Last update: 29 January 2014
Definition. Let be a Coxeter-Matrix over Let be a subset such that the of in possess a common multiple. Then the uniquely determined least common multiple of the letters of in is called the fundamental element for in
For the sake of simplicity we call the images of letters of in 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 is irreducible and if there exists a fundamental word then or generates the centre of The condition for the existence of is very strong: exists exactly when 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 be a finite set for which a fundamental element in exists. Then we have:
(i) If were not left-divisible by an with then by 3.2 it could be represented by a chain from to an and thus it would not be right-divisible by in contradiction to its definition. Hence is divisible by Analogously one shows that is divisible by and hence both these elements of are equivalent to one another.
(ii) The assertion (ii) follows trivially from (1) and 4.3.
5.2. Let be a certain Coxeter-matrix over and the corresponding Artin-semigroup. If is an arbitrary subset, then we denote by the subsemigroup of which is generated by the letters Of course, is canonically isomorphic to where is the Coxeter-matrix over obtained by restriction of
If a fundamental element in exists for then there is a uniquely determined involutionary automorphism of with the following properties:
|(i)||sends letters to letters, i.e. for all Hence is a permutation of with and|
is left-divisible by by the same argument as in the proof of 5.1. So by 2.3 there is a uniquely determined such that (ii) holds. From 2.3 it also follows immediately that is an automorphism of Since preserves lengths it takes letters to letters, and hence arises from a permutation of From it follows by application of rev that The right hand side is positive equivalent to and hence from 2.3, Thus is an involution and clearly is too. Finally, since for all it follows that Thus (i) is proved.
Remark. The converse of 5.2 also holds: let be irreducible, a permutation and a nontrivial element such that for all letters of Then there exists a fundamental element
It suffices to show that is a common multiple of the letters of At least, one letter from divides Hence let If is any letter of with so by the reduction lemma 2.1, we have that is divisible by and thus is a divisor of Hence is divisible by all the letters of since the Coxeter-graph of is assumed connected. By 4.1 the existence of then follows.
5.3. The first part of the following lemma follows immediately from 5.2 (ii).
Suppose there exists a fundamental element Then for all
|(i)||left-divides exactly when it right-divides|
|(ii)||If divides the product then each letter for either right-divides the factor or left-divides|
(ii) If neither right-divides nor left-divides then, by 3.2, one can represent by a chain with target and by a chain with source Thus one can represent by a chain [Ed: a word in the letters of ] which, by 3.1, is not divisible by its source, and hence neither by
5.4. The following lemma contains an important characterization of fundamental elements.
If a fundamental element exists for the following hold:
|(i)||is square free if and only if is a divisor of|
|(ii)||The least common multiple of square free elements of is square free.|
(i) From 3.5 it follows immediately by induction on the number of elements of that is square free; and consequently so are its divisors. The converse is shown by induction on the length of Let By the induction assumption there exists a with Since does not right-divide it left divides by 5.3 and hence is a divisor of
(ii) The assertion (ii) follows trivially from (i).
5.5. Let be a Coxeter-matrix over The Artin semigroups with fundamental element can be described by the type of embedding in the corresponding Artin group Instead of resp. we will write simply resp. when there is no risk of confusion.
For a Coxeter-matrix the following statements are equivalent:
|(i)||There is a fundamental element in|
|(ii)||Every finite subset of has a least common multiple.|
|(iii)||The canonical map is injective, and for each there exist with|
|(iv)||The canonical map is injective, and for each there exist with where the image of lies in the centre of|
We denote elements of and their images in by the same letters.
[Ed: In this proof is written for simplicity]
We will show first of all the equivalence of (i) and (ii), where clearly (ii) trivially implies (i). Let or according to whether or not. Then is, by 5.2, a central element in and for each letter there is by 5.1 a with Now, if is an arbitrary element of then Hence is divisible by each element of with 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 there exists a common multiple, and thus with From this and cancellativity, 2.3, it follows by a general theorem of Öre that embeds in a group. Thence follows the injectivity of and also that each element can be represented in the form or also with That can moreover be chosen to be central follows from the fact that — as shown above — to every with there exists with so that, therefore,
[Ed: As an alternative to applying Öre’s condition we provide the following proof of the injectivity of when there exists a fundamental element
By (5.2) it is clear that is a central element in Let be positive words such that in Then there is some sequence of words in the letters of and their inverses such that where at each step is obtained from either by a positive transformation (cf 1.3.) or (so-called trivial) insertion or deletion of a subword or for some letter Note that the number of inverse letters appearing in any word is bounded by
Let denote the central element of Then we may define positive words for such that as follows. Write for a positive word, a letter. Then so if we let denote the unique element of such that where is positive. Repeating this step for successive inverses in yields a positive word equal in to for some Put Essentially, is obtained from by replacing each occurrence of with the word and then attaching unused copies of to the front.
Now we check that
If differs from by a positive transformation, then is with some positive subword switched with a positive subword and so the same transformation applied to gives the word so they are positive equivalent.
If is obtained from by insertion of or Then or for positive words where By the centrality of and the fact that and are positive equivalent.
If is obtained by a trivial deletion, then the proof is identical as above, but with the roles of and reversed.
