# Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Even-Degree Indices for Orbits

- the size of an orbit of any Coxeter group is always limited;
- the points of an orbit have only real numbers as their coordinates;
- the product of several orbits can always be decomposed into the sum of orbits of smaller sizes.

**Definition**

**1.**

**Remark**

**1.**

**Remark**

**2.**

**Proposition**

**1.**

**Proof.**

**Definition**

**2.**

**Definition**

**3.**

**Remark**

**3.**

**Example**

**1.**

**Proposition**

**2.**

**Proof.**

**Remark**

**4.**

**Example**

**2.**

**Proposition**

**3.**

**Proof.**

## 3. Odd-Degree Indices for Orbits

**Definition**

**4.**

**Example**

**3.**

**Remark**

**5.**

- The anomaly numbers ${A}_{(a,b)}^{1}\left({H}_{2}\right)={A}_{(a,b)}^{3}\left({H}_{2}\right)=0,\mathit{for}\mathit{any}a,b\in \mathbb{R}.$
- The odd-order indices ${A}_{(a,b)}^{2p-1}\left({H}_{2}\right)\ne 0$, for $a\ne b$ and $p>2$.
- For the Coxeter groups ${H}_{3}$ and ${H}_{4}$, as any orbit contains the elements with positive and negative signs, the anomaly numbers obtained for any orbit are equal to zero:$${A}_{\lambda}^{2p-1}\left({H}_{n}\right)=\sum _{\mu \in {O}_{\lambda}\left({H}_{n}\right)}{\langle \mu ,v\rangle}^{2p-1}=0,\phantom{\rule{1.em}{0ex}}n=3,4.$$

**Definition**

**5.**

**Proposition**

**4.**

**Remark**

**6.**

## 4. Embedding Index

**Definition**

**6.**

**Remark**

**7.**

**Theorem**

**1.**

**Proof.**

- consider the points of an orbit ${O}_{\lambda}\left({H}_{3}\right)$;
- remove the first coordinate of each point in the case of ${H}_{2}$, and the third one for the crystallographic group ${A}_{2}$;
- among all the points in ${\mathbb{R}}^{2}$ select those with non-negative coordinates; such points provide the orbits of ${G}^{\prime}$ in ${\mathbb{R}}^{2}$.

## 5. Lower Orbits of ${H}_{2}$ and ${H}_{3}$

- determination of the highest weight;
- subtraction of simple roots from the highest weight;
- an algorithm that describes the subtraction path.

- $\left(i\right)$
- determine a dominant point $\lambda =({l}_{1},\dots ,{l}_{i})$, ${l}_{i}={a}_{i}+{b}_{i}\tau \in \mathbb{Z}{\left[\tau \right]}^{>0}$, $i\in \{1,\cdots ,n\}$;
- $\left(ii\right)$
- establish a correspondence between the coordinates of a dominant point $\lambda $ and the index$i\in \{1,\dots ,n\}$ of a simple root ${\alpha}_{i}$: $i\to {l}_{i}$;
- $\left(iii\right)$
- if at least one of ${l}_{i}>0$, $i\in \{1,\dots ,n\}$, then proceed the following subtraction:
- if ${b}_{i}=0$, then ${\mu}_{i}=\lambda -j\xb7{\alpha}_{i}$, $j\in \{1,\dots ,{a}_{i}\};$
- if ${b}_{i}\ge 1$:
- -
- and ${a}_{i}=0$, then ${\mu}_{i}=\lambda -k\tau \xb7{\alpha}_{i}$, $k\in \{1,\dots ,{b}_{i}\};$
- -
- and ${a}_{i}\ge 1$, then ${\mu}_{i}=\lambda -k\frac{{l}_{i}}{gcd({a}_{i},{b}_{i})}\xb7{\alpha}_{i}$, $k\in \{1,2,\dots ,gcd({a}_{i},{b}_{i})\};$

- $\left(iv\right)$
- replace a point $\lambda $ in $\left(i\right)$ with ${\mu}_{i}$;
- $\left(v\right)$
- repeat the steps $\left(ii\right)$–$\left(iv\right)$ until at least one of the coordinates ${\mu}_{i}$ is greater than zero.

