Artin groups and Coxeter groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 29 January 2014

The Fundamental Element

Definition. Let M be a Coxeter-Matrix over I. Let JI be a subset such that the letters1 of J in GM+ possess a common multiple. Then the uniquely determined least common multiple of the letters of J in GM+ is called the fundamental element ΔJ for J in GM+.

1For 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 GM is irreducible and if there exists a fundamental word ΔI, then ΔI or ΔI2 generates the centre of GM. The condition for the existence of ΔI is very strong: ΔI exists exactly when GM 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 JI be a finite set J={j1,,jk}, for which a fundamental element ΔJ in GM+ exists. Then we have:

(i) ΔJ [aj1,,ajk]l [aj1,,ajk]r.
(ii) revΔJΔJ.


(i) If [aj1,,ajk]r were not left-divisible by an aj with jJ then by 3.2 it could be represented by a chain from aj to an aj, jJ and thus it would not be right-divisible by aj in contradiction to its definition. Hence [aj1,,ajk]r is divisible by [aj1,,ajk]l. Analogously one shows that [aj1,,ajk]l is divisible by [aj1,,ajk]r and hence both these elements of G+ are equivalent to one another.

(ii) The assertion (ii) follows trivially from (1) and 4.3.

5.2. Let M be a certain Coxeter-matrix over I, and G+ the corresponding Artin-semigroup. If JI is an arbitrary subset, then we denote by GJ+ the subsemigroup of G+ which is generated by the letters aj, jJ. Of course, GJ+ is canonically isomorphic to GM+, where MJ is the Coxeter-matrix over J obtained by restriction of M.

If a fundamental element ΔJ in G+ exists for JI, then there is a uniquely determined involutionary automorphism σJ of GJ+ with the following properties:

(i) σJ sends letters to letters, i.e. σJ(aj)=aσ(j) for all jJ. Hence σ is a permutation of J with σ2=id and mσ(i)σ(j)=mij.
(ii) For all WGJ+, WΔJΔJ σJ(W).


WΔJ is left-divisible by ΔJ by the same argument as in the proof of 5.1. So by 2.3 there is a uniquely determined σJ(W) such that (ii) holds. From 2.3 it also follows immediately that σJ is an automorphism of GJ+. Since σJ preserves lengths it takes letters to letters, and hence arises from a permutation σ of J. From σ(a)ΔJΔJσ2(a) it follows by application of rev that σ2(a)ΔJΔJσ(a). The right hand side is positive equivalent to aΔJ and hence from 2.3, σ2(a)a. Thus σ is an involution and clearly σJ is too. Finally, since for all i, j, σ(ai)σ(aj) mij σ(aj)σ(aj) mij it follows that mij=mσ(i)σ(j). Thus (i) is proved.

Remark. The converse of 5.2 also holds: let GJ+ be irreducible, σ:JJ a permutation and ΔGJ+ a nontrivial element such that aΔΔσ(a) for all letters a of J. Then there exists a fundamental element ΔJ.


It suffices to show that Δ is a common multiple of the letters of J. At least, one letter a from J divides Δ. Hence let ΔaΔ. If b is any letter of J with mab>2, baΔΔσ(b)σ(a)aΔσ(b)σ(a), so by the reduction lemma 2.1, we have that aΔ is divisible by abmab-1 and thus b is a divisor of Δ. Hence Δ is divisible by all the letters of J since the Coxeter-graph of MJ is assumed connected. By 4.1 the existence of ΔJ then follows.

5.3. The first part of the following lemma follows immediately from 5.2 (ii).

Suppose there exists a fundamental element ΔJ. Then for all U,V,WGJ+:

(i) ΔJ left-divides U exactly when it right-divides U.
(ii) If ΔJ divides the product VW, then each letter aj, for jJ, either right-divides the factor V or left-divides W.


(ii) If aj neither right-divides V nor left-divides W then, by 3.2, one can represent V by a chain with target aj and W by a chain with source aj. 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 ΔJ.

5.4. The following lemma contains an important characterization of fundamental elements.

If a fundamental element ΔJ exists for JI, the following hold:

(i) UGJ+ is square free if and only if U is a divisor of ΔJ.
(ii) The least common multiple of square free elements of GJ+ is square free.


(i) From 3.5 it follows immediately by induction on the number of elements of J that ΔJ is square free; and consequently so are its divisors. The converse is shown by induction on the length of U. Let UVa. By the induction assumption there exists a W with ΔJVW. Since a does not right-divide V, it left divides W by 5.3 and hence U is a divisor of ΔJ.

(ii) The assertion (ii) follows trivially from (i).

5.5. Let M be a Coxeter-matrix over I. The Artin semigroups GM+ with fundamental element ΔI can be described by the type of embedding in the corresponding Artin group GM. Instead of ΔI, resp. σI, we will write simply Δ, resp. σ, when there is no risk of confusion.

