Specht Polytopes and Specht Matroids

0.31 MB·

by  John D. Wiltshire-Gordon

Specht Polytopes and Specht Matroids
JohnD.Wiltshire-Gordon,AlexanderWoo,andMagdalenaZajaczkowska
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Abstract The generators of the classical Specht module satisfy intricate relations.
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We introduce the Specht matroid, which keeps track of these relations, and the
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Spechtpolytope,whichalsokeepstrackofconvexityrelations.Weestablishbasic

] facts about the Specht polytope, for example, that the symmetric group acts tran-
O
sitively on its vertices and irreducibly on its ambient real vector space. A similar
C construction builds a matroid and polytope for a tensor product of Specht mod-
. ules,giving“Kroneckermatroids”and“Kroneckerpolytopes”insteadoftheusual
h
t Kroneckercoefficients.Wedubthisprocessofupgradingnumberstomatroidsand
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polytopes “matroidification,” giving two more examples. In the course of describ-
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ingtheseobjects,wealsogiveanelementaryaccountoftheconstructionofSpecht
[ modulesdifferentfromthestandardone.Finally,weprovidecodetocomputewith

1 SpechtmatroidsandtheirChowrings.
v
7 The irreducible representations of the symmetric group S were worked out by
n
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YoungandSpechtintheearly20thcentury,andtheyremainomnipresentinalge-
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braic combinatorics. The symmetric group S has a unique irreducible representa-
5 n
0 tionforeachpartitionofn.Forexample,S4 hasexactlyfiveirreduciblerepresenta-
. tionscorrespondingtothepartitions
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(4) (3,1) (2,2) (2,1,1) (1,1,1,1).
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JohnD.Wiltshire-Gordon
r
University of Wisconsin, Madison; Department of Mathematics; Van Vleck Hall; 480 Lincoln
a
Drive;Madison,WI53706;USA,e-mail:[email protected]
AlexanderWoo
UniversityofIdaho;DepartmentofMathematics;UniversityofIdaho;875PerimeterDriveMS
1103;Moscow,ID83844;USA,e-mail:[email protected]
MagdalenaZajaczkowska
UniversityofWarwick;MathematicsInstitute;GibbetHillRd;CoventryCV47AL;UK,e-mail:
[email protected]
1
2 JohnD.Wiltshire-Gordon,AlexanderWoo,andMagdalenaZajaczkowska
Young and Specht constructed these irreducible representations, which are now
called Specht modules. Young [21] gave a matrix representation and Specht [19]
gaveacombinatorialspanningset.Garnir[7]laterexplainedhowtorewriteSpecht’s
spanningsetintermsofYoung’sbasis.TheserewritingrulesarenowcalledGarnir
relations. Modern accounts of these constructions can be found in Sagan [17] or
JamesandKerber[10].
Thisclassicalapproach,however,privilegesYoung’sbasisoverotherbasesand
theGarnirrelationsoverotherlineardependencies.FocusingonYoung’sbasisand
theGarnirrelationsimmediatelybreaksthesymmetryofSpecht’sspanningsetand
ignoresitsothercombinatorialproperties.Certainly,therearelinearrelationsother
thanthosegivenbyGarnirandbasesotherthanthosegivenbyYoungtoinvestigate!
Tothisend,weintroducetheSpechtmatroid,whichencodesallthelineardepen-
dencies among the vectors of Specht’s spanning set. We also introduce the Specht
polytope,whichprovidesawaytovisualizetheSpechtmodule,sincethepolytope
sitsinsidethecorrespondingrealvectorspacewithpositivevolume,andtheaction
ofthesymmetricgrouptakesthepolytopetoitself.
Inthecaseofthepartition(n−1,1),werecoverbothclassicalconstructionsand
objects of current research interest. The corresponding Specht polytope is essen-
tially the root polytope of type A . In Theorem 6.5, we record a result of Ardila,
n
Beck, Hosten, Pfeifle and Seashore [2] describing the faces of this polytope. The
Spechtmatroidforthepartition(n−1,1)isthematroidofthebraidarrangement,
and hence its Chow ring is the cohomology ring for the moduli space M of n
0,n
markedpointsonthecomplexprojectiveline[4,5].
We compute further examples of Chow rings in §4, including the solution to
Problem1onGrassmanniansin[20],whichpartiallyinspiredthisproject.Westate
a combinatorial conjecture for the graded dimensions of the Chow rings for the
Specht matroid for the partition (2,1n−1). However, we do not study any further
connectionswithmodulispaces.