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

**Example**

**7.**

## 6. Concluding Remarks

- The decomposition of a tensor product of representations of a simple Lie algebra into a direct sum of irreducible components given by Young tableaux symmetries plays an essential role in physics. As the indices of the representations help to determine such a decomposition [31], we demonstrate that their definitions can be extended to orbits of the non-crystallographic Coxeter groups. As a result, the notation of the even- and odd-order indices of representations are reformulated for the orbits of ${H}_{n}$, $n\in \{2,3,4\}$.
- It would be useful to generalize the properties of higher-order indices and anomaly numbers of orbits, similarly to [18,23]. Along with these properties, one could potentially obtain the formulas for the explicit forms of higher even-order indices of a tensor product of orbits. Moreover, the expressions for the even-order indices, anomaly numbers and embedding indices could be reformulated and adapted to orbits of any finite reflection group of crystallographic type.
- Even though the Coxeter groups of non-crystallographic types do not have underlying Lie algebras, the recursive algorithm introduced in Section 5 is shown to be similar to the algorithm developed for the weight multiplicities of simple Lie groups [34]. It is important to mention that our algorithm also provides the seed points of orbits that are smaller in radius than an initial orbit (referred to as ‘lower orbits’). The geometrical construction of sets of lower orbits results in the structures of nested polytopes. Since the recursive rules are only applied to a dominant point $\lambda $ of the non-crystallographic groups ${H}_{2}$ and ${H}_{3}$, one could consider applying them to any seed point of the ${H}_{4}$ group as well. As the size of an orbit $|{O}_{(a,b,c,d)}\left({H}_{4}\right)|={120}^{2}$, for $a,b,c,d>0$, the generalization of the formulas for the coordinates of the seed points of lower orbits is considered as future research. Moreover, it would be an interesting task to generalize the formulas given in Table 4 for any $a\in \mathbb{N}$, as it was done for the ${H}_{2}$ case.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The Coxeter–Dynkin diagrams of the non-crystallographic groups ${H}_{n}$, $n\in \{2,3,4\}$. The nodes correspond to the simple roots ${\alpha}_{k}$, $k\in \{1,\dots ,n\}$.

**Figure 2.**The root system of the Coxeter group ${H}_{2}$. The dashed lines ${r}_{1}$ and ${r}_{2}$ correspond to the reflecting hyperplanes orthogonal to the simple roots ${\alpha}_{1}$ and ${\alpha}_{2}$, respectively. The root $\xi $ denotes the highest root of ${H}_{2}$. The coordinates of the points of an orbit with a dominant point $\lambda =(a,b)$ of ${H}_{2}$ are listed. The orbits of the reflection group ${A}_{1}$ are depicted by green segments.

**Figure 3.**The tree-diagram for the orbits of ${H}_{2}$. (

**a**) ${O}_{(1,0)}\left({H}_{2}\right)$; (

**b**) ${O}_{(0,1)}\left({H}_{2}\right)$; (

**c**) ${O}_{(1,1)}\left({H}_{2}\right)$. The dominant points are displayed in boxes. The points that do not belong to ${O}_{(1,1)}\left({H}_{2}\right)$ are depicted by blue colour.

**Figure 4.**The tree-diagram for the orbit ${O}_{(\tau ,1)}\left({H}_{2}\right)$. The dominant points are displayed in boxes. The points that do not belong to ${O}_{(\tau ,1)}\left({H}_{2}\right)$ are depicted by blue colour.

**Figure 5.**The tree-diagrams constructed for the orbits of ${H}_{3}$. (

**a**) ${O}_{(1,0,0)}\left({H}_{3}\right)$; (

**b**) ${O}_{(0,0,1)}\left({H}_{3}\right)$. The repetitive coordinates and subtraction paths are marked in blue.

**Figure 6.**(

**a**) The tree-diagram for the orbit ${O}_{(2,0,0)}\left({H}_{3}\right)$; (

**b**) the corresponding nested polytopes. The orbits ${O}_{(2,0,0)}\left({H}_{3}\right)$, ${O}_{(0,1,0)}\left({H}_{3}\right)$ and ${O}_{(0,-{\tau}^{\prime},0)}\left({H}_{3}\right)$ are presented in green, black and bold colours, respectively.

