---- > [!definition] Definition. ([[root system]]) > > A **(crystallographic) root system** is a finite subset $\Phi$ of a Euclidean space $\big( E, (-,-) \big)$, the elements of which are called **roots**, that satisfies the following four axioms: > > **(R1)** $\Phi$ [[submodule generated by a subset|spans]] $E$ and $0$ is not a root: $0 \notin \Phi$ > > **(R2)** If $\alpha \in \Phi$ and $c \in \mathbb{R}$, then $c \alpha \in \Phi \iff c=\pm 1$ > > **(R3)** $\Phi$ is stable under [[reflection|reflections]]: for all $\alpha \in \Phi$, $w_{\alpha}(\Phi) \subset \Phi$ > > **(R4)** For all $\alpha, \beta \in \Phi$, one has $\langle \beta, \check{\alpha} \rangle \in \mathbb{Z}$, where [[reflection|recall]] that $\langle \beta, \check{\alpha} \rangle=\frac{2(\beta, \alpha)}{(\alpha, \alpha)}$. In other words, one has that $(\beta, \alpha)/(\alpha, \alpha)$ is an integer or a half-integer for all $\alpha, \beta \in \Phi$.[^1] > > The **rank** of $\Phi$ is $\dim_{\mathbb{R}}E$. > [!note] Remark. > - Geometrically, **(R4)** is specifying that the projection of $\beta$ onto the line segment spanned by $\alpha$ is an integer or half-integer multiple of $\alpha$. It is called the **crystallographic axiom** and is omitted in some definitions of $\Phi$. > - In some contexts (such as the study of [[Coxeter group|Coxeter groups]]) **(R1)** is relaxed so that $\Phi$ is not required to span. > - Graph paper for the [[reducible root system|irreducible]] rank-2 root systems: https://personal.math.ubc.ca/~cass/frivs/frivolities.html ^note > [!basicexample] (Rank-1) > The subset $\Phi=\{ \pm 1 \}$ of $(\mathbb{R}, \text{std})$ is, up to isomorphism, the unique rank-1 root system. It's called $A_{1}$. We note that $\mathfrak{sl}_{2}(\mathbb{C})$ with the [[Cartan subalgebra]] $\mathfrak{t}=\text{span}(h)$ [[root system of a Lie algebra|gives rise to]] this rank-1 root system. ^basic-example > [!basicexample] (Rank-2) > The following picture displays (up to isomorphism) all root systems of rank 2.![[Pasted image 20250417151638.png]] > ![[Pasted image 20250507234255.png|1500]] > Note: $C_{2}$ is isomorphic to $B_{2}$ via scaling by $\sqrt{ 2 }$ and a 45-degree rotation. Pictured above is $B_{2}$. Here is $C_{2}$: ![[Pasted image 20250514183223.png|250]] ^basic-example > [!basicexample] (Type $A_{n}$) > - [ ] bring over ^basic-example [^1]: Geometrically, this is saying that the projection of $\beta$ onto the line spanned by $\alpha$ is an integer of half-integer multiple of $\alpha$. induction - [ ] bring over more pictorial examples ---- #### $\frac{(\beta, \alpha)}{(\alpha, \alpha)}= \frac{\|\alpha\| \|\beta\| \cos \theta}{\|\alpha\|^{2}}=\frac{\|\beta\|\cos \theta}{\|\alpha\|}$ ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` *Which* 'canonical isomorphism', of course, depends on how the vanilla Laplacian $\boldsymbol L_{F}$ is constructed: 1. If $\boldsymbol L_{F}=\boldsymbol L_{F}^{\text{Roos}}=(d^{0}_{\text{Roos}})^{*}d^{0}_{\text{Roos}}$ is the Laplacian of the Roos complex of $G$, then $\Gamma(G, F) = H^{0}_{\text{Roos}}(G ; F) \xrightarrow[\text{DHT}]{\cong}\operatorname{ker }\boldsymbol L_{F}$ 2. If $\boldsymbol L_{F}=\boldsymbol L_{F}^{\text{cell}}=(d_{\text{cell}}^{0})^{*} d_{\text{cell}}^{0}$ is the cellular sheaf Laplacian of $G$, then $\Gamma(G, F) \xrightarrow[\toref]{\cong} H^{0}_{\text{cell}}(G ; F) \xrightarrow[\torefDHT]{\cong}\operatorname{ker }\boldsymbol L_{F}.$ 3. If $\boldsymbol L_{F}=\boldsymbol L_{F}^{\text{Duta}}$ is the sheaf hypergraph Laplacian of Duta et al. \tocite, then $\Gamma(G, F) \xrightarrow[\toref]{\cong}\operatorname{ker } \boldsymbol L_{F}.$ In this proof, we are working with diffusion with the Laplacian in $(2)$ or $(3)$. We will refer to elements of $H^{0}_{\text{cell}}$ and of $\operatorname{ker }\boldsymbol L_{F}$ freely as 'global sections' in light of these identifications, even though — unlike in $(1)$ — these are elements of $\mathbb{R}^{|V|}$ rather than $\mathbb{R}^{|V|} \oplus \mathbb{R}^{|E|}$.