Worpitzky partitions for root systems and characteristic quasi-polynomials

0.27 MB·English

by  Masahiko Yoshinaga

WORPITZKY PARTITIONS FOR ROOT SYSTEMS AND
CHARACTERISTIC QUASI-POLYNOMIALS
5
1
0 MASAHIKOYOSHINAGA
2

c
ABSTRACT. For a given irreducible root system, we introduce a par-
e
D tition of (coweight) lattice points inside the dilated fundamentalparal-
lelepipedinto thoseof partiallyclosedsimplices. Thispartitioncanbe

1 considered as a generalization and a lattice points interpretationof the
3
classicalformulaofWorpitzky.

Thispartition,andthegeneralizedEulerianpolynomial,recentlyin-
]
O troducedby Lam andPostnikov,can be usedto describethe character-
C istic (quasi)polynomialsof Shi and Linial arrangements. As an appli-
. cation, we provethat the characteristic quasi-polynomialof the Shi ar-
h
rangementturnsouttobeapolynomial. Wealsopresentseveralresults
t
a on thelocationofzerosofcharacteristicpolynomials,relatedto a con-
m
jectureofPostnikovandStanley.Inparticular,weverifythe“functional
[ equation”ofthecharacteristicpolynomialoftheLinialarrangementfor

any root system, and give partial affirmative results on “Riemann hy-
2
v pothesis”fortherootsystemsoftypeE6,E7,E8,andF4.
5
5
9
4
0 CONTENTS
.
1
1. Introduction 2
0
5 1.1. Arrangements ofhyperplanes 2
1
1.2. Main Results 3
:
v 1.3. Outlineoftheproof 4
i
X 2. Background 6
r 2.1. Quasi-polynomialswithgcd-property 6
a
2.2. Arrangements andcharacteristicquasi-polynomials 6
2.3. Root systems 7
2.4. Partitionofthefundamentalparallelepiped 8
2.5. Shift operatorand “Riemannhypothesis” 9
2.6. Eulerian polynomial 11
3. Ehrhartquasi-polynomialforthefundamentalalcove 12
3.1. Ehrhart quasi-polynomial 12
Date:January1,2016.
2010MathematicsSubjectClassification. Primary52C35,Secondary20F55.
Keywordsandphrases. Rootsystem,Shiarrangement,Linialarrangement,character-
isticquasi-polynomial,Worpitzkyidentity,Eulerianpolynomial.
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2 MASAHIKOYOSHINAGA
3.2. Ehrhart quasi-polynomialforA◦ 12
3.3. Characteristic quasi-polynomial 15
4. Generalized Eulerianpolynomial 17
4.1. Definitionand basicproperty 17
4.2. Worpitzkypartition 19
5. Shiand Linialarrangements 21
5.1. Shi arrangements 21
5.2. Linialarrangements 23
5.3. Thefunctionalequation 24
5.4. Partial resultsonthe“Riemann hypothesis” 25
References 26
1. INTRODUCTION
1.1. Arrangements of hyperplanes. Let A = {H ,...,H } be an ar-
1 n
rangementofaffinehyperplanesinavectorspaceV. Theintersectionposet
of A is the set L(A) = {∩S | S ⊂ A} of intersections of A. The inter-
section poset L(A) is partially ordered by reverse inclusion, which has a
unique minimal element ˆ0 = V. The arrangement A is called essential if
the maximal elements of L(A) are 0-dimensional subspaces. The charac-
teristicpolynomialofA isdefined by
χ(A,q) = µ(X)qdimX,
X
X∈L(A)
whereµis theMo¨biusfunctiononL(A), defined by
1, ifX = ˆ0
µ(X) =
(cid:26) − µ(Y), otherwise.
Y<X
P
Thecharacteristicpolynomialχ(A,q)ofA isoneofthemostfundamental
invariants. Indeed, χ(A,q) captures combinatorial and topological proper-
tiesofAas follows.
