Combinatorial Gelfand Models

15 Pages·2017·0.42 MB·English

Combinatorial Gelfand Models
Ron M. Adin, Alex Postnikov, Yuval Roichman
To cite this version:
Ron M. Adin, Alex Postnikov, Yuval Roichman. Combinatorial Gelfand Models. 20th Annual Inter-
national Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), 2008, Viña
del Mar, Chile. pp.717-730, ￿10.46298/dmtcs.3612￿. ￿hal-01185146￿
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FPSAC2008,Valparaiso-Vin˜adelMar,Chile DMTCSproc.AJ,2008,717–730
Combinatorial Gelfand Models
Ron M. Adin1 and Alex Postnikov2 and Yuval Roichman3
1DepartmentofMathematics,Bar-IlanUniversity,Ramat-Gan52900,Israel [email protected]
2Dept.ofAppliedMathematics,MassachusettsInstituteofTechnology,MA02139,USA [email protected]
3DepartmentofMathematics,Bar-IlanUniversity,Ramat-Gan52900,Israel [email protected]
Abstract.AcombinatorialconstructionofGelfandmodelsforthesymmetricgroup, foritsIwahori-Heckealgebra
andforthehyperoctahedralgroupispresented.
Keywords: symmetricgroup,hyperoctahedralgroup,Iwahori-Heckealgebra,descents,inversions,characterformu-
las,Gelfandmodel
1 Introduction
AcomplexrepresentationofagrouporanalgebraAiscalledaGelfandmodelforA,orsimplyamodel,
ifitisequivalenttothemultiplicityfreedirectsumofallA-irreduciblerepresentations.
Models(forcompactLiegroups)werefirstconstructedbyBernstein,GelfandandGelfand[8]. Con-
structionsofmodelsforthesymmetricgroup,usinginducedrepresentationsfromcentralizers,werefound
by Klyachko[13, 14] and by Inglis, Richardsonand Saxl [11]; see also [6, 21, 4, 3, 5]. Our goalis to
determineanexplicitandsimplecombinatorialactionwhichgivesamodelforthesymmetricgroupand
its Iwahori-Heckealgebra and for the hyperoctahedralgroup. The descent set of a permutationplays a
crucialroleintheconstructionandinaresultingcharacterformula.
Therestofthepaperisorganizedasfollows.MainresultsaregiveninSections2–4.Proofsaresketched
inSections5–7. Section8concludeswithremarksandopenproblems.
Thisisanextendedabstract. Foramoredetailedversionsee[1].
2 The Symmetric Group
Let S be the symmetric group on n letters, S = {s ,...,s } the set of simple reflections in S ,
n 1 n−1 n
I = {π ∈ S |π2 = id}itsthesetofinvolutions,andV := span {C |w ∈ I }avectorspaceover
n n n Q w n
Qformallyspannedbytheinvolutions.
Recallthestandardlengthfunctiononthesymmetricgroup
ℓ(π):=min{ℓ|π=s s ···s , s ∈S(∀j)},
i1 i2 iℓ ij
thedescentset
Des(π):={s∈S|ℓ(πs)<ℓ(π)},
1365–8050(cid:13)c 2008DiscreteMathematicsandTheoreticalComputerScience(DMTCS),Nancy,France
718 RonM.Adin andAlexPostnikov andYuvalRoichman
andthedescentnumberdes(π):=#Des(π).
Defineamapρ:S →GL(V )by
n
ρ(s)C :=sign(s; w)·C (∀s∈S,w ∈I )
w sws n
where
−1, ifsws=wands∈Des(w);
sign(s; w):= (1)
(1, otherwise.
Theorem2.1 ρdeterminesanS -representation.
n
Theorem2.2 ρdeterminesaGelfandmodelforS .
n
3 Hecke Algebra Action
ConsiderH (q),theHeckealgebraofthesymmetricgroupS (sayoverthefieldQ(q1/2)),generatedby
n n
{T |1≤i<n}withthedefiningrelations
i
(T +q)(T −1)=0 (∀i),
i i
T T =T T if |i−j|>1,
i j j i
T T T =T T T (1≤i<n−1).
i i+1 i i+1 i i+1
Notethatsomeauthorsuseaslightlydifferentnotation,withT consistentlyreplacedby−T . Thecurrent
i i
definitiongivesamoreconvenientdescriptionofthecharacters,seeProposition3.3below.
In orderto constructan extendedsigned conjugation,whichgivesa modelforH (q), we extendthe
n
standardnotionsoflengthandweakorder.Recallthatthe(left)weakorder≤ onS isthereflexiveand
L n
transitiveclosureoftherelation:w≺ wsifs∈S andℓ(w)+1=ℓ(sw).
L
