arXiv:1001.0028v2  [math.CO]  28 Feb 2012CYCLIC SIEVING FOR GENERALISED NON-CROSSING
PARTITIONS ASSOCIATED WITH COMPLEX REFLECTION
GROUPS OF EXCEPTIONAL TYPE
Christian Krattenthaler†andThomas W. M ¨uller‡
†Fakult¨ at f¨ ur Mathematik, Universit¨ at Wien,
Nordbergstraße 15, A-1090 Vienna, Austria.
WWW:http://www.mat.univie.ac.at/ ~kratt
‡School of Mathematical Sciences,
Queen Mary & Westfield College, University of London,
Mile End Road, London E1 4NS, United Kingdom.
WWW:http://www.maths.qmw.ac.uk/ ~twm/
Dedicated to the memory of Herb Wilf
Abstract. We prove that the generalised non-crossing partitions associated with
well-generated complex reflection groups of exceptional type obe y two different cyclic
sieving phenomena, as conjectured by Armstrong, and by Bessis a nd Reiner. The
computational details are provided in the manuscript “Cyclic sieving for generalised
non-crossing partitions associated with complex reflectio n groups of exceptional type
— the details” [arχiv:1001.0030 ].
1.Introduction
In his memoir [2], Armstrong introduced generalised non-crossing partitions asso-
ciated with finite (real) reflection groups, thereby embedding Krew eras’ non-crossing
partitions [22], Edelman’s m-divisible non-crossing partitions [12], thenon-crossing par-
titions associated with reflection groups due to Bessis [6] and Brady and Watt [10] into
one uniform framework. Bessis and Reiner [9] observed that Arms trong’s definition can
be straightforwardly extended to well-generated complex reflection groups (see Section 2
for the precise definition). These generalised non-crossing partit ions possess a wealth
of beautiful properties, and they display deep and surprising relat ions to other combi-
natorial objects defined for reflection groups (such as the gene ralised cluster complex
2000Mathematics Subject Classification. Primary 05E15; Secondary 05A10 05A15 05A18 06A07
20F55.
Key words and phrases. complex reflection groups, unitary reflection groups, m-divisible non-
crossing partitions, generalised non-crossing partitions, Fuß–Ca talan numbers, cyclic sieving.
†Research partially supported by the Austrian Science Foundation F WF, grants Z130-N13 and
S9607-N13, the latter in the framework of the National Research Network “Analytic Combinatorics
and Probabilistic Number Theory.”
‡Research supported by the Austrian Science Foundation FWF, Lise Meitner grant M1201-N13.
12 C. KRATTENTHALER AND T. W. M ¨ULLER
of Fomin and Reading [13], or the extended Shi arrangement and the geometric multi-
chains of filters of Athanasiadis [4, 5]); see Armstrong’s memoir [2] and the references
given therein.
Ontheotherhand, cyclic sieving isaphenomenonbroughttolightbyReiner, Stanton
and White [30]. It extends the so-called “( −1)-phenomenon” of Stembridge [34, 35].
Cyclic sieving can be defined in three equivalent ways (cf. [30, Prop. 2.1]). The one
which gives the name can be described as follows: given a set Sof combinatorial
objects, an action on Sof a cyclic group G=/an}bracketle{tg/an}bracketri}htwith generator gof ordern, and
a polynomial P(q) inqwith non-negative integer coefficients, we say that the triple
(S,P,G)exhibits the cyclic sieving phenomenon , if the number of elements of Sfixed
bygkequalsP(e2πik/n). In [30] it is shown that this phenomenon occurs in surprisingly
many contexts, and several further instances have been discov ered since then.
In [2, Conj. 5.4.7] (also appearing in [9, Conj. 6.4]) and [9, Conj. 6.5], Ar mstrong,
respectively Bessis and Reiner, conjecture that generalised non- crossing partitions for
irreducible well-generated complex reflection groups exhibit two diffe rent cyclic sieving
phenomena (see Sections 3 and 7 for the precise statements).
According to the classification of these groups due to Shephard an d Todd [32], there
are two infinite families of irreducible well-generated complex reflectio n groups, namely
the groups G(d,1,n) andG(e,e,n), wheren,d,eare positive integers, and there are 26
exceptional groups. For the infinite families of types G(d,1,n) andG(e,e,n), the two
cyclic sieving conjectures follow from the results in [19].
Thepurposeofthepresent articleistopresent aproofofthecyc licsieving conjectures
of Armstrong, and of Bessis and Reiner, for the 26 exceptional ty pes, thus completing
the proof of these conjectures. Since the generalised non-cros sing partitions feature a
parameterm, from the outset this is nota finite problem. Consequently, we first need
several auxiliary results to reduce the conjectures for each of t he 26 exceptional types
to afiniteproblem. Subsequently, we use Stembridge’s Maplepackagecoxeter [36]
and theGAPpackageCHEVIE[14, 28] to carry out the remaining finitecomputations.
The details of these computations are provided in [21]. In the presen t paper, we con-
tent ourselves with exemplifying the necessary computations by go ing through some
representative cases. It is interesting to observe that, for the verification of the type
E8case, it is essential to use the decomposition numbers in the sense o f [17, 18, 20] be-
cause, otherwise, the necessary computations would not be feas ible in reasonable time
with the currently available computer facilities. We point out that, fo r the special case
where the aforementioned parameter mis equal to 1, the first cyclic sieving conjecture
has been proved in a uniform fashion by Bessis and Reiner in [9]. (See [3 ] for a —
non-uniform — proof of cyclic sieving for non-crossing partitions as sociated with real
reflection groups under the action of the so-called Kreweras map, a special case of the
second cyclic sieving phenomenon discussed in the present paper.) T he crucial result on
which the proof of Bessis and Reiner is based is (5.5) below, and it plays an important
rolein our reduction of the conjectures forthe 26 exceptional gr oupsto a finite problem.
Our paper is organised as follows. In the next section, we recall the definition of
generalised non-crossing partitions for well-generated complex re flection groups and of
decomposition numbers in the sense of [17, 18, 20], and we review so me basic facts.
The first cyclic sieving conjecture is subsequently stated in Section 3. In Section 4, we
outline an elementary proof that the q-Fuß–Catalan number, which is the polynomial
Pin the cyclic sieving phenomena concerning the generalised non-cros sing partitionsCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 3
for well-generated complex reflection groups, is always a polynomial with non-negative
integer coefficients, as required by the definition of cyclic sieving. (F ull details can be
found in [21, Sec. 4]. The reader is referred to the first paragraph of Section 4 for
comments on other approaches for establishing polynomiality with no n-negative coeffi-
cients.) Section 5 contains the announced auxiliary results which, fo r the 26 exceptional
types, allow a reduction of the conjecture to a finite problem. In Se ction 6, we discuss
a few cases which, in a representative manner, demonstrate how t o perform the re-
maining case-by-case verification of the conjecture. For full det ails, we refer the reader
to [21, Sec. 6]. The second cyclic sieving conjecture is stated in Sect ion 7. Section 8
contains the auxiliary results which, for the 26 exceptional types, allow a reduction of
the conjecture to a finite problem, while in Section 9 we discuss some r epresentative
cases of the remaining case-by-case verification of the conjectu re. Again, for full details
we refer the reader to [21, Sec. 9].
2.Preliminaries
Acomplex reflection group isa groupgeneratedby(complex) reflections in Cn. (Here,
a reflection is a non-trivial element of GLn(C) which fixes a hyperplane pointwise and
which hasfiniteorder.) Wereferto[24]foranin-depthexpositionof thetheorycomplex
reflection groups.
Shephard and Todd provided a complete classification of all finitecomplex reflection
groups in [32] (see also [24, Ch. 8]). According to this classification, a n arbitrary
complex reflection group Wdecomposes into a direct product of irreducible complex
reflection groups, acting on mutually orthogonal subspaces of th e complex vector space
onwhichWisacting. Moreover, thelistofirreduciblecomplexreflectiongroups consists
of the infinite family of groups G(m,p,n), wherem,p,nare positive integers, and 34
exceptional groups, denoted G4,G5,...,G 37by Shephard and Todd.
In this paper, we are only interested in finite complex reflection grou ps which are
well-generated . A complex reflection group of rank nis called well-generated if it is
generated by nreflections.1Well-generation can be equivalently characterised by a
duality property due to Orlik and Solomon [29]. Namely, a complex reflec tion group of
ranknhastwo sets ofdistinguished integers d1≤d2≤ ··· ≤dnandd∗
1≥d∗
2≥ ··· ≥d∗
n,
called its degreesandcodegrees , respectively (see [24, p. 51 and Def. 10.27]). Orlik and
Solomon observed, using case-by-case checking, that an irreduc ible complex reflection
groupWof ranknis well-generated if and only if its degrees and codegrees satisfy
di+d∗
i=dn
for alli= 1,2,...,n. The reader is referred to [24, App. D.2] for a table of the degree s
and codegrees of all irreducible complex reflection groups. Togeth er with the classi-
fication of Shephard and Todd [32], this constitutes a classification o f well-generated
complex reflection groups: the irreducible well-generated complex r eflection groups are
— the two infinite families G(d,1,n) andG(e,e,n), whered,e,nare positive inte-
gers,
— the exceptional groups G4,G5,G6,G8,G9,G10,G14,G16,G17,G18,G20,G21of
rank 2,
1We refer to [24, Def. 1.29] for the precise definition of “rank.” Roug hly speaking, the rank of a
complex reflection group Wis the minimal nsuch that Wcan be realized as reflection group on Cn.4 C. KRATTENTHALER AND T. W. M ¨ULLER
— the exceptional groups G23=H3,G24,G25,G26,G27of rank 3,
— the exceptional groups G28=F4,G29,G30=H4,G32of rank 4,
— the exceptional group G33of rank 5,
— the exceptional groups G34,G35=E6of rank 6,
— the exceptional group G36=E7of rank 7,
— and the exceptional group G37=E8of rank 8.
In this list, we have made visible the groups H3,F4,H4,E6,E7,E8which appear as
exceptional groups in the classification of all irreducible realreflection groups (cf. [16]).
LetWbe a well-generated complex reflection group of rank n, and letT⊆Wdenote
theset of all(complex) reflections inthegroup. Let ℓT:W→Zdenotethewordlength
in terms of the generators T. This word length is called absolute length orreflection
length. Furthermore, we define a partial order ≤TonWby
u≤Twif and only if ℓT(w) =ℓT(u)+ℓT(u−1w). (2.1)
This partial order is called absolute order orreflection order . As is well-known and
easy to see, the equation in (2.1) is equivalent to the statement tha t every shortest
representation of uby reflections occurs as an initial segment in some shortest produc t
representation of wby reflections.
Now fix a (generalised) Coxeter element2c∈Wand a positive integer m. The
m-divisible non-crossing partitions NCm(W) are defined as the set
NCm(W) =/braceleftbig
(w0;w1,...,w m) :w0w1···wm=cand
ℓT(w0)+ℓT(w1)+···+ℓT(wm) =ℓT(c)/bracerightbig
.
A partial order is defined on this set by
(w0;w1,...,w m)≤(u0;u1,...,u m) if and only if ui≤Twifor 1≤i≤m.
We have suppressed the dependence on c, since we understand this definition up to
isomorphism of posets. To be more precise, it can be shown that any two Coxeter
elements are related to each other by conjugation and (possibly) a n automorphism on
the field of complex numbers (see [33, Theorem 4.2] or [24, Cor. 11.2 5]), and hence the
resulting posets NCm(W) are isomorphic to each other. If m= 1, thenNC1(W) can
be identified with the set NC(W) of non-crossing partitions for the (complex) reflection
groupWasdefined byBessis andCorran(cf.[8]and[7, Sec.13]; theirdefinit ionextends
the earlier definition by Bessis [6] and Brady and Watt [10] for real r eflection groups).
The following result has been proved by a collaborative effort of seve ral authors (see
[7, Prop. 13.1]).
