Growth functions for Artin monoids
Kyoji Saito ∗Abstract
In [S1], we showed that the growth function PM (t) for an Artin monoid offinite type M is a rational function of the form 1/(1−t)NM (t), and formulated
three conjectures on the zeros of the denominator polynomial NM (t) of a finite
type.1 In the present note, we observe that the same formula holds for arbitraly
Artin monoids, and formulate slightly modified conjectures for the denominator
polynomials of the growth function of affine type. The new conjectures areverified for affine types Ã2 ,· · ·, Ã8 , C̃2 ,· · ·, C̃8 , D̃4 , Ẽ7 , Ẽ8 , F̃4 , G̃2 among others.
In Appendix, we formally define the denominator polynomials for elliptic rootsystems ([S2]). Then, the new conjectures are verified also for elliptic types
(1,1)
(1,1)
(1,1)
(1,1)
(1,1)
(1,1)
(1,1)
A2
,· · ·, A7
, D4
, E6
, E7
, E8
and G2.

1

Growth function for an Artin monoid

We recall the definition (1.3) of a growth function for an Artin monoid, and introduce
the denominator polynomial (1.6) in the following Lemma-Definition 1 (see 2 ).
Let M = (mij )i,j∈I be a Coxeter matrix ([B]). The Artin monoid G+
M ([B-S,§1.2]) associated with M (or, of type M ) is a monoid generated by the letters
ai , i ∈ I which are subordinate to the relation generated by(1.1)

ai aj ai · · · = aj ai aj · · ·i, j ∈ I,

where both hand sides of (1.1) are words of alternating sequences of letters ai
and aj of the same length mij = mji with the initials ai and aj , respectively.
More precisely, G+
M is the quotient of the free monoid generated by the lettersai (i ∈ I) by the equivalence relation: two words U and V in the letters areequivalent, if there exists a sequence U0 := U, U1 , · · · , Um := V such that theword Uk (k = 1, · · · , m) is obtained by replacing a phrase in Uk−1 of the formon LHS of (1.1) by RHS of (1.1) for some i, j ∈ I. We write by U =V if U and•
V are equivalent. The equivalence class (i.e. an element of G+M ) of a word W isdenoted by the same notation W . By the definition, equivalent words have thesame length. Hence, we define the degree homomorphism:(1.2)

deg : G+M −→ Z≥0by assigning to each equivalence class of words the length of the words.
∗ Institute for the Physics and Mathematics of the Universe, the University of Tokyo
1 Here M is a Coxeter matrix [B]. We shall refer to M as the type of the Coxeter group Ḡ ,M
Artin group GM , Artin monoid G+
M , growth function PM (t), etc. associated with M . In thepresent note, we shall say a Coxeter matrix M is of finite (resp. affine) type if the associated
Coxeter group ([B, Ch.IV §1]) is finite (resp. affine), or equivalently, if the associated bilinear
form BM [B] is positive definite (resp. semi-positive with rank 1 kernel).
2 The definition is a slight generalization of that for finite type in [S1, (2.2),(2.3)]. A similar
generalization is made by Albenque and Nadeau ([A-N, (1.2)]) for cancellative monoids suchthat if a subset of atomic generators has a common multiple then it admits a least common
multiple. This class covers the Artin monoids, which we discuss in the present paper.

1The growth function PG+ ,I (t) for the Artin monoid G+
M is defined by
M

(1.3)

PG+ ,I (t) :=M

Xn#{W ∈ G+M | deg(W ) ≤ n} t .

n∈Z≥0The spherical growth function of the monoid G+
M of type M is defined byX(1.4)
ṖG+ ,I (t) :=#(deg−1 (n)) tn ,
M

n∈Z≥0

so that one has the obvious relation: PG+ ,I (t) = ṖG+ ,I (t)/(1 − t).
M

MLemma-Definition 1. Let G+
M be the Artin monoid of any type M . Then thespherical growth function of the monoid is given by the Taylor expansion of therational function of the form
(1.5)

