Characteristic and Ehrhart quasi polynomials
for root systems-I,Introduction

-1.

2.

Characteristic andand supersolvable

Free

Geometry

I,Ehrhartandquasi-

polynomialsarrangements

Enumeration

onWeyl arrangements#I2

Worpitzky compatible arrangements-.

A Eulerian-.

polynomialsconsideration on type A

HI ,

-12.

Compatible arrangements andcomparability graphs
Application to graph polynomials of comparability graphs
coco.

Main

references

-A.

A. U.

Ashraf,M

T N Tran,.

.

Yoshinaga.

Eulerianpolynomials for subarrangements of Weyl arrangements
Adv2.

ppl

A-.

sub arr.

Mathand.

Tsuchiya Worpitzkcy compatible
arxiv 2007of brand arr and co comp graphs

T N Tran.

.

A.
.

-.

.

.

01248

.

.

I,

Introduction-LandCharacteristic.

Ehrhart quasi-pot.

Det :C Quasi pot )-•A function
F'

•":,

calledThe above
The smallest

i:c:D :f- Skg)

9=8 mode1¥

cdcopgld Cd 4) oldis

canp

-'t--

rt

Coop

periodic function with integral periodaCicq )§CIE KE p ):c:,

Z -7 IQa--1=i. e. ,is) E ④ It)
Ck constituentsiii.

EquivalentlyCi:>ogcq )•Z→ Qguasi-polynom.itandPEZ
fkctg=Cicqt Ti )be chosen
iscalled

asp-lanesthemed,.

.

,Td )

of g

Thin ( Ehrhart 19027¥t =let

Zai

=

21

a

lattice,

P beletpolytope with vertices

aIrlin

¥in.

ai.

,

9 E Z>o , thenIf↳ Cq): =#Cq Ph P )pot in q ofcoefficients in ①

is a quasi-dim CPD withdegree.

↳ Cg )

theis calledEhrhart quasi pot-.

-of•ThePDCP )We.

.

the latticedenominator DCP)

k£220

•w rtfor which
isaP.

the vertices

periodPofofis theKP

smallest

arein17.

of Lp

say that periodcolapse
is strictly less than DCP).

when themin.

periodExe ( standard simplex )P =22{ cnet.az ) Elle ) Kidz 30 LEE

D=,KHNL

qP={ Cat xD Elk l Kidz,>0 ,El

94

Rita ←:3
÷

D= con v { ( 0,0 ) d. 07,19171,Lpl g)More↳ cap

l9tDgt2

P={ Caagenerally ,Nit

then=C---=theEl,.

. .

.ae )

and all( Atl)

EIRE /RizoYIRZ

17=22

P=EI 'z3x[ ILpc

92
(9h32EI

Exe ( McAllister=--,{Wood

D= cow 90,07
DCP ) =DThen↳ Cop=,,

med 292DI) ) EIR)

CD o ) Cl ,-1

Dig -11

comfy ,o) # D d }) }DCP ) =3 and
497=19-115
P--'while

¥92,

EE

9=1 mode

2005,,,,.

D> 2Def ( hyperplane•lett#Zaibea=,•Llet

subgroup arr )

and.

2hE

free abelian grouplost Cmultiset ) infiniteT.

.

-••

9 E Ezo
For21g,a =It: =ai di21/92
EL,.hn/pevplane-Hqr:=frElRl i&aiai-€c07El
Rls
ubgr,aizi=J4E2ql
Ha:

•21g

-

: =LUR ) :arr.

-zq

fzezql I

L determines

The lostIR,L ( 297arr.

{ Hair I AEL Y={ Ha ,zq I AEL }: =Def :C Combinatorics of hyp.

.

•arr.

)

intersecteonposetlyqpgi-gafslta.ir

I SEL )

Mibiasfiuctoon µ Lying -72:

MURG =L::), Mex:=-XSMH)EYE IRL

characteristicaLary

The ( Kamiya
If q E 2- soxayuasicop
isa: =monkin Z,Ct)

E: =X-µ Cx )

tdomx

ELL

Tarawera-Tera2008o)then#( age I Left za ),quasi-

pot in q of deg L with coefficients.

.

Its called thecharactensh-cguasi-pol.GL
writ

Thin ("theFinite field method"

group 17

coincides.

with theftyct)=.

)

first constituent ofThei.e.