2An element of an irreducible well-generated complex reflection group Wof ranknis called a
Coxeter element if it isregularin the sense of Springer [33] (see also [24, Def. 11.21]) and of order dn.
An element of Wis called regular if it has an eigenvector which lies in no reflecting hyperp lane of a
reflection of W. It follows from an observation of Lehrer and Springer, proved un iformly by Lehrer
and Michel [23] (see [24, Theorem 11.28]), that there is always a regu lar element of order dnin an
irreducible well-generated complex reflection group Wof rankn. More generally, if a well-generated
complex reflection group Wdecomposes as W∼=W1×W2×···×Wk, where the Wi’s are irreducible,
then a Coxeter element of Wis an element of the form c=c1c2···ck, whereciis a Coxeter element of
Wi,i= 1,2,...,k. IfWis arealreflection group, that is, if all generators in Thave order 2, then the
notion of generalised Coxeter element given above reduces to that of a Coxeter element in the classical
sense (cf. [16, Sec. 3.16]).CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 5
Theorem 1. LetWbe an irreducible well-generated complex reflection group, and let
d1≤d2≤ ··· ≤dnbe its degrees and h:=dnits Coxeter number. Then
|NCm(W)|=n/productdisplay
i=1mh+di
di. (2.2)
Remark1.(1) The number in (2.2) is called the Fuß–Catalan number for the reflection
groupW.
(2) Ifcis a Coxeter element of a well-generated complex reflection group Wof rank
n, thenℓT(c) =n. (This follows from [7, Sec. 7].)
We conclude this section by recalling the definition of decomposition nu mbers from
[17, 18, 20]. Although we need them here only for (very small) real re flection groups,
and although, strictly speaking, they have been only defined for re al reflection groups in
[17, 18, 20], this definition can be extended to well-generated comple x reflection groups
without any extra effort, which we do now.
Given a well-generated complex reflection group Wof rankn, typesT1,T2,...,T d(in
the sense of the classification of well-generated complex reflection groups) such that the
sumoftheranksofthe Ti’sequalsn, andaCoxeter element c, thedecompositionnumber
NW(T1,T2,...,T d) is defined as the number of “minimal” factorisations c=c1c2···cd,
“minimal” meaning that ℓT(c1) +ℓT(c2) +···+ℓT(cd) =ℓT(c) =n, such that, for
i= 1,2,...,d, the type of cias a parabolic Coxeter element is Ti. (Here, the term
“parabolic Coxeter element” means a Coxeter element in some parab olic subgroup. It
follows from [31, Prop.6.3] that any element ciis indeed a Coxeter element in a unique
parabolic subgroup of W.3By definition, the type of ciis the type of this parabolic
subgroup.) Since any two Coxeter elements are related to each oth er by conjugation
plus field automorphism, the decomposition numbers are independen t of the choice of
the Coxeter element c.
The decomposition numbers for real reflection groups have been c omputed in [17,
18, 20]. To compute the decomposition numbers for well-generated complex reflection
groups is a task that remains to be done.
3.Cyclic sieving I
In this section we present the first cyclic sieving conjecture due to Armstrong [2,
Conj. 5.4.7], and to Bessis and Reiner [9, Conj. 6.4].
Letφ:NCm(W)→NCm(W) be the map defined by
(w0;w1,...,w m)/mapsto→/parenleftbig
(cwmc−1)w0(cwmc−1)−1;cwmc−1,w1,w2,...,w m−1/parenrightbig
.(3.1)
It is indeed not difficult to see that, if the ( m+ 1)-tuple on the left-hand side is an
element ofNCm(W), then so is the ( m+1)-tuple on the right-hand side. For m= 1,
this action reduces to conjugation by the Coxeter element c(applied to w1). Cyclic
sieving arising from conjugation by chas been the subject of [9].
3The uniqueness can be argued as follows: suppose that ciwere a Coxeter element in two parabolic
subgroups of W, sayU1andU2. Then it must also be a Coxeter element in the intersection U1∩U2.
On the other hand, the absolute length of a Coxeter element of a co mplex reflection group Uis always
equal to rk( U), the rank of U. (This follows from the fact that, for each element uofU, we have
ℓT(u) = codim/parenleftbig
ker(u−id)/parenrightbig
, with id denoting the identity element in U; see e.g. [31, Prop. 1.3]). We
conclude that ℓT(ci) = rk(U1) = rk(U2) = rk(U1∩U2), This implies that U1=U2.6 C. KRATTENTHALER AND T. W. M ¨ULLER
It is easy to see that φmhacts as the identity, where his the Coxeter number of W
(see (5.1) and Lemma 29 below). By slight abuse of notation, let C1be the cyclic group
of ordermhgenerated by φ. (The slight abuse consists in the fact that we insist on C1
to be a cyclic group of order mh, while it may happen that the order of the action of
φgiven in (3.1) is actually a proper divisor of mh.)
Given these definitions, we are now in the position to state the first c yclic sieving
conjecture of Armstrong, respectively of Bessis and Reiner. By t he results of [19] and
of this paper, it becomes the following theorem.
Theorem 2. For an irreducible well-generated complex reflection group Wand any
m≥1, the triple (NCm(W),Catm(W;q),C1), whereCatm(W;q)is theq-analogue of
the Fuß–Catalan number defined by
Catm(W;q) :=n/productdisplay
i=1[mh+di]q
[di]q, (3.2)
exhibits the cyclic sieving phenomenon in the sense of Reine r, Stanton and White [30].
Here,nis the rank of W,d1,d2,...,d nare the degrees of W,his the Coxeter number
ofW, and[α]q:= (1−qα)/(1−q).
Remark2.We write Catm(W) for Catm(W;1).
By definition of the cyclic sieving phenomenon, we have to prove that Catm(W;q) is
a polynomial in qwith non-negative integer coefficients, and that
|FixNCm(W)(φp)|= Catm(W;q)/vextendsingle/vextendsingle
q=e2πip/mh, (3.3)
for allpin the range 0 ≤p<mh. The first fact is established in the next section, while
the proof of the second is achieved by making use of several auxiliar y results, given
in Section 5, to reduce the proof to a finite problem, and a subseque nt case-by-case
analysis. Alldetails ofthisanalysiscanbefoundin[21, Sec. 6]. Inthe present paper, we
content ourselves with discussing the cases where W=G24and whereW=G37=E8,
since these suffice to convey the flavour of the necessary comput ations.
4.Theq-Fusz–Catalan numbers Catm(W;q)
The purpose of this section is to provide an elementary, self-conta ined proof of the
fact that, for all irreducible complex reflection groups W, theq-Fuß–Catalan number
Catm(W;q) is a polynomial in qwith non-negative integer coefficients. For most of
the groups, this is a known property. However, aside from the fac t that, for many of
the known cases, the proof is very indirect and uses deep algebraic results on rational
Cherednik algebras, there still remained some cases where this pro perty had not been
formally established. The reader is referred to the “Theorem” in Se ction 1.6 of [15],
whichsaysthat, undertheassumptionofacertainrankcondition( [15, Hypothesis2.4]),
theq-Fuß–Catalan number Catm(W;q) is a Hilbert series of a finite-dimensional quo-
tient of the ring of invariants of Wand also the graded character of a finite-dimensional
irreducible representation of a spherical rational Cherednik algeb ra associated with
W. At present, this rank condition has been proven for all irreducible well-generated
complex reflection groups apart from G17,G18,G29,G33,G34; see [26, Tables 8 and 9,
column “rank”], and the recent paper [27], which establishes the res ult in the case of
G32.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 7
In the sequel, aside from the standard notation [ α]q= (1−qα)/(1−q) forq-integers,
we shall also use the q-binomial coefficient, which is defined by
/bracketleftbigg
n
k/bracketrightbigg
q:=/braceleftBigg
1, ifk= 0,
[n]q[n−1]q···[n−k+1]q
[k]q[k−1]q···[1]q,ifk>0.
We begin with several auxiliary results.
Proposition 3. For all non-negative integers nandk, theq-binomial coefficient [n
k]q
is a polynomial in qwith non-negative integer coefficients.
Proof.This is a well-known fact, which can be derived either from the recurr ence rela-
tion(s) satisfied by the q-binomial coefficients (generalising Pascal’s recurrence relation
for binomial coefficients; cf. [1, eqs. (3.3.3) and (3.3.4)]), or from th e fact that the q-
binomial coefficient [n
k]qis the generating function for (integer) partitions with at most
kparts all of which are at most n−k(cf. [1, Theorem 3.1]). /square
Proposition 4. For all non-negative integers mandn, theq-Fuß–Catalan number of
typeAn,
1
[(m+1)n+1]q/bracketleftbigg
(m+1)n+1
n/bracketrightbigg
q,
is a polynomial in qwith non-negative integer coefficients.
Proof.In [25, Sec. 3.3], Loehr proves that
1
[(m+1)n+1]q/bracketleftbigg
(m+1)n+1
n/bracketrightbigg
q
=/summationdisplay
v∈V(m)
nqm(n
2)+/summationtext
i≥0(m(vi
2)−ivi)/productdisplay
i≥1qvi/summationtextm
j=1(m−j)vi−j/bracketleftbigg
vi+vi−1+···+vi−m−1
vi/bracketrightbigg
q,(4.1)
whereV(m)
ndenotes the set of all sequences v= (v0,v1,...,v s) (for some s) of non-
negative integers with v0>0,vs>0, andv0+v1+···+vs=n, and such that there
is never a string of mor more consecutive zeroes in v. By convention, vi= 0 for all
negativei. His proof works by showing that the expressions on both sides of ( 4.1)
satisfy the same recurrence relation and initial conditions, using cla ssicalq-binomial
identities. We refer the reader to [25] for details. By Proposition 3, the expression on
the right-hand side of (4.1) is manifestly a polynomial in qwith non-negative integer
coefficients. /square
Lemma 5. Ifaandbare coprime positive integers, then
[ab]q
[a]q[b]q(4.2)
is a polynomial in qof degree (a−1)(b−1), all of whose coefficients are in {0,1,−1}.
Moreover, if one disregards the coefficients which are 0, then+1’s and(−1)’s alternate,
and the constant coefficient as well as the leading coefficient o f the polynomial equal +1.
Proof.LetΦn(q)denotethe n-thcyclotomicpolynomialin q. Usingtheclassicalformula
1−qn=/productdisplay
d|nΦd(q),8 C. KRATTENTHALER AND T. W. M ¨ULLER
we see that
(1−q)(1−qab)
(1−qa)(1−qb)=/productdisplay
d1|a,d1/ne}ationslash=1
d2|a,d2/ne}ationslash=1Φd1d2(q),
so that, manifestly, the expression in (4.2) is a polynomial in q. The claim concerning
the degree of this polynomial is obvious.
In order to establish the claim on the coefficients, we start with a sub -expression of
(4.2),
(1−qab)
(1−qa)(1−qb)=/parenleftbiggb−1/summationdisplay
i=0qia/parenrightbigg/parenleftbigg∞/summationdisplay
j=0qjb/parenrightbigg
=∞/summationdisplay
k=0Ckqk, (4.3)
say. The assumption that aandbare coprime implies that 0 ≤Ck≤1 fork≤
(a−1)(b−1). Multiplying both sides of (4.3) by 1 −q, we obtain the equation
[ab]q
[a]q[b]q= (1−q)(a−1)(b−1)/summationdisplay
k=0Ckqk+(1−q)∞/summationdisplay
k=(a−1)(b−1)+1Ckqk. (4.4)
By our previous observation on the coefficients Ckwithk≤(a−1)(b−1), it is obvious
that the coefficients of the first expression on the right-hand side of (4.4) are alternately
+1 and−1, when 0’s are disregarded. Since we already know that the left-ha nd side is
a polynomial in qof degree (a−1)(b−1), we may ignore the second expression.
The proof is concluded by observing that the claims on the constant and leading
coefficients are obvious. /square
Corollary 6. Letaandbbe coprime positive integers, and let γbe an integer with
γ≥(a−1)(b−1). Then the expression
[γ]q[ab]q
[a]q[b]q
is a polynomial in qwith non-negative integer coefficients.