Theorem 1.1. (i) (Zaslavsky [28]). Suppose that V is a real vector space.
ThenthenumberofconnectedcomponentsofVr∪Aisequalto(−1)ℓχ(A,−1).
If A is essential, then the number of bounded connected components of
V r∪A isequalto (−1)ℓχ(A,1).
(ii)(OrlikandSolomon[17]). SupposethatV isacomplexvectorspace.
Then thePoincare´ polynomialofV r∪Ais equalto(−t)ℓχ(A,−1).
t
WORPITZKYPARTITION 3
1.2. MainResults. LetΦbearootsystemofrankℓ,withexponentse ,...,e
1 ℓ
and Coxeter number h. Fix a set of positiveroots Φ+ ⊂ Φ. The structures
oftruncated affineWeyl arrangements
A[a,b] = {H | α ∈ Φ+,a ≤ k ≤ b}
Φ α,k
(see also §2.3 for the notation) have been intensively studied, because of
their intriguing combinatorial properties [20, 2, 21]. The characteristic
polynomial of the Coxeter arrangement A = A[0,0] was computed in [7].
Φ Φ
Later,thiswasgeneralizedtotheextendedCatalanarrangementA[−k,k]and
Φ
the extended Shi arrangement A[1−k,k]. The characteristic polynomial of
Φ
thesearrangements factoras follows
ℓ
[−k,k]
χ(A ,t) = (t−e −kh),
Φ i
Yi=1
χ(A[1−k,k],t) = (t−kh)ℓ,
Φ
([9, 1, 4, 27]). For other parameters a ≤ b, e.g., the Linial arrangement
A[1,n], the characteristic polynomial χ(A[a,b],t) does not factor in general.
Φ Φ
Howeverthereareanumberofbeautifulconjecturesconcerningχ(A[a,b],t).
Φ
Amongothers, PostnikovandStanley [19]conjecturedthat
(a) χ(A[1−k,n+k],t) = χ(A[1,n],t−kh) (“h-shiftreduction”).
Φ Φ
(b) χ(A[1,n],nh−t) = (−1)ℓχ(A[1,n],t) (“Functional equation”).
Φ Φ
(c) All the roots of the polynomial χ(A[1,n],t) have the same real part
Φ
nh (“Riemann hypothesis”).
2
Postnikov and Stanley verified these assertions for Φ = A in [19]. Later,
ℓ
Athanasiadisgaveproofs forΦ = A ,B ,C ,D in [1, 3].
ℓ ℓ ℓ ℓ
Recently,Kamiya,TakemuraandTerao[15]introducedthenotionofthe
characteristic quasi-polynomialfor an arrangement A defined overQ. The
characteristic quasi-polynomial χ (A,t) is a periodic polynomial (see
quasi
§2.2fordetails)thatmaybeconsideredasarefinementofthecharacteristic
polynomialχ(A,t).
Our main result concerns the characteristic quasi-polynomial for A[a,b]:
Φ
“h-shift reduction” and the “functional equation” hold at the level of char-
acteristicquasi-polynomials.
Theorem 1.2. Let Φ beanarbitraryirreduciblerootsystem.
(i) The characteristic quasi-polynomial of the extended Shi arrange-
mentisapolynomial,χ (A[1−k,k],t) = (t−kh)ℓ (Theorem 5.1).
quasi Φ
4 MASAHIKOYOSHINAGA
(ii) Thecharacteristicquasi-polynomialsatisfies“h-shiftreduction”χ (A[1−k,n+k],t) =
quasi Φ
χ (A[1,n],t−kh)(Theorem5.3). Inparticular,thisholdsforthe
quasi Φ
characteristicpolynomial(Corollary5.4).
(iii) Thecharacteristicquasi-polynomialsatisfies“Functionalequation”
χ (A[1,n],nh−t) = (−1)ℓχ (A[1,n],t)(Theorem5.6). Inpar-
quasi Φ quasi Φ
ticular,thisholdsforthecharacteristicpolynomial(Corollary5.7).