Xoluyasiqy

characteristicXycCt)

pot of LGR)

L

Exe

={ l 1,17 , ( 0,27 10,417 E 22-,Ldr) ={ 9-211-22=04 , { xz=oYSaz,-094

90,079 I^-I{ Nyt age of-s-{ KEO }-L

IK IXLGR)
L( 21g)=t2-2tt1X=219£98icq)} f-Zeitz -54 } Zzz -54,442-2-5/4,#=Rem ( T Yoshinaga-.

potconstituent can be-"ofan arr.

Liu T-,-.

If I--GTZ 22222,422¥0 med q¥2{94=Every{ZE}9=43 model
9=2 mod 4

9=-4 mod 4Yoshinaga )described

asthe"

characteristic

Thin ( Kamiya Tarawera-ForSEL ,subletator.subgn.CZ/lKss )Tercio 2008-)

supposeiQ Mds iz

Ens > 0,lads i ldsiiti,

Thenis8y:=,periodaQuestionIs[ True,km Cdsofspy

if L

=

X9YLsi.

the miniumngl SEL )( LCM

period-of

It positive system ofperiod )X9ym& ?
arootsystem )Ifstudynot ,"theperiod collapse¥:":÷asi:÷÷".

.in#sai*.: : : ?. O99kasinpo/.

Mim.

withperioddans←

2ndForEhrhart theoryviaa -_¥3ai aiEL.

write ca :=Can .ae )T,⇐Use( 0,9392I2gGenco,.

.

ze

we can write.gg?*.?I!rad=#fzlnfgx(laisegY.ysie ir9ae kDD
xqgasicp==#,934¥E Ehrhart quasi pot of rational-

hast open polytopes-=I

Ehrhart quasi pot of-.

inside-outarational

Beck-

Zaslvasvsky

2006

2Free and.

Supersolrable arr.

.

-let k be afield end••hyperplaneA•Acentrala:

Def ( free•SayS:•For.

--=arn.

Dee }basisV-, neOff g)

-.

===.

.

=Eof V.

H=ker2H*-.

deke ¥0Ik

ai Elkf : SSSlinearmapffcg ) t golf ) tf yes:a-,Dera )¥ Zais,( Q.

Home Volk )*s t

The set of derivationsDerk)

Cookin Subspace]

9121 tderivation of S

As t

•1980.

-)

Tera oat.

.

HEA , fix AH=ais

It=Ca finite setof hyp ) in VarrFlat ,AH•.

in VUQcnilfa)

isafree S module-The•

modedeofA-d-eriva.tw#DCA7.=fOEDersIoCaH)EhtSt tEt7c- Der

A is a•freearriDet ( supersolvableBjorner••rank CA)

BE A

anySt.

Ais0.

.

,

Stanley 1972

Edelman--,Zigler 1990 )HEAis.

arr-Cookin A Ha① rank CB )(2) forif DCA) is a free S headed.

of A tf

modulator=ranket )
'H #H

-Ie

Al B ,'"Hh HEH

F H " C- B

called supsoliable if 7 chain of arr.

=Ao EA , E

Ai l's a modular--Etrr=A

atom of kit,coOE 're rtThin ( Jamba Tercio1989)-•••••VV=If A B

superIRL

standard

IOI

with

321 , ,ae9=( Et 3).

.

-'IE ETB called

1317132

,

•Agt:

an.

wantE&÷7

ideal if

Be, KEEF

Weyl{ NEV I Hn) O }
'Hal
HEE
HY
{==a c-Weyl

AH affine.

subarrequivalent.

44112)

It.

> odi

KEI

P1 EI'Hal
LEE}=3HaAy( yo)pot , Pee It

B > Be ⇐ Bt B,•

Astraproductbase:root poset:

,

inner

then

positive system of OI:ET ZA,

root system

irreducible:

OI 2 Itsolvable.

arr.

•IE'ETThin ( AbeIf,-AI

Barakat'I EET

F : freeideal slab arr:

-is anSS :Gentz-.

HugeTercio 2010)-

ideal , then AI is freesolvablesuper.

I sidled
Subaru.

Geometry andI,

Worpitzky

I-

EnumerationcompatibleonWeyl arr.

arr.

-

REV 2 OI 2 It ? I•Q•¥=beforeas:root lattice

Zai :,HE ET E QCOI )•Xyoguasicq):the characteristic quasi pot of H
W htroot lattice:-.

.

.

-EIOI:Az =3 II=,Idz , I Kita)

2h22

E' CAD47£← Kitana),=Azxquasi¢Cg )XFL ayG)rxquasi (9)
Eft

#=

==

Zf

#=}

=-

2cL ,yayof{ zE2g4 zit ZEIT }

9cg D
-

24427,442222-21=07--6-134-27# { Ze•••

•hfW

I

do

==

Co -1Gt-Cixi=

highest root w.r.to#zD

Co : =L,-Csi H-tCeai EA

Sai=:,# f OE i El l=Si
•

72=74

E':

Coxeter number:-49

Ci7

index of connection:

Weyl group:

simple reflection.