Proof.Let
[ab]q
[a]q[b]q=(a−1)(b−1)/summationdisplay
k=0Dkqk.
We then have
[γ]q[ab]q
[a]q[b]q=(a−1)(b−1)+γ−1/summationdisplay
N=0qNN/summationdisplay
k=max{0,N−γ+1}Dk. (4.5)
IfN≤γ−1, then, by Lemma 5, the sum over kon the right-hand side of (4.5) equals
1−1+1−1+···, which is manifestly non-negative. On the other hand, if N >γ−1,
then we may rewrite the sum over kon the right-hand side of (4.5) as
N/summationdisplay
k=max{0,N−γ+1}Dk=(a−1)(b−1)/summationdisplay
k=N−γ+1Dk=(a−1)(b−1)+γ−1−N/summationdisplay
k=0D(a−1)(b−1)−k.
Again, by Lemma 5, this sum equals 1 −1 + 1−1 +···, which is manifestly non-
negative. /squareCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 9
The next lemmas all have a very similar flavour, and so do their proofs . In order to
avoid repetition, proof details are only provided for Lemmas 7 and 16 ; the proofs of
Lemmas 9–15, 22–24 follow the pattern exhibited in the proof of Lem ma 7, while the
proofs of Lemmas 17–21 follow that of the proof of Lemma 15. Full d etails are found
in [21, Sec. 4].
Lemma 7. Letαandβbe positive integers with α≥6andβ≥8. Then the expression
[α]q3[β]q4[72]q[3]q[4]q
[8]q[9]q[12]q
is a polynomial in qwith non-negative integer coefficients.
Proof.We have
[72]q[3]q[4]q
[8]q[9]q[12]q
= (1−q3+q9−q15+q18)(1−q4+q8−q12+q16−q20+q24−q28+q32).
It should be observed that both factors on the right-hand side ha ve the property that
coefficients are in {0,1,−1}and that (+1)’s and ( −1)’s alternate, if one disregards the
coefficients which are 0. If we now apply the same idea as in the proof o f Corollary 6,
then we see that [ α]q3times the first factor is a polynomial in qwith non-negative
integer coefficients, as is [ β]q4times the second factor. Taken together, this establishes
the claim. /square
Lemma 8. Letαandβbe positive integers with α≥26andβ≥8. Then the expression
[α]q[β]q4[15]q
[3]q[5]q[72]q[3]q[4]q
[8]q[9]q[12]q
is a polynomial in qwith non-negative integer coefficients.
Lemma 9. Letαandβbe positive integers with α≥18andβ≥3. Then the expression
[α]q3[β]q4[90]q[3]q[4]q
[5]q[6]q[9]q
is a polynomial in qwith non-negative integer coefficients.
Lemma 10. Letαandβbe positive integers with α≥20andβ≥18. Then the
expression
[α]q[β]q3[90]q[3]q
[5]q[6]q[9]q
is a polynomial in qwith non-negative integer coefficients.
Lemma 11. Letαbe a positive integer with α≥26. Then the expression
[α]q[15]q
[3]q[5]q[12]q3
[3]q3[4]q3
is a polynomial in qwith non-negative integer coefficients.10 C. KRATTENTHALER AND T. W. M ¨ULLER
Lemma 12. Letαbe a positive integer with α≥14. Then the expression
[α]q[15]q
[3]q[5]q[6]q3
[2]q3[3]q3
is a polynomial in qwith non-negative integer coefficients.
Lemma 13. Letαandβbe positive integers with α≥30andβ≥20. Then the
expression
[α]q[β]q2[84]q[2]q
[4]q[6]q[7]q
is a polynomial in qwith non-negative integer coefficients.
Lemma 14. Letαandβbe positive integers with α≥24andβ≥68. Then the
expression
[α]q[β]q[105]q
[3]q[5]q[7]q
is a polynomial in qwith non-negative integer coefficients.
Lemma 15. Letαandβbe positive integers with α≥24andβ≥34. Then the
expression
[α]q[β]q[70]q
[2]q[5]q[7]q
is a polynomial in qwith non-negative integer coefficients.
Lemma 16. Letαandβbe positive integers with α≥4andβ≥2. Then the expression
[α]q2[β]q5[30]q[2]q[3]q[5]q
[6]q[10]q[15]q
is a polynomial in qwith non-negative integer coefficients.
Proof.We have
[30]q[2]q[3]q[5]q
[6]q[10]q[15]q= 1+q−q3−q4−q5+q7+q8.
If we multiply this expression by [ α]q2, then, forα= 4 we obtain
1+q+q2−q5−q9+q12+q13+q14,
forα= 5 we obtain
1+q+q2−q5+q8−q11+q14+q15+q16,
and, forα≥6, we obtain
1+q+q2−q5+q8+q10+p1(q)+q2α−4+q2α−2−q2α+1+q2α+4+q2α+5+q2α+6,
wherep1(q) is a polynomial in qwith non-negative coefficients of order at least 11 and
degree at most 2 α−5. In all cases it is obvious that the product of the result and [ β]q5,
withβ≥2, is a polynomial in qwith non-negative coefficients. /squareCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 11
Lemma 17. Letαandβbe positive integers with α≥14andβ≥2. Then the
expression
[α]q[β]q5[14]q
[2]q[7]q[30]q[2]q[3]q[5]q
[6]q[10]q[15]q
is a polynomial in qwith non-negative integer coefficients.
Lemma 18. Letαandβbe positive integers with α≥32andβ≥12. Then the
expression
[α]q[β]q2[35]q
[5]q[7]q[30]q[2]q[3]q[5]q
[6]q[10]q[15]q
is a polynomial in qwith non-negative integer coefficients.
Lemma 19. Letαandβbe positive integers with α≥16andβ≥2. Then the
expression
[α]q2[β]q5[60]q[2]q[3]q[5]q
[10]q[12]q[15]q
is a polynomial in qwith non-negative integer coefficients.
Lemma 20. Letαandβbe positive integers with α≥56andβ≥4. Then the
expression
[α]q[β]q2[35]q
[5]q[7]q[60]q[2]q[3]q[5]q
[10]q[12]q[15]q
is a polynomial in qwith non-negative integer coefficients.
Lemma 21. Letαandβbe positive integers with α≥38andβ≥2. Then the
expression
[α]q[β]q5[14]q
[2]q[7]q[60]q[2]q[3]q[5]q
[10]q[12]q[15]q
is a polynomial in qwith non-negative integer coefficients.
Lemma 22. Letαandβbe positive integers with α≥30andβ≥26. Then the
expression
[α]q[β]q3[126]q[3]q
[6]q[7]q[9]q
is a polynomial in qwith non-negative integer coefficients.
Lemma 23. Letαandβbe positive integers with α≥66andβ≥54. Then the
expression
[α]q[β]q3[252]q[3]q
[7]q[9]q[12]q
is a polynomial in qwith non-negative integer coefficients.
Lemma 24. Letαandβbe positive integers with α≥54andβ≥34. Then the
expression
[α]q[β]q2[140]q[2]q
[4]q[7]q[10]q
is a polynomial in qwith non-negative integer coefficients.12 C. KRATTENTHALER AND T. W. M ¨ULLER
We are now ready for the proof of the main result of this section.
Theorem 25. For all irreducible well-generated complex reflection grou ps and posi-
tive integers m, theq-Fuß–Catalan number Catm(W;q)is a polynomial in qwith non-
negative integer coefficients.
Proof.First, letW=An. In this case, the degrees are 2 ,3,...,n+1, and hence
Catm(An;q) =1
[(m+1)n+1]q/bracketleftbigg
(m+1)n+1
n/bracketrightbigg
q,
which, by Proposition 4, is a polynomial in qwith non-negative integer coefficients.
Next, letW=G(d,1,n). In this case, the degrees are d,2d,...,nd , and hence
Catm(G(d,1,n);q) =/bracketleftbigg
(m+1)n
n/bracketrightbigg
qd,
which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients.
Now, letW=G(e,e,n). In this case, the degrees are e,2e,...,(n−1)e,n, and hence
Catm(G(e,e,n);q) =[m(n−1)e+n]q
[n]qn−1/productdisplay
i=1[m(n−1)e+ie]q
[ie]q
=/bracketleftbigg
(m+1)(n−1)
n−1/bracketrightbigg
qe+qn[e]qn/bracketleftbigg
(m+1)(n−1)
n/bracketrightbigg
qe,
which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients.
It remains to verify the claim for the exceptional groups.
For the groups W=G6,G9,G14,G17,G21,and partially for the groups W=G20,G23,
G28,G30,G33,G35,G36,G37(depending on congruence properties of the parameter m),
polynomiality and non-negativity of coefficients of the correspondin gq-Fuß–Catalan
number can be directly read off by a proper rearrangement of the t erms in the defining
expression; for example, for W=G21(with degrees given by 12 ,60) we have
Catm(G21;q) =[60m+12]q[60m+60]q
[12]q[60]q= [5m+1]q12[m+1]q60,
which is manifestly a polynomial in qwith non-negative integer coefficients.
For the groups G5,G10,G18,G26,G27,G29,G34, the terms in the defining expres-
sion of the corresponding q-Fuß–Catalan number can be arranged in a manner so
that aq-binomial coefficient appears; polynomiality and non-negativity of co efficients
then follow from Proposition 3. For example, for W=G34(with degrees given by
6,12,18,24,30,42) we have
Catm(G34;q) =[42m+6]q[42m+12]q[42m+18]q[42m+24]q[42m+30]q[42m+42]q
[6]q[12]q[18]q[24]q[30]q[42]q
= [m+1]q42/bracketleftbigg
7m+5
5/bracketrightbigg
q6,
which, written in this form, is obviously a polynomial in qwith non-negative integer
coefficients.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 13
On the other hand, for the groups G4,G8,G16,G25,G32, the terms in the defining
expression of the corresponding q-Fuß–Catalan number can be arranged in a manner so
that aq-Fuß–Catalannumber of type Aappears andProposition 4 applies; for example,
forW=G32(with degrees given by 12 ,18,24,30) we have
Catm(G32;q) =[30m+12]q[30m+18]q[30m+24]q[30m+30]q
[12]q[18]q[24]q[30]q
=1
[5m+6]q6/bracketleftbigg
5m+6
5/bracketrightbigg
q6,
which indeed fits into the framework of Proposition 4 and, hence, is a polynomial in q
with non-negative integer coefficients.
In the other cases, the more “specialised” auxiliary results given in C orollary 6 and
Lemmas7–24havetobeapplied. Forthesakeofillustration, weexhib it oneexample for
each of them below, with full details being provided in [21, Sec. 4]. In ge neral, the idea
is that, given a rational expression consisting of cyclotomic factor s, as in the definition
oftheq-Fuß–Catalannumbers, onetriestoplacedenominator factorsbe lowappropriate
numerator factors so that one can divide out the denominator fac tor completely. For
example, if we were to encounter the expression
[30m+12]q·(other terms)
[12]q·(other terms)
and know that mis even, then we would try to simplify this to
/bracketleftbig5m+2
2/bracketrightbig
q12·(other terms)
(other terms),
where [5m+2
2]q12is manifestly a polynomial in qwith non-negative integer coefficients.
On the other hand, in a situation where twodenominator factors “want” to divide a
singlenumerator factor, we “extract” as much as we can from the nume rator factor and
compensate by additional “fudge” factors. To be more concrete , if we encounter the
expression
[14m+14]q·(other terms)
[6]q[14]q·(other terms)
and we know that m≡0 (mod 3), then we would try the rewriting
/bracketleftbigm+1
3/bracketrightbig
q42[21]q2
[3]q2[7]q2[2]q·(other terms)
(other terms),
with the idea that we might find somewhere else a term [2 α]q, which could be combined
with the term[2] qin the denominator into [2 α]q/[2]q= [α]q2, andthen apply Corollary6
to see that
[α]q2[21]q2
[3]q2[7]q2
is a polynomial in qwith non-negative integer coefficients (provided αis at least 12),
with/bracketleftbigm+1
3/bracketrightbig
q42being such a polynomial in any case.