Definition3.1 Definetheinvolutivelengthofaninvolutionw ∈I withcycletype(1n−2k2k)as
n
ℓˆ(w):=min{ℓ(v)|w=vs s ···s v−1},
1 3 2k−1
whereℓ(v)isthestandardlengthofv ∈S .
n
Definetheinvolutiveweakorder≤ onI asthereflexiveandtransitiveclosureoftherelation: w ≺
I n I
swsifs∈S andℓˆ(w)+1=ℓˆ(sws).
Defineamapρ :S →GL(V )by
q n
−qC , ifsws=wands∈Des(w);
w
C , ifsws=wands6∈Des(w);
ρ (T )C := w (2)
q s w (1−q)Cw+qCsws, ifw ≺I sws;
C , ifsws≺ w.
sws I
Theorem3.2 ρ isaGelfandmodelforH (q)(qindeterminate);namely,
q n
CombinatorialGelfandModels 719
(1) ρ isanH (q)-representation.
q n
(2) ρ isequivalenttothemultiplicityfreesumofallirreducibleH (q)-representations.
q n
The proof involves Lusztig’s version of Tits’ deformation theorem [17]. For other versions of this
theoremsee[9,§4],[10,§68.A]and[7].
Letµ=(µ ,µ ,...,µ )beapartitionofnandleta := j µ (0≤j ≤t). Apermutationπ ∈S
1 2 t j i=1 i n
isµ-unimodalifforevery0≤j <tthereexists1≤d ≤µ suchthat
j j+1
P
π <π <···<π >π >π >···>π .
aj+1 aj+2 aj+dj aj+dj+1 aj+dj+2 aj+1
Thecharacterofρ maybeexpressedasageneratingfunctionforthedescentnumberoverµ-unimodal
q
involutions.
Proposition3.3
Tr(ρ (T ))= (−q)des(w)
q µ
{w∈In|wXisµ-unimodal}
whereT := T T ···T T ···T isthesubproductofT T ···T omittingthefac-
µ 1 2 µ1−1 µ1+1 µ1+...+µt−1 1 2 n−1
torsT forall1≤i<t.
µ1+···+µi
4 The Hyperoctahedral Group
LetB betheWeylgroupoftypeB,SB -thesetofsimplereflectionsinB ,IB -thesetofinvolutions
n n n
inB ,andVB := span {C |w ∈ IB}avectorspaceoverQspannedbytheinvolutions. Recallthat
n n Q w n
B = Z ≀S ,sothateachelementw ∈ B isidentifiedwithapair(v,σ),wherev ∈ Zn andσ ∈ S .
n 2 n n 2 n
Denote|w|:=σ.
DefineamapρB :SB →GL(V )by
n
ρB(s)C :=sign(s; w)·C (∀s∈SB,w ∈IB)
w sws n
where,fors=s =((1,0,...,0),id),theexceptionalCoxetergenerator,thesignis
0
−1, ifsws=wands ∈Des(w);
0
sign(s ; w):=
0
(1, otherwise,
andfors6=s thesignis
0
−1, ifsws=wands∈Des(|w|);
sign(s; w):=
(1 otherwise.
Theorem4.1 ρB isaGelfandmodelforB .
n
Foraproofandgeneralizationssee[2].
720 RonM.Adin andAlexPostnikov andYuvalRoichman
5 Characters
5.1 Character Formula
The followingclassical resultfollowsfromthe work ofFrobeniusand Schur, see [12, §4]and [23, §7,
Ex. 69].
Theorem5.1 Let G be a finite group, for which every complex representation is equivalent to a real
representation.Thenforeveryw ∈G
χ(w)=#{u∈G|u2 =w},
χ∈G∗
X
whereG∗ denotesthesetoftheirreduciblecharactersofG.
Itiswellknown[22]thatallcomplexrepresentationsofaWeylgroupareequivalenttorationalrepre-
sentations. Inparticular,Theorem5.1holdsforG=S . Oneconcludes
n
Corollary5.2 Letπ ∈Snhavecyclestructure1d12d2···ndn. Then
n
χ(π)= f(r,d ),
r
χX∈Sn∗ rY=1
where
0, ifrisevenandd isodd;
r
f(r,d )= 2,d..r.,2 ·rdr/2, ifranddr areeven;
r ⌊(cid:0)dr/2⌋(cid:1) dr ·rk, ifrisodd.
dr−2k,2,2,...,2
k=0
Inparticular,f(r,0)=1forallrP. (cid:0) (cid:1)
Foraproofsee[1].
5.2 Proof of Theorem 2.2
WeshallcomputethecharacteroftherepresentationρandcompareitwithCorollary5.2.
Recallthedefinitionoftheinversionsetofapermutationπ ∈S ,
n
Inv(π):={{i,j}|(j−i)·(π(j)−π(i))<0}.
Definition5.3 Foraninvolutionw ∈I letPair(w)bethesetof2-cyclesofw(consideredasunordered
n
2-sets). Forapermutationπ ∈S andaninvolutionw ∈I let
n n
Inv (π):=Inv(π)∩Pair(w)
w
and
inv (π):=#Inv (π).
w w
CombinatorialGelfandModels 721
Proposition5.4 [1,equation(4)]
ρ(π)Cw =(−1)invw(π)·Cπwπ−1 (∀π ∈Sn,w ∈In).
Itfollowsthat
Tr(ρ(π))= (−1)invw(π),
w∈InX∩Stn(π)
whereSt (π)isthecentralizerofπinS (i.e.,thestabilizerofπunderconjugation).
n n
Observation5.5 Letπ ∈S ,w∈I ∩St (π)anda ∈[n]anyletter. Thenoneofthefollowingholds:
n n n 1
(1) (a ,a ,...,a )isacycleinπ(r ≥1);a ,a ,...,a arefixedpointsofw.
1 2 r 1 2 r
(2) (a ,a ,...,a )and(a ,...,a )arecyclesinπ(r ≥1);(a ,a ),(a ,a ),...,(a ,a )
1 2 r r+1 2r 1 r+1 2 r+2 r 2r
arecyclesinw.
(3) (a ,a ,...,a )isacycleinπ(m≥1);(a ,a ),(a ,a ),...,(a ,a )arecyclesinw.
1 2 2m 1 m+1 2 m+2 m 2m
For every A ⊆ [n] let S := {π ∈ S |Supp(π) ⊆ A} be the subgroup of S consisting of all
A n n
permutationswhosesupportiscontainedinA.
Foreveryπ ∈ S and1 ≤ r ≤ nletA(π,r) ⊆ [n]bethesetofallletterswhichappearincyclesof
n
lengthrinπ. Inotherwords,
A(π,r):={i∈[n]|πr(i)=iand(∀j <r)πj(i)6=i}
Forexample,A(π,1)isthesetoffixedpointsofπ.
Denotebyπ therestrictionofπtoA(π,r). Thenπ maybeconsideredasapermutationinS .
|r |r A(π,r)
ByObservation5.5,
Corollary5.6 Fixπ ∈S . Eachw∈I ∩St (π)hasauniquedecomposition
n n n
w = w
r
r≥1
Y
where
w ∈I ∩St (π ) (∀r)
r SA(π,r) SA(π,r) |r
withA(π,r),π andS asdefinedabove;and
|r A(π,r)
Inv (π)= Inv (π ),
w wr |r
r≥1
[
adisjointunion.
Hence,itsufficestoprovethatTr(ρ(π))isequaltotherighthandsideoftheformulainCorollary5.2,
forπofcycletypern/r. Sinceρisaclassfunction,wemayassumethat
π =(1,2,...,r)(r+1,...,2r)···(n−r+1,n−r+2,...,n). (3)
722 RonM.Adin andAlexPostnikov andYuvalRoichman
Observation5.7 Letrbeapositiveinteger.
(1) Ifiandj aredistinctnonnegativeintegers, π asin(3)above,andw = (ir+1,jr+σ(1))(ir+
2,jr+σ(2))···(ir+r,jr+σ(r))(whereσissomepowerofthecyclicpermutatation(1,2,...,r)),
then
(−1)invw(π) =1.
(2) Ifr =2miseven,πasin(3)above,andw =(1,m+1)(2,m+2)···(m,2m),then
(−1)invw(π) =−1.
Lemma5.8 Foreveryoddrandapermutationπasin(3)above,
⌊n/2r⌋
n/r
(−1)invw(π) =#(I ∩St (π))= ·rk.
n n
n/r−2k,2,2,...,2
w∈InX∩Stn(π) Xk=0 (cid:18) (cid:19)
ProofofLemma 5.8. If r is odd thenonly cases (1)and (2) in Observation5.5 are possible. The first
equalityinthestatementofthelemmathenfollowsfromObservation5.7(1).Thesecondequalityfollows
fromObservation5.5(1)(2),countingtheinvolutionsw∈I ∩St (π)with#Supp(w)=2rk.
n n
2
Lemma5.9 Foreveryevenrandapermutationπasin(3)above,
0, ifn/risodd;
(−1)invw(π) =
( n/r ·rn/2r, ifn/riseven.
w∈InX∩Stn(π) 2,...,2
(cid:0) (cid:1)
Proof of Lemma 5.9. Let c = (ir +1,ir +2,...,ir +r) be one of the cycles of π, as in (3). By
i
Observation5.5,aninvolutionw ∈I ∩St (π)hasoneofthefollowingthreetypeswithrespecttoc :
n n i
Type(1): Eachelementofc isafixedpointofw.
i
Type(2): wmapsc ontoadifferentcycleofπ.
i
Type(3): r=2miseven,andc isaunionof2-cyclesofw:
i
{ir+t,ir+t+m}∈Pair(w) (1≤t≤m).
Denote
P :={w∈I ∩St (π)|wisoftype(2)w.r.t.allcyclesofπ}.
2 n n
Foranyw ∈(I ∩St (π))\P ,let
n n 2
i(w):=min{i|wisoftype(1)or(3)w.r.t.thecyclec }.
i
Denote
P :={w∈(I ∩St (π))\P |wisoftype(1)w.r.t.thecyclec }
1 n n 2 i(w)
CombinatorialGelfandModels 723
and
P :={w∈(I ∩St (π))\P |wisoftype(3)w.r.t.thecyclec }.
3 n n 2 i(w)
The map ϕ : P → P which changes the action of w on c from type (1) to type (3) is clearly a
1 3 i(w)
well-definedbijection;and,byObservation5.7(2),itreversesthesignof(−1)invw(π). Thecontributions
ofP andP tothesumthereforecanceleachother. EachelementoftheremainingsetP contributes1,
1 3 2
byObservation5.7(1).Lemma5.9follows.
2
Lemmas5.8and5.9completetheproofofTheorem2.2.
2
6 The Involutive Length
Recallthe definitionofthe involutivelengthℓˆ(Definition3.1). In orderto proveTheorem3.2we need
thefollowingcombinatorialinterpretationoftheinvolutivelengthℓˆ.
Lemma6.1 Letw ∈S beaninvolutionofcycletype2k1n−2k. Then
n
2k+1 1
ℓˆ(w):= t− + inv(w )−k . (4)
 2  2 |Supp(w)
t∈SXupp(w) (cid:18) (cid:19) (cid:2) (cid:3)
 