ṖG+ ,I (t) =M

1.
NM (t)

Here, NM (t) is called the denominator polynomial and is given byX(1.6)
NM (t) :=(−1)#(J ) tdeg(∆J ) ,J ⊂I

where the summation index J runs over subsets of I such that the restricted
Coxeter matrix M |J is of finite type,3 and ∆J is the fundamental element inG+
M associated with J ( [B-S, §5 Definition]. See also Lemma-Definition 2 andRemark 1.2 of the present note).
Proof. The proof is achieved by a recursion formula (1.12) on the coefficientsof the growth function. For the proof of the formula, we use the method used
to solve the word problem for the Artin monoid ([B-S, §6.1]), which we recall
below. We first recall the fact that an Artin monoid satisfies the cancellation
condition in the following sense ([B-S, Prop.2.3]).
AY B. Then X =Y.
Lemma 1.1. Let A, B, X, Y ∈ G+
M . If AXB =••A word U is said to be divisible (from the left) by a word V , and denotedby V |U , if there exists a word W such that U =V W . Since V =V ′, U =U ′ and•••
V |U implies V ′ |U ′ , we use the notation “|” of divisibility also between elementsof the monoid G+
M . We have the following basic concepts ([B-S, §5 Definitionand §6.1]).
Lemma-Definition 2. Let M be a Coxeter matrix of any type, and let J ⊂ I
be a subset of I such that M |J is of finite type (which may not necessarily beindecomposable). Then, there exists a unique element ∆J ∈ G+
M , called thefundamental element, such that i) ai |∆J for all i ∈ J, and ii) if W ∈ G+
M and
ai |W for all i ∈ J, then ∆J |W .
3. To an element W ∈ G+
M , we associate the subset of I:(1.7)

I(W ) := {i ∈ I | ai |W }.

The restricted Coxeter matrix M |I(W ) is of finite type for any W ∈ G+
M.
Proof. 2. This follows from the fact that the existence of ∆J is achieved under aweaker assumption than MJ is of finite type, rather that there exists a common
multiple of aj for j ∈ J in G+
2
M (see [B-S, Prop. (4.1)].
By the definition (1.7), one has ∆I(W ) |W and if ∆J |W then J ⊂ I(W ).
3 For a Coxeter matrix M = (m )ij i,j∈I and a subset J of I, we define the restricted Coxeter
matrix by M |J := (mij )i,j∈J , which, obviously, is again a Coxeter matrix.

2

We return to the proof of Theorem.
For n ∈ Z≥0 and for any subset J ⊂ I, put:= {W ∈ G+
M | deg(W ) = n}

G+
n

(1.8)

+
G+
n,J := {W ∈ Gn | I(W ) = J}.
+
We note that Gn,J = ∅ if M |J is not of finite type. By the definition, we have

(1.9)

the disjoint decomposition:(1.10)+
G+n = ∐J ⊂I Gn,J ,where J runs over all subsets of I. Note that G+= ∅ if n > 0 but G+= {∅} 6= ∅.
n,∅0,∅
For any subset J of I, the union ∐J ⊂K⊂I G+
n,K , where the index K runs overall subsets of I containing J, is equal to the subset of G+n consisting of elements
divisible by aj for j ∈ J. That is, one has∆ J · G+
if M |J is of finite type,n−deg(∆J )∐J ⊂K⊂I G+
n,K =∅if M |J is not of finite type.
Thus, if M |J is of finite type, due to the cancellation condition Lemma 1.1, themultiplication map of ∆J is injective and we obtain a bijection: G+≃
n−deg(∆ )∐J ⊂K⊂I G+
n,K . This implies a numerical relation:
P
+
#(G+
) =
(1.11)
J ⊂K⊂I #(Gn,K ).
n−deg(∆ )JJ

If M |J is not of finite type, still the formula (1.11) holds formally, by putting+deg(∆J ) := ∞ and G+−∞ := ∅, i.e. #(Gn−deg(∆J ) ) := 0. Then, for n > 0, using(1.11), we get the recursion relation:X(1.12)(−1)#(J ) #(G+) = 0,
n−deg(∆ )J

J ⊂I

where the index J may run either over all subsets of I, or, over only subset J
such that the restricted Coxeter matrix M |J is of finite type. Together with#(G+0 ) = 1 for n = 0, this is equivalent to the formula:(1.13)ṖG+ ,I (t)NM (t) = 1.
M