In situations where threedenominator factors “want” to divide a singlenumerator
factor, one has to perform more complicated rearrangements, in order to be able to
apply one of the Lemmas 7–24.14 C. KRATTENTHALER AND T. W. M ¨ULLER
For example, for W=G24, the degrees are 4 ,6,14, and hence
Catm(G24;q) =[14m+4]q[14m+6]q[14m+14]q
[4]q[6]q[14]q.
We have
Catm(G24;q) =

/bracketleftbig7m
2+1/bracketrightbig
q4/bracketleftbig14m
6+1/bracketrightbig
q6[m+1]q14,ifm≡0 (mod 6),/bracketleftbig7m+2
3/bracketrightbig
q6/bracketleftbig7m+3
2/bracketrightbig
q4[m+1]q14, ifm≡1 (mod 6),
/bracketleftbig7m
2+1/bracketrightbig
q4[7m+3]q2/bracketleftbigm+1
3/bracketrightbig
q42[21]q2
[3]q2[7]q2,ifm≡2 (mod 6),
[7m+2]q2/bracketleftbig7m
3+1/bracketrightbig
q6/bracketleftbigm+1
2/bracketrightbig
q28[14]q2
[2]q2[7]q2,ifm≡3 (mod 6),
/bracketleftbig7m+2
6/bracketrightbig
q12[6]q2
[2]q2[3]q2[7m+3]q2[m+1]q14,ifm≡4 (mod 6),
[7m+2]q2/bracketleftbig7m+3
2/bracketrightbig
q4/bracketleftbigm+1
3/bracketrightbig
q42[21]q2
[3]q2[7]q2,ifm≡5 (mod 6),
which, by Corollary 6, are polynomials in qwith non-negative integer coefficients in all
cases.
ForW=G30=H4, the degrees are 2 ,12,20,30, and hence
Catm(H4;q) =[30m+2]q[30m+12]q[30m+20]q[30m+30]q
[2]q[12]q[20]q[30]q.
Ifmis odd, then we may write
Catm(H4;q) =/bracketleftbig15m+1
2/bracketrightbig
q4[5m+2]q6[3m+2]q10/bracketleftbigm+1
2/bracketrightbig
q60[30]q2[2]q2[3]q2[5]q2
[6]q6[10]q2[15]q2,
which, by Lemma 16, is a polynomial in qwith non-negative integer coefficients.
ForW=G35=E6, the degrees are 2 ,5,6,8,9,12, and hence
Catm(E6;q) =[12m+2]q[12m+5]q[12m+6]q[12m+8]q[12m+9]q[12m+12]q
[2]q[5]q[6]q[8]q[9]q[12]q.
Ifm≡5 (mod 30),then we have
Catm(E6;q) = [6m+1]q2/bracketleftbig12m+5
5/bracketrightbig
q5[2m+1]q6
×[3m+2]q4[4m+3]q3/bracketleftbigm+1
6/bracketrightbig
q72[72]q[3]q[4]q
[8]q[9]q[12]q,
which, by Lemma 7, is a polynomial in qwith non-negative integer coefficients.
Ifm≡7 (mod 30),then we have
Catm(E6;q) =/bracketleftbig6m+1
2/bracketrightbig
q4[12m+5]q/bracketleftbig2m+1
15/bracketrightbig
q90
×[90]q[3]q[4]q
[5]q[6]q[9]q[3m+2]q4[4m+3]q3/bracketleftbigm+1
2/bracketrightbig
q24[6]q4
[2]q4[3]q4,
which, by Corollary 6 and Lemma 9, is a polynomial in qwith non-negative integer
coefficients.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 15
Ifm≡8 (mod 30),then we have
Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6/bracketleftbig3m+2
2/bracketrightbig
q8
×/bracketleftbig4m+3
5/bracketrightbig
q15[15]q
[3]q[5]q/bracketleftbigm+1
3/bracketrightbig
q36[12]q3
[3]q3[4]q3,
which, by Lemma 11, is a polynomial in qwith non-negative integer coefficients.
Ifm≡13 (mod 30) ,then we have
Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1
3/bracketrightbig
q18[6]q3
[2]q3[3]q3
×[3m+2]q4/bracketleftbig4m+3
5/bracketrightbig
q15[15]q
[3]q[5]q/bracketleftbigm+1
2/bracketrightbig
q24[6]q4
[2]q4[3]q4,
which, by Lemma 12, is a polynomial in qwith non-negative integer coefficients.
Ifm≡22 (mod 30) ,then we have
Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1
15/bracketrightbig
q90[90]q[3]q
[5]q[6]q[9]q
×/bracketleftbig3m+2
2/bracketrightbig
q8[4m+3]q3[m+1]q12,
which, by Lemma 10, is a polynomial in qwith non-negative integer coefficients.
Ifm≡23 (mod 30) ,then we have
Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6
×[3m+2]q4/bracketleftbig4m+3
5/bracketrightbig
q15[15]q
[3]q[5]q/bracketleftbigm+1
6/bracketrightbig
q72[72]q[3]q[4]q
[8]q[9]q[12]q,
which, by Lemma 8, is a polynomial in qwith non-negative integer coefficients.
ForW=G36=E7, the degrees are 2 ,6,8,10,12,14,18, and hence
Catm(E7;q) =[18m+2]q[18m+6]q[18m+8]q[18m+10]q
[2]q[6]q[8]q[10]q
×[18m+12]q[18m+14]q[18m+18]q
[12]q[14]q[18]q.
Ifm≡18 (mod 140) ,then we have
Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
5/bracketrightbig
q30[15]q2
[3]q2[5]q2
×/bracketleftbig9m+4
2/bracketrightbig
q4[9m+5]q2/bracketleftbig3m+2
28/bracketrightbig
q168[84]q2[2]q2
[4]q2[6]q2[7]q2[9m+7]q2[m+1]q18,
which, by Corollary 6 and Lemma 13, is a polynomial in qwith non-negative integer
coefficients.16 C. KRATTENTHALER AND T. W. M ¨ULLER
Ifm≡23 (mod 140) ,then we have
Catm(E7;q) =/bracketleftbig9m+1
4/bracketrightbig
q8/bracketleftbig3m+1
35/bracketrightbig
q210[105]q2
[3]q2[5]q2[7]q2[9m+4]q2[9m+5]q2
×[3m+2]q6[9m+7]q2/bracketleftbigm+1
2/bracketrightbig
q36[6]q6
[2]q6[3]q6,
which, by Corollary 6 and Lemma 14, is a polynomial in qwith non-negative integer
coefficients.
Ifm≡54 (mod 140) ,then we have
Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
70/bracketrightbig
q140[70]q2
[2]q2[5]q2[7]q2[9m+5]q2
×/bracketleftbig3m+2
4/bracketrightbig
q24[6]q4
[2]q4[3]q4[9m+7]q2[m+1]q18.
Ifonedecomposes[9 m+7]q2as[9m
2+4]q4+q2[9m
2+3]q4, thenoneseesthat, byCorollary6
and Lemma 15, this is a polynomial in qwith non-negative integer coefficients.
ForW=G37=E8, the degrees are 2 ,8,12,14,18,20,24,30, and hence
Catm(E7;q) =[30m+2]q[30m+8]q[30m+12]q[30m+14]q
[2]q[8]q[12]q[14]q
×[30m+18]q[30m+20]q[30m+24]q[30m+30]q
[18]q[20]q[24]q[30]q.
Ifm≡3 (mod 84),then we have
Catm(E8;q) =/bracketleftbig15m+1
2/bracketrightbig
q4/bracketleftbig15m+4
7/bracketrightbig
q14[5m+2]q6/bracketleftbig15m+7
4/bracketrightbig
q8/bracketleftbig5m+3
6/bracketrightbig
q36[6]q6
[2]q6[3]q6
×[3m+2]q10[5m+4]q6/bracketleftbigm+1
4/bracketrightbig
q120[60]q2[2]q2[3]q2[5]q2
[10]q2[12]q2[15]q2,
which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer
coefficients.
Ifm≡8 (mod 84),then we have
Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
4/bracketrightbig
q8/bracketleftbig5m+2
42/bracketrightbig
q252[126]q2[3]q2
[6]q2[7]q2[9]q2[15m+7]q2[5m+3]q6
×/bracketleftbig3m+2
2/bracketrightbig
q20/bracketleftbig5m+4
4/bracketrightbig
q24[m+1]q30,
which, by Lemma 22, is a polynomial in qwith non-negative integer coefficients.
Ifm≡11 (mod 84) ,then we have
Catm(E8;q) =/bracketleftbig15m+1
2/bracketrightbig
q4[15m+4]q2/bracketleftbig5m+2
3/bracketrightbig
q18/bracketleftbig15m+7
4/bracketrightbig
q8/bracketleftbig5m+3
2/bracketrightbig
q12
×/bracketleftbig3m+2
7/bracketrightbig
q70[35]q2
[5]q2[7]q2[5m+4]q6/bracketleftbigm+1
4/bracketrightbig
q120[60]q2[2]q2[3]q2[5]q2
[10]q2[12]q2[15]q2,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 17
which, by Corollary 6 and Lemma 20, is a polynomial in qwith non-negative integer
coefficients.
Ifm≡16 (mod 84) ,then we have
Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
4/bracketrightbig
q8/bracketleftbig5m+2
2/bracketrightbig
q12[15m+7]q2[5m+3]q6
×/bracketleftbig3m+2
2/bracketrightbig
q20/bracketleftbig5m+4
84/bracketrightbig
q504[252]q2[3]q2
[7]q2[9]q2[12]q2[m+1]q30,
which, by Lemma 23, is a polynomial in qwith non-negative integer coefficients.
Ifm≡18 (mod 84) ,then we have
Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
2/bracketrightbig
q4/bracketleftbig5m+2
4/bracketrightbig
q24[15m+7]q2/bracketleftbig5m+3
3/bracketrightbig
q18
/bracketleftbig3m+2
28/bracketrightbig
q280[140]q2[2]q2
[4]q2[7]q2[10]q2/bracketleftbig5m+4
2/bracketrightbig
q12[m+1]q30,
which, by Lemma 24, is a polynomial in qwith non-negative integer coefficients.
Ifm≡21 (mod 84) ,then we have
Catm(E8;q) =/bracketleftbig15m+1
4/bracketrightbig
q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7
14/bracketrightbig
q28[14]q2
[2]q2[7]q2/bracketleftbig5m+3
12/bracketrightbig
q72[12]q6
[3]q6[4]q6
×[3m+2]q10[5m+4]q6/bracketleftbigm+1
2/bracketrightbig
q60[30]q2[2]q2[3]q2[5]q2
[6]q2[10]q2[15]q2,
which, by Corollary 6 and Lemma 17, is a polynomial in qwith non-negative integer
coefficients.
Ifm≡25 (mod 84) ,then we have
Catm(E8;q) =/bracketleftbig15m+1
4/bracketrightbig
q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7
2/bracketrightbig
q4/bracketleftbig5m+3
4/bracketrightbig
q24
×/bracketleftbig3m+2
7/bracketrightbig
q70[35]q2
[5]q2[7]q2/bracketleftbig5m+4
3/bracketrightbig
q18/bracketleftbigm+1
2/bracketrightbig
q60[30]q2[2]q2[3]q2[5]q2
[6]q2[10]q2[15]q2,
which, by Lemma 18, is a polynomial in qwith non-negative integer coefficients.
Ifm≡27 (mod 84) ,then we have
Catm(E8;q) =/bracketleftbig15m+1
14/bracketrightbig
q28[14]q2
[2]q2[7]q2[15m+4]q2[5m+2]q6/bracketleftbig15m+7
4/bracketrightbig
q8/bracketleftbig5m+3
6/bracketrightbig
q36[6]q6
[2]q6[3]q6
×[3m+2]q10[5m+4]q6/bracketleftbigm+1
4/bracketrightbig
q120[60]q2[2]q2[3]q2[5]q2
[10]q2[12]q2[15]q2,
which, by Corollary 6 and Lemma 21, is a polynomial in qwith non-negative integer
coefficients.
