\top}$, which
translates into

$\operatorname{dec}(s_{\beta}(x))=\begin{matrix}-1&0&1&2&3&4&3&2&1\\\
&&&&&2\end{matrix}.$

To find real roots, we consider the orbits under $W(\mathsf{A}_{n-1})$ (Remark
4.1). The numbers of $W(\mathsf{A}_{n-1})$-orbits of real roots and almost
real roots are listed in Table 6, and the total numbers of real roots and
almost real roots are listed in Tables 7 and 8 respectively.

##  5\. Comparison with Manin’s hyperbolic construction

In [Manin] Manin gave a construction of the root system $\mathsf{E}_{8}$
inside a hyperbolic space. His construction works as follows.

Consider a $9$-dimensional space $V$ equipped with the inner product of
signature $(1,8)$. This means that there exists an orthogonal basis
$f_{0},f_{1},\ldots,f_{8}$ of $V$ such that $(f_{0},f_{0})=1$,
$(f_{i},f_{i})=-1$ for $i\geqslant 1$. Set $\omega=-3f_{0}+f_{1}+\ldots+f_{8}$
and define the lattice $L=\mathbb{Z}f_{0}+\ldots+\mathbb{Z}f_{8}$. Then the
set

$R=\\{f\in L\mid(f,\omega)=0,\ (f,f)=-2\\}$

is the root system of type $\mathsf{E}_{8}$ [Manin, Proposition 25.2 and
Theorem 25.4].

This realisation is related to the structure of del Pezzo surfaces. If a del
Pezzo surface $V$ of degree $d$ is not isomorphic to
$\mathbb{P}^{1}\times\mathbb{P}^{1}$, then its Picard group
$\operatorname{Pic}(V)$ is isomorphic to the odd unimodular lattice
$L=I_{1,9-d}$, in which the root system is realised.

The complete enumeration of roots of $\mathsf{E}_{8}$ is provided by [Manin,
Proposition 25.5.3]. It states that if $(a,b_{1},\ldots,b_{8})$ are the
coordinates of a root with respect to the basis $f_{0},f_{1},\ldots,f_{8}$,
then these coordinates can be obtained from the rows of the following table by
a permutation of the last $8$ entries $b_{1},\ldots,b_{8}$ and, possibly, a
simultaneous change of the sign for all $9$ entries :

$a$ | $b_{1}$ | $b_{2}$ | $b_{3}$ | $b_{4}$ | $b_{5}$ | $b_{6}$ | $b_{7}$ | $b_{8}$
---|---|---|---|---|---|---|---|---
$0$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-1$
$1$ | $1$ | $1$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$
$2$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $0$ | $0$
$3$ | $2$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$

Comparing this with the content of Table 1 reveals that this construction
coincides with our presentation of $\mathsf{E}_{8}$-roots insise
$\mathsf{J}_{4,9}$. The correspondence, from presentation in the $x_{i}$ to
presentation in the $f_{i}$, is as follows

$x=(x_{1},\ldots,x_{8})\rightsquigarrow(\deg(x),x_{1},\ldots,x_{8}),$

(extending the presentations from Table 1 by $0$s at the end where needed).
For degree $0$, see Remark 2.5. This is exactly the inclusion
$\mathsf{J}_{3,8}\hookrightarrow\mathsf{J}_{4,9}$ described in Remark 3.2.

Moreover, the exceptional curves on $V$ are parametrised by the following
elements of $\operatorname{Pic}(V)$ (with respect to
$f_{0},f_{1},\ldots,f_{n})$:

$a$ | $b_{1}$ | $b_{2}$ | $b_{3}$ | $b_{4}$ | $b_{5}$ | $b_{6}$ | $b_{7}$ | $b_{8}$
---|---|---|---|---|---|---|---|---
$0$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$1$ | $1$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$2$ | $1$ | $1$ | $1$ | $1$ | $1$ | $0$ | $0$ | $0$
$3$ | $2$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $0$
$4$ | $2$ | $2$ | $2$ | $1$ | $1$ | $1$ | $1$ | $1$
$5$ | $2$ | $2$ | $2$ | $2$ | $2$ | $2$ | $1$ | $1$
$6$ | $3$ | $2$ | $2$ | $2$ | $2$ | $2$ | $2$ | $2$

together with all obtained by permuting $b_{1},\ldots,b_{8}$. The calculation
of these elements in [Manin, Proposition 26.1] is done by introducing an
auxiliary parameter $b_{9}$ and then specifying it to $1$. Note, however, that
for such values of $b_{i}$ the vector $(a,b_{1},\ldots,b_{8},1)$ coincides
with one of the roots of $\mathsf{J}_{4,10}$ coming from the affine subsystem
$\mathsf{J}_{3,9}$, given by formulas (A0)–(A3) with $m=1$. Namely, the image
$\delta$ of $\delta_{3,9}$ in $\mathsf{J}_{4,10}$ is
$(3,1,1,1,1,1,1,1,1,1,1)^{\top}$, so

(A3$-$)$\colon$ | $-\underbracket{3}_{1}\underbracket{2}_{1}\underbracket{1\ldots 1}_{7}+\delta=(0,-1,0,0,0,0,0,0,0,1)^{\top},$
---|---
(A2$-$)$\colon$ | $-\underbracket{2}_{1}\underbracket{1\ldots 1}_{6}+\delta=(1,0,0,0,0,0,0,1,1,1)^{\top},$
(A1$-$)$\colon$ | $-\underbracket{1\ldots 1}_{4}+\delta=(2,0,0,0,1,1,1,1,1,1)^{\top},$
(A0)$\colon$ | $\vphantom{\underbracket{1}_{1}}e_{2}-e_{8}+\delta=(3,2,1,1,1,1,1,1,0,1)^{\top},$
(A1$+$)$\colon$ | $\underbracket{1\ldots 1}_{4}+\delta=(4,2,2,2,1,1,1,1,1,1)^{\top},$
(A2$+$)$\colon$ | $\underbracket{2}_{1}\underbracket{1\ldots 1}_{6}+\delta=(5,2,2,2,2,2,2,1,1,1)^{\top},$
(A3$+$)$\colon$ | $\underbracket{3}_{1}\underbracket{2}_{1}\underbracket{1\ldots 1}_{7}+\delta=(6,3,2,2,2,2,2,2,2,1)^{\top}.$

##  6\. Connection with cluster algebras

Jensen, King and Su [JKS] have given an additive categorification of the
cluster algebra structure on the coordinate ring $\mathbb{C}[{\rm Gr}_{k,n}]$
of the Grassmannian of $k$-subspaces in $n$-space, by considering the category
$\mathrm{CM}(B_{k,n})$ of Cohen-Macaulay modules over a quotient $B_{k,n}$ of
the preprojective algebra of type $\widetilde{\mathsf{A}}$.

Jensen, King and Su point out that in [JKS, Section 8] the finite types, the
indecomposable module corresponds to a real root in the associated root system
and that the number of indecomposable rank $d$ modules is $d$ times the number
of real roots of degree $d$. They observe that the evidence suggest that rigid
indecomposable modules correspond to roots (as classes in the Grothendieck
group) and that for every real root of degree $d$ there are $d$ rigid
indecomposable objects of rank $d$.

Every $B_{k,n}$-module of rank $1$ can be characterized by a $k$-element
subset of $\\{1,\ldots,n\\}$, see [JKS, Definition 5.1 and Proposition 5.2].
These in turn correspond to real roots in degree $1$.

The rank $1$ modules can be viewed as building blocks for the category as
every module in ${\rm CM}(B_{k,n})$ has a filtration with factors which are
rank $1$ modules, as pointed out in a private communication by A. King and M.
Pressland. If $M$ is an arbitrary module in ${\rm CM}(B_{k,n})$, one can
consider homormophisms $L_{I}\hookrightarrow M$ such that the quotient
$M/L_{I}$ is also in ${\rm CM}(B_{k,n})$. Such homomorphisms always exist and
allow to reduce the rank of $M$. Such a filtration is not unique in general.
Let $M$ be a rank $n$ module in ${\rm CM}(B_{k,n})$ with factors
$L_{I_{1}},\dots,L_{I_{d}}$ in its filtration, where $L_{I_{d}}$ is a
submodule of $M$. We write

