Last update: 29 January 2014
Definition. Let $M$ be a Coxeter-Matrix over $I\text{.}$ Let $J\subset I$ be a subset such that the ${\text{letters}}^{1}$ of $J$ in ${G}_{M}^{+}$ possess a common multiple. Then the uniquely determined least common multiple of the letters of $J$ in ${G}_{M}^{+}$ is called the fundamental element ${\Delta}_{J}$ for $J$ in ${G}_{M}^{+}\text{.}$
${}^{1}$For the sake of simplicity we call the images of letters of ${F}^{+}$ in ${G}^{+}$ also letters, and we also denote them as such.
The word “fundamental”, introduced by Garside, refers to the fundamental role which these elements play. We will show for example that if ${G}_{M}$ is irreducible and if there exists a fundamental word ${\Delta}_{I},$ then ${\Delta}_{I}$ or ${\Delta}_{I}^{2}$ generates the centre of ${G}_{M}\text{.}$ The condition for the existence of ${\Delta}_{I}$ is very strong: ${\Delta}_{I}$ exists exactly when ${G}_{M}$ is of finite type (cf 5.6).
5.1. The lemmas of the following sections are proven foremost because they will be required in later proofs, but they already indicate the important properties of fundamental elements.
Let $J\subset I$ be a finite set $J=\{{j}_{1},\cdots ,{j}_{k}\},$ for which a fundamental element ${\Delta}_{J}$ in ${G}_{M}^{+}$ exists. Then we have:
(i) | ${\Delta}_{J}\u2a66{[{a}_{{j}_{1}},\cdots ,{a}_{{j}_{k}}]}_{l}\u2a66{[{a}_{{j}_{1}},\cdots ,{a}_{{j}_{k}}]}_{r}\text{.}$ |
(ii) | $\text{rev}\hspace{0.17em}{\Delta}_{J}\u2a66{\Delta}_{J}\text{.}$ |
Proof. | |
(i) If ${[{a}_{{j}_{1}},\cdots ,{a}_{{j}_{k}}]}_{r}$ were not left-divisible by an ${a}_{j}$ with $j\in J$ then by 3.2 it could be represented by a chain from ${a}_{j}$ to an ${a}_{j}^{\prime},$ $j\prime \in J$ and thus it would not be right-divisible by ${a}_{j}^{\prime}$ in contradiction to its definition. Hence ${[{a}_{{j}_{1}},\cdots ,{a}_{{j}_{k}}]}_{r}$ is divisible by ${[{a}_{{j}_{1}},\cdots ,{a}_{{j}_{k}}]}_{l}\text{.}$ Analogously one shows that ${[{a}_{{j}_{1}},\cdots ,{a}_{{j}_{k}}]}_{l}$ is divisible by ${[{a}_{{j}_{1}},\cdots ,{a}_{{j}_{k}}]}_{r}$ and hence both these elements of ${G}^{+}$ are equivalent to one another. (ii) The assertion (ii) follows trivially from (1) and 4.3. $\square $ |
5.2. Let $M$ be a certain Coxeter-matrix over $I,$ and ${G}^{+}$ the corresponding Artin-semigroup. If $J\subset I$ is an arbitrary subset, then we denote by ${G}_{J}^{+}$ the subsemigroup of ${G}^{+}$ which is generated by the letters ${a}_{j},$ $j\in J\text{.}$ Of course, ${G}_{J}^{+}$ is canonically isomorphic to ${G}_{M}^{+},$ where ${M}_{J}$ is the Coxeter-matrix over $J$ obtained by restriction of $M\text{.}$
If a fundamental element ${\Delta}_{J}$ in ${G}^{+}$ exists for $J\subset I,$ then there is a uniquely determined involutionary automorphism ${\sigma}_{J}$ of ${G}_{J}^{+}$ with the following properties:
(i) | ${\sigma}_{J}$ sends letters to letters, i.e. ${\sigma}_{J}\left({a}_{j}\right)={a}_{\sigma \left(j\right)}$ for all $j\in J\text{.}$ Hence $\sigma $ is a permutation of $J$ with ${\sigma}^{2}=\text{id}$ and ${m}_{\sigma \left(i\right)\sigma \left(j\right)}={m}_{ij}\text{.}$ |