All other cases are disposed of in a similar fashion. /square
5.Auxiliary results I
This section collects several auxiliary results which allow us to reduce the problem
of proving Theorem 2, or the equivalent statement (3.3), for the 2 6 exceptional groups
listed in Section 2 to a finite problem. While Lemmas 27 and 28 cover spec ial choices
of the parameters, Lemmas 26 and 30 afford an inductive procedur e. More precisely,18 C. KRATTENTHALER AND T. W. M ¨ULLER
if we assume that we have already verified Theorem 2 for all groups o f smaller rank,
then Lemmas 26 and 30, together with Lemmas 27 and 31, reduce th e verification of
Theorem 2 for the group that we are currently considering to a finit e problem; see
Remark 3. The final lemma of this section, Lemma 32, disposes of com plex reflection
groups with a special property satisfied by their degrees.
Letp=am+b, 0≤b<m. We have
φp/parenleftbig
(w0;w1,...,w m)/parenrightbig
= (∗;ca+1wm−b+1c−a−1,ca+1wm−b+2c−a−1,...,ca+1wmc−a−1,
caw1c−a,...,cawm−bc−a/parenrightbig
,(5.1)
where∗stands for the element of Wwhich is needed to complete the product of the
components to c.
Lemma 26. It suffices to check (3.3)forpa divisor of mh. More precisely, let pbe
a divisor of mh, and letkbe another positive integer with gcd(k,mh/p) = 1, then we
have
Catm(W;q)/vextendsingle/vextendsingle
q=e2πip/mh= Catm(W;q)/vextendsingle/vextendsingle
q=e2πikp/mh (5.2)
and
|FixNCm(W)(φp)|=|FixNCm(W)(φkp)|. (5.3)
Proof.For (5.2), this follows immediately from
lim
q→ζ[α]q
[β]q=/braceleftBigg
α
βifα≡β≡0 (modd),
1 otherwise ,(5.4)
whereζis ad-th root of unity and α,βare non-negative integers such that α≡β
(modd).
In order to establish (5.3), suppose that x∈FixNCm(W)(φp), that is,x∈NCm(W)
andφp(x) =x. It obviously follows that φkp(x) =x, so thatx∈FixNCm(W)(φkp).
To establish the converse, note that, if gcd( k,mh/p) = 1, then there exists k′with
k′k≡1 (modmh
p). It follows that, if x∈FixNCm(W)(φkp), that is, if x∈NCm(W) and
φkp(x) =x, thenx=φk′kp(x) =φp(x), whencex∈FixNCm(W)(φp). /square
Lemma 27. Letpbe a divisor of mh. Ifpis divisible by m, then(3.3)is true.
Proof.According to (5.1), the action of φponNCm(W) is described by
φp/parenleftbig
(w0;w1,...,w m)/parenrightbig
= (∗;cp/mw1c−p/m,...,cp/mwmc−p/m/parenrightbig
.
Hence, if (w0;w1,...,w m) is fixed by φp, then each individual wimust be fixed under
conjugation by cp/m.
Using the notation W′= Cent W(cp/m), theprevious observationmeans that wi∈W′,
i= 1,2,...,m. Springer [33, Theorem 4.2] (see also [24, Theorem 11.24(iii)]) prove d
thatW′is a well-generated complex reflection group whose degrees coincide with those
degrees ofWthat are divisible by mh/p. It was furthermore shown in [9, Lemma 3.3]
that
NC(W)∩W′=NC(W′). (5.5)CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 19
Hence, the tuples ( w0;w1,...,w m) fixed byφpare in fact identical with the elements of
NCm(W′), which implies that
|FixNCm(W)(φp)|=|NCm(W′)|. (5.6)
Application of Theorem 1 with Wreplaced by W′and of the “limit rule” (5.4) then
yields that
|NCm(W′)|=/productdisplay
1≤i≤n
mh
p|dimh+di
di= Catm(W;q)/vextendsingle/vextendsingle
q=e2πip/mh. (5.7)
Combining (5.6) and (5.7), we obtain (3.3). This finishes the proof of t he lemma. /square
Lemma 28. Equation (3.3)holds for all divisors pofm.
Proof.Using (5.4) and the fact that the degrees of irreducible well-genera ted complex
reflection groups satisfy di<hfor alli<n, we see that
Catm(W;q)/vextendsingle/vextendsingle
q=e2πip/mh=/braceleftBigg
m+1 ifm=p,
1 ifm/ne}ationslash=p.
On the other hand, if ( w0;w1,...,w m) is fixed by φp, then, because of the action (5.1),
we must have w1=wp+1=···=wm−p+1andw1=cwm−p+1c−1. In particular,
w1∈CentW(c). By the theorem of Springer cited in the proof of Lemma 27, the
subgroup Cent W(c) is itself a complex reflection group whose degrees are those degre es
ofWthat are divisible by h. The only such degree is hitself, hence Cent W(c) is the
cyclic group generated by c. Moreover, by (5.5), we obtain that w1=ε, the identity
element of W, orw1=c. Therefore, for m=pthe set Fix NCm(W)(φp) consists of the
m+1 elements ( w0;w1,...,w m) obtained by choosing wi=cfor a particular ibetween
0 andm, all otherwj’s being equal to ε, while, for m/ne}ationslash=p, we have
FixNCm(W)(φp) =/braceleftbig
(c;ε,...,ε)/bracerightbig
,
whence the result. /square
Lemma 29. LetWbe an irreducible well-generated complex reflection group a ll of
whose degrees are divisible by d. Then each element of Wis fixed under conjugation by
ch/d.
Proof.By the theorem of Springer cited in the proof of Lemma 27, the subg roupW′=
CentW(ch/d) is itself a complex reflection group whose degrees are those degre es ofW
that are divisible by d. Thus, by our assumption, the degrees of W′coincide with the
degrees ofW, and hence W′must be equal to W. Phrased differently, each element of
Wis fixed under conjugation by ch/d, as claimed. /square
Lemma 30. LetWbe an irreducible well-generated complex reflection group o f rankn,
and letp=m1h1be a divisor of mh, wherem=m1m2andh=h1h2. Without loss of
generality, we assume that gcd(h1,m2) = 1. Suppose that Theorem 2has already been
verified for all irreducible well-generated complex reflect ion groups with rank <n. Ifh2
does not divide all degrees di, then Equation (3.3)is satisfied.20 C. KRATTENTHALER AND T. W. M ¨ULLER
Proof.Let us write h1=am2+b, with 0 ≤b < m 2. The condition gcd( h1,m2) = 1
translates into gcd( b,m2) = 1. From (5.1), we infer that
φp/parenleftbig
(w0;w1,...,w m)/parenrightbig
= (∗;ca+1wm−m1b+1c−a−1,ca+1wm−m1b+2c−a−1,...,ca+1wmc−a−1,
caw1c−a,...,cawm−m1bc−a/parenrightbig
.(5.8)
Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
wi=ca+1wi+m−m1bc−a−1, i= 1,2,...,m 1b,
wi=cawi−m1bc−a, i=m1b+1,m1b+2,...,m,
which, after iteration, implies in particular that
wi=cb(a+1)+(m2−b)awic−b(a+1)−(m2−b)a=ch1wic−h1, i= 1,2,...,m.
It is at this point where we need gcd( b,m2) = 1. The last equation shows that each wi,
i= 1,2,...,m, and thus also w0, lies in Cent W(ch1). By the theorem of Springer cited
in the proof of Lemma 27, this centraliser subgroup is itself a complex reflection group,
W′say, whose degrees are those degrees of Wthat are divisible by h/h1=h2. Since,
by assumption, h2does not divide alldegrees,W′has rank strictly less than n. Again
by assumption, we know that Theorem 2 is true for W′, so that in particular,
|FixNCm(W′)(φp)|= Catm(W′;q)/vextendsingle/vextendsingle
q=e2πip/mh.
The arguments above together with (5.5) show that Fix NCm(W)(φp) = Fix NCm(W′)(φp).
On the other hand, using (5.4) it is straightforward to see that
Catm(W;q)/vextendsingle/vextendsingle
q=e2πip/mh= Catm(W′;q)/vextendsingle/vextendsingle
q=e2πip/mh.
This proves (3.3) for our particular p, as required. /square
Lemma 31. LetWbe an irreducible well-generated complex reflection group o f rank
n, and letp=m1h1be a divisor of mh, wherem=m1m2andh=h1h2. We assume
thatgcd(h1,m2) = 1. Ifm2>nthen
FixNCm(W)(φp) =/braceleftbig
(c;ε,...,ε)/bracerightbig
.
Proof.Let us suppose that ( w0;w1,...,w m)∈FixNCm(W)(φp) and that there exists a
j≥1 such that wj/ne}ationslash=ε. By (5.8), it then follows for such a jthat alsowk/ne}ationslash=εfor
allk≡j−lm1b(modm), where, as before, bis defined as the unique integer with
h1=am2+band 0≤b < m 2. Since, by assumption, gcd( b,m2) = 1, there are
exactlym2suchk’s which are distinct mod m. However, this implies that the sum of
the absolute lengths of the wi’s, 0≤i≤m, is at least m2> n, a contradiction to
Remark 1.(2). /square
Remark 3.(1) If we put ourselves in the situation of the assumptions of Lemma 30,
then we may conclude that equation (3.3) only needs to be checked f or pairs (m2,h2)
subject to the following restrictions:
m2≥2,gcd(h1,m2) = 1,andh2divides all degrees of W. (5.9)
Indeed, Lemmas 27 and 30 together imply that equation (3.3) is alway s satisfied in all
other cases.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 21
(2) Still putting ourselves in the situation of Lemma 30, if m2>nandm2h2does not
divide any of the degrees of W, then equation (3.3) is satisfied. Indeed, Lemma 31 says
thatinthiscasetheleft-handsideof (3.3)equals1,whileastraightf orwardcomputation
using (5.4) shows that in this case the right-hand side of (3.3) equals 1 as well.
(3)It shouldbeobserved that thisleaves afinitenumber of choices form2to consider,
whence a finite number of choices for ( m1,m2,h1,h2). Altogether, there remains a finite
number of choices for p=h1m1to be checked.
Lemma 32. LetWbe an irreducible well-generated complex reflection group o f rankn
with the property that di|hfori= 1,2,...,n. Then Theorem 2is true for this group
W.
Proof.By Lemma 26, we may restrict ourselves to divisors pofmh.
Suppose that e2πip/mhis adi-th rootof unity for some i. In other words, mh/pdivides
di. Sincediis a divisor of hby assumption, the integer mh/palso divides h. But this
is equivalent to saying that mdividesp, and equation (3.3) holds by Lemma 27.
Now assume that mh/pdoes not divide any of the di’s. Then, by (5.4), the right-
hand side of (3.3) equals 1. On the other hand, ( c;ε,...,ε) is always an element of
FixNCm(W)(φp). To see that there are no others, we make appeal to the classific a-
tion of all irreducible well-generated complex reflection groups, whic h we recalled in
Section 2. Inspection reveals that all groups satisfying the hypot heses of the lemma
have rank n≤2. Except for the groups contained in the infinite series G(d,1,n)
andG(e,e,n) for which Theorem 2 has been established in [19], these are the grou ps
G5,G6,G9,G10,G14,G17,G18,G21. We now discuss these groups case by case, keeping
the notation of Lemma 30. In order to simplify the argument, we not e that Lemma 31
implies that equation (3.3) holds if m2>2, so that in the following arguments we
always may assume that m2= 2.
CaseG5. The degrees are 6 ,12, and therefore Remark 3.(1) implies that equa-
tion (3.3) is always satisfied.