This completes the proof of Lemma-Definition 1.
Remark 1.2. We have the equality ([B-S, §5.7]):deg(∆J ) = #{reflections in GM |J } = the length of the longest element of GM |J .
By the definition (1.6) of the denominator
polynomial, one has
P#J
NM (1) =J ⊂I,M |J is (−1)
of finite typeThis, in particular, impliesi) NM (t) has the factor 1 − t if M contains a component of finite type, andii) NM (1) = (−1)l if M is of indecomposable affine type of rank l (i.e. M isindecomposable and affine such that #(I) = l + 1 ).4
We refer to [S1] and http://www.kurims.kyoto-u.ac.jp/∼saito/FFST/ for examples of finite type (the author express his gratitude to S. Tsuchioka for making
this page). Here, we give a few examples of affine type.
Example. There are three types of indecomposable affine Coxeter matrices ofrank 2. In the following, for each type, we associate the Coxeter diagram ΓM
and the denominator polynomial NM (t).
◦1. Ã2

ΓÃ = / \NÃ (t) = 1 − 3t + 3t3

2. C̃2

ΓC̃ = ◦ 4 ◦ 4 ◦

NC̃ (t) = 1 − 3t + t2 + 2t4

3. G̃2

ΓG̃ = ◦

2

◦

◦

2

2

◦ 6 ◦

2

2

NG̃ (t) = 1 − 3t + t2 + t3 + t6
2

4 The discrepancy between the rank l and the number #(I) = l + 1 for a Coxeter matrix

M of indecomposable affine type comes from the fact that the associated affine Coxeter group
acts on a semi-positive R-vector space of signature (l, 1, 0).

3

2 A bound on the zeros of the denominator polynomial NM (t) of affine typeThe following lemma gives a numerical bound on the zeros of the denominator
polynomials for indecomposable affine type.
Lemma 2.1. Let M be a Coxeter matrix of indecomposable affine type of rankl. Then, all the roots of NM (t) = 0 are contained in the open disc of radius r
centered at the origin, where r is give by“ 2l+1 − s − 1 ”1/(deg(∆M |
)−d)
I\{v}
,
(2.1)
r :=
s
and deg(∆M |I\{v} ), d, s are invariants of M explained in the proof.
Proof. In the affine Coxeter graph ΓM (whose vertex set is identified with I,
and hence #(ΓM ) = #(I) = l + 1), there is a vertex v, called special [B, p.87]
such that ΓM \{v} is the Coxeter graph of the finite Coxeter group isomorphic
to the radical quotient of the affine Coxeter group GM . Let s be the number of
special vertices in ΓM . For types Ãl , B̃l , C̃l , D̃l , Ẽ6 , Ẽ7 , Ẽ8 , F̃4 , G̃2 , the number
s is given by l+1, 2, 2, 4, 3, 2, 1, 1, 1, respectively.
Noting the fact that the type of ΓM \{v} (and, hence, deg(∆M |I\{v} )) does
not depend on the choice of a special vertex v, we see that the monomial N (t) :=deg(∆M |)I\{v}(−1)l s · t(v a special vertex) is the leading term of NM (t), one
has |NM (t) − N (t)| ≤ (2l+1 − s − 1)|t|d for t ∈ C with |t| > 1 (strict inequality
holds except for the type Ã1 ), where we put

d := max{deg(∆J ) | J ⊂ I such that I \J is not a single special vertex}.
l+1d−deg(∆M |)−s−1
I\{v} . If r ∈ R|t|>1 satisfiessd−deg(∆)
M |I\{v}2−s−1
an inequalityr≤ 1, then, due to Rouche’s theorem,sthe number of zeros of NM (t) = 0 in the disc of radius r is equal to that ofHence |NM (t)−N (t)|/|N (t)| ≤ 2
l+1

N (t) = 0, which has zeros only at 0 of multiplicity deg(N (t)) = deg(NW (t)).
That is, all roots of NM (t) = 0 are in the disc {|t| < r} for r given in (2.1).