Our approach allows us to upgrade familiar combinatorial coefficients to ma-
troidsandpolytopes.Byanalogywithcategorification,whichsometimesupgrades
numberstovectorspaces,wecallthisprocessmatroidification,orpolytopification
when working over the reals. This is the subject of Section 7. In Theorem 7.3,
we polytopify the Kronecker coefficients, building the Kronecker polytopes. We
alsoconstructtheKroneckermatroidsencodingtheGarnir-stylerewritingrulesthat
govern linear dependence in a tensor product of Specht modules. An analogue of
Young’s basis for the Kronecker matroid would provide a combinatorial rule for
computingKroneckercoefficients.InTheorems7.6and7.9,wegivesimilarresults
forLittlewood-Richardsoncoefficientsandplethysmcoefficients.
The outline of the article is as follows. We begin in Section 1 with a self-
contained construction of the Specht module that is suited to our purposes. This
construction is a bit unusual in that it makes no mention of tabloids, polytabloids,
standardtableaux,orthegroupalgebra.InSections3,4,and5,weintroducerespec-
tively the Specht matroid, its Chow ring, and the Specht polytope; we then prove
somebasicgeneralfactsaboutthem.Section6isdevotedtothepartitions(n−1,1)
SpechtPolytopesandSpechtMatroids 3
and(2,1n−1),forwhichtheSpechtmatroidsandpolytopescoincidewithotherwell-
studiedobjects.WedescribeKronecker,Littlewood–Richardson,andplethysmma-
troidsandpolytopesinSection7.Section8includescodeforcalculatingtheobjects
describedinthispaper.
1 IntroductiontoSpechtModules
Our aim in this section is to give an exposition of the representation theory of the
symmetric group that is motivated from elementary combinatorial considerations.
ThereaderwhowishestostartwiththemainstatementsshouldfirstlookatDefini-
tions1.15and1.17andTheorem1.19.
Webeginwithanelementarycombinatoricsproblem:Inhowmanydistinctways
can the letters in a word TENNESSEE be rearranged? There are 9! ways to move
thelettersaround,butsincesomelettersarerepeated,someoftheserearrangements
give the same string. For example, the four Es can be rearranged to appear in any
orderwithoutaffectingthestring.Thisreasoninggivesthefollowinganswer:
9!
#{rearrangementsofTENNESSEE}= .
1!·4!·2!·2!
The idea of rearranging letters can be formalized as an action of the symmetric
group. In ourexample S acts on the wordTENNESSEE. The stabilizer subgroup
9
of S with respect to the word TENNESSEE is isomorphic to S ×S ×S ×S .
9 1 4 2 2
HencethepreviousargumentactuallyprovidesanisomorphismofS -sets,whichis
9
tosay,abijectionthatcommuteswiththegroupaction:
S
{rearrangementsofTENNESSEE}(cid:39) 9 .
S ×S ×S ×S
1 4 2 2
Usingtheorbit-stabilizertheorem,werecoverthenumericalanswerabove.
Nowweaddalayerofcomplexity.SupposewewishtounderstandtheS -set
9
{rearrangementsofTENNESSEE}
×
{rearrangementsofSASSAFRAS},
where S acts diagonally (i.e. simultaneously) on the two factors. This diago-
9
nal action makes sense because SASSAFRAS has the same number of letters as
TENNESSEE. Each factor in this Cartesian product has a single S -orbit, but the
9
productcertainlydoesnot!Forexample,thepairs
(cid:18) (cid:19) (cid:18) (cid:19)
EEEENNSST TENNESSEE
and
SSSSAAAFR SASSAFRAS
4 JohnD.Wiltshire-Gordon,AlexanderWoo,andMagdalenaZajaczkowska
cannot be in the same orbit because the upper Es and lower Ss always appear to-
getherinthefirstpairbutnotinthesecond.Anotherproofisthattheirorbitshave
different sizes. Indeed, if we consider a column such as E as a compound letter,
S
thenthetotalnumberofdistinctrearrangementsofthefirstcompoundwordinthese
compoundlettersis9!/(4!·2!·1!·1!·1!)=7560,whichdoesnotequalthenumber
9!/(1!·3!·2!·1!·1!·1!)=30240ofrearrangementsofthesecondcompoundword.