**Figure 7.**The nested polytopes provided by the algorithm of root-subtraction. $\left(\mathbf{a}\right)$ ${O}_{(3,1,0)}\left({H}_{3}\right)$; $\left(\mathbf{b}\right)$ ${O}_{(0,1,3)}\left({H}_{3}\right)$.

**Table 1.**The sizes of orbits ${O}_{\lambda}\left({H}_{n}\right)$ of the non-crystallographic groups ${H}_{n}$, $n\in \{2,3,4\}$ provided for each type of a dominant point $\lambda $ with the coefficients $a,b,c,d\in {\mathbb{R}}^{>0}$.

$\mathit{\lambda}$ | $|{\mathit{O}}_{\mathit{\lambda}}\left({\mathit{H}}_{2}\right)|$ |

$(a,0)$ | 5 |

$(0,b)$ | 5 |

$(a,b)$ | 10 |

$\mathbf{\lambda}$ | $|{\mathbf{O}}_{\mathbf{\lambda}}\left({\mathbf{H}}_{\mathbf{3}}\right)|$ |

$(a,0,0)$ | 12 |

$(0,b,0)$ | 30 |

$(0,0,c)$ | 20 |

$(a,b,0)$ | 60 |

$(a,0,c)$ | 60 |

$(0,b,c)$ | 60 |

$(a,b,c)$ | 120 |

$\mathbf{\lambda}$ | $|{\mathbf{O}}_{\mathbf{\lambda}}\left({\mathbf{H}}_{\mathbf{4}}\right)|$ |

$(a,0,0,0)$ | 120 |

$(0,b,0,0)$ | 720 |

$(0,0,c,0)$ | 1200 |

$(0,0,0,d)$ | 600 |

$(a,b,0,0)$ | 1440 |

$(a,0,c,0)$ | 3600 |

$(a,0,0,d)$ | 2400 |

$(0,b,c,0)$ | 3600 |

$(0,b,0,d)$ | 3600 |

$(0,0,c,d)$ | 2400 |

$(a,b,c,0)$ | 7200 |

$(a,b,0,d)$ | 7200 |

$(a,0,c,d)$ | 7200 |

$(0,b,c,d)$ | 7200 |

$(a,b,c,d)$ | 14,400 |

**Table 2.**The Cartan matrices and their inverses for the non-crystallographic groups ${H}_{2}$, ${H}_{3}$ and ${H}_{4}$.

${C}_{{H}_{2}}=\left(\begin{array}{cc}2& -\tau \\ -\tau & 2\end{array}\right)$ | ${C}_{{H}_{2}}^{-1}=\frac{1}{3-\tau}\left(\begin{array}{cc}2& \tau \\ \tau & 2\end{array}\right)$ |

${C}_{{H}_{3}}=\left(\begin{array}{ccc}2& -1& 0\\ -1& 2& -\tau \\ 0& -\tau & 2\end{array}\right)$ | ${C}_{{H}_{3}}^{-1}=\frac{1}{2}\left(\begin{array}{ccc}2+\tau & 2+2\tau & 1+2\tau \\ 2+2\tau & 4+4\tau & 2+4\tau \\ 1+2\tau & 2+4\tau & 3+3\tau \end{array}\right)$ |

${C}_{{H}_{4}}=\left(\begin{array}{cccc}2& -1& 0& 0\\ -1& 2& -1& 0\\ 0& -1& 2& -\tau \\ 0& 0& -\tau & 2\end{array}\right)$ | ${C}_{{H}_{4}}^{-1}=\left(\begin{array}{cccc}2+2\tau & 3+4\tau & 4+6\tau & 3+5\tau \\ 3+4\tau & 6+8\tau & 8+12\tau & 6+10\tau \\ 4+6\tau & 8+12\tau & 12+18\tau & 9+15\tau \\ 3+5\tau & 6+10\tau & 9+15\tau & 8+12\tau \end{array}\right)$ |

**Table 3.**The embedding index $\gamma $ provided for the non-crystallographic groups ${H}_{n}$, $n\in \{2,3,4\}$ and their maximal subgroups ${G}^{\prime}$.