$P_{M}=\begin{aligned} I_{1}\\\ \hline\cr\\\\[-9.0pt]
\stackrel{{\scriptstyle\vdots{\phantom{a}}}}{{\phantom{.}}}\\\
\hline\cr\\\\[-9.0pt] I_{d}\end{aligned}\quad\text{or}\quad
P_{M}=I_{1}|\cdots|I_{d},$

and $P_{M}$ is called the _profile_ of $M$. The number $d$ is called the rank
of the module $M$.

For every module $M$ with a profile $P_{M}$ of $d$ rows, one associates the
element $\varphi(M)=\varphi(P_{M}):=(x_{1},\ldots,x_{n})^{\top}$ in
$\mathbb{Z}\Delta$ where $x_{i}$ is the number of occurrences of $i$ in the
profile of $M$.

Indeed, since each of these $d$ rows has size $k$, the total number of entries
is $x_{1}+\ldots+x_{n}=kd$. We have $0\leqslant x_{i}\leqslant d$ for each
$i\in\\{1,\ldots,n\\}$.

Conversely, for any element $x=(x_{1},\ldots,x_{n})^{\top}\in\mathbb{Z}\Delta$
with $0\leqslant x_{i}\leqslant d$ for all $i$, one can construct a profile
mapping to $x$. To do this, take the sequence

$a=\big{(}\underbrace{1,\ldots,1}_{x_{1}},\underbrace{2,\ldots,2}_{x_{2}},\ldots,\underbrace{n,\ldots,n}_{x_{n}}\big{)}$

of length $kd$ and set

$I_{i}=\\{a_{k-i+1},a_{k-i+1+d},\ldots,a_{k-i+1+(k-1)d}\\}.$

Then define $P_{x}:=I_{1}|\cdots|I_{d}$. This is a profile with
$\varphi(P_{x})=x$. For example, for the root $x=(2,1,1,1,1,1,1,1)$ in
$\mathsf{J}_{3,8}$, this produces

$P_{x}=\begin{aligned} 258\\\ \hline\cr\\\\[-9.0pt] 147\\\
\hline\cr\\\\[-9.0pt] 136\end{aligned}.$

Now given a profile $P$ with $d$ rows, we order the entries increasingly in
each row and write this as $P=(P_{ij})$, $1\leq i\leq d$, $1\leq j\leq k$. So
$(P_{ij})_{j}$ is the $i$th row of the profile and $P_{ij}<P_{ij^{\prime}}$
for $j<j^{\prime}$. The profile $P$ is called weakly column decreasing if for
every $j\in[k]$ and for every $i\in[d-1]$, we have $P_{i,j}\geq P_{i+1,j}$. If
$P$ is weakly column decreasing and in addition, we have $P_{d,j}\geq
P_{1,j-1}$ for all $j\in[2,k]$, we say that $P$ is _canonical_.

The profile $P_{x}$ corresponding to $x\in\mathbb{Z}\Delta$ constructed above
is a canonical profile. In [BBGL, Theorem 5.7], it is shown that the profile
of any rigid indecomposable module of rank $3$ such that $\varphi(M)$ is a
real root is a cyclic permutation of a canonical profile. For example, the
profile

$P=\begin{aligned} 258\\\ \hline\cr\\\\[-9.0pt] 147\\\ \hline\cr\\\\[-9.0pt]
136\end{aligned}$

is a canonical profile of rank $3$ and $\varphi(P)$ is a real root in
$\mathsf{J}_{3,8}$. The cyclic permutations of $P$ are

$\begin{aligned} 258\\\ \hline\cr\\\\[-9.0pt] 147\\\ \hline\cr\\\\[-9.0pt]
136\end{aligned},\quad\begin{aligned} 147\\\ \hline\cr\\\\[-9.0pt] 136\\\
\hline\cr\\\\[-9.0pt] 258\end{aligned}\quad\text{and}\quad\begin{aligned}
136\\\ \hline\cr\\\\[-9.0pt] 258\\\ \hline\cr\\\\[-9.0pt] 147\end{aligned}.$

The modules with these profiles are all rigid indecomposable. We note that it
is conjectured that whenever $M$ in ${\rm CM}(B_{k,n})$ is rigid
indecomposable and $\varphi(M)$ is a real root in $\mathsf{J}_{k,n}$, then the
profile $P_{M}$ is a cyclic permutation of a canonical profile, [BBGL,
Conjecture 5.8].