FortheconstructionoftheSpechtmodule,weareinterestedinfreeorbits.(Recall
thatanorbitisfreeifeachofitspointshastrivialstabilizer.)Inourcontext,anon-
trivialstabilizercomesfromrepeatedcolumns.Soapairisinafreeorbitifandonly
ifallofitscolumnsaredistinct.Forexample,
(cid:18) (cid:19)
SNETNEESE
SASSSAFAR
hasnorepeatedcolumns,soitsorbitisfree.Weclaim:
• thereisonlyonefreeorbit,andtherefore,
• wehavealreadyfoundit.
• Theproofisbasicallyapicture,andevenbetter,
• theproof-picture-ideaisstrongenoughtoconstructacompletesetofirreducible
representations over C for the symmetric group S . These are the Specht Mod-
n
ules.(Thestorywouldbethesameforanyfieldofcharacteristiczero.)
Hereisthepicture.
Fig.1:Asimultaneoushistogram
Theboxesprovideasimultaneoushistogramtallyingthelettermultiplicitiesfor
each word. From the picture, we see that the letter E from the word TENNESSEE
must appear once with each of the letters S, A, F, and R. Indeed, in order to keep
distinctthefourcolumnsinwhichEappears,Emustbepairedwitheachofthefour
availablelettersinthebottomrow.
RemovingthefourEsfromthepoolalongwithonecopyofeachofthelettersS,
A,F,andR,wemayproceedtopairoffNwiththetwolettersSandA.Continuing
SpechtPolytopesandSpechtMatroids 5
inductively, we see that the combinations that appear in a valid pair of rearrange-
mentsgivetheboxesinthediagram.
Wegivesomedefinitionsthatencodethesepictures.
Definition1.1.Apartitionofn∈Nisanintegervectorλ =(λ ,...,λ )suchthat
1 (cid:96)
λ ≥λ ≥···≥λ >0withλ +···+λ =n.Thenumber(cid:96)=(cid:96)(λ)isthelengthof
1 2 (cid:96) 1 (cid:96)
thepartition.
Definition1.2.AdiagramisafinitesubsetofN2 .Theelementsofadiagramare
≥1
calledboxes.
Definition1.3.Givenapartitionλ,thediagramassociatedtoλ is
D(λ)={(i,j)|1≤ j≤λ},
i
where,byconvention,λ =0ifi>(cid:96)(λ).
i
Thepartitioninourrunningexampleis(4,3,1,1);itsassociateddiagramis
{(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(3,1),(4,1)}.
Hereandeverywhereelse,wewillusematrixcoordinates,so(2,3)denotesthebox
inthesecondrowandthirdcolumn.
Thefollowingpropositionisimmediate.
Proposition1.4.AdiagramDisthediagramofsomepartitionλ ifandonlyifDis
closedundercoordinate-wise≤.Inotherwords,D=D(λ)forsomeλ ifandonly
if,forany(i,j)∈Dandany(a,b)with1≤a≤iand1≤b≤ j,(a,b)∈D.
Definition1.5.We say two words w ,w of (the same) length n have complemen-
1 2
taryrearrangementsifthediagonalactionofS ontheproduct
n
{rearrangementsofw }×{rearrangementsofw }
1 2
hasauniquefreeorbit.If,furthermore,(w ,w )isinthisfreeorbit,thenwesayw
1 2 1
andw arecomplementary.
2
For example, our diagram above shows that TENNESSEE and SASSAFRAS
have complementary rearrangements. The two words are not complementary, but
TENENEESSandSASSAFRASarecomplementary.
Theorem1.6.Twowordsw ,w havecomplementaryrearrangementsifandonlyif
1 2
thereexistsaparititondiagramDwiththe“simultaneoushistogram”property
th
#occurrencesinw ofits j -most-commonletter=#{(d ,d )∈D|d = j}.
i 1 2 i
Proof. Wehavealreadyarguedtheharddirection.IfwehaveapartitiondiagramD
withthesimultaneoushistogramproperty,wecanputinthebox(d ,d )thed -th
1 2 1
6 JohnD.Wiltshire-Gordon,AlexanderWoo,andMagdalenaZajaczkowska
most common letter in w and the d -th most common letter in w (breaking ties
1 2 2
arbitrarily). The boxes have distinct pairs, so there exists at least one free orbit;
thisorbitisuniquebytheiterativeargumentbelowFigure1.Intheotherdirection,
rewritethewordsusingnumbersinN sothat(ineachword)kappearsatleastas
≥1
oftenask+1.Thentake
D={(d ,d )|d appearsinacolumnwithd intheuniquefreeorbit}.