Hence we have a sequence of words such that each is positive equivalent to its predecessor, so that is positive equivalent to But and so by cancellativity, is positive equivalent to
So embeds in ]
Assuming (iv), (iii) follows trivially. And from (iii), (ii) follows easily. Since for there exist with and thus and consequently so by 4.1 and have a least common multiple. Thus 5.5 is proved.
5.6. Let be the set of square free elements of For the canonical map defined by composition of inclusion and the residue class map it follows immediately from Theorem 3 of Tits in [Tit1968] that
Let be a Coxeter-matrix. Then there exists a fundamental element in if and only if is finite.
By Tits, is finite exactly when is finite. By 5.4 and 3.5 this is the case if and only if exists. Since, if exists, by 5.4 consists of the divisors of And if does not exist, by 3.5 there exists a sequence of infinitely many distinct square free elements.
5.7. By the length of an element in a Coxeter group we mean the minimum of the lengths of all positive words which represent The image of a positive word or an element of in we denote by The theorem of Tits already cited immediately implies the following: The square free elements of are precisely those with
If an element is not square free, then it is represented by a word which contains a square. But this is clearly not a reduced word for the Coxeter element and so (the square cancels).
Conversely, suppose that represents a square free element of By definition of length there is a such that and Then by above is square free. But and hence by Tits theorem and
Let be finite. The following hold for the fundamental element of
|(i)||is the uniquely determined square free element of maximal length in|
|(ii)||There exists a uniquely determined element of maximal length in namely The fundamental element is represented by the positive words with and|
(i) By 5.4, the elements of are the divisors of A proper divisor of clearly has Thus is the unique square free element of maximal length.
(ii) By the theorem of Tits and (i) there is also in only one unique element of maximal length, namely A positive word with and is according to Tits square-free and it has maximal length, so by (i) it represents That only such positive words can represent is clear.
5.8. Let be a finite Coxeter group and for simplicity let the Coxeter system be irreducible.
[Note: Bourbaki defines a Coxeter system to be a group and a set of elements of order in such that the following holds: For in let be the order of Let be the set of pairs such that is finite. The generating set and relations for in form a presentation of the group
A Coxeter system is irreducible if the associated Coxeter graph is connected and non empty.
Note also that Bourbaki, Groupes et Algèbres de Lie, IV §1 ex 9 gives an example of a group and two subsets and of elements of order such that and are non-isomorphic Coxeter systems, one of which is irreducible, the other reducible. Hence the notion of irreducibility depends on not just on the underlying group Bourbaki says: when is a Coxeter system, and also says, by abuse of language, that 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 of maximal length in 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 is equal to the number of reflections of and thus where is the Coxeter number and the rank, i.e. the cardinality of the generating system Explicit representations of by suitable words are also known and from this we now obtain quite simple corresponding expressions for
Let be an irreducible Coxeter system of finite type over A pair of subsets of is a decomposition of if is the disjoint union of and and for all and all Obviously there are exactly two decompositions of which are mapped into each other by interchanging and
By Bourbaki, V §4 number 8 corollary to proposition 8, or from the classification of finite Coxeter groups we know that if is irreducible and finite then its graph is a tree. So the statement about decompositions boils down to the following statement about trees: if is a tree with a finite set of vertices then there exists a unique partition (up to interchange of and of into two sets such that no two elements of and no two elements of are joined by an edge.
We prove this by induction on the number of vertices of For a graph on one vertex it is clear that the only suitable partitions are and Now let be an arbitrary tree with a finite set of vertices and let be a terminal vertex. Then applying the assumption to the subgraph of whose vertices are those vertices of we see that there exists a unique partition (up to interchange of of such that no two elements of and no two elements of are joined by an edge of Now, by definition of a tree, is joined to exactly one vertex of Without loss of generality let Then it is easy to see that is a partition of satisfying the above conditions and that it is unique up to interchanging and
Definition. Let be a Coxeter matrix over and a decomposition of The following products of generators in are associated to the decomposition:
Let be a Coxeter-matrix over irreducible and of finite type. Let and be the products of generators of defined by a decomposition of and let be the Coxeter number. Then:
According to Bourbaki, V §6 ex 2 (6) the corresponding equations for hold. Since, in addition, the elements on the right hand sides of the equations have length the statement follows from Proposition 5.7 (ii).
Remark. The Coxeter number is odd only for types and When is even, it is by no means necessary for to hold where 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 is not central in thus in the irreducible cases of types for and [See end of §7.]
Proposition. Let be the generating letters for the Artin semigroup of finite type. Then:
|(i)||For the product of the generators in an arbitrary order|
|(ii)||If is central in then in fact for the product of the generators in an arbitrary order,|
|(iii)||If is not central and is even, there is an ordering of the generators such that, for the product of the generators in this order|
By [Bou1968] V §6.1 Lemma 1, all products of generators in are conjugate to one another. Thus is conjugate to and is conjugate to if is even. If is central and is even then is central and hence in and thus in Likewise it follows immediately that since is always central. Hence (i) and (ii) are shown. [Note: we are using the fact that is injective here.]
(iii) Suppose is not central, i.e. and let be even. If for all products we were to have the equation then this also would be true for the product which arises from it by cyclic permutation of the factors. Now, in if were such a product with then we would have and thus in contradiction to
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.