CaseG6. The degrees are 4 ,12, and therefore, according to Remark 3.(1), we need
only consider the casewhere h2= 4andm2= 2, that is, p= 3m/2. Then (5.8) becomes
φp/parenleftbig
(w0;w1,...,w m)/parenrightbig
= (∗;c2wm
2+1c−2,c2wm
2+2c−2,...,c2wmc−2,cw1c−1,...,cw m
2c−1/parenrightbig
.
(5.10)
If (w0;w1,...,w m) isfixed by φpandnot equal to ( c;ε,...,ε), there must exist an iwith
1≤i≤m
2such thatℓT(wi) =ℓT(wm
2+i) = 1,wm
2+i=cwic−1,wiwm
2+i=wicwic−1=c,
and allwj, withj/ne}ationslash=i,m
2+i, equalε. However, with the help of the GAPpackage
CHEVIE[14, 28], one verifies that there is no wiinG6such that
ℓT(wi) = 1 and wicwic−1=c
are simultaneously satisfied. Hence, the left-hand side of (3.3) is eq ual to 1, as required.
CaseG9. The degrees are 8 ,24, and therefore, according to Remark 3.(1), we need
only consider the case where h2= 8 andm2= 2, that is, p= 3m/2. This is the same p
as forG6. Again, CHEVIEfinds no solution. Hence, the left-hand side of (3.3) is equal
to 1, as required.
CaseG10. The degrees are 12 ,24, and therefore Remark 3.(1) implies that equa-
tion (3.3) is always satisfied.22 C. KRATTENTHALER AND T. W. M ¨ULLER
CaseG14. The degrees are 6 ,24, and therefore Remark 3.(1) implies that equa-
tion (3.3) is always satisfied.
CaseG17. The degrees are 20 ,60, and therefore, according to Remark 3.(1), we need
only consider the cases where h2= 20 orh2= 4. In the first case, p= 3m/2, which is
the samepas forG6. Again,CHEVIEfinds no solution. In the second case, p= 15m/2.
Then (5.8) becomes
φp/parenleftbig
(w0;w1,...,w m)/parenrightbig
= (∗;c8wm
2+1c−8,c8wm
2+2c−8,...,c8wmc−8,c7w1c−7,...,c7wm
2c−7/parenrightbig
.(5.11)
By Lemma 29, every element of NC(W) is fixed under conjugation by c3, and, thus, on
elements fixed by φp, the above action of φpreduces to the one in (5.10). This action
was already discussed in the first case. Hence, in both cases, the le ft-hand side of (3.3)
is equal to 1, as required.
CaseG18. The degrees are 30 ,60, and therefore Remark 3.(1) implies that equa-
tion (3.3) is always satisfied.
CaseG21. The degrees are 12 ,60, and therefore, according to Remark 3.(1), we need
only consider the cases where h2= 12 orh2= 4. In the first case, p= 5m/2, so that
(5.8) becomes
φp/parenleftbig
(w0;w1,...,w m)/parenrightbig
= (∗;c3wm
2+1c−3,c3wm
2+2c−3,...,c3wmc−3,c2w1c−2,...,c2wm
2c−2/parenrightbig
.(5.12)
If (w0;w1,...,w m) is fixed by φpand not equal to ( c;ε,...,ε), there must exist an i
with 1≤i≤m
2such thatℓT(wi) = 1 andwic2wic−2=c. However, with the help of
theGAPpackageCHEVIE[14, 28], one verifies that there is no such solution to this
equation. In the second case, p= 15m/2. Then (5.8) becomes the action in (5.11).
By Lemma 29, every element of NC(W) is fixed under conjugation by c5, and, thus,
on elements fixed by φp, the action of φpin (5.11) reduces to the one in the first case.
Hence, in both cases, the left-hand side of (3.3) is equal to 1, as re quired.
This completes the proof of the lemma. /square
6.Exemplification of case-by-case verification of Theorem 2
It remains to verify Theorem 2 for the groups G4,G8,G16,G20,G23=H3,G24,G25,
G26,G27,G28=F4,G29,G30=H4,G32,G33,G34,G35=E6,G36=E7,G37=E8. All
details can be found in [21, Sec. 6]. We content ourselves with illustra ting the type of
computation that is needed here by going through the case of the g roupG24, and by
discussing some of the arguments needed for the group G37=E8.
In the sequel we write ζdfor a primitive d-th root of unity.
CaseG24.The degrees are 4 ,6,14, and hence we have
Catm(G24;q) =[14m+14]q[14m+6]q[14m+4]q
[14]q[6]q[4]q.
Letζbe a 14m-th root of unity. In what follows, we abbreviate the assertion tha t “ζis
a primitive d-th root of unity” as “ ζ=ζd.” The following cases on the right-hand sideCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 23
of (3.3) occur:
lim
q→ζCatm(G24;q) =m+1,ifζ=ζ14,ζ7, (6.1a)
lim
q→ζCatm(G24;q) =7m+3
3,ifζ=ζ6,ζ3,3|m, (6.1b)
lim
q→ζCatm(G24;q) =7m+2
2,ifζ=ζ4,2|m, (6.1c)
lim
q→ζCatm(G24;q) = Catm(G24),ifζ=−1 orζ= 1, (6.1d)
lim
q→ζCatm(G24;q) = 1,otherwise. (6.1e)
We must now prove that the left-handside of (3.3) in each case agre es with the values
exhibited in (6.1). The only cases not covered by Lemma 27 are the on es in (6.1b),
(6.1c), and (6.1e). (In both (6.1a) and (6.1d) we have d|h.)
We first consider (6.1b). By Lemma 26, we are free to choose p= 7m/3 ifζ=ζ6,
respectively p= 14m/3 ifζ=ζ3. In both cases, mmust be divisible by 3.
We start with the case that p= 7m/3. From (5.1), we infer
φp/parenleftbig
(w0;w1,...,w m)/parenrightbig
= (∗;c3w2m
3+1c−3,c3w2m
3+2c−3,...,c3wmc−3,c2w1c−2,...,c2w2m
3c−2/parenrightbig
.
Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
wi=c3w2m
3+ic−3, i= 1,2,...,m
3, (6.2a)
wi=c2wi−m
3c−2, i=m
3+1,m
3+2,...,m. (6.2b)
There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
are equal to ε, or there is an iwith 1≤i≤m
3such that
ℓT(wi) =ℓT(wi+m
3) =ℓT(wi+2m
3) = 1.
Writingt1,t2,t3forwi,wi+m
3,wi+2m
3, respectively, the equations (6.2) reduce to
t1=c3t3c−3, (6.3a)
t2=c2t1c−2, (6.3b)
t3=c2t2c−2. (6.3c)
One of these equations is in fact superfluous: if we substitute (6.3b ) and (6.3c) in
(6.3a), then we obtain t1=c7t1c−7which is automatically satisfied due to Lemma 29
withd= 2.
Since (w0;w1,...,w m)∈NCm(G24), we must have t1t2t3=c. Combining this with
(6.3), we infer that
t1(c2t1c−2)(c4t1c−4) =c. (6.4)
With the help of CHEVIE, one obtains 7 solutions for t1in this equation, each of them
giving rise to m/3 elements of Fix NCm(G24)(φp) sincei(inwi) ranges from 1 to m/3.
In total, we obtain 1 + 7m
3=7m+3
3elements in Fix NCm(G24)(φp), which agrees with
the limit in (6.1b).
The case where p= 14m/3 can be treated in a similar fashion. In the end, it
turns out that we have to solve the same enumeration problem as fo rp= 7m/3, and,24 C. KRATTENTHALER AND T. W. M ¨ULLER
consequently, the number of elements of Fix NCm(G24)(φp) is the same, namely7m+3
3, as
required.
Our next case is (6.1c). Proceeding in a similar manner as before, we s ee that there is
againthe trivial possibility ( c;ε,...,ε), and otherwise we have to find t1withℓT(t1) = 1
satisfying the inequality
t1(c3t1c−3)≤Tc. (6.5)
With the help of CHEVIE, one obtains 7 solutions for t1in this relation, each of them
giving rise to m/2 elements of Fix NCm(G24)(φp) sincei(inwi) ranges from 1 to m/2.
In total, we obtain 1 + 7m
2=7m+2
2elements in Fix NCm(G24)(φp), which agrees with
the limit in (6.1c).
Finally, we turn to (6.1e). By Remark 3, the only choices for h2andm2to be consid-
ered areh2= 1 andm2= 3,h2=m2= 2, andh2= 2 andm2= 3. These correspond
to the choices p= 14m/3,p= 7m/2, respectively p= 7m/3, all of which have already
been discussed as they do not belong to (6.1e). Hence, (3.3) must n ecessarily hold, as
required.
CaseG37=E8.The degrees are 2 ,8,12,14,18,20,24,30, and hence we have
Catm(E8;q) =[30m+30]q[30m+24]q[30m+20]q[30m+18]q
[30]q[24]q[20]q[18]q
×[30m+14]q[30m+12]q[30m+8]q[30m+2]q
[14]q[12]q[8]q[2]q.
Letζbe a 30m-th root of unity. The cases occurring on the right-hand side of (3 .3)
not covered by Lemma 27 are:
lim
q→ζCatm(E8;q) =5m+4
4,ifζ=ζ24,4|m, (6.6a)
lim
q→ζCatm(E8;q) =3m+2
2,ifζ=ζ20,2|m, (6.6b)
lim
q→ζCatm(E8;q) =5m+3
3,ifζ=ζ18,ζ9,3|m, (6.6c)
lim
q→ζCatm(E8;q) =15m+7
7,ifζ=ζ14,ζ7,7|m, (6.6d)
lim
q→ζCatm(E8;q) =(5m+4)(5m+2)
8,ifζ=ζ12,2|m, (6.6e)
lim
q→ζCatm(E8;q) =(5m+4)(15m+4)
16,ifζ=ζ8,4|m, (6.6f)
lim
q→ζCatm(E8;q) =(5m+4)(3m+2)(5m+2)(15m+4)
64,ifζ=ζ4,2|m,(6.6g)
lim
q→ζCatm(E8;q) = Catm(E8),ifζ=−1 orζ= 1, (6.6h)
lim
q→ζCatm(E8;q) = 1,otherwise. (6.6i)
We now have to prove that the left-hand side of (3.3) in each case ag rees with the
values exhibited in (6.6). Since the corresponding computations in th e various cases are
very similar, we concentrate here only on the cases (6.6f) and (6.6g ), these two being
representative of the types of arguments arising. As before, we refer the reader to [21,
Sec. 6] for full details.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 25
Letusconsiderthecasein(6.6f)first. ByLemma26, wearefreeto choosep= 15m/4.
In particular, mmust be divisible by 4. From (5.1), we infer
φp/parenleftbig
(w0;w1,...,w m)/parenrightbig
= (∗;c4wm
4+1c−4,c4wm
4+2c−4,...,c4wmc−4,c3w1c−3,...,c3wm
4c−3/parenrightbig
.
Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
wi=c4wm
4+ic−4, i= 1,2,...,3m
4, (6.7a)
wi=c3wi−3m
4c−3, i=3m
4+1,3m
4+2,...,m. (6.7b)
There are several distinct possibilities for choosing the wi’s, 1≤i≤m, which we
summarise as follows:
(i) all thewi’s are equal to ε(andw0=c),
(ii) there is an iwith 1≤i≤m
4such that
1≤ℓT(wi) =ℓT(wi+m
4) =ℓT(wi+2m
4) =ℓT(wi+3m
4)≤2, (6.8a)
and the other wj’s, 1≤j≤m, are equal to ε,
(iii) there are i1andi2with 1≤i1<i2≤m
4such that
ℓT(wi1) =ℓT(wi2) =ℓT(wi1+m
4) =ℓT(wi2+m
4)
=ℓT(wi1+2m
4) =ℓT(wi2+2m
4) =ℓT(wi1+3m
4) =ℓT(wi2+3m
4) = 1,(6.8b)
and all other wjare equal to ε.
