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# 数学代写|几何组合代写Geometric Combinatorics代考|MATH7435 Other “Finite Type” Classifications

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## 数学代写|几何组合代写Geometric Combinatorics代考|Other “Finite Type” Classifications

The classification of root systems is similar or identical to several other classifications of objects of “finite type,” briefly reviewed below.

Non-crystallographic root systems
Lifting the crystallographic restriction does not allow very many additional root systems. The only non-crystallographic irreducible finite root systems are those of types $H_3, H_4$ and $I_2(m)$ for $m=5$ or $m \geq 7$. See [34].
Coxeter groups and reflection groups
By Theorems $2.10$ and 2.11, the classification of finite Coxeter groups is parallel to the classification of reflection groups and is essentially the same as the classification of root systems. The difference is that the root systems $B_n$ and $C_n$ correspond to the same Coxeter group $B_n$. A Coxeter group is encoded by its Coxeter diagram, a graph whose vertex set is $S$, with an edge $s$ – $t$ whenever $m_{s t}>2$. If $m_{s t}>3$, the edge is labeled by $m_{s t}$. Figure $2.2$ shows the Coxeter diagrams of the finite irreducible Coxeter systems, including the non-crystallographic Coxeter groups $\mathrm{H}_3$, $H_4$ and $I_2(m)$. The group $G_2$ appears as $I_2(6)$. See $[34]$ for more details.

Regular polytopes
By Theorem 1.5, the symmetry group of a regular polytope is a reflection group. In fact, it is a Coxeter group whose Coxeter diagram is linear: the underlying graph is a path with no branching points. This narrows down the possibilities, leading to the conclusion that there are no other regular polytopes besides the ones described in Section 1.2. In particular, there are no “exceptional” regular polytopes beyond dimension 4: only simplices, cubes, and crosspolytopes.
Lie algebras
The original motivation for the Cartan-Killing classification of root systems came from Lie theory. Complex finite-dimensional simple Lie algebras correspond naturally, and one-to-one, to finite irreducible crystallographic root systems. There exist innumerable expositions of this classical subject; see, e.g.,

## 数学代写|几何组合代写Geometric Combinatorics代考|Reduced Words and Permutohedra

Each element $w \in W$ can be written as a product of elements of $S$ :
$$w=s_{i_1} \cdots s_{i_{\ell}} .$$
A shortest factorization of this form (or the corresponding sequence of subscripts $\left.\left(i_1, \ldots, i_{\ell}\right)\right)$ is called a reduced word for $w$; the number of factors $\ell$ is called the length of $w$.

Any finite Coxeter group has a unique element $w_{\circ}$ of maximal length. In the symmetric group $\mathcal{S}{n+1}=A_n$, this is the permutation $w{\circ}$ that reverses the order of the elements of the set ${1, \ldots, n+1}$.

Example 2.12. Let $W=\mathcal{S}_4$ be the Coxeter group of type $A_3$. The standard choice of simple reflections yields $S=\left{s_1, s_2, s_3\right}$, where $s_1, s_2$ and $s_3$ are the transpositions which interchange 1 with 2,2 with 3 , and 3 with 4 , respectively. (Cf. Example 1.7.)

The word $s_1 s_2 s_1 s_3 s_2 s_3$ is a non-reduced word for the permutation that interchanges 1 with 3 and 2 with 4 . This permutation has two reduced words $s_2 s_1 s_3 s_2$ and $s_2 s_3 s_1 s_2$.

An example of a reduced word for $w_{\circ}$ is $s_1 s_2 s_1 s_3 s_2 s_1$. There are 16 such reduced words altogether. (Cf. Example $2.14$ and Theorem 2.15.)

Recall from Section $1.2$ that we label the regions $R_w$ of the Coxeter arrangement by the elements of the reflection group $W$, so that $R_w$ is the image of $R_1$ under the action of $w$. More generally, $R_{u v}=u\left(R_v\right)$.

## MATLAB代写

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