Root System Vectors: Foundations

• Root system vectors are a finite set of vectors in Euclidean space that encode discrete reflection symmetries, forming the basis of Lie theory and invariant theory.
• They are characterized by properties such as spanning the space, unique directionality, and invariance under reflections, which enable their classification into types like A, B, and G.
• Root frames constructed from these vectors provide tight and scalable frames used in spectral analysis and integrable models, linking algebraic structure with geometric insights.

A root system consists of a finite set of vectors in a real Euclidean space that encodes discrete symmetry through reflection, forming the mathematical foundation for Lie theory, Coxeter groups, and related areas such as invariant theory and integrable systems. Root system vectors, sometimes identified as root frames, play a central role in the study of reflection groups, Lie algebras, and denominator formulae for Weyl groups and Kac–Moody algebras.

1. Definitions and Fundamental Properties

Let $V$ be a real Euclidean space of dimension $d$, with inner product $\langle\,,\,\rangle$. A (reduced) root system $R \subset V \setminus\{0\}$ is characterized by:

• $R$ spans $V$.
• For each $\alpha \in R$, $R \cap \mathbb{R}\alpha = \{\pm\alpha\}$ (no multiple roots).
• Reflections: For $\alpha, \beta \in R$, the reflection $$\sigma_\alpha(\beta) = \beta - 2\frac{\langle \beta, \alpha\rangle}{\|\alpha\|^2}\alpha \in R$$.
• Irreducibility: $R$ is regular (irreducible) if it cannot be decomposed as a union of two orthogonal, nonempty root systems (Maslouhi et al., 2022).

The set of positive roots $R_+$ is defined via a separating linear functional (a “dominant direction”), and consists of all $\alpha \in R$ with $\langle \alpha, \beta\rangle > 0$ for a suitably chosen $\beta$. The positive roots form the set of root system vectors or “root frame” as used in frame theory.

A generalized root system (GRS) weakens the full Weyl-invariance requirement but maintains strict compatibility conditions on angle and closure under sum/difference of elements (Cuntz et al., 30 Mar 2024):

• For all $\alpha,\beta\in R$, precisely one of the following holds:
  • $(\alpha,\beta)<0 \implies \alpha+\beta\in R$,
  • $(\alpha,\beta)>0 \implies \alpha-\beta\in R$,
  • $(\alpha,\beta)=0 \implies \alpha\pm\beta\notin R$.
• $R = -R$, and all elements have positive norm.

2. Classification and Explicit Realizations

The main classes of irreducible root systems, up to orthogonal transformation and scaling, are classified as follows (Maslouhi et al., 2022, Cuntz et al., 30 Mar 2024):

Type	Ambient space	Positive roots $R_+$	Total roots $|R|$	Root lengths (squared)
$A_n$	$\{x\in\mathbb{R}^{n+1} : \sum x_i=0\}$	$\{e_i-e_j:1\le i<j\le n+1\}$	$2\binom{n+1}{2}$	Uniform (2)
$B_n$	$\mathbb{R}^n$	$\{e_i\}$, $\{e_i\pm e_j: 1\le i<j\le n\}$	$2n+2n(n-1)$	1 (short), 2 (long)
$C_n$	$\mathbb{R}^n$	$\{2e_i\}$, $\{e_i\pm e_j\}$	$2n+2n(n-1)$	2 (short), 4 (long)
$D_n$	$\mathbb{R}^n$	$\{e_i\pm e_j: 1\le i<j\le n\}$	$2n(n-1)$	Uniform (2)
$G_2$	$\mathbb{R}^2$	6 short, 6 long (multiples of $30^\circ$)	12	2 (short), 6 (long)
$F_4$	$\mathbb{R}^4$	See (Cuntz et al., 30 Mar 2024) for details	48	1, 2
$E_6$	$\mathbb{R}^6$	Constructed from $E_8$ restriction	72	Uniform (2)
$E_7$	$\mathbb{R}^7$	Constructed from $E_8$ restriction	126	Uniform (2)
$E_8$	$\mathbb{R}^8$	See (Cuntz et al., 30 Mar 2024)	240	Uniform (2)

Every irreducible finite GRS of rank $\ge2$ is equivalent (as a set up to isometry) to a restriction of a Weyl arrangement corresponding to these root systems.

3. Structural and Spectral Analysis: Frames and Operators

Root system vectors can be organized as unit–norm “frames.” A frame in $\mathbb{R}^d$ is a family $\Phi = \{\varphi_i\}_{i=1}^N$ for which there exist $0 < A \leq B < \infty$ with

$$A\,\|x\|^2 \le \sum_{i=1}^N |\langle x, \varphi_i\rangle|^2 \le B\,\|x\|^2, \quad \forall x\in\mathbb{R}^d\,.$$

A root frame $\Phi_R$ is the unit-norm vector set corresponding to positive roots $R_+$ of a given root system $R$, provided $R_+$ spans $\mathbb{R}^d$ (Maslouhi et al., 2022).

The frame operator $S$ associated with a root system satisfies:

$$S = \sum_{i=1}^{N} \alpha_i \alpha_i^T, \qquad S(x) = \sum_{i=1}^N \langle x, \alpha_i\rangle \alpha_i\,,$$

where $\Phi_R = \{\alpha_i\}$. Each $\alpha \in R_+$ is an eigenvector of $S$ with eigenvalue $$\lambda_\alpha = \sum_{\beta\in R_+}\langle\alpha,\beta\rangle^2$$.