Moreover, since ( w0;w1,...,w m)∈NCm(E8), we must have
wiwi+m
4wi+2m
4wi+3m
4≤Tc,
or
wi1wi2wi1+m
4wi2+m
4wi1+2m
4wi2+2m
4wi1+3m
4wi2+3m
4=c.
Together with equations (6.7)–(6.8), this implies that
wi=c15wic−15andwi(c11wic−11)(c7wic−7)(c3wic−3)≤Tc, (6.9)
or that
wi1=c15wi1c−15, wi1=c15wi2c−15,
andwi1wi2(c11wi1c−11)(c11wi2c−11)(c7wi1c−7)(c7wi2c−7)(c3wi1c−3)(c3wi2c−3) =c.
(6.10)
Here, the first equation in (6.9) and the first two equations in (6.10) are automatically
satisfied due to Lemma 29 with d= 2.
With the help of Stembridge’s Maplepackagecoxeter [36], one obtains 30 solutions
forwiin (6.9) with ℓT(wi) = 1, 45 solutions for wiwithℓT(wi) = 2 and wiof type
A2
1(as a parabolic Coxeter element; see the end of Section 2), and 20 s olutions for
wiwithℓT(wi) = 2 and wiof typeA2. Each of them gives rise to m/4 elements of
FixNCm(E8)(φp) sinceiranges from 1 to m/4.
The number of solutions in Case (iii) can be computed from our knowled ge of the
solutions in Case (ii) according to type, using some elementary count ing arguments.
Namely, the number of solutions of (6.10) is equal to
45·2+20·3 = 150,26 C. KRATTENTHALER AND T. W. M ¨ULLER
since an element of type A2
1can be decomposed in two ways into a product of two
elements of absolute length 1, while for an element of type A2this can be done in 3
ways.
In total, we obtain 1 + (30 + 45 + 20)m
4+ 150/parenleftbigm/4
2/parenrightbig
=(5m+4)(15m+4)
16elements in
FixNCm(E8)(φp), which agrees with the limit in (6.6f).
Next, we discuss the case in (6.6g). By Lemma 26, we are free to cho osep= 15m/2.
In particular, mmust be divisible by 2. From (5.1), we infer
φp/parenleftbig
(w0;w1,...,w m)/parenrightbig
= (∗;c8wm
2+1c−8,c8wm
2+2c−8,...,c8wmc−8,c7w1c−7,...,c7wm
2c−7/parenrightbig
.
Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
wi=c8wm
2+ic−8, i= 1,2,...,m
2, (6.11a)
wi=c7wi−m
2c−7, i=m
2+1,m
2+2,...,m. (6.11b)
There are several distinct possibilities for choosing the wi’s, 1≤i≤m:
(i) all thewi’s are equal to ε(andw0=c),
(ii) there is an iwith 1≤i≤m
2such that
1≤ℓT(wi) =ℓT(wi+m
2)≤4, (6.12a)
and the other wj’s, 1≤j≤m, are equal to ε,
(iii) there are i1andi2with 1≤i1<i2≤m
2such that
ℓ1:=ℓT(wi1) =ℓT(wi1+m
2)≥1, ℓ2:=ℓT(wi2) =ℓT(wi2+m
2)≥1,andℓ1+ℓ2≤4,
(6.12b)
and the other wj’s, 1≤j≤m, are equal to ε,
(iv) there are i1,i2,i3with 1≤i1<i2<i3≤m
2such that
ℓ1:=ℓT(wi1) =ℓT(wi1+m
2)≥1, ℓ2:=ℓT(wi2) =ℓT(wi2+m
2)≥1,
ℓ3:=ℓT(wi3) =ℓT(wi3+m
2)≥1,andℓ1+ℓ2+ℓ3≤4,(6.12c)
and the other wj’s, 1≤j≤m, are equal to ε,
(v) there are i1,i2,i3,i4with 1≤i1<i2<i3<i4≤m
2such that
ℓT(wi1) =ℓT(wi2) =ℓT(wi3) =ℓT(wi4)
=ℓT(wi1+m
2) =ℓT(wi2+m
2) =ℓT(wi3+m
2) =ℓT(wi4+m
2) = 1,(6.12d)
and all other wj’s are equal to ε.
Moreover, since ( w0;w1,...,w m)∈NCm(E8), we must have wiwi+m
2≤Tc, respec-
tivelywi1wi2wi1+m
2wi2+m
2≤Tc, respectively
wi1wi2wi3wi1+m
2wi2+m
2wi3+m
2≤Tc,
respectively
wi1wi2wi3wi4wi1+m
2wi2+m
2wi3+m
2wi4+m
2=c.
Together with equations (6.11)–(6.12), this implies that
wi=c15wic−15andwi(c7wic−7)≤Tc, (6.13)
respectively that
wi1=c15wi1c−15, wi2=c15wi2c−15,andwi1wi2(c7wi1c−7)(c7wi2c−7)≤Tc,(6.14)CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 27
respectively that
wi1=c15wi1c−15, wi2=c15wi2c−15, wi3=c15wi3c−15,
andwi1wi2wi3(c7wi1c−7)(c7wi2c−7)(c7wi3c−7)≤Tc,(6.15)
respectively that
wi1=c15wi1c−15, wi2=c15wi2c−15, wi3=c15wi3c−15, wi4=c15wi4c−15,
andwi1wi2wi3wi4(c7wi1c−7)(c7wi2c−7)(c7wi3c−7)(c7wi4c−7) =c.(6.16)
Here, the first equation in (6.13), the first two in (6.14), the first t hree in (6.15), and
the first four in (6.16), are all automatically satisfied due to Lemma 2 9 withd= 2.
With the help of Stembridge’s Maplepackagecoxeter [36], one obtains
— 45 solutions for wiin (6.13) with ℓT(wi) = 1,
— 150 solutions for wiin (6.13) with ℓT(wi) = 2 andwiof typeA2
1,
— 100 solutions for wiin (6.13) with ℓT(wi) = 2 andwiof typeA2,
— 75 solutions for wiin (6.13) with ℓT(wi) = 3 andwiof typeA3
1,
— 165 solutions for wiin (6.13) with ℓT(wi) = 3 andwiof typeA1∗A2,
— 90 solutions for wiin (6.13) with ℓT(wi) = 3 andwiof typeA3,
— 15 solutions for wiin (6.13) with ℓT(wi) = 4 andwiof typeA2
1∗A2,
— 45 solutions for wiin (6.13) with ℓT(wi) = 4 andwiof typeA1∗A3;
— 5 solutions for wiin (6.13) with ℓT(wi) = 4 andwiof typeA2
2,
— 18 solutions for wiin (6.13) with ℓT(wi) = 4 andwiof typeA4,
— 5 solutions for wiin (6.13) with ℓT(wi) = 4 andwiof typeD4.
Each of them gives rise to m/2 elements of Fix NCm(E8)(φp) sinceiranges from 1 to m/2.
There are no solutions for wiin (6.13) with wiof typeA4
1.
Letting the computer find all solutions in cases (iii)–(v) would take ye ars. However,
the number of these solutions can be computed from our knowledge of the solutions
in Case (ii) according to type, if this information is combined with the de composition
numbers in the sense of [17, 18, 20] (see the end of Section 2) and some elementary
(multiset) permutation counting. The decomposition numbers for A2,A3,A4, andD4
of which we make use can be found in the appendix of [18].
To begin with, the number of solutions of (6.14) with ℓ1=ℓ2= 1 is equal to
n1,1:= 150·2+100·NA2(A1,A1) = 600,
since an element of type A2
1can be decomposed in two ways into a product of two
elements of absolute length 1, while for an element of type A2this can be done in
NA2(A1,A1) = 3 ways. Similarly, the number of solutions of (6.14) with ℓ1= 2 and
ℓ2= 1 is equal to
n2,1:= 75·3+165·(1+NA2(A1,A1))+90·NA3(A2,A1) = 1425,
the number of solutions of (6.14) with ℓ1= 3 andℓ2= 1 is equal to
n3,1:= 15·(2+NA2(A1,A1))+45·(1+NA3(A2,A1))+5·(2NA2(A1,A1))
+18·(NA4(A3,A1)+NA4(A1∗A2,A1))+5·(ND4(A3,A1)+ND4(A3
1,A1)) = 660,28 C. KRATTENTHALER AND T. W. M ¨ULLER
the number of solutions of (6.14) with ℓ1=ℓ2= 2 is equal to
n2,2:= 15·(2+2NA2(A1,A1))+45·(2NA3(A2,A1))+5·(2+NA2(A1,A1)2)
+18·(NA4(A2,A2)+NA4(A2
1,A2
1)+2NA4(A2,A2
1))
+5·(ND4(A2,A2)+2ND4(A2,A2
1)) = 1195,
the number of solutions of (6.15) with ℓ1=ℓ2=ℓ3= 1 is equal to
n1,1,1:= 75·3!+165·(3NA2(A1,A1))+90NA3(A1,A1,A1) = 3375,
the number of solutions of (6.15) with ℓ1= 2 andℓ2=ℓ3= 1 is equal to
n2,1,1:= 15·(2+NA2(A1,A1)+2·2·NA2(A1,A1))+45·(2NA3(A2,A1)+NA3(A1,A1,A1))
+5·(2NA2(A1,A1)+2NA2(A1,A1)2)+18·(NA4(A2,A1,A1)+NA4(A2
1,A1,A1))
+5·(ND4(A2,A1,A1)+ND4(A2
1,A1,A1)) = 2850,
and the number of solutions of (6.16) is equal to
n1,1,1,1:= 15·(12NA2(A1,A1))+45·(4NA3(A1,A1,A1))+5·(6NA2(A1,A1)2)
+18·NA4(A1,A1,A1,A1)+5·ND4(A1,A1,A1,A1) = 6750.
In total, we obtain
1+(45+150+100+75+165+90+15+45+5+18+5)m
2+(n1,1+2n2,1+2n3,1+n2,2)/parenleftbiggm/2
2/parenrightbigg
+(n1,1,1+3n2,1,1)/parenleftbiggm/2
3/parenrightbigg
+n1,1,1,1/parenleftbiggm/2
4/parenrightbigg
=(5m+4)(3m+2)(5m+2)(15m+4)
64
elements in Fix NCm(E8)(φp), which agrees with the limit in (6.6g).
7.Cyclic sieving II
In this section we present the second cyclic sieving conjecture due to Bessis and
Reiner [9, Conj. 6.5].
Letψ:NCm(W)→NCm(W) be the map defined by
(w0;w1,...,w m)/mapsto→/parenleftbig
cwmc−1;w0,w1,...,w m−1/parenrightbig
. (7.1)
Form= 1, we have w0=cw−1
1, so that this action reduces to the inverse of the
Kreweras complement Kc
idas defined by Armstrong [2, Def. 2.5.3].
It is easy to see that ψ(m+1)hacts as the identity, where his the Coxeter number of
W(see (8.1) below). By slight abuse of notation as before, let C2be the cyclic group
of order (m+1)hgenerated by ψ.
Given these definitions, we are now in the position to state the secon d cyclic sieving
conjecture of Bessis and Reiner. By the results of [19] and of this p aper, it becomes the
following theorem.
Theorem 33. For an irreducible well-generated complex reflection group Wand any
m≥1, the triple (NCm(W),Catm(W;q),C2), whereCatm(W;q)is theq-analogue of
the Fuß–Catalan number defined in (3.2), exhibits the cyclic sieving phenomenon.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 29
By definition of the cyclic sieving phenomenon, we have to prove that
|FixNCm(W)(ψp)|= Catm(W;q)/vextendsingle/vextendsingle
q=e2πip/(m+1)h, (7.2)
for allpin the range 0 ≤p<(m+1)h.
8.Auxiliary results II
This section collects several auxiliary results which allow us to reduce the problem of
proving Theorem 33, respectively the equivalent statement (7.2), for the 26 exceptional
groups listed in Section 2 to a finite problem. The corresponding lemma s, Lemmas 34–
39, are analogues of Lemmas 26–28 and 30–32 in Section 5.