The results about real roots in $\mathsf{J}_{k,n}$ in this paper are thus
expected to help with the characterisation of rigid indecomposable modules in
${\rm CM}(B_{k,n})$ corresponding to real roots.

|  degree | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
---|---|---|---|---|---|---|---|---|---|---|---|---
$(k,n)=(\infty,\infty)$ | real roots | 1 | 1 | 3 | 8 | 17 | 37 | 72 | 139 | 253 | 439 | 722
almost r. r. | 0 | 0 | 0 | 2 | 6 | 20 | 65 | 153 | 390 | 878 | 1888
$(k,n)=(3,\infty)$ | real roots | 1 | 1 | 1 | 2 | 3 | 5 | 7 | 13 | 17 | 28 | 37
almost r. r. | 0 | 0 | 0 | 0 | 1 | 1 | 4 | 7 | 16 | 27 | 52
$(k,n)=(4,\infty)$ | real roots | 1 | 1 | 2 | 4 | 8 | 15 | 26 | 44 | 76 | 115 | 183
almost r. r. | 0 | 0 | 0 | 1 | 2 | 5 | 15 | 31 | 64 | 131 | 250
$(k,n)=(5,\infty)$ | real roots | 1 | 1 | 3 | 6 | 11 | 24 | 45 | 81 | 143 | 236 | 372
almost r. r. | 0 | 0 | 0 | 1 | 3 | 9 | 26 | 53 | 133 | 266 | 529
$(k,n)=(3,10)$ | real roots | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 5 | 5 | 7 | 9
almost r. r. | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$(k,n)=(3,11)$ | real roots | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 10 | 14 | 18
almost r. r. | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 2 | 1 | 3
$(k,n)=(3,12)$ | real roots | 1 | 1 | 1 | 2 | 3 | 4 | 6 | 10 | 13 | 20 | 27
almost r. r. | 0 | 0 | 0 | 0 | 1 | 1 | 2 | 2 | 5 | 5 | 9
$(k,n)=(4,9)$ | real roots | 1 | 1 | 2 | 2 | 3 | 5 | 7 | 9 | 14 | 17 | 22
almost r. r. | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$(k,n)=(4,10)$ | real roots | 1 | 1 | 2 | 3 | 6 | 8 | 15 | 20 | 34 | 44 | 70
almost r. r. | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 3 | 1 | 8 | 4
$(k,n)=(4,11)$ | real roots | 1 | 1 | 2 | 4 | 7 | 12 | 20 | 31 | 52 | 74 | 117
almost r. r. | 0 | 0 | 0 | 1 | 1 | 2 | 4 | 8 | 10 | 24 | 32
$(k,n)=(5,10)$ | real roots | 1 | 1 | 3 | 4 | 6 | 12 | 21 | 31 | 52 | 76 | 110
almost r. r. | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2
Table 6. Number of $W(\mathsf{A}_{n-1})$-orbits of real roots and almost real roots in $\mathsf{J}_{k,n}$. |  | $d$
---|---|---
$k$ |  $n$ | 1 | 2 | 3 | 4 | 5 | 6 | 7
3 | 6 | 20 | 1 | 0 | 0 | 0 | 0 | 0
7 | 35 | 7 | 0 | 0 | 0 | 0 | 0
8 | 56 | 28 | 8 | 0 | 0 | 0 | 0
9 | 84 | 84 | 72 | 84 | 84 | 72 | 84
10 | 120 | 210 | 360 | 850 | 1680 | 3870 | 7560
11 | 165 | 462 | 1320 | 4730 | 13860 | 42240 | 106260
12 | 220 | 924 | 3960 | 19140 | 73932 | 267300 | 802164
13 | 286 | 1716 | 10296 | 62920 | 300456 | 1235520 | 4241952
14 | 364 | 3003 | 24024 | 178178 | 1010100 | 4618628 | 17669652
15 | 455 | 5005 | 51480 | 450450 | 2948400 | 14774970 | 61861800