(ii) | For all $W\in {G}_{J}^{+},$ $$W{\Delta}_{J}\u2a66{\Delta}_{J}{\sigma}_{J}\left(W\right)\text{.}$$ |
Proof. | |
$W{\Delta}_{J}$ is left-divisible by ${\Delta}_{J}$ by the same argument as in the proof of 5.1. So by 2.3 there is a uniquely determined ${\sigma}_{J}\left(W\right)$ such that (ii) holds. From 2.3 it also follows immediately that ${\sigma}_{J}$ is an automorphism of ${G}_{J}^{+}\text{.}$ Since ${\sigma}_{J}$ preserves lengths it takes letters to letters, and hence arises from a permutation $\sigma $ of $J\text{.}$ From $\sigma \left(a\right){\Delta}_{J}\u2a66{\Delta}_{J}{\sigma}^{2}\left(a\right)$ it follows by application of rev that ${\sigma}^{2}\left(a\right){\Delta}_{J}\u2a66{\Delta}_{J}\sigma \left(a\right)\text{.}$ The right hand side is positive equivalent to $a{\Delta}_{J}$ and hence from 2.3, ${\sigma}^{2}\left(a\right)\u2a66a\text{.}$ Thus $\sigma $ is an involution and clearly ${\sigma}_{J}$ is too. Finally, since for all $i,$ $j,$ $${\u27e8\sigma \left({a}_{i}\right)\sigma \left({a}_{j}\right)\u27e9}^{{m}_{ij}}\u2a66{\u27e8\sigma \left({a}_{j}\right)\sigma \left({a}_{j}\right)\u27e9}^{{m}_{ij}}$$ it follows that ${m}_{ij}={m}_{\sigma \left(i\right)\sigma \left(j\right)}\text{.}$ Thus (i) is proved. $\square $ |
Remark. The converse of 5.2 also holds: let ${G}_{J}^{+}$ be irreducible, $\sigma :J\u27f6J$ a permutation and $\Delta \in {G}_{J}^{+}$ a nontrivial element such that $a\Delta \u2a66\Delta \sigma \left(a\right)$ for all letters $a$ of $J\text{.}$ Then there exists a fundamental element ${\Delta}_{J}\text{.}$
Proof. | |
It suffices to show that $\Delta $ is a common multiple of the letters of $J\text{.}$ At least, one letter $a$ from $J$ divides $\Delta \text{.}$ Hence let $\Delta \u2a66a\Delta \prime \text{.}$ If $b$ is any letter of $J$ with ${m}_{ab}>2,$ $ba\Delta \u2a66\Delta \sigma \left(b\right)\sigma \left(a\right)\u2a66a\Delta \prime \sigma \left(b\right)\sigma \left(a\right),$ so by the reduction lemma 2.1, we have that $a\Delta $ is divisible by ${\u27e8ab\u27e9}^{{m}_{ab}-1}$ and thus $b$ is a divisor of $\Delta \text{.}$ Hence $\Delta $ is divisible by all the letters of $J$ since the Coxeter-graph of ${M}_{J}$ is assumed connected. By 4.1 the existence of ${\Delta}_{J}$ then follows. $\square $ |
5.3. The first part of the following lemma follows immediately from 5.2 (ii).
Suppose there exists a fundamental element ${\Delta}_{J}\text{.}$ Then for all $U,V,W\in {G}_{J}^{+}\text{:}$
(i) | ${\Delta}_{J}$ left-divides $U$ exactly when it right-divides $U\text{.}$ |
(ii) | If ${\Delta}_{J}$ divides the product $VW,$ then each letter ${a}_{j},$ for $j\in J,$ either right-divides the factor $V$ or left-divides $W\text{.}$ |
Proof. | |
(ii) If ${a}_{j}$ neither right-divides $V$ nor left-divides $W$ then, by 3.2, one can represent $V$ by a chain with target ${a}_{j}$ and $W$ by a chain with source ${a}_{j}\text{.}$ Thus one can represent $VW$ by a chain [Ed: a word in the letters of $J$] which, by 3.1, is not divisible by its source, and hence neither by ${\Delta}_{J}\text{.}$ $\square $ |