For a Coxeter-matrix M the following statements are equivalent:

(i) There is a fundamental element Δ in GM+.
(ii) Every finite subset of GM+ has a least common multiple.
(iii) The canonical map GM+GM is injective, and for each AGM there exist B,CGM+ with A=BC-1.2
(iv) The canonical map GM+GM is injective, and for each AGM there exist B,CGM+ with A=BC-1, where the image of C lies in the centre of GM.

2We denote elements of GM+ and their images in GM by the same letters.


[Ed: In this proof GM+ is written G+ for simplicity]

We will show first of all the equivalence of (i) and (ii), where clearly (ii) trivially implies (i). Let ΛΔ or ΛΔ2 according to whether σ=1 or not. Then Λ is, by 5.2, a central element in G+ and for each letter ai, iI, there is by 5.1 a Λi with Λ=aiΛi. Now, if Aai1aim is an arbitrary element of G+ then Λmaim Λimai1 Λi1A Λim Λi1. Hence Λm is divisible by each element A of G+ with L(A)m. 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,CG+ there exists a common multiple, and thus B,C,G+ with BCCB. 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+G and also that each element AG can be represented in the form A=C-1B or also BC-1 with B,B,C,CG+. That C can moreover be chosen to be central follows from the fact that — as shown above — to every C with L(C)m there exists DG+ with ΛmCD so that, therefore, C-1=Λ-mD=DΛ-m.

[Ed: As an alternative to applying Öre’s condition we provide the following proof of the injectivity of G+G when there exists a fundamental element Δ.

By (5.2) it is clear that Δ2 is a central element in G+. Let W,W be positive words such that W=W in G. Then there is some sequence W1,W2,,Wk of words in the letters of I and their inverses such that WW1=W2==WkW where at each step Wi+1 is obtained from Wi either by a positive transformation (cf 1.3.) or (so-called trivial) insertion or deletion of a subword aa-1 or a-1a for some letter a. Note that the number of inverse letters appearing in any word is bounded by k.

Let C denote the central element Δ2 of G+. Then we may define positive words Vi for i=1,,k such that Vi=CkWi as follows. Write WiUa-1U for U a positive word, a a letter. Then CUUC, so if we let Ca denote the unique element of G+ such that CaaC, CWi= CUa-1U= UCa-1U= UCa-1U= UCaU where UCa is positive. Repeating this step for successive inverses in U yields a positive word Vi equal in G to CrWi for some rk. Put ViCk-rVi. Essentially, Vi is obtained from Wi by replacing each occurrence of a-1 with the word Ca, and then attaching unused copies of C to the front.

Now we check that ViVi+1.

If Wi+1 differs from Wi by a positive transformation, then Wi+1 is Wi with some positive subword U switched with a positive subword U, and so the same transformation applied to Vi gives the word Vi+1, so they are positive equivalent.

If Wi+1 is obtained from Wi by insertion of aa-1 or a-1a Then Vi+1CrUCaaV or CrUaCaV for positive words U,V, where ViCr+1UV. By the centrality of C and the fact that CaCaCaa, Vi and Vi+1 are positive equivalent.

If Wi+1 is obtained by a trivial deletion, then the proof is identical as above, but with the roles of Wi+1 and Wi reversed.

Hence we have a sequence of words V1,V2,,Vk such that each is positive equivalent to its predecessor, so that V1 is positive equivalent to Vk. But V1CkW and VkCkW so by cancellativity, W is positive equivalent to W.

So G+ embeds in G. ]

Assuming (iv), (iii) follows trivially. And from (iii), (ii) follows easily. Since for B,CG+ there exist B,CG+ with C-1B=BC-1, and thus BC=CB and consequently BCCB so by 4.1 B and C have a least common multiple. Thus 5.5 is proved.

5.6. Let QFGM+ be the set of square free elements of GM+. For the canonical map QFGM+GM defined by composition of inclusion and the residue class map it follows immediately from Theorem 3 of Tits in [Tit1968] that QFGM+GM is bijective.

Let M be a Coxeter-matrix. Then there exists a fundamental element Δ in GM+ if and only if GM is finite.


By Tits, GM is finite exactly when QFGM+ is finite. By 5.4 and 3.5 this is the case if and only if Δ exists. Since, if Δ exists, by 5.4 QFGM+ consists of the divisors of Δ. And if Δ does not exist, by 3.5 there exists a sequence of infinitely many distinct square free elements.

5.7. By the length l(w) of an element w in a Coxeter group GM we mean the minimum of the lengths L(W) of all positive words W which represent w. The image of a positive word W or an element W of GM+ in GM we denote by W. The theorem of Tits already cited immediately implies the following: The square free elements of GM+ are precisely those W with L(W)=l(W)

[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 W, and so l(W)L(W)-2 (the square cancels).

Conversely, suppose that W represents a square free element of GM+. By definition of length there is a VGM+ such that V=W and L(W)=l(V)=l(W). Then by above V is square free. But V=W and hence by Tits theorem VW and L(W)=L(V)=l(W).