1 2 1 2
(cid:116)(cid:117)
Wewillproceedtousetheideaofcomplementarywordstoconstructirreducible
representations of the symmetric group S . Before doing so, we recall some basic
n
definitionsinrepresentationtheory.
Definition1.7.Let G be a group. A (complex) representation of G is a C-vector
spaceV alongwithalinearactionofGonV,meaningthat,foranyvectorsv,w∈V,
anyscalarc∈C,andanygroupelementg∈G,
• g·v+g·w=g·(v+w),and
• c(g·v)=g·(cv).
Alternatively,thedataofarepresentationcanbeencodedinagrouphomomorphism
G→GL(V),whereGL(V)isthegroupofinvertiblelinearautomorphismsofV.
Definition1.8.IfV andW arerepresentationsofG,alineartransformationϕ:V →
W isamapofGrepresentationsifϕ commuteswiththeactionofG,meaningthat
ϕ(g·v)=g·(ϕ(v))forallg∈Gandallv∈V.
Definition1.9.IfV andW arerepresentationsofG,thenthetensorproductV⊗W
isarepresentationofGundertheaction
g·(v⊗w)=(g·v)⊗(g·w).
Definition1.10.IfV isarepresentationofG,thenV∨=Hom(V,C)isarepresen-
tation of G under the action where, for f ∈V∨ and g∈G, g· f is the functional
definedby
(g·f)(v)= f(g−1·v)
foranyv∈V.
WeneedtwodefinitionsspecifictothegroupS .
n
Definition1.11.Foranyn,therepresentationε istheonedimensionalvectorspace
onwhichS actsbysign.Thismeans,forv∈ε,g·v=vifgisanevenpermutation
n
andg·v=−vifgisanoddpermutation.
IfV isanyrepresentationofS ,thenV⊗ε isisomorphictoV asavectorspace,
n
buttheactionofS differsinthattheactionofanoddpermutationpicksupasign.
n
Definition1.12.Given aword w, therepresentationV(w)is thevector spacewith
basisgivenbytherearrangementsofw,withS actingbypermutingourbasisac-
n
cordingtohowitrearrangeswords.
SpechtPolytopesandSpechtMatroids 7
The representation V(w) is special in that the action of S is actually induced
n
from a combinatorial action of S on a basis ofV(w). This property has a useful
n
consequence.
Lemma1.13.Givenanywordwoflengthn,wehaveacanonicalisomorphismof
S representationsV(w)(cid:39)V(w)∨givenbyidentifyingourbasisofrearrangements
n
withitsdualbasis.
Proof. SinceV(w ) has a canonical basis {v }, where r is an arbitrary rearrange-
1 r
ment,V(w )∨hasadualbasis{f },and
1 r
(cid:26)1 ifr(cid:48)=σ·r
(σ·fr)(vr(cid:48))= 0 ifr(cid:48)(cid:54)=σ·r.
Henceσ·f = f ,andthemapsendingv to f isanisomorphismofS represen-
r σ·r r r n
tations. (cid:116)(cid:117)
WearenowreadytoconstructrepresentationsofS usingthecombinatoricsof
n
complementarywordsdiscussedabove(seeDefinition1.5).
Corollary1.14.Supposew andw arewordsoflengthnwithcomplementaryre-
1 2
arrangements.Thenthereisaunique-up-to-scalingmapofS representations
n
ϕ:V(w )⊗ε −→V(w ),
2 1
andtheimageofϕ isanirreduciblerepresentation.
Proof. Consider an arbitrary map of S representations (meaning a linear mapΨ
n
whereσ·Ψ(v)=Ψ(σ·v)forallv)
Ψ :V(w )⊗V(w )∨→ε.
2 1
Anysuchmapmustfactorthroughthequotient
V(w )⊗V(w )∨/W,
2 1
whereW isthesubspacespannedbyelementsoftheform(sign(σ)v−σv),forany
v.Inthisquotient,anyelementofS actsontheimageofanyvectorbysign.Pairs
n
ofrearrangementsformabasisforthetensorproduct,sotheimagesofthesebasis
vectorsstillspanthequotient.Supposesomepairofrearrangementshasarepeated
column; then swapping those columns fixes the pair. The vectors indexed by such
pairsbecomezerointhequotientbecausetranspositionsareodd.