3 Conjectures on the zeros of the denominator polynomial NM (t) of affine type
Motivated by a study of the author on certain limit functions associated withfinitely generated monoids or groups (see §11 and 12 of [S] in the reference [S1] of
the present note), some discussions and examples at the end of §1 lead us to thefollowing three conjectures on the distribution of the zeros of the denominator
polynomial NM (t) of indecomposable finite or affine type.5
Conjecture 1. i) The polynomial ÑM (t) := NM (t)/(1 − t) is irreducible over Z
for any indecomposable finite type M . ii) The polynomial NM (t) is irreducible
over Z for any indecomposable affine type M .
Conjecture 2. There are l mutually distinct real roots of NM (t) = 0 on theinterval (0, 1] where l is the rank of positive definite part of the root latticeassociated with M .
Conjecture 3. Let rM be the smallest of the roots on the interval (0, 1]. Then,the absolute values of the other roots of NM (t) = 0 are strictly larger than rW .
5 As we shall observe in Appendix, these conjectures are (formally) valid also for elliptic
root systems [S2]. After a suitable modification, the conjectures seems to be valid also for some
Artin monoids of hyperbolic type (see Remark 3.1 and 3.4). It is interesting to clarify the range
where the conjecture works, and to develop a unified understanding of them (hopefully, inconnection with the original motivation to study the limit functions associated with monoids).

4

Conjectures on the denominator polynomials of finite type were alreadystated in [S1] and verified by computer calculations for the types Al , Bl , Cl , Dl(l ≤ 30), E6 , E7 , E8 , F4 , G2 , H3 , H4 and I2 (p) (p ∈ Z≥3 ) by M. Fuchiwaki,
S. Tsuchioka and others (see http://www.kurims.kyoto-u.ac.jp/˜ saito/FFST/).
Some theoretical approach on the conjectures is in progress by S. Yasuda.
Conjectures on the denominator polynomial of affine type are positively confirmed directly for the three types Ã2 , C̃2 and G̃2 of rank 2 from the explicit
expressions in §2 Example. Further cases, include Ã3 ,· · ·, Ã8 , C̃3 ,· · ·, C̃8 , D̃4 , Ẽ7 ,Ẽ8 and F̃4 , are calculated affirmatively by S. Tsuchioka by use of computer.
Remark 3.1. As we observed, a denominator polynomial NX (t) of finite typehas zeros of order 1 at t = 1, and that of affine type does not vanish there. In
Appendix, we observe that a denominator polynomial of elliptic type does notvanish there either. It is an immediate calculation to show that a denominator
polynomial of hyperbolic type (2,3,7), (2,4,5), (3,3,4), (2,3,8), (3,3,5), (2,5,5),(2,3,9), (2,4,7), (2,5,6), (3,4,5) or (4,4,4) has zeros at t = 1. It is interesting tofind a formula of the order d of zeros at t = 1 in general and to ask more precise
question than Conjecture 1: whether or when is NX (t)/(1 − t)d irreducible?
Remark 3.2. The definition of rank in Conjecture 2 is restrictive in the sensethat some Coxeter systems (which are not cristalographic such as type H3 , H4
and I2 (p)) do not admit a root system. Therefore, we should read here theword “the rank of positive part of the root lattice” loosely in the sense “therank of the positive part of the real quadratic form BM ([B])” associated with
M . On the other side, in the following Appedix, we study formally denominator
polynomials associated with elliptic root systems, which are not associated with
Coxeter systems. However their root systems and root lattices make sense of thedefinition of the rank in Cojecture 2 (cf. footnote 5).
Remark 3.3. In Conjecture 3, the fact that rM is less than or equal the absolutevalues of any other roots of NW (t) = 0 is trivially true, since rM is equal to
the radius of convergence of the series PM (t) of non-negative real coefficients.
Therefore, the true question here is that there are no other roots of NW (t) = 0
whose absolute value is equal to rW . This question is motivated from a study of
the author on certain limit functions associated with the monoid G+
M (see [S1, §5]).

Appendix.
Pursuing formal analogy (i.e. without an explicit relation with the growth
functions6 ), let us introduce the denominator polynomial NX (t) of elliptic type:
let (R, G) be an irreducible marked elliptic root system of type X and let ΓX :=Γ(R, G) be the associated Dynkin diagram [S2, I,§8]. Then, similar to (1.6), we
define the elliptic denominator polynomial of type X by
X(−1)#(J ) tdeg(∆J ) ,NX (t) :=J ⊂ΓX