5.4. The following lemma contains an important characterization of fundamental elements.
If a fundamental element ${\Delta}_{J}$ exists for $J\subset I,$ the following hold:
(i) | $U\in {G}_{J}^{+}$ is square free if and only if $U$ is a divisor of ${\Delta}_{J}\text{.}$ |
(ii) | The least common multiple of square free elements of ${G}_{J}^{+}$ is square free. |
Proof. | |
(i) From 3.5 it follows immediately by induction on the number of elements of $J$ that ${\Delta}_{J}$ is square free; and consequently so are its divisors. The converse is shown by induction on the length of $U\text{.}$ Let $U\u2a66Va\text{.}$ By the induction assumption there exists a $W$ with ${\Delta}_{J}\u2a66VW\text{.}$ Since $a$ does not right-divide $V,$ it left divides $W$ by 5.3 and hence $U$ is a divisor of ${\Delta}_{J}\text{.}$ (ii) The assertion (ii) follows trivially from (i). $\square $ |
5.5. Let $M$ be a Coxeter-matrix over $I\text{.}$ The Artin semigroups ${G}_{M}^{+}$ with fundamental element ${\Delta}_{I}$ can be described by the type of embedding in the corresponding Artin group ${G}_{M}\text{.}$ Instead of ${\Delta}_{I},$ resp. ${\sigma}_{I},$ we will write simply $\Delta ,$ resp. $\sigma ,$ when there is no risk of confusion.
For a Coxeter-matrix $M$ the following statements are equivalent:
(i) | There is a fundamental element $\Delta $ in ${G}_{M}^{+}\text{.}$ |
(ii) | Every finite subset of ${G}_{M}^{+}$ has a least common multiple. |
(iii) | The canonical map ${G}_{M}^{+}\u27f6{G}_{M}$ is injective, and for each $A\in {G}_{M}$ there exist $B,C\in {G}_{M}^{+}$ with $A=B{C}^{-1}{\text{.}}^{2}$ |
(iv) | The canonical map ${G}_{M}^{+}\u27f6{G}_{M}$ is injective, and for each $A\in {G}_{M}$ there exist $B,C\in {G}_{M}^{+}$ with $A=B{C}^{-1},$ where the image of $C$ lies in the centre of ${G}_{M}\text{.}$ |
${}^{2}$We denote elements of ${G}_{M}^{+}$ and their images in ${G}_{M}$ by the same letters.
Proof. | |
[Ed: In this proof ${G}_{M}^{+}$ is written ${G}^{+}$ for simplicity] We will show first of all the equivalence of (i) and (ii), where clearly (ii) trivially implies (i). Let $\Lambda \u2a66\Delta $ or $\Lambda \u2a66{\Delta}^{2}$ according to whether $\sigma =1$ or not. Then $\Lambda $ is, by 5.2, a central element in ${G}^{+}$ and for each letter ${a}_{i},$ $i\in I,$ there is by 5.1 a ${\Lambda}_{i}$ with $\Lambda ={a}_{i}{\Lambda}_{i}\text{.}$ Now, if $A\u2a66{a}_{{i}_{1}}\dots {a}_{{i}_{m}}$ is an arbitrary element of ${G}^{+}$ then $${\Lambda}^{m}\u2a66{a}_{{i}_{m}}{\Lambda}_{{i}_{m}}\cdots {a}_{{i}_{1}}{\Lambda}_{{i}_{1}}\u2a66A{\Lambda}_{{i}_{m}}\cdots {\Lambda}_{{i}_{1}}\text{.}$$ Hence ${\Lambda}^{m}$ is divisible by each element $A$ of ${G}^{+}$ with $L\left(A\right)\le m\text{.}$ In particular, a finite set of elements always has a common multiple and thus by 4.1 a least common multiple. This proves the equivalence of (i) and (ii). If (ii), then (iv). Since to all $B,C\in {G}^{+}$ there exists a common multiple, and thus $B\prime ,C\prime ,\in {G}^{+}$ with $BC\prime \u2a66CB\prime \text{.}$ From this and cancellativity, 2.3, it follows by a general theorem of Öre that ${G}^{+}$ embeds in a group. Thence follows the injectivity of ${G}^{+}\to G$ and also that each element $A\in G$ can be represented in the form $A={C}^{-1}B$ or also $B\prime {C}^{-1}$ with $B,B\prime ,C,C\prime \in {G}^{+}\text{.