Letp=a(m+1)+b, 0≤b<m+1. We have
ψp/parenleftbig
(w0;w1,...,w m)/parenrightbig
= (ca+1wm−b+1c−a−1;ca+1wm−b+2c−a−1,...,ca+1wmc−a−1,
caw0c−a,...,cawm−bc−a/parenrightbig
.(8.1)
Lemma 34. It suffices to check (7.2)forpa divisor of (m+1)h. More precisely, let pbe
a divisor of (m+1)h, and letkbe another positive integer with gcd(k,(m+1)h/p) = 1,
then we have
Catm(W;q)/vextendsingle/vextendsingle
q=e2πip/(m+1)h= Catm(W;q)/vextendsingle/vextendsingle
q=e2πikp/(m+1)h (8.2)
and
|FixNCm(W)(ψp)|=|FixNCm(W)(ψkp)|. (8.3)
Proof.For (8.3), this follows in the same way as (5.3) in Lemma 26.
For (8.2), we must argue differently than in Lemma 26. Let us write ζ=e2πip/(m+1)h.
For a given group W, we writeS1(W) for the set of all indices isuch thatζdi−h= 1,
and we write S2(W) for the set of all indices isuch thatζdi= 1. By the rule of de
l’Hospital, we have
Catm(W;q)/vextendsingle/vextendsingle
q=e2πip/(m+1)h=

0 if |S1(W)|>|S2(W)|,/producttext
i∈S1(W)(mh+di)/producttext
i∈S2(W)di/producttext
i/∈S1(W)(1−ζdi−h)
/producttext
i/∈S2(W)(1−ζdi),if|S1(W)|=|S2(W)|.
(8.4)
Since, by Theorem 25, Catm(W;q) is a polynomial in q, the case |S1(W)|<|S2(W)|
cannot occur.
We claim that, for the case where |S1(W)|=|S2(W)|, the factors in the quotient of
products/producttext
i/∈S1(W)(1−ζdi−h)/producttext
i/∈S2(W)(1−ζdi)
cancel pairwise. If we assume the correctness of the claim, it is obv ious that we get
the same result if we replace ζbyζk, where gcd( k,(m+1)h/p) = 1, hence establishing
(8.2).
In order to see that our claim is indeed valid, we proceed in a case-by- case fash-
ion, making appeal to the classification of irreducible well-generated complex reflection
groups, which werecalled inSection2. Firstofall, since dn=h, thesetS1(W)isalways
non-empty as it contains the element n. Hence, if we want to have |S1(W)|=|S2(W)|,30 C. KRATTENTHALER AND T. W. M ¨ULLER
the setS2(W) must be non-empty as well. In other words, the integer ( m+ 1)h/p
must divide at least one of the degrees d1,d2,...,d n. In particular, this implies that,
for each fixed reflection group Wof exceptional type, only a finite number of values of
(m+1)h/phas to be checked. Writing Mfor (m+1)h/p, what needs to be checked is
whether the multisets (that is, multiplicities of elements must be taken into account)
{(di−h) modM:i /∈S1(W)}and{dimodM:i /∈S2(W)}
are the same. Since, for a fixed irreducible well-generated complex r eflection group,
thereisonlyafinitenumber ofpossibilities for M, thisamountstoaroutineverification.
/square
Lemma 35. Letpbe a divisor of (m+ 1)h. Ifpis divisible by m+ 1, then(7.2)is
true.
We leave the proof to the reader as it is completely analogous to the p roof of
Lemma 27.
Lemma 36. Equation (7.2)holds for all divisors pofm+1.
Proof.We have
Catm(W;q)/vextendsingle/vextendsingle
q=e2πip/(m+1)h=/braceleftBigg
0 ifp<m+1,
m+1 ifp=m+1.
Here, the first case follows from (8.4) and the fact that we have S1(W)⊇ {n}and
S2(W) =∅ifp|(m+1) andp<m+1.
Ontheother hand, if ( w0;w1,...,w m) is fixed by ψp, then onecanapply anargument
similar to that in Lemma 28 with any witaking the role of w1, 0≤i≤m. It follows
that ifp=m+1, the set Fix NCm(W)(ψp) consists of the m+1 elements ( w0;w1,...,w m)
obtained by choosing wi=cfor a particular ibetween 0 and m, all otherwj’s being
equal toε. Ifp<m+1, then there is no element in Fix NCm(W)(ψp). /square
Lemma 37. LetWbe an irreducible well-generated complex reflection group o f rank
n, and letp=m1h1be a divisor of (m+1)h, wherem+1 =m1m2andh=h1h2. We
assume that gcd(h1,m2) = 1. Suppose that Theorem 33has already been verified for
all irreducible well-generated complex reflection groups w ith rank< n. Ifh2does not
divide all degrees di, then equation (7.2)is satisfied.
We leave the proof to the reader as it is completely analogous to the p roof of
Lemma 30.
Lemma 38. LetWbe an irreducible well-generated complex reflection group o f rank
n, and letp=m1h1be a divisor of (m+1)h, wherem+1 =m1m2andh=h1h2. We
assume that gcd(h1,m2) = 1. Ifm2>nthen
FixNCm(W)(ψp) =∅.
We leave the proof to the reader as it is analogous to the proof of Le mma 31.
Remark 4.By applying the same reasoning as in Remark 3 with Lemmas 30 and 31
replaced by Lemmas 37 and 38, respectively, it follows that we only ne ed to check (7.2)
for pairs (m2,h2) satisfying (5.9) and m2≤n. This reduces the problem to a finite
number of choices.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 31
Lemma 39. LetWbe an irreducible well-generated complex reflection group o f rankn
with the property that di|hfori= 1,2,...,n. Then Theorem 33is true for this group
W.
Proof.Proceeding in a fashion analogous to the beginning of the proof of Le mma 32, we
mayrestricttothecasewhere p|(m+1)hand(m+1)h/pdoesnotdivideanyofthe di’s.
Inthiscase, itfollowsfrom(8.4)andthefactthatwehave S1(W)⊇ {n}andS2(W) =∅
that the right-hand side of (7.2) equals 0. Inspection of the classifi cation of all irre-
ducible well-generated complex reflection groups, which we recalled in Section 2, reveals
that all groups satisfying the hypotheses of the lemma have rank n≤2. Except for the
groups contained in the infinite series G(d,1,n) andG(e,e,n) for which Theorem 2 has
been established in [19], these are the groups G5,G6,G9,G10,G14,G17,G18,G21. The
verification of (7.2) can be done in a similar fashion as in the proof of Le mma 32. We
illustrate this by going through the case of the group G6. In analogy with the earlier
situation, we note that Lemma 38 implies that equation (7.2) holds if m2>2, so that
in the following arguments we may assume that m2= 2.
CaseG6. The degrees are 4 ,12, and therefore, according to Remark 4, we need only
consider the case where h2= 4 andm2= 2, that is, p= 3(m+1)/2. Then the action
ofψpis given by
ψp/parenleftbig
(w0;w1,...,w m)/parenrightbig
= (c2wm+1
2c−2;c2wm+3
2c−2,...,c2wmc−2,cw0c−1,...,cw m−1
2c−1/parenrightbig
.
(8.5)
If (w0;w1,...,w m) is fixed by ψp, there must exist an iwith 0≤i≤m−1
2such that
ℓT(wi) = 1,wicwic−1=c, and allwj,j/ne}ationslash=i,m+1
2+i, equalε. However, with the help of
CHEVIE, one verifies that there is no such solution to this equation. Hence, the left-hand
side of (7.2) is equal to 0, as required.
This completes the proof of the lemma. /square
9.Exemplification of case-by-case verification of Theorem 3 3
It remains to verify Theorem 33 for the groups G4,G8,G16,G20,G23=H3,G24,G25,
G26,G27,G28=F4,G29,G30=H4,G32,G33,G34,G35=E6,G36=E7,G37=E8. All
details can be found in [21, Sec. 9]. We content ourselves with discuss ing the case of
the groupG24, as this suffices to convey the flavour of the necessary computat ions.
In order to simplify our considerations, it should be observed that t he action of ψ
(given in(7.1)) is exactly the same as the actionof φ(given in (3.1)) with mreplaced by
m+1on the components w1,w2,...,w m+1, that is, if we disregard the 0-th component
of the elements of the generalised non-crossing partitions involved . The only difference
which arises is that, while the ( m+ 1)-tuples ( w0;w1,...,w m) in (7.1) must satisfy
w0w1···wm=c, forw1,w2,...,w m+1in (3.1) we only must have w1w2···wm+1≤Tc.
Consequently, we may use the counting results from Section 6, exc ept that we have to
restrict our attention to those elements ( w0;w1,...,w m,wm+1)∈NCm+1(W) for which
w1w2···wm+1=c, or, equivalently, w0=ε.
CaseG24.The degrees are 4 ,6,14, and hence we have
Catm(G24;q) =[14m+14]q[14m+6]q[14m+4]q
[14]q[6]q[4]q.32 C. KRATTENTHALER AND T. W. M ¨ULLER
Letζbe a 14(m+ 1)-th root of unity. The following cases on the right-hand side of
(7.2) occur:
lim
q→ζCatm(G24;q) =m+1,ifζ=ζ14,ζ7, (9.1a)
lim
q→ζCatm(G24;q) =7m+7
3,ifζ=ζ6,ζ3,3|(m+1), (9.1b)
lim
q→ζCatm(G24;q) = Catm(G24),ifζ=−1 orζ= 1, (9.1c)
lim
q→ζCatm(G24;q) = 0,otherwise. (9.1d)
We must now prove that the left-handside of (7.2) in each case agre es with the values
exhibited in (9.1). The only cases not covered by Lemma 35 are the on es in (9.1b) and
(9.1d). On the other hand, the only cases left to consider accordin g to Remark 4 are
the cases where h2= 1 andm2= 3,h2= 2 andm2= 3, andh2=m2= 2. These
correspond to the choices p= 14(m+1)/3,p= 7(m+1)/3, respectively p= 7(m+1)/2.
The first two cases belong to (9.1b), while p= 7(m+1)/2 belongs to (9.1d).
In the case that p= 7(m+1)/3, the action of ψpis given by
ψp/parenleftbig
(w0;w1,...,w m)/parenrightbig
= (c3w2m+2
3c−3;c3w2m+5
3c−3,...,c3wmc−3,c2w0c−2,...,c2w2m−1
3c−2/parenrightbig
.
Hence, for an iwith 0≤i≤m−2
3, we must find an element wi=t1, wheret1satisfies
(6.4), so that we can set wi+m+1
3=c2t1c−2,wi+2m+2
3=c4t1c−4, and all other wj’s equal
toε. We have found seven solutions to the counting problem (6.4), and e ach of them
gives rise to ( m+1)/3 elements in Fix NCm(G24)(ψp) since the index iranges from 0 to
(m−2)/3.
On the other hand, if p= 14(m+1)/3, then the action of ψpis given by
ψp/parenleftbig
(w0;w1,...,w m)/parenrightbig
= (c5wm+1
3c−5;c5wm+4
3c−5,...,c5wmc−5,c4w0c−4,...,c4wm−2
3c−4/parenrightbig
.
By Lemma 29, every element of NC(W) is fixed under conjugation by c7, and, thus, the
equations for t1in this case are the same as in the previous one where p= 7(m+1)/3.
Hence, in either case, we obtain 7m+1
3=7m+7
3elements in Fix NCm(G24)(ψp), which
agrees with the limit in (9.1b).
Ifp= 7(m+ 1)/2, the relevant counting problem is (6.5). However, no element
(w0;w1,...,w m)∈FixNCm(G24)(ψp) can be produced in this way since the counting
problem imposes the restriction that ℓT(w0) +ℓT(w1) +···+ℓT(wm) be even, which
contradicts the fact that ℓT(c) =n= 3. This is in agreement with the limit in (9.1d).
Acknowledgements
The authors thank an anonymous referee for a very careful rea ding of the original
manuscript, and for numerous pertinent suggestions which have h elped to considerably
improve the original manuscript.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 33
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