.

HI k =f KEV 165k) KY
affine hyperplane
A connected component of V l U
Fak

KEI ,LEI

:,=,.

•REIT

is called•{ DI.

.

.

,

Dev Y

anKEG

alcove-:thedeathsof AHi Dj 7=8,

•ZCOIV )

20T=.
.

,

•PD

=& co Boi,

fundamental parallelepiped.

it.

Paz #⇐

aweigh latticecows o Ian,.FI#:s:i!:e7".

Last g )•:=# (Ehrhartgain IE ) )of AT

quasi pot-writ.

coweighttattoo

EI:

I=

W =L s ,sz1sf=sE=CsAzI2¥ )AT==,,ZOE

corn.

key53
ZOE

{ 0,01 , BEY

4*47=(9-1174-12)
y:standard simplex
In

KOI AIRE

DefA:

The

alcove:ceilings

wails
supportingA- are theofwhich

hyp to facet.

do notthea

pass through the origin and havetheoriginonsameside as.fi#yeeikng

floors0The

upper closure

AG:

=AU

( ceiling facets of A)

Thin ( Worpitzky partitionPG=WA : alcove

A E PD,

AHumphreys Yoshinaga ).

P"realblue✓Def( AshrafA:Yoshinaga)
40£ It Is said-

T2- EH

thatfloors-Haman:

oto beWorpitzky compatible--(orcompatbe)
Aceilings:

ifeither empty

isorcontainedinZHp mp,

If 44 is compatible,ceilingof A withthen we call

Ap compatible

aaswell.

BETMPEZ

Def( T Tsuchiya)-:

A-subset

HEE is said to be strongly (Worpitzk )'compatible if for any one Y and for every choiceofBts.

.

,Bm EET's

there exists k with"themwelet.

t.AE?Zsopi thenIE KEM

there exists "obtainansSCt→equivalent def

I be the set ofC.

13k£ 4?

.

""

Sora'llof.

all ideals

ideals.

of #compatible sets

-oft

It

strongly compatible sets-of IoTThy ( Ashraf T Yoshinaga--I E
obviousSC)

2020

EC

EI:Ca )If(b)

0 , It C- C

SE Et

He -129
then

0I=Bz ,,

thenwhen4,a#C.

4=051987 EC=Et didi

with all dial,

4=44+22,22+22,7 ¢ C
2h22"ru2¥42

EE:

4--922,24+2<7 ECISC

Io=Bz

41=14+22,2 },Question:

class ofIs there
arr.

orC-SCLI""

any significant
contained in FACIIFASCIa2A.

-Eulerian polynomial-I€=

highest root

Cixi.

,

do

Def

=-,

( Ashraf:T-Yoshinaga)

-'let

4C

HEE ,WE W•Co = )

I

dsei

#:

40=Cw)

:

E=Ci

White

-

Y

'Osiel
•A-Eu6nanpo#of H

EyTha::=is¥defined byEth-deep cue)

WEW

Eqdt) has positive integer coefficient
withconstant termnoThin ( Euler 1236 )Eulerianpot
E971.

isgetThe

the9=.

classical l th-polynomialAdt)

FettAdt)

Rem:

-

H=oIt

•dsc¥tCw,

EET

=4=0

•Lam

-Postnikov

Roi
TE

Def-_:

Ae#¥

Ego

=

the

RECD

=polynomial

EulerianAdt).

II I-Ii
,

,

thenf
P

:

REICH

Z →=b- we W

1=0=I-tAdt)

aftarskshifter (PCs )f )EICEDnZakat K)-:=F- I

Thin :( ATT )TFAE't:compatible( ))(LATg
Eqx9¥asicq )Ci )HEEis=Ciii)

qxqya.iq

Ey> t9-=TT Cl971them:40

1=0

twoThisrecovers

Et

Athanasia dis=-Sater

Lam

42=0

EI

E:

Eq---,

known results-KIT

Posnikovtht=Esa,*z¥= tht=2+3:

Blass Sagan ,,,Az=EET Ct)

tci )

-Adt)

Yoshinaga¥Cq7=C9tDCgexqasicgzof-4-zI-9II-fcs.IS#oKD
quasiOEC

97=94 D-satay⇐3+04*-143=4219-27+9=

94-17-11

fatal etc

similarly,

OITEC.