}$ That $C$ can moreover be chosen to be central follows from the fact that — as shown above — to every $C$ with $L\left(C\right)\le m$ there exists $D\in {G}^{+}$ with ${\Lambda}^{m}\u2a66CD$ so that, therefore, ${C}^{-1}={\Lambda}^{-m}D=D{\Lambda}^{-m}\text{.}$ [Ed: As an alternative to applying Öre’s condition we provide the following proof of the injectivity of ${G}^{+}\to G$ when there exists a fundamental element $\Delta \text{.}$ By (5.2) it is clear that ${\Delta}^{2}$ is a central element in ${G}^{+}\text{.}$ Let $W,W\prime $ be positive words such that $W=W\prime $ in $G\text{.}$ Then there is some sequence ${W}_{1},{W}_{2},\dots ,{W}_{k}$ of words in the letters of $I$ and their inverses such that $W\equiv {W}_{1}={W}_{2}=\cdots ={W}_{k}\equiv W\prime $ where at each step ${W}_{i+1}$ is obtained from ${W}_{i}$ either by a positive transformation (cf 1.3.) or (so-called trivial) insertion or deletion of a subword $a{a}^{-1}$ or ${a}^{-1}a$ for some letter $a\text{.}$ Note that the number of inverse letters appearing in any word is bounded by $k\text{.}$ Let $C$ denote the central element ${\Delta}^{2}$ of ${G}^{+}\text{.}$ Then we may define positive words ${V}_{i}$ for $i=1,\dots ,k$ such that ${V}_{i}={C}^{k}{W}_{i}$ as follows. Write ${W}_{i}\equiv U{a}^{-1}U\prime $ for $U$ a positive word, $a$ a letter. Then $CU\u2a66UC,$ so if we let ${C}_{a}$ denote the unique element of ${G}^{+}$ such that ${C}_{a}a\u2a66C,$ $C{W}_{i}=CU{a}^{-1}U\prime =UC{a}^{-1}U\prime =UC{a}^{-1}U\prime =U{C}_{a}U\prime $ where $U{C}_{a}$ is positive. Repeating this step for successive inverses in $U\prime $ yields a positive word ${V}_{i}^{\prime}$ equal in $G$ to ${C}^{r}{W}_{i}$ for some $r\le k\text{.}$ Put ${V}_{i}\equiv {C}^{k-r}{V}_{i}^{\prime}\text{.}$ Essentially, ${V}_{i}$ is obtained from ${W}_{i}$ by replacing each occurrence of ${a}^{-1}$ with the word ${C}_{a},$ and then attaching unused copies of $C$ to the front. Now we check that ${V}_{i}\u2a66{V}_{i+1}\text{.}$ If ${W}_{i+1}$ differs from ${W}_{i}$ by a positive transformation, then ${W}_{i+1}$ is ${W}_{i}$ with some positive subword $U$ switched with a positive subword $U\prime ,$ and so the same transformation applied to ${V}_{i}$ gives the word ${V}_{i+1},$ so they are positive equivalent. If ${W}_{i+1}$ is obtained from ${W}_{i}$ by insertion of $a{a}^{-1}$ or ${a}^{-1}a$ Then ${V}_{i+1}\equiv {C}^{r}U{C}_{a}aV$ or ${C}^{r}Ua{C}_{a}V$ for positive words $U,V,$ where ${V}_{i}\equiv {C}^{r+1}UV\text{.}$ By the centrality of $C$ and the fact that $C\u2a66a{C}_{a}\u2a66{C}_{a}a,$ ${V}_{i}$ and ${V}_{i+1}$ are positive equivalent. If ${W}_{i+1}$ is obtained by a trivial deletion, then the proof is identical as above, but with the roles of ${W}_{i+1}$ and ${W}_{i}$ reversed. Hence we have a sequence of words ${V}_{1},{V}_{2},\dots ,{V}_{k}$ such that each is positive equivalent to its predecessor, so that ${V}_{1}$ is positive equivalent to ${V}_{k}\text{.}$ But ${V}_{1}\equiv {C}_{k}W$ and ${V}_{k}\equiv {C}_{k}W\prime $ so by cancellativity, $W$ is positive equivalent to $W\prime \text{.}$ So ${G}^{+}$ embeds in $G\text{.}$ ] Assuming (iv), (iii) follows trivially. And from (iii), (ii) follows easily. Since for $B,C\in {G}^{+}$ there exist $B\prime ,C\prime \in {G}^{+}$ with ${C}^{-1}B=B\prime {C\prime}^{-1},$ and thus $BC\prime =CB\prime $ and consequently $BC\prime \u2a66CB\prime $ so by 4.1 $B$ and $C$ have a least common multiple. Thus 5.5 is proved. $\square $ |