where the summation index J runs over all subdiagrams of ΓX which is of finitetype.
For these denominator polynomials of elliptic type, we ask again whether
Conjectures 1.ii) (replacing indecomposable affine type by irreducible marked
elliptic type), 2. and 3. hold or not. Then, the conjectures are affirmatively
(1,1)
(1,1)
(1,1)
(1,1)
(1,1)
(1,1)
(1,1)
verified for the types A2,· · ·, A7
, D4
, E6
, E7
, E8
and G2using computer by S. Tsuchioka.
Remark 3.4. The above observation suggests that it seems of interest to defineformally the denominator polynomial associated with the Gabrielov diagram ofthe basis of the lattice of vanishing cycles associated with 14 regular systems ofweights with ε = −1 and to compare them with the denominator polynomials of
the Artin monoids associated with the Picard lattice of the K3 surfaces obtainedby Pinkham compatification of the Milnor fiber (see [S3, §13 and §18]).
6 Associated with elliptic root systems, there are concepts of elliptic Weyl groups, elliptic
Lie groups, elliptic Hecke algebra,. . . etc. However, at present, there is no clear descriptionof the growth function associated with the elliptic Artin monoid (since they are no longer
Coxeter groups or Kac-Moody groups,. . . etc), which is a subject to be studied yet.

5

Example (S. Tsuchioka). We illustrate the zero loci of the denominator polyno(1,1)
mials of finite type E8 , affine type Ẽ8 and elliptic type E8. In the following
figures, zero-loci are indicated by crosses “+”.
Type E8NE8 (t) = 1−8t+21t2 −14t3 −21t4 +28t5 −7t6 +12t7 −8t8 −10t9 +10t1 0−12t1 1+
7t1 2+2t13 −t14−3t15+2t16 −2t20+6t21 −t22−t23−t28+t30+t36−t37 −t42−t63+t120 .
1.5
cos(t), sin(t)
"E_8"

1

0.5

0

-0.5

-1

-1.5
-1.5

-1

-0.5

0

0.5

1

1.5

Type Ẽ8NẼ (t) = 1−9t+28t2 −28t3 −22t4 +54t5 −20t6 +10t7 −17t8 −13t9 +21t10 −8
11
23t +19t12 +7t13 −5t14 −3t15 +4t16 −3t17 −3t18 +t19 −t20 +9t21 −4t22 −3t23 +26
28
t −3t +t29 +t30 −t31 +2t36 −2t37 +t39 −t42 +t56 −t63 +t64 +t120 .
1.5
cos(t), sin(t)"E^(1)_8"

1

0.5

0

-0.5

-1

-1.5
-1.5

-1

-0.5

0

6

0.5

1

1.5

(1,1)

Type E8N

2 −32t3 −35t4 +73t5 −23t6 +21t7 −30t8 −28t9 +36t10 −(1,1) (t) = 1−10t+33t

E8

38t11 +34t12 +12t13 −8t14 −5t15 +5t16 −4t17 −5t18 +t19 −2t20 +18t21 −8t22 −6t23 +2t26 −6t28 +2t29 +2t30 −2t31 +4t36 −4t37 +2t39 −2t42 +2t56 −2t63 +2t64 +2t120 .
1.5
cos(t), sin(t)"E^(1,1)_8"

1

0.5

0

-0.5

-1

-1.5
-1.5

-1

-0.5

0

0.5

1

1.5

Acknowledgement. The author expresses his gratitude to Shunsuke
Tsuchioka for help calculating examples as well as drawing figures bycomputer, and also for informing him of [A-N]. He is also grateful to
Mikael Pichot for his interest and discussions on the subjects, and to Ken
Shackleton for carefully reading an earlier version of this note.

References
[A-N]

[B]
[B-S]

Marie Albenque and Philppe Nadeau: Growth function for aclass of monoids, FPSAC 2009, DMTCS proc. (subm.), by theauthors, 1-14.
Nicolas Bourbaki: Groups et algébres de Lie, Chapitres 4,5 et 6.
Éléments de Mathèmatique XXXIV. Paris: Hermann 1968.
Egbert Brieskorn and Kyoji Saito: Artin-Gruppen und CoxeterGruppen, Invent. Math. 17 (1972), 245–271.

[S1]

Kyoji Saito: Growth functions associated with Artin monoids offinite type, Proc. Japan Acad., 84, Ser. A (2008).

[S2]

Kyoji Saito: Extended affine root systems I (Coxeter transformations), III (Elliptic Weyl groups), Publ. RIMS,Kyoto-Univ.,21(1985), 75-179, ibid., 33 (1997), 301-329.

[S3]

Kyoji Saito: Towards Categorical Construction of Lie Algebras,
Advanced Studies in Pure Mathematics 50 (2008), Algebraic geometry in East Asia–Hanoi 2005, pp.101-175.

7