G | ${\mathit{G}}^{\prime}$ | $\mathit{\gamma}$ |
---|---|---|

${H}_{2}$ | ${A}_{1}$ | 2 |

${H}_{3}$ | ${A}_{1}\times {A}_{1}\times {A}_{1}$ | 1 |

${H}_{3}$ | ${A}_{2}$ | $3/2$ |

${H}_{3}$ | ${H}_{2}$ | $3/2$ |

${H}_{4}$ | ${A}_{2}\times {A}_{2}$ | 1 |

${H}_{4}$ | ${H}_{2}\times {H}_{2}$ | 1 |

${H}_{4}$ | ${A}_{1}\times {A}_{1}\times {A}_{1}\times {A}_{1}$ | 1 |

${H}_{4}$ | ${H}_{3}\times {A}_{1}$ | 1 |

${H}_{4}$ | ${A}_{4}$ | 1 |

${H}_{4}$ | ${D}_{4}$ | 1 |

**Table 4.**Dominant points for lower orbits obtained by subtraction of the simple roots ${\alpha}_{1},{\alpha}_{2},{\alpha}_{3}$ of ${H}_{3}$ are listed for any type of a dominant point of the initial orbit: $(a,0,0)$, $(0,a,0)$, $(0,0,a)$, $(a,a,0)$, $(0,a,a)$, $(a,0,a)$. The coefficients are provided by the values $a\in \{1,2,\dots ,9\}$.

$(\mathit{a},0,0)$: | |

$(a-2k,k,0),\phantom{\rule{4pt}{0ex}}k\in \left\{0,\dots ,\left[\frac{a}{2}\right]\right\}$ | any a |

$\left(0,\frac{a}{2}(\tau -1),0\right)$ | even a |

$\left(0,\left[\frac{a}{2}\right]\tau -\left[\frac{a+2}{2}\right],\tau \right)$ | odd $a>3$ |

$(\mathbf{0},\mathit{a},\mathbf{0})$: | |

$(k,a-2k,k\tau ),\phantom{\rule{4pt}{0ex}}k\in \left\{0,\dots ,\left[\frac{a}{2}\right]\right\}$ | any a |

$(0,0,0),\left(\frac{a}{2}(\tau -1),0,\frac{a}{2}\right),\left(a,\frac{a}{2}(\tau -1),0\right)$ | even a |

$\left(\left[\frac{a}{2}\right]\tau -\left[\frac{a+2}{2}\right],\tau +1,\left[\frac{a}{2}\right]-\tau \right),\left(a,\left[\frac{a}{2}\right]\tau -\left[\frac{a+2}{2}\right],\tau \right)$ | odd $a>3$ |

$(\mathbf{0},\mathbf{0},\mathit{a})$: | |

$(0,k\tau ,a-2k),\phantom{\rule{4pt}{0ex}}k\in \left\{0,\dots ,\left[\frac{a}{2}\right]\right\}$ | any a, |

$\left(0,\frac{a}{2}(\tau -1),0\right),\left(\frac{a}{2}\tau ,0,\frac{a}{2}(\tau -1)\right)$ | even a |

$\left(\left[\frac{a}{2}\right]\tau ,\tau ,\left[\frac{a}{2}\right]\tau -\left[\frac{a+2}{2}\right]\right),\left(\tau +1,\left[\frac{a}{2}\right]\tau -\left[\frac{a+2}{2}\right],0\right)$ | odd $a>3$ |

$(\mathit{a},\mathit{a},\mathbf{0})$: | |

$(a,a,0),(0,0,a\tau ),(a,a(\tau -1),0)$ | any a |

$\left(a-2k,a+k,0\right),(a+k,a-2k,k\tau ),\phantom{\rule{4pt}{0ex}}k\in \left\{1,\dots \left[\frac{a}{2}\right]\right\}$ | $a>1$ |

$\frac{a}{2}\left(2\tau -1,0,2-\tau \right),\frac{a}{2}\left(0,\tau -1,0\right),\frac{a}{2}\left(4,\tau -1,0\right),\frac{a}{2}\left(0,2-\tau ,a\right)$ | even a |

$(a,(a-1)\tau -(a+1),2\tau )$ | $a>4$ |

$\left(2a,\left[\frac{a}{2}\right]\tau -\left[\frac{a}{2}+1\right],\tau \right),\left(0,\left[\frac{a}{2}\right]\tau -\left[\frac{a}{2}+1\right],\tau \right)$ | odd $a>3$ |