Let GM be finite. The following hold for the fundamental element Δ of GM+:

(i) Δ is the uniquely determined square free element of maximal length in GM+.
(ii) There exists a uniquely determined element of maximal length in GM, namely Δ. The fundamental element Δ is represented by the positive words W with W=Δ and L(W)=l(Δ).


(i) By 5.4, the elements of QFGM+ are the divisors of Δ. A proper divisor W of Δ clearly has L(W)<L(Δ). Thus Δ is the unique square free element of maximal length.

(ii) By the theorem of Tits and (i) there is also in GM only one unique element of maximal length, namely Δ. A positive word with W=Δ and L(W)=l(Δ) is according to Tits square-free and it has maximal length, so by (i) it represents Δ. That only such positive words can represent Δ is clear.

5.8. Let GM be a finite Coxeter group and for simplicity let the Coxeter system (GM,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 in S, let m(s,s) be the order of ss. Let I be the set of pairs (s,s) such that m(s,s) is finite. The generating set S and relations (ss)m(s,s)=1 for (s,s) in I form a presentation of the group W.

A Coxeter system (W,S) is irreducible if the associated Coxeter graph Γ 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 of elements of order 2 such that (W,S) and (W,S) 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. 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 Δ of maximal length in 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(Δ) is equal to the number of reflections of G and thus L(Δ)=nh2 where h is the Coxeter number and n the rank, i.e. the cardinality of the generating system I. Explicit representations of Δ by suitable words are also known and from this we now obtain quite simple corresponding expressions for Δ.

Let M be an irreducible Coxeter system of finite type over I. A pair (I,I) of subsets of I is a decomposition of I if I is the disjoint union of I and I and mij2 for all i,jI and all i,jI. Obviously there are exactly two decompositions of I which are mapped into each other by interchanging I and I.

[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 Γ is a tree with a finite set S of vertices then there exists a unique partition (S,S) (up to interchange of S and S), of S into two sets such that no two elements of S and no two elements of S are joined by an edge.

We prove this by induction on the number of vertices of Γ. For a graph on one vertex a it is clear that the only suitable partitions are ({a},ϕ) and (ϕ,{a}). Now let Γ 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 Γ whose vertices are those vertices na of Γ we see that there exists a unique partition (S1,Sl) (up to interchange of S1,S1) of S\{a} such that no two elements of S1 and no two elements of S1 are joined by an edge of Γ. Now, by definition of a tree, a is joined to exactly one vertex b of Γ. Without loss of generality let bS1. Then it is easy to see that (S1,S1{a}) is a partition of S satisfying the above conditions and that it is unique up to interchanging S1 and S1{a}.


Definition. Let M be a Coxeter matrix over I and (I,I) a decomposition of I. The following products of generators in GM+ are associated to the decomposition: Π iIai, Π iIai, ΠΠ Π.

Let M be a Coxeter-matrix over I, irreducible and of finite type. Let Π,Π and Π be the products of generators of GM+ defined by a decomposition of I and let h be the Coxeter number. Then: Δ Πh/2 ifhis even, Δ Πh-1/2Π Π Πh-1/2 ifhis odd, Δ2 Πh always.


According to Bourbaki, V §6 ex 2 (6) the corresponding equations for Δ, Π, Π, Π 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).

Remark. The Coxeter number h is odd only for types A2k and I2(2q+1). When h is even, it is by no means necessary for ΔPh/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 Δ is not central in G+, thus in the irreducible cases of types An for n2, D2k+1, E6 and I2(2q+1). [See end of §7.]

Proposition. Let a1,,an be the generating letters for the Artin semigroup GM+ of finite type. Then:

(i) For the product Pai1ain of the generators in an arbitrary order Δ2Ph.
(ii) If Δ is central in GM+ then in fact for the product P of the generators in an arbitrary order, ΔPh/2.
(iii) If Δ 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 ΔPh/2.


By [Bou1968] V §6.1 Lemma 1, all products P of generators in GM are conjugate to one another. Thus Ph is conjugate to Πh and Ph/2 is conjugate to Πh/2 if h is even. If Δ is central and h is even then Δ=Πh/2 is central and hence Ph/2=Πh/2 in GM and thus Ph/2Πh/2Δ in GM+. Likewise it follows immediately that PhΠhΔ2 since Δ2Πh is always central. Hence (i) and (ii) are shown. [Note: we are using the fact that GM+GM is injective here.]

(iii) Suppose Δ is not central, i.e. σid and let h be even. If for all products Pai1ain we were to have the equation Ph/2Δ then this also would be true for the product ainPain-1 which arises from it by cyclic permutation of the factors. Now, in if P were such a product with σ(ain)ain then we would have ainPh/2ain-1Δ and thus ainΔΔain in contradiction to ainΔΔσ(ain).

Notes and references

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.

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