5.6. Let $QF{G}_{M}^{+}$ be the set of square free elements of ${G}_{M}^{+}\text{.}$ For the canonical map $QF{G}_{M}^{+}\u27f6{\stackrel{\u203e}{G}}_{M}$ defined by composition of inclusion and the residue class map it follows immediately from Theorem 3 of Tits in [Tit1968] that $$QF{G}_{M}^{+}\u27f6{\stackrel{\u203e}{G}}_{M}\phantom{\rule{1em}{0ex}}\text{is bijective.}$$
Let $M$ be a Coxeter-matrix. Then there exists a fundamental element $\Delta $ in ${G}_{M}^{+}$ if and only if ${\stackrel{\u203e}{G}}_{M}$ is finite.
Proof. | |
By Tits, ${\stackrel{\u203e}{G}}_{M}$ is finite exactly when $QF{G}_{M}^{+}$ is finite. By 5.4 and 3.5 this is the case if and only if $\Delta $ exists. Since, if $\Delta $ exists, by 5.4 $QF{G}_{M}^{+}$ consists of the divisors of $\Delta \text{.}$ And if $\Delta $ does not exist, by 3.5 there exists a sequence of infinitely many distinct square free elements. $\square $ |
5.7. By the length $l\left(w\right)$ of an element $w$ in a Coxeter group ${\stackrel{\u203e}{G}}_{M}$ we mean the minimum of the lengths $L\left(W\right)$ of all positive words $W$ which represent $w\text{.}$ The image of a positive word $W$ or an element $W$ of ${G}_{M}^{+}$ in ${\stackrel{\u203e}{G}}_{M}$ we denote by $\stackrel{\u203e}{W}\text{.}$ The theorem of Tits already cited immediately implies the following: The square free elements of ${G}_{M}^{+}$ are precisely those $W$ with $L\left(W\right)=l\left(\stackrel{\u203e}{W}\right)$
[Ed: Proof. | |
If an element is not square free, then it is represented by a word $W$ which contains a square. But this is clearly not a reduced word for the Coxeter element $\stackrel{\u203e}{W},$ and so $l\left(\stackrel{\u203e}{W}\right)\le L\left(W\right)-2$ (the square cancels). Conversely, suppose that $W$ represents a square free element of ${G}_{M}^{+}\text{.}$ By definition of length there is a $V\in {G}_{M}^{+}$ such that $\stackrel{\u203e}{V}=\stackrel{\u203e}{W}$ and $L\left(W\right)=l\left(\stackrel{\u203e}{V}\right)=l\left(\stackrel{\u203e}{W}\right)\text{.}$ Then by above $V$ is square free. But $\stackrel{\u203e}{V}=\stackrel{\u203e}{W}$ and hence by Tits theorem $V\u2a66W$ and $L\left(W\right)=L\left(V\right)=l\left(\stackrel{\u203e}{W}\right)\text{.}$ $\square \phantom{\rule{1em}{0ex}}\text{]}$ |
Let ${\stackrel{\u203e}{G}}_{M}$ be finite. The following hold for the fundamental element $\Delta $ of ${G}_{M}^{+}\text{:}$
(i) | $\Delta $ is the uniquely determined square free element of maximal length in ${G}_{M}^{+}\text{.}$ |
(ii) | There exists a uniquely determined element of maximal length in $\stackrel{\u203e}{{G}_{M}},$ namely $\stackrel{\u203e}{\Delta}\text{.}$ The fundamental element $\Delta $ is represented by the positive words $W$ with $\stackrel{\u203e}{W}=\stackrel{\u203e}{\Delta}$ and $L\left(W\right)=l\left(\stackrel{\u203e}{\Delta}\right)\text{.}$ |
Proof. | |