(iv) Suppose Φ ∈ {E ,E ,E ,F }. Let n be the period of the Ehrhart
6 7 8 4
quasi-polynomialofthefundamentalalcove(see§3.2 andTable1).
If n ≡ −1 mod rad(n), then the “Reiemann hypothesis”holds for
A[1,n] (Theorem 5.8).
Φ e
1.3. Outline of the proof. We follow the strategy adopted in [6, 1, 4,
15] for the computation of χ (A ,q), where A := A[0,0] is the Cox-
quasi Φ Φ Φ
eter arrangement (see §3.3). The idea is to relate the characteristic quasi-
polynomial to the Ehrhart quasi-polynomial L (q) of the fundamental al-
A◦
cove A◦. Consider the associated hyperplane arrangement A in the quo-
tientZ(Φ)/qZ(Φ),whereZ(Φ)isthecoweightlattice. Then,bydefinition,
χ (A ,q) is the number of points in the complement of A, for q ≫ 0.
quasi Φ
If we define P♦ = ℓ (0,1]̟ , (where ̟∨ is the basis dual to the sim-
k=1 i i
ple basis), then therPe is a bijective correspondence between the points in
Z(Φ)/qZ(Φ) and the lattice points in the dilated parallelepiped qP♦. The
parallelepiped P♦ is dissected by the affine Weyl arrangement into open
simplices (alcoves). Thus χ (A ,q) can be expressed as the sum of
quasi Φ
Ehrhart quasi-polynomials of these alcoves. Since A is Weyl group in-
Φ
variant, the above dissection is into simplices of the same size (Figure 1),
whichyieldsthesimpleformula
f
(1) χ (A ,q) = ·L (q)
quasi Φ |W| A◦
(seeCorollary3.5,Corollary3.6andProposition3.7). Theformula(1)first
appeared explicitly in [6], where it was proved using the classification of
root systems. A case free proof was given in [1]. The argument was later
extendedtothecaseA[−m,m] in [4].
Φ
IfweapplythesamestrategyforthecaseofShiandLinialarrangements,
then χ (A[a,b],q) can again be expressed as the sum of Ehrhart quasi-
quasi Φ
polynomials. However the sizes of simplices are no longer uniform (see
Figure4). Thisdifficultycan beovercomeby lookingatadisjointpartition
ofP♦ intopartiallyclosedalcoves
(2) P♦ = A♦,
ξ
ξG∈Ξ
WORPITZKYPARTITION 5
(see§2.4 fordetails). Thenobviouslywehavea partitionoflatticepoints
(3) qP♦ ∩Z(Φ) = (qA♦ ∩Z(Φ)),
ξ
ξG∈Ξ
whichwewillcallaWorpitzkypartition. Thenumberoflatticepointscon-
tainedinqA♦ is expressedas
ξ
(4) L (q) = L (q −asc(A◦)),
A♦ A◦ ξ
ξ
(Lemma 4.9), where asc(A◦) is a certain integer (Definition 4.1 and (35)).
ξ
The key result (Theorem 4.7) in the proof of our main results is that the
distribution of the quantity asc(A◦) is given by the generalized Eulerian
ξ
polynomial R (t) (Definition 4.4) introduced by Lam and Postnikov [16].
Φ
UsingtheshiftoperatorS (§2.5), thepartition(3)impliestheformula,
(5) qℓ = (R (S)L )(q).
Φ A◦
In the case Φ = A , thepolynomialR (t) is equal to the classical Eulerian
ℓ Φ
polynomial. Then the above formula (5) is known as the Worpitzky iden-
tity [26, 8]. Hence (5) can be considered as a generalization of Worpitzky
identityand (3)as itslatticepointsinterpretation.