Proof: Denote the righthand side of (4) by f(w). It is easy to verifythat f(w) = 0 when ℓˆ(w) = 0,
i.e., whenw = s s ···s . Letuand v = s us be involutionsin S with ℓˆ(v) = ℓˆ(u)+1. Then
1 3 2k−1 i i n
|{i,i+1}∩Supp(u)|>0. If|{i,i+1}∩Supp(u)|=1then
t− t=±1
t∈SXupp(v) t∈SXupp(u)
andinv(v )=inv(u ). If|{i,i+1}∩Supp(u)|=2then
|Supp(v) |Supp(u)
t= t
t∈SXupp(v) t∈SXupp(u)
andinv(v )−inv(u )∈{2,0,−2}.Thusinbothcases|f(v)−f(u)|≤1. Thisproves,by
|Supp(v) |Supp(u)
inductiononℓˆ,thatf(w)≤ℓˆ(w)foreveryinvolutionw.
On the other hand, if w is an involution with f(w) > 0 then either t > 2k+1 , or
t∈Supp(w) 2
t = 2k+1 and inv(w ) > k. In the first case there exists i+1 ∈ Supp(w) such
t∈Supp(w) 2 |Supp(w) P (cid:0) (cid:1)
that i 6∈ Supp(w). Then f(s ws ) = f(w)−1. In the second case Supp(w) = {1,...,2k}. Since
P (cid:0) (cid:1) i i
inv(w )>k,w 6=s s ···s . Thustheremustexistaminimalisatisfyingw(i)>i+1.De-
|Supp(w) 1 3 2k−1
notingj :=w(i)−1>iwecannothavew(j)<i(byminimalityofi),andthereforew(j)>i=w(j+1)
andf(s ws )=f(w)−1. Weconcludethatℓˆ(w)≤f(w)foreveryinvolutionw.
j j
2
724 RonM.Adin andAlexPostnikov andYuvalRoichman
7 The Hecke Algebra
7.1 Proof of Theorem 3.2(1).
Weprovethatρ isanH (q)-representationbyverifyingthedefiningrelations.
q n
First, consider the braid relation T T T = T T T . To verify this relation observe that there
i i+1 i i+1 i i+1
are six possibletypesof orbitsof aninvolutionw underconjugationby hs ,s i - the subgroupin S
i i+1 n
generatedbys ands . Theseorbitsdifferbytheactionofwonthelettersi,i+1,i+2:
i i+1
1. i,i+1,i+26∈Supp(w).
2. Exactlyoneofthelettersi,i+1,i+2isinSupp(w).
3. Exactlytwoofthelettersi,i+1,i+2areinSupp(w),andthesetwolettersforma2-cycleinw.
4. Exactlytwoofthelettersi,i+1,i+2areinSupp(w),andthesetwolettersdonotforma2-cycle
inw.
5. i,i+1,i+2∈Supp(w),andtwooftheselettersforma2-cycleinw.
6. i,i+1,i+2∈Supp(w),andnotwooftheselettersforma2-cycleinw.
Notethatanorbitofthefirsttypeisoforderone;orbitsofthesecond,thirdandfifthtypeareoforder
three;andorbitsofthefourthandsixthtypeareofordersix. Moreover,byLemma6.1,orbitsofthesame
orderformisomorphicintervalsintheweakinvolutiveorder(seeDefinition3.1). Inparticular,allorbits
ofordersixhavearepresentativewofminimalinvolutivelength,suchthattheorbithastheform:
s s s ws s s
i i+1 i i i+1 i
ւ ց
s s ws s s s ws s
i i+1 i+1 i i+1 i i i+1
↓ ↓ (5)
s ws s ws
i+1 i+1 i i
ց ւ
w
Allorbitsoforderthreearelinearposets:
w ≺ s ws ≺ s s ws s , (6)
I i i I i+1 i i i+1
or
w ≺ s ws ≺ s s ws s . (7)
I i+1 i+1 I i i+1 i+1 i
Thustheanalysisisreducedintothreecases.
Case(a). Anorbitofordersix. By(2)and(5),therepresentationmatricesofthegeneratorswithrespect
totheorderedbasisC ,C ,C ,C ,C ,C are:
w siwsi si+1siwsisi+1 sisi+1siwsisi+1si si+1wsi+1 sisi+1wsi+1si
1−q 1 0 0 0 0
q 0 0 0 0 0
 