(i) By 5.4, the elements of $QF{G}_{M}^{+}$ are the divisors of $\Delta \text{.}$ A proper divisor $W$ of $\Delta $ clearly has $L\left(W\right)<L\left(\Delta \right)\text{.}$ Thus $\Delta $ is the unique square free element of maximal length. (ii) By the theorem of Tits and (i) there is also in ${\stackrel{\u203e}{G}}_{M}$ only one unique element of maximal length, namely $\stackrel{\u203e}{\Delta}\text{.}$ A positive word with $\stackrel{\u203e}{W}=\stackrel{\u203e}{\Delta}$ and $L\left(W\right)=l\left(\stackrel{\u203e}{\Delta}\right)$ is according to Tits square-free and it has maximal length, so by (i) it represents $\Delta \text{.}$ That only such positive words can represent $\Delta $ is clear. $\square $ |
5.8. Let ${\stackrel{\u203e}{G}}_{M}$ be a finite Coxeter group and for simplicity let the Coxeter system $({\stackrel{\u203e}{G}}_{M},\stackrel{\u203e}{I})$ be irreducible.
[Note: Bourbaki defines a Coxeter system $(W,S)$ to be a group $W$ and a set $S$ of elements of order $2$ in $W$ such that the following holds: For $s,s\prime $ in $S,$ let $m(s,s\prime )$ be the order of $ss\prime \text{.}$ Let $I$ be the set of pairs $(s,s\prime )$ such that $m(s,s\prime )$ is finite. The generating set $S$ and relations ${\left(ss\prime \right)}^{m(s,s\prime )}=1$ for $(s,s\prime )$ in $I$ form a presentation of the group $W\text{.}$
A Coxeter system $(W,S)$ is irreducible if the associated Coxeter graph $\Gamma $ is connected and non empty.
Note also that Bourbaki, Groupes et Algèbres de Lie, IV §1 ex 9 gives an example of a group $W$ and two subsets $S$ and $s\prime $ of elements of order $2$ such that $(W,S)$ and $(W,S\prime )$ are non-isomorphic Coxeter systems, one of which is irreducible, the other reducible. Hence the notion of irreducibility depends on $S,$ not just on the underlying group $W\text{.}$ Bourbaki says: when $(W,S)$ is a Coxeter system, and also says, by abuse of language, that $W$ is a Coxeter group. However one can check that, in the example cited, the two systems do have distinct Artin groups, which may be distinguished by their centres (see §7). ]
The existence of the unique word $\stackrel{\u203e}{\Delta}$ of maximal length in $\stackrel{\u203e}{G}$ and its properties are well known (see [Bou1968], Bourbaki, Groupes et Algèbres de Lie, IV, §1, ex. 22; V, §4, ex. 2 and 3; V, §6, ex. 2). For example we know that the length $l\left(\stackrel{\u203e}{\Delta}\right)$ is equal to the number of reflections of $\stackrel{\u203e}{G}$ and thus $$L\left(\Delta \right)=\frac{nh}{2}$$ where $h$ is the Coxeter number and $n$ the rank, i.e. the cardinality of the generating system $I\text{.}$ Explicit representations of $\stackrel{\u203e}{\Delta}$ by suitable words are also known and from this we now obtain quite simple corresponding expressions for $\Delta \text{.}$
Let $M$ be an irreducible Coxeter system of finite type over $I\text{.}$ A pair $(I\prime ,I\u2033)$ of subsets of $I$ is a decomposition of $I$ if $I$ is the disjoint union of $I\prime $ and $I\u2033$ and ${m}_{ij}\le 2$ for all $i,j\in I\prime $ and all $i,j\in I\u2033\text{.}$ Obviously there are exactly two decompositions of $I$ which are mapped into each other by interchanging $I\prime $ and $I\u2033\text{.}$
[Ed: Proof. | |