Usingtheseresults,thecharacteristicquasi-polynomialsforShiandLinial
arrangements have expressions similar to the Worpitzky identity (5). We
have
χ (A[1−k,k],q) =(SkhR (S)L )(q) = (q −kh)ℓ,
quasi Φ Φ A◦
(6)
χ (A[1−k,n+k],q) =(SkhR (Sn+1)L )(q),
quasi Φ Φ A◦
(Theorem5.1, Theorem5.3). Usingtheseexpressions,thefunctionalequa-
tionisobtainedfromthedualityofthegeneralized Eulerian polynomial
1
(7) thR ( ) = R (t),
Φ Φ
t
(Proposition4.5).
If n ≡ −1 mod rad(n), then 1 + n is divisible by rad(n). Hence
gcd(q,n) = 1 implies gcd(q − k(n + 1),n) = 1 for k ∈ Z. This enables
us to simplify the expressieon of the characteristic polynomial χe(A[1,n],t).
e e Φ
Using techniques similar to those in Postnikov, Stanley and Athanasiadis
[19, 3], wecan verify the“Riemann hypothesis”forsuch parameters n.
The paper is organized as follows. §2 contains background materials
on root systems,characteristic quasi-polynomialsand the Eulerian polyno-
mial. The partition of the fundamental parallelepiped, which will play an
important role later, is introduced in §2.4 (Definition 2.3). In §3 the rela-
tion between the Ehrhart quasi-polynomial of the fundamental alcove and
the characteristic quasi-polynomial is discussed. In §4 we first summarize
6 MASAHIKOYOSHINAGA
basic properties of the generalized Eulerian polynomial R (t) introduced
Φ
byLam and Postnikov[16]. ThenweintroduceWorpitzkypartitionsofthe
lattice points which provide a Worpitzky-type identity (Theorem 4.8). We
also givean explicitexampleofthe Worpitzkypartitionfor Φ = B . In §5,
2
we obtain formulae for characteristic quasi-polynomials by modifying the
Worpitzky-typeidentity. Usingtheseformulae, weproveourmainresults.
2. BACKGROUND
2.1. Quasi-polynomials with gcd-property. A function f : Z −→ Z is
calledaquasi-polynomialifthereexistn > 0andpolynomialsg (t),g (t),...,g (t) ∈
1 2 n
Z[t]such that e
e
f(q) = g (q), ifq ≡ r mod n,
r
(1 ≤ r ≤ n). The minimal such n is called the period of the quasi-
e
polynomialf.
Moreover,ethefunctionf : Z −→ Zeissaidtobeaquasi-polynomialwith
gcd-property if the polynomial g (t) depends on r only through gcd(r,n).
r
Inotherwords,g (t) = g (t) ifgcd(r ,n) = gcd(r ,n).
r1 r2 1 2
e
2.2. Arrangements and characteristicequasi-polynoemials. Let L ≃ Zℓ
be a lattice and L∨ = Hom (L,Z) be the dual lattice. Given α ,...,α ∈
Z 1 n
L∨ andintegersk ,...,k ∈ Z,wecanassociateahyperplanearrangement
1 n
A = {H ,...,H }inRℓ ≃ L⊗ R,withH = {x ∈ L⊗R | α (x) = k }.
1 n A i i i
Fora positiveintegerq > 0, define
(8) M(A;q) := {x ∈ L/qL | ∀i,α(x) 6≡ k mod q}.
i
Kamiya,TakemuraandTerao provedthefollowing.
Theorem2.1. ([13,14])Thereexistq > 0andaquasi-polynomialχ (A,t)
0 quasi
withgcd-propertysuchthat#M(A,q) = χ (A,q)forq > q .
quasi 0
Moreprecisely, thereexistsaperiod n and apolynomialg (t) ∈ Z[t] for
d
each divisord|nsuchthat
e
#M(A;q) = g (q),
e d
forq > q , where d = gcd(n,q).
0
One of the most important invariants of a hyperplane arrangement A is
thecharacteristicpolynomiaelχ(A,t) ∈ A[t](see[18]forthedefinitionand
basic properties). The characteristic polynomial is one of the polynomials
givenbyTheorem 2.1(see also[3, Theorem2.1]), specifically,
(9) χ(A,t) = g (t).