4 | 8 | 70 | 56 | 70 | 56 | 70 | 56 | 70
9 | 126 | 252 | 702 | 1764 | 4914 | 9828 | 24390
10 | 210 | 840 | 3870 | 15960 | 55020 | 159480 | 419460
11 | 330 | 2310 | 15510 | 87890 | 355740 | 1276110 | 3626040
12 | 495 | 5544 | 50490 | 361680 | 1683990 | 6965640 | 21521610
13 | 715 | 12012 | 141570 | 1221792 | 6456606 | 29673072 | 99664422
14 | 1001 | 24024 | 354354 | 3571568 | 21191352 | 105921816 | 385453068
15 | 1365 | 45045 | 810810 | 9339330 | 61637940 | 330720600 | 1297836540
5 | 10 | 252 | 1260 | 7020 | 30492 | 117180 | 330120 | 950220
11 | 462 | 4620 | 39930 | 243012 | 1113420 | 3903240 | 12134760
12 | 792 | 13860 | 166320 | 1292412 | 6763680 | 27642780 | 92038320
13 | 1287 | 36036 | 563706 | 5305872 | 31081050 | 142573860 | 506859210
14 | 2002 | 84084 | 1645644 | 18138120 | 117466440 | 590545956 | 2235937704
15 | 3003 | 180180 | 4285710 | 54029976 | 383439420 | 2079637560 | 8363775420
6 | 12 | 924 | 18480 | 239580 | 1899744 | 10308144 | 41888880 | 143037840
13 | 1716 | 60060 | 1055340 | 10249096 | 63075012 | 288004860 | 1057150380
14 | 3003 | 168168 | 3777774 | 43259216 | 295387092 | 1482785304 | 5793796008
15 | 5005 | 420420 | 11621610 | 152912760 | 1143127440 | 6211345140 | 25687061400
7 | 14 | 3432 | 210210 | 4924920 | 57028972 | 396203808 | 1987088532 | 7851283440
15 | 6435 | 630630 | 18648630 | 249909660 | 1917115200 | 10417968990 | 43770406680
Table 7. Number of real roots of a given degree in $\mathsf{J}_{k,n}$. |  | $d$
---|---|---
$k$ |  $n$ | 4 | 5 | 6 | 7
3 | 10 | 0 | 0 | 0 | 0
11 | 0 | 55 | 0 | 462
12 | 0 | 660 | 1320 | 13464
13 | 0 | 4290 | 17160 | 148434
14 | 0 | 20020 | 120120 | 1021020
15 | 0 | 75075 | 600600 | 5225220
4 | 9 | 0 | 0 | 0 | 0
10 | 120 | 0 | 1260 | 840
11 | 1320 | 3960 | 41580 | 138600
12 | 7920 | 48180 | 445500 | 1953864
13 | 34320 | 317460 | 2925780 | 15038452
14 | 120120 | 1501500 | 14294280 | 82496414
15 | 360360 | 5705700 | 56936880 | 360751755
5 | 10 | 0 | 0 | 0 | 0
11 | 1320 | 6930 | 62832 | 274890
12 | 15840 | 130680 | 1197504 | 5959800
13 | 102960 | 1162590 | 11052756 | 60911136
14 | 480480 | 6906900 | 68757689 | 412876464
15 | 1801800 | 31531500 | 329924595 | 2135223090
6 | 12 | 15840 | 166320 | 1507968 | 8149680
13 | 137280 | 1930500 | 18666648 | 110630520
14 | 840840 | 14434420 | 148420272 | 943518576
15 | 3963960 | 80029950 | 871065195 | 5889723840
7 | 14 | 960960 | 18018000 | 187675488 | 1224431208
15 | 5405400 | 121696575 | 1356755400 | 9474134670
Table 8. Number of almost real roots of a given degree in $\mathsf{J}_{k,n}$.

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