0 0 1−q 1 0 0
ρ (T )=
q i  0 0 q 0 0 0
 
 0 0 0 0 1−q 1
 
 0 0 0 0 q 0
 
 
CombinatorialGelfandModels 725
and
1−q 0 0 0 1 0
0 1−q 1 0 0 0
 
0 q 0 0 0 0
ρ (T )=
q i+1  0 0 0 0 0 q 
 
 q 0 0 0 0 0 
 
 0 0 0 1 0 1−q
 
 
Itiseasytoverifythatindeed
ρ (T )ρ (T )ρ (T )=ρ (T )ρ (T )ρ (T ).
q i q i+1 q i q i+1 q i q i+1
Case (b). An orbit of order three. With no loss of generality, the orbit is of type (6); the analysis of
type (7) is analogous. Then s ws = w and s (s s ws s )s = s s ws s . It is shown
i+1 i+1 i i+1 i i i+1 i i+1 i i i+1
in[1]thats ∈Des(w)ifandonlyifs ∈Des(s s ws s ).
i+1 i i+1 i i i+1
Giventheabove,by(2),therepresentationmatricesofthegeneratorswithrespecttotheorderedbasis
w ≺ s ws ≺ s s ws s are
I i i I i+1 i i i+1
1−q 1 0
ρ (T )= q 0 0
q i
 
0 0 x
 
and
x 0 0
ρ (T )= 0 1−q 1 ,
q i+1
 
0 q 0
 
wherex∈{1,−q}.Thesematricessatisfythebraidrelation.
Case (c). An orbit of order one. Then s ws = w, s ws = w and s ,s 6∈ Des(w). By (2),
i i i+1 i+1 i i+1
ρ (T )ρ (T )ρ (T )C = ρ (T )ρ (T )ρ (T )C = C , completingtheproofofthethirdrela-
q i q i+1 q i w q i+1 q i q i+1 w w
tion.
Theproofoftheothertworelationsiseasierandwillbelefttothereader.
2
Remark7.1 Substitutingq =1provesTheorem2.1.
7.2 Proof of Theorem 3.2(2).
WeapplyLusztig’sversionofTits’deformationtheoremtoprovethatρ isaGelfandmodel.
q
Consider the Hecke algebra H (q) as the algebra over Q(q1/2) spanned by {T |v ∈ S } with the
n v n
multiplicationrules
T T =T ifℓ(vu)=ℓ(v)+ℓ(u)
v u vu
and
(T +q)(T −1)=0 (∀s∈S).
s s