1
WORPITZKYPARTITION 7
2.3. Rootsystems. LetV = Rℓ betheEuclideanspacewithinnerproduct
(·,·). Let Φ ⊂ V be an irreducible root system with exponents e ,...,e ,
1 ℓ
Coxeternumberh and Weyl group W. Forany integerk ∈ Z and α ∈ Φ+,
theaffine hyperplaneH isdefined by
α,k
(10) H = {x ∈ V | (α,x) = k}.
α,k
FixapositivesystemΦ+ ⊂ Φandthesetofsimpleroots∆ = {α ,...,α } ⊂
1 ℓ
Φ+. Thehighestroot,denotedbyα ∈ Φ+,canbeexpressedasalinearcom-
bination α = ℓ c α (c ∈ Z ). We also set α := −α and c := 1.
i=1 i i i >0 0 0
e
ThenwehavePthelinearrelation
e e
(11) c α +c α+···+c α = 0.
0 0 1 ℓ ℓ
The coweight lattice Z(Φ) and the coroot lattice Qˇ(Φ) are defined as fol-
lows.
Z(Φ) = {x ∈ V | (α ,x) ∈ Z,α ∈ ∆},
i i
2α
Qˇ(Φ) = Z· .
(α,α)
Xα∈Φ
The coroot lattice Qˇ(Φ) is a finite index subgroup of the coweight lattice
Z(Φ). Theindex#Z(Φ) = f iscalled theindexof connection.
Qˇ(Φ)
Let ̟∨ ∈ Z(Φ) be the dual basis to the simple roots α ,...,α , that is,
i 1 ℓ
(α ,̟∨) = δ . ThenZ(Φ)isafreeabeliangroupgeneratedby̟∨,...,̟∨.
i j ij 1 ℓ
Wealsohavec = (̟∨,α).
i i
A connected component of V r H is called an alcove. Let us
α,k
e α∈SΦ+
k∈Z
definethefundamentalalcoveA◦ by
(α ,x) > 0, (1 ≤ i ≤ ℓ)
A◦ = x ∈ V i
(cid:26) (cid:12) (α,x) < 1 (cid:27)
(cid:12)
(cid:12)
(α ,x) > 0, (1 ≤ i ≤ ℓ)
= x ∈ V (cid:12) ei .
(cid:26) (cid:12) (α ,x) > −1 (cid:27)
0
(cid:12)
(cid:12)
(cid:12)
The closure A◦ = {x ∈ V | (α ,x) ≥ 0 (1 ≤ i ≤ ℓ), (α,x) ≤ 1} is the
i
̟∨ ̟∨
convexhullof 0, 1 ,..., ℓ ∈ V. Theclosed alcoveA◦ is a simplex. The
c1 cℓ e
supportinghyperplanesof facets of A◦ are H ,...,H ,H . We note
α1,0 αℓ,0 α,1
thatA◦isafundamentaldomainoftheaffineWeylgroupW =eW⋉Qˇ(Φ).
aff
8 MASAHIKOYOSHINAGA
Let P♦ denotethefundamentaldomain ofthecoweightlatticeZ(Φ) de-
fined by
ℓ
P♦ = (0,1]̟∨
(12) i
Xi=1
= {x ∈ V | 0 < (α ,x) ≤ 1,i = 1,...,ℓ}.
i
Herewesummarizewithoutproofssomeuseful facts onroot systems[12].
Proposition2.2. (i) c +c +···+c = h.
0 1 ℓ
(ii) |W| = vol(P♦) = l!·c ·c ···c .
f vol(A◦) 1 2 ℓ
(iii) |Φ+| = ℓh.
2
2.4. Partitionofthefundamentalparallelepiped. Letusconsidertheset
of alcoves contained in P♦, denoted by {A◦ | ξ ∈ Ξ}, where Ξ is a finite
ξ
setwith|Ξ| = |W| (by Proposition2.2 (ii)). In otherwords,
f
(13) P♦ r H = A◦.