23Consideration.

ontype A-1.

Compatible

andarr.

cocomparability graphs-•{ Ez,.

U:•Set.

.

Ee Y=xij{E.
Ei=TEC Ae 1)

orthonormal basis for V

an=-Irieiev¥-04

ro-Ejfeta; )

KEG

IE

f ai

=

•OI

=

Ae i ; then-HE 91,.

it ,I

systemtypeAe,KieltyUzA§tcae.LT. .

rootkick ly

Braid arr•aof

TEK Ae -17=9 xij I-eI,

C- IU

Al Ae DIRL-.

)
=BrCl

's fsubarr of Brush{ subsets of E' CAet ) 4£I. I-{ simple graphsLij•EYRA) ⇐HajE B ⇐-EKG467ACG): =:,E),letoncosACA )I , E)=: =Aqapf aij IS isjl E E Ccj ) Ygraphic arr.

EY

Sis j 7 E E
G-ForLIRL-1

DefDef:

Agraphif

it's

A graphcalledisCheechordatgrapha-calledfree

cocomparability-graph.it
coutpkmeut
:

ishas

its

aa

transitive orientation-

U→ oifEquivalentlyiscocompordering
5. t.

.

a → weG-@, E) with=graphan:thenandifuusIV ) =L

it hasordering 01C

an---Umbriel
Coeofitsvertices@igoj7E theueither9oisorheEor3oksoj7E Norboth.r

KkLj,---

yOi

-

ok--.

og'

Def:

A graphcalledcan beeach vertex

onisthe real line,

adjacent if theanonempty

U
Co :

SIf

interval graphanaunit intervalinterval/

withassoc.

anassoc.

intervals

intersection=interval

arehave.

Ch n coCh :U:

ifa unit interval

and two vertices

CocomparabilityI:graph ).

chordalunit interval.

/Tbim

C. Stanley:

let G-1872

Edelman,Cts E))

ACG)

isfreeCii)

ACE)

issupersolvable

Ciii)

G- ischordal

Tha ( folklore )G- hasagraphideal

is anideal-⇐arrGas

TFAE :( VI =L,graphic.

.

Ged E)

labeling using Cl)

anis)

Reiner 1994

be a graph,Cia-

so thati. ea.

.

.

ACG)

HCG) EET CA e D

unit interval-graph-

Thd

G-

CT Tsuchiya )-.

CV,\GHI =LE)labeling from Its so thatG- has a

ACA is=.

compatible
compatible and freeB

arta

cocomparability graphintervalgraph.

.

::¥i÷÷¥µE

I:c-chordalfree ⇐supersolvable)

Stanley E- R.

interval

compatible n freeT T-

interval
parallelThin

If oI=Aeconcepts intype A,

thenSC = C.

2Application.

tograph pot.

-G:

CqThe

simple graph.

chromatic:

graphicpot.

Walt?

Eulerian pot.

-E

cqCg7t9=¥+1930The

reducedWatt)graphic Eulerian pot-E. GII
Fact Walt ,:Yq⑦t'=

both haveinteger coefficientYott)'¥¥epositive

DefG-:

IT ==IT,@e3.E)--

ite E Te( w rt f) at

itIi > Titiand.

.

•simple graph:has-ofHittin 7 E ECE )•A- descenta(Tet ;=Tz)

T

Definee.

ComplementFatt)i=Efk tk
Kafief):-_Le#IT has l
1
SIEGE-A descents-Prof:Fqct)=K

A-Eulerian

pot of44ft{ Ei Ej 14,5 HE Kj }
-,)

/

The:

G-

=

( El] E),TFAE.

.

G- is a co comparability graph andCi)

K2 CCii ) 122C.

--Ciii)HCG)"DCatt)d)Fatt )E.

-Clalisumbrella

l 'sumbrella free-

E' C Ae D--=free-iscompatible ⇐ strongly )a)tMatildeft if-YAH)

wqlt7-tll.tl#fqCt3-tlFGCt)T( Vi )t IEKEL ,⇐Questionuokcat-kfpla-7-ll-ktdfk.IE)( Brent 19923

IT Watt) log concave Cor just unimodal ) ?
-Propr:If Yott ) is log-concave

Watt ) B
alsolog concave-Conj⇐If:

G- isacocomparability graph ,thenYfct) is log

IfG- isKKthen.

-coClisislogFqRed CT T)
:.

afor

concavecompconj--.

.

graph andumbrella-

free-concave.

holds truecocomp graphs.

on

I 8 vertices

17--11
Wq⑦

is

not real roofed-2h2211✓✓l l

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