By Bourbaki, V §4 number 8 corollary to proposition 8, or from the classification of finite Coxeter groups we know that if $(W,S)$ is irreducible and finite then its graph is a tree. So the statement about decompositions boils down to the following statement about trees: if $\Gamma $ is a tree with a finite set $S$ of vertices then there exists a unique partition $(S\prime ,S\u2033)$ (up to interchange of $S\prime $ and $S\u2033\text{),}$ of $S$ into two sets such that no two elements of $S\prime $ and no two elements of $S\u2033$ are joined by an edge. We prove this by induction on the number of vertices of $\Gamma \text{.}$ For a graph on one vertex $a$ it is clear that the only suitable partitions are $(\left\{a\right\},\varphi )$ and $(\varphi ,\left\{a\right\})\text{.}$ Now let $\Gamma $ be an arbitrary tree with a finite set of vertices and let $a$ be a terminal vertex. Then applying the assumption to the subgraph of $\Gamma $ whose vertices are those vertices $n\ne a$ of $\Gamma $ we see that there exists a unique partition $({S}_{1}^{\prime},{S}_{l}^{\u2033})$ (up to interchange of ${S}_{1}^{\prime},{S}_{1}^{\u2033}\text{)}$ of $S\backslash \left\{a\right\}$ such that no two elements of ${S}_{1}^{\prime}$ and no two elements of ${S}_{1}^{\u2033}$ are joined by an edge of $\Gamma \prime \text{.}$ Now, by definition of a tree, $a$ is joined to exactly one vertex $b$ of $\Gamma \prime \text{.}$ Without loss of generality let $b\in {S}_{1}^{\prime}\text{.}$ Then it is easy to see that $({S}_{1}^{\prime},{S}_{1}^{\u2033}\cup \left\{a\right\})$ is a partition of $S$ satisfying the above conditions and that it is unique up to interchanging ${S}_{1}^{\prime}$ and ${S}_{1}^{\u2033}\cup \left\{a\right\}\text{.}$ $\square \phantom{\rule{1em}{0ex}}\text{]}$ |
Definition. Let $M$ be a Coxeter matrix over $I$ and $(I\prime ,I\u2033)$ a decomposition of $I\text{.}$ The following products of generators in ${G}_{M}^{+}$ are associated to the decomposition: $$\Pi \prime \u2a66\prod _{i\in I\prime}{a}_{i},\phantom{\rule{2em}{0ex}}\Pi \u2033\u2a66\prod _{i\in I\u2033}{a}_{i},\phantom{\rule{2em}{0ex}}\Pi \u2a66\Pi \prime \Pi \u2033\text{.}$$
Let $M$ be a Coxeter-matrix over $I,$ irreducible and of finite type. Let $\Pi \prime ,\Pi \u2033$ and $\Pi $ be the products of generators of ${G}_{M}^{+}$ defined by a decomposition of $I$ and let $h$ be the Coxeter number. Then: $$\begin{array}{cccc}\Delta & \u2a66& {\Pi}^{h/2}& \text{if}\hspace{0.17em}h\hspace{0.17em}\text{is even,}\\ \Delta & \u2a66& {\Pi}^{h-1/2}\Pi \prime \u2a66\Pi \u2033{\Pi}^{h-1/2}& \text{if}\hspace{0.17em}h\hspace{0.17em}\text{is odd,}\\ {\Delta}^{2}& \u2a66& {\Pi}^{h}& \text{always.}\end{array}$$
Proof. | |
According to Bourbaki, V §6 ex 2 (6) the corresponding equations for $\stackrel{\u203e}{\Delta},$ $\stackrel{\u203e}{\Pi},$ $\stackrel{\u203e}{\Pi}\prime ,$ $\stackrel{\u203e}{\Pi}\u2033$ hold. Since, in addition, the elements on the right hand sides of the equations have length $nh/2$ the statement follows from Proposition 5.7 (ii). $\square $ |
Remark. The Coxeter number $h$ is odd only for types ${A}_{2k}$ and ${I}_{2}(2q+1)\text{.}$ When $h$ is even, it is by no means necessary for $\Delta \u2a66{P}^{h/2}$ to hold where $P$ is a product of the generators in an arbitrary order. In any case, the following result show that this dependence on the order plays a role when $\Delta $ is not central in ${G}^{+},$ thus in the irreducible cases of types ${A}_{n}$ for $n\ge 2,$ ${D}_{2k+1},$ ${E}_{6}$ and ${I}_{2}(2q+1)\text{.}$ [See end of §7.]