α,k ξ
α∈Φ[+,k∈Z ξG∈Ξ
Each A◦ can bewrittenuniquelyas
ξ
(α,x) > k forα ∈ I
(14) A◦ = x ∈ V α ,
ξ (cid:26) (cid:12) (β,x) < kβ forβ ∈ J (cid:27)
(cid:12)
(cid:12)
forsomepositiverootsI,J ⊂(cid:12)Φ+ with|I⊔J| = ℓ+1,andk ,k ∈ Z(α ∈
α β
I,β ∈ J). By definition, the facets of A◦ are supported by the hyperplanes
ξ
H (α ∈ I)and H (β ∈ J).
α,kα β,kβ
Definition 2.3. With notation as above, let us define the partially closed
alcoveA♦ by
ξ
(α,x) > k forα ∈ I
(15) A♦ := x ∈ V α .
ξ (cid:26) (cid:12) (β,x) ≤ kβ forβ ∈ J (cid:27)
(cid:12)
(cid:12)
Obviously,the interiorof A♦(cid:12) is A◦. Although A♦ is not a closureof A◦,
ξ ξ ξ ξ
A♦ maybeconsidered asthepartial closureofA◦.
ξ ξ
Proposition 2.4. Let ρ = ℓ ̟∨. Then x ∈ A♦ if and only if for suffi-
i=1 i ξ
ciently small 0 < ε ≪ 1, xP−ε·ρ ∈ A◦ξ, (that is, there exists ε0 > 0 such
thatif0 < ε < ε , thenx−ε·ρ ∈ A◦).
0 ξ
Proof. Straightforward. (cid:3)
From Proposition2.4, wehaveapartitionofP♦.
WORPITZKYPARTITION 9
Proposition2.5.
(16) P♦ = A♦.
ξ
ξG∈Ξ
Proof. It is enough to show that each x ∈ P♦ is contained in the unique
A♦. Let x ∈ P♦. Then for sufficiently small ε > 0, (α,x − ε · ρ) ∈/ Z
ξ
for all α ∈ Φ+, hence x − ε · ρ is contained in the unique alcove A◦. By
ξ
Proposition2.4,x iscontainedin thecorrespondingA♦. (cid:3)
ξ
2.5. Shift operator and “Riemann hypothesis”. Let a,b ∈ Z beintegers
witha ≤ b. Let usdenotebyA[a,b] thehyperplanearrangement
Φ
A[a,b] = {H | α ∈ Φ+,k ∈ Z,a ≤ k ≤ b}.
Φ α,k
By Proposition 2.2 (iii), we have |A[a,b]| = ℓ·h·(b−a+1). For special cases,
Φ 2
thecharacteristicpolynomialχ(A[a,b],t) factors.
Φ
Theorem 2.6. (i) If k ≥ 0,then χ(A[−k,k],t) = ℓ (t−e −kh).
Φ i=1 i
(ii) Ifk ≥ 1, thenχ(A[1−k,k],t) = (t−kh)ℓ. Q
Φ
The aboveresult had been conjectured by Edelman and Reiner [9]. The-
orem 2.6 (i) was proved in [4] by using lattice point counting techniques,
which will be developed further in this paper. Theorem 2.6 (ii) was proved
in[27]byuseofthetheoryoffree arrangements([9,18, 24, 25]).
For an interval [a,b] 6= [−k,k],[1 − k,k], the characteristic polynomial
χ(A[a,b],t) does not factor in general. Postnikov and Stanley pose the fol-
Φ
lowing“Riemann hypothesis”.
Conjecture 2.7. ([19, Conjecture 9.14]) Let a,b ∈ Z with a ≤ 1 ≤ b.
Suppose a+b ≥ 1. Then every root t ∈ C of the equation χ(A[a,b],t) = 0
Φ
satisfiesRet = h(b−a+1).