$(a,(a-2)\tau -(a+2),4\tau )$ | $a>8$ |

$(\mathit{a},\mathbf{0},\mathit{a})$: | |

$(a,0,a),(a\tau ,0,0)$ | any a |

$(a-2k,k,a),(a,k\tau ,a-2k),\phantom{\rule{4pt}{0ex}}k\in \left\{1,\dots \left[\frac{a}{2}\right]\right\}$ | $a>1$, |

$\left(0,(a-2k-1)\tau -\left[\frac{a}{2}+k+1\right],(2k+1)(\tau +1)\right),\phantom{\rule{4pt}{0ex}}k\in \left\{0,\dots \left[\frac{a-2}{4}\right]\right\}$ | $a>1$ |

$\frac{a}{2}(0,1,0),\frac{a}{2}(\tau +2,0,\tau -1),\frac{a}{2}(1,0,2-\tau ),\frac{a}{2}(\tau -1,0,2\tau -1)$ | even a |

$\left(0,(a-2k)\tau -\frac{a}{2}-k,2k(\tau +1)\right),\phantom{\rule{4pt}{0ex}}k\in \left\{0,\dots \left[\frac{a}{4}\right]\right\}$ | |

$\left(\tau +2,\left[\frac{a}{2}\right]\tau -1,0\right)$ | odd $a>1$ |

$\left(\left[\frac{a}{2}\right]\tau +a,\tau ,\left[\frac{a}{2}\right]\tau -\left[\frac{a}{2}+1\right]\right),\left(\left[\frac{a}{2}\right]\tau -\left[\frac{a}{2}+1\right],\tau +1,(a-1)\tau -\left[\frac{a}{2}+1\right]\right)$ | odd $a>3$ |

$\left(2\tau +4,\left(\frac{a}{2}-1\right)\tau -2,0\right)$ | even $a>4$ |

$\left(3\tau +6,\left[\frac{a}{2}-1\right]\tau -3,0\right)$ | odd $a>5$ |

$(\mathbf{0},\mathit{a},\mathit{a})$: | |

$(0,a,a),\left(a\right(\tau +1),0,0)$ | any a |

$\left(k,a-2k,k\tau +a\right),\left(0,k\tau +a,a-2k\right),\phantom{\rule{4pt}{0ex}}k\in \left\{1,\dots ,\left[\frac{a}{2}\right]\right\}$ | $a>1$ |

$\left(0,0,a\right),\frac{a}{2}\left(2\tau -1,0,\tau \right),\left(0,\frac{a}{2}(\tau -1),0\right)$ | even a |

$\left(\left(\frac{a}{2}-k\right)(\tau +1),2k(\tau +1),(a-2k)\tau -\left[\frac{a}{2}+k\right]\right)$ | even a |

$\left(a,(a-2k)\tau -\frac{a}{2}-k,2k(\tau +1)\right),\phantom{\rule{4pt}{0ex}}k\in \left\{0,\dots ,\left[\frac{a}{4}\right]\right\}$ | |

$\left(\left[\frac{a}{2}-k\right](\tau +1),(2k+1)(\tau +1),(a-2k-1)\tau -\left[\frac{a}{2}+k+1\right]\right)$ | odd $a>1$ |

$\left(a,(a-2k-1)\tau -\left[\frac{a}{2}+k+1\right],(2k+1)(\tau +1)\right),\phantom{\rule{4pt}{0ex}}k\in \left\{0,\dots ,\left[\frac{a-3}{4}\right]\right\}$ | |

$\left((a-1)\tau -\left[\frac{a}{2}+1\right],2\tau +1,\left[\frac{a}{2}-1\right]\tau -1\right)$ | odd $a>3$ |

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Myronova, M.; Patera, J.; Szajewska, M.
Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type. *Symmetry* **2020**, *12*, 1737.
https://doi.org/10.3390/sym12101737

**AMA Style**

Myronova M, Patera J, Szajewska M.
Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type. *Symmetry*. 2020; 12(10):1737.
https://doi.org/10.3390/sym12101737

**Chicago/Turabian Style**

Myronova, Mariia, Jiří Patera, and Marzena Szajewska.
2020. "Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type" *Symmetry* 12, no. 10: 1737.
https://doi.org/10.3390/sym12101737