Proposition. Let ${a}_{1},\cdots ,{a}_{n}$ be the generating letters for the Artin semigroup ${G}_{M}^{+}$ of finite type. Then:
(i) | For the product $P\u2a66{a}_{{i}_{1}}\cdots {a}_{{i}_{n}}$ of the generators in an arbitrary order ${\Delta}^{2}\u2a66{P}^{h}\text{.}$ |
(ii) | If $\Delta $ is central in ${G}_{M}^{+}$ then in fact for the product $P$ of the generators in an arbitrary order, $\Delta \u2a66{P}^{h/2}\text{.}$ |
(iii) | If $\Delta $ is not central and $h$ is even, there is an ordering of the generators such that, for the product of the generators in this order $\Delta \ne {P}^{h/2}\text{.}$ |
Proof. | |
By [Bou1968] V §6.1 Lemma 1, all products $P$ of generators in ${G}_{M}$ are conjugate to one another. Thus ${P}^{h}$ is conjugate to ${\Pi}^{h}$ and ${P}^{h/2}$ is conjugate to ${\Pi}^{h/2}$ if $h$ is even. If $\Delta $ is central and $h$ is even then $\Delta ={\Pi}^{h/2}$ is central and hence ${P}^{h/2}={\Pi}^{h/2}$ in ${G}_{M}$ and thus ${P}^{h/2}\u2a66{\Pi}^{h/2}\u2a66\Delta $ in ${G}_{M}^{+}\text{.}$ Likewise it follows immediately that ${P}^{h}\u2a66{\Pi}^{h}\u2a66{\Delta}^{2}$ since ${\Delta}^{2}\u2a66{\Pi}^{h}$ is always central. Hence (i) and (ii) are shown. [Note: we are using the fact that ${G}_{M}^{+}\u27f6{G}_{M}$ is injective here.] (iii) Suppose $\Delta $ is not central, i.e. $\sigma \ne \text{id}$ and let $h$ be even. If for all products $P\u2a66{a}_{{i}_{1}}\cdots {a}_{{i}_{n}}$ we were to have the equation ${P}^{h/2}\u2a66\Delta $ then this also would be true for the product ${a}_{{i}_{n}}P{a}_{{i}_{n}}^{-1}$ which arises from it by cyclic permutation of the factors. Now, in if $P$ were such a product with $\sigma \left({a}_{{i}_{n}}\right)\ne {a}_{{i}_{n}}$ then we would have ${a}_{{i}_{n}}{P}^{h/2}{a}_{{i}_{n}}^{-1}\u2a66\Delta $ and thus ${a}_{{i}_{n}}\Delta \u2a66\Delta {a}_{{i}_{n}}$ in contradiction to ${a}_{{i}_{n}}\Delta \u2a66\Delta \sigma \left({a}_{{i}_{n}}\right)\text{.}$ $\square $ |
This is a translation, with notes, of the paper, Artin-Gruppen und Coxeter-Gruppen, Inventiones math. 17, 245-271, (1972).
Translated by: C. Coleman, R. Corran, J. Crisp, D. Easdown, R. Howlett, D. Jackson and A. Ram at the University of Sydney, 1996.