2
Conjecture 2.7 has been proved by Stanley, Postnikov and Athanasiadis
in[3, 19] forΦ ∈ {A ,B ,C ,D ,G }. We recall theirresults.
ℓ ℓ ℓ ℓ 2
Let f : N −→ R be a partial function, that is, a function defined on a
subsetofN. Define theaction oftheshiftoperator S by
(Sf)(t) = f(t−1).
More generally, for a polynomial P(S) = a Sk in S, the action is
k k
defined by P
(P(S)f)(t) = a f(t−k).
k
Xk
Proposition 2.8. Let g(S) ∈ R[S] and f(t) ∈ R[t]. Suppose degf = n.
Then g(S)f = 0if andonlyif (1−S)n+1|g(S).
10 MASAHIKOYOSHINAGA
Proof. Firstnotethatsince(1−S)f(t) = f(t)−f(t−1)isthedifference
operator, deg((1 − S)f) = n − 1. Suppose (1 − S)n+1|g(S). Then by
induction,itiseasily seen that(1−S)n+1f = 0. Henceg(S)f = 0.
Conversely, suppose g(S)f = 0. Consider the Taylor expansion of g(S)
at S = 1. Set g(S) = b +b (S −1)+b (S −1)2 +···+ b (S −1)n +
0 1 2 n
(S −1)n+1g(S). Since(S −1)n+1f = 0, wehave
(17) (b +b (S −1)+b (S −1)2 +···+b (S −1)n)f = 0.
0e 1 2 n
Setf(t) = r tn+r tn−1+···+r withr 6= 0. Thecoefficientoftheterm
0 1 n 0
of degree n in (17) is b r . Hence b = 0. Similarly, b = ··· = b = 0,
0 0 0 1 n
andwehaveg(S) = (S −1)n+1g(S). (cid:3)
Theshiftoperatorcan beused toexpresscharacteristicpolynomials.
e
Theorem 2.9. ([3, 19]).
(1) Let n ≥ 1 and k ≥ 0. For the cases Φ ∈ {A ,B ,C ,D },
ℓ ℓ ℓ ℓ
χ(A[1−k,n+k],t) = χ(A[1,n],t−kh).
Φ Φ
(2) Let n ≥ 1. Then the characteristic polynomial χ(A[1,n],t) has the
Φ
followingexpression.
(i) ForΦ = A ,
ℓ
1+S +S2 +···+Sn ℓ+1
(18) χ(A[1,n],t) = tℓ.
Aℓ (cid:18) 1+n (cid:19)
(ii) ForΦ = B orC ,
ℓ ℓ
(19)
4S(1+S2+S4+···+S2n)ℓ−1(1+S2+S4+···+Sn−1)2tℓ, ifnodd,
(1+n)ℓ+1
χ(A[1,n],t) =
Φ 
 (1+S2+S4+···+S2n)ℓ−1(1+S2+S4+···+Sn)2tℓ, ifneven.
(1+n)ℓ+1

(iii) ForΦ = D ,
ℓ
(20)
8S(1+S2)(1+S2+S4+···+S2n)ℓ−3(1+S2+S4+···+Sn−1)4tℓ, ifnodd,
(1+n)ℓ+1
χ(A[1,n],t) = 
Dℓ  (1+S2+S4+···+S2n)ℓ−3(1+S2+S4+···+Sn)4tℓ, ifneven.
(1+n)ℓ+1

Owing to the next result, the above expressions implies Conjecture 2.7
forA[a,b] withΦ = A ,B ,C , orD and a+b ≥ 2.
Φ ℓ ℓ ℓ ℓ
Lemma 2.10. ([19, Lemma 9.13]) Let f(t) ∈ C[t]. Suppose all the roots
of the equation f(t) = 0 have real part equal to a. Let g(S) ∈ C[S]
be a polynomial such that every root of the equation g(z) = 0 satisfies
|z| = 1. Then allrootsoftheequation(g(S)f)(t) = 0haverealpartequal
toa+ degg.
2