Tightness and scalability:

• $\Phi_R$ is tight (i.e., $S = \frac{|R_+|}{d} I$) iff $R$ is irreducible.
• The rescaled collection $$\{\psi_\alpha = \alpha / \sqrt{\lambda_\alpha} : \alpha \in R_+\}$$ is Parseval ($\sum \psi_\alpha\psi_\alpha^T = I_d$).

Root frames are thus examples of scalable frames and of a broader class called eigenframes, where all vectors are eigenvectors of the frame operator (Maslouhi et al., 2022).

4. Denominator Formulae and Geometric Characterization

The denominator formula provides a powerful tool for characterizing root system vectors algebraically and geometrically (Aoki et al., 5 Mar 2025). Given a finite subset $S \subset V$ (with $V$ a Euclidean space), and $m: V \rightarrow \mathbb{N}_0$ a multiplicity function with support $S(m)$, form the group ring expression:

$$F(m) = \prod_{s\in S(m)} (1 - e^s)^{m(s)} \in \mathbb{Z}[V],\quad A(m) = \mathrm{supp}\;F(m)\,.$$

Finite case: $S(m)\cap(-S(m)) = \emptyset$ and $R = S(m) \cup (-S(m))$ is a reduced finite root system of rank $N$, and $m(s) = 1$ for all $s\in S(m)$ $\iff$ $A(m)$ is contained in a sphere.

Affine case: A corresponding result holds for affine root systems, replacing the sphere by a paraboloid in extended $(N+1)$-dimensional space, with precise conditions on the support (Aoki et al., 5 Mar 2025).

This formulation provides a converse to the classical Weyl denominator formula, and allows one to certify the positive roots of a finite/affine root system by checking the geometric configuration of the support in the group ring.

5. Connections with V-systems and Supersymmetric Integrable Models

Root system vectors span the classical Coxeter root systems, forming the combinatorial backbone for V-systems. A V-system is a finite collection of vectors for which the potential $$F(x) = \sum_{\alpha \in \mathcal{A}} (\alpha\cdot x)^2\log(\alpha\cdot x)$$ satisfies generalized WDVV equations. For Coxeter root systems, choosing $\mathcal{A} = R_+$ gives the standard root system potential, and accordingly determines the corresponding Calogero–Moser integrable system, including all classical series and models such as $A_n$, $B_n$, $C_n$ and $D_n$ (Antoniou et al., 2018).

Supersymmetric generalizations to the Calogero–Moser–Sutherland system exploit the same vector structure, assigning multiplicities according to orbit type (e.g., short, medium, long roots in $BC_n$). The root system structure thus directly informs potential terms, spectral properties, and symmetry classes of quantum integrable models.

6. Geometric and Combinatorial Features

Key geometric properties of root system vectors include strict control of angles and lengths:

• In simply laced cases ($A$, $D$, $E$), all roots share equal length and the possible angles between positive roots are in $\{60^\circ, 90^\circ, 120^\circ\}$.
• Non-simply laced cases ($B$, $C$, $F_4$, $G_2$) possess two distinct root lengths, with angle restrictions reflecting the underlying symmetry (Cuntz et al., 30 Mar 2024).

Any GRS must arise, up to equivalence, from projection/restriction of an ordinary root system. No genuinely new finite types beyond the ADE, BCFG (classical and exceptional) families arise in the finite case. The structure of the corresponding hyperplane arrangement is simplicial, with chambers forming simplicial cones (Cuntz et al., 30 Mar 2024).

7. Representative Examples

Type $A_2$ in $\mathbb{R}^2$: $$R = \{\pm(1,0), \pm(-\frac12, \frac{\sqrt{3}}{2}), \pm(-\frac12, -\frac{\sqrt{3}}{2})\}$$; $R_+$ is any positively oriented half. The frame operator $S = \frac{3}{2} I_2$, so the collection forms a $3/2$–tight frame of three unit vectors (Maslouhi et al., 2022).

Type $B_3$ in $\mathbb{R}^3$: $$R_+ = \{e_i : 1\le i\le 3\} \cup \{e_i\pm e_j: 1\le i < j \le 3\}$$, six positive roots, frame operator $S = 3 I_3$, giving a tight frame (redundancy $2$) (Maslouhi et al., 2022).

Type $G_2$ in $\mathbb{R}^2$: Twelve vectors equally spaced at multiples of $30^\circ$; two root lengths in the ratio $\sqrt{3}:1$; arises both as a classical root system and as a GRS (Cuntz et al., 30 Mar 2024).

Table: Low-Rank Root Systems and Geometric Features

Type	Number of positive roots	Root lengths	Angles (deg)
$A_2$	3	1	60, 120
$B_2$	4	$1, \sqrt{2}$	45, 90, 135
$G_2$	6	$1, \sqrt{3}$	30, 60, 90, 120, 150

The tabular data emphasizes the uniformity and constraints dictating root system vector arrangements.

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Root system vectors function as organizing principles for discrete symmetries, spectral operators, algebraic relations, and integrable model constructions, and their role is foundational across Lie theory, algebraic combinatorics, and mathematical physics (Aoki et al., 5 Mar 2025, Maslouhi et al., 2022, Cuntz et al., 30 Mar 2024, Antoniou et al., 2018).