By Theorem 1.6, the action on pairs has a unique free orbit. Any two vectors
in the free orbit are related by a unique permutation, and so any vector spans the
quotient,whichmustthereforebeone-dimensional.Usingtensor-homadjunction,
Hom(V(w )⊗V(w )∨/W , ε)
2 1
(cid:39)Hom(V(w )⊗V(w )∨, ε)
2 1
(cid:39)Hom(V(w )⊗ε ,V(w )),
2 1
8 JohnD.Wiltshire-Gordon,AlexanderWoo,andMagdalenaZajaczkowska
where the first space is one-dimensional by the previous argument. (Note that all
isomorphismsarenatural.)Wemaytakeϕ tobeanynonzerovectorinthelasthom-
space.Thisshowswehaveaunique-up-to-scalinglinearmap
ϕ:V(w )⊗ε −→V(w ).
2 1
Now we show V =imϕ is irreducible. Suppose U ⊂V is a proper subrepre-
sentation. By Maschke’s theorem, there exists a complementary subrepresentation
U(cid:48)⊆V withthepropertythatV =U⊕U(cid:48).Letπ:V →V denotetheprojectionwith
kernelU(cid:48)andimageU.Butnowthecomposite
π
V(w )⊗ε −→V −→V −→V(w )
2 1
cannotbeascalarmultipleofϕ sinceitisnonzeroandhasadifferentimage. (cid:116)(cid:117)
Thefollowingdefinitionwillhelpuswriteanexplicitexampleofthelinearmapϕ.
Definition1.15.Letw ,w befixedwordsoflengthn,andletr andr bearbitrary
1 2 1 2
rearrangementsofw andw respectively.DefineYoung’scharacter
1 2
Y (r ,r )=∑sign(σ),
w1,w2 1 2
σ
whereσ ∈S rangesoverallpermutationssuchthatσ·w =r andσ·w =r .
n 1 1 2 2
Proposition1.16.Young’s character takes values in {−1,0,1}. Whenever writing
r ontopofr hasarepeatedcolumn,wehaveY (r ,r )=0.Ifw andw are
2 1 w1,w2 1 2 1 2
complementary,thenY (r ,r )(cid:54)=0exactlywhenallcolumnsaredistinct.
w1,w2 1 2
Proof. If there is a repeated column, then flipping those columns does not change
thevalueofY(sincetheinputsarethesame),butitalsointroducesasignchange
(since flipping two columns is an odd permutation). It follows that Y=0 in this
case.Ifallthecolumnsaredistinct,thenthereisatmostonepermutationcarrying
eachrowbacktow,andsothesumeitherisemptyorhasasingleterm.Intheevent
i
thatw andw arecomplementary,thesumisnonempty. (cid:116)(cid:117)
1 2
Definition1.17.Ifw ,w arecomplementarywordsoflengthn,theSpechtmatrix
1 2
ϕ(w ,w )isthe
1 2
{rearrangementsofw }×{rearrangementsofw }
1 2
matrix with (r ,r )-entry Y (r ,r ). If w and w are not complementary but
1 2 w1,w2 1 2 1 2
havecomplementaryrearrangements,wechoosecomplementaryrearrangementsw(cid:48)
1
and w(cid:48) of w and w respectively and define the Specht matrix ϕ(w ,w ) to be
2 1 2 1 2
ϕ(w(cid:48),w(cid:48)). The column-span of the Specht matrix (as a subspace ofV(w )) is the
1 2 1
SpechtmoduleV(w ,w ).
1 2
Note that, in the case where w and w are not themselves complementary, the
1 2
Specht matrix ϕ(w ,w ) is only defined up to a global choice of sign (depending
1 2
SpechtPolytopesandSpechtMatroids 9
onwhichcomplementaryrearrangementsarechosen),buttheSpechtmoduleisthe
sameregardlessofthischoice.
ThesymmetricgroupactsonV(w ,w )bypermutingtherearrangementsofw .
1 2 1
Example1.18.Letw =1122andw =1212.ThentheSpechtmatrixϕ(1122,1212)
1 2
isshowninTable1.
112212121221211222112121
1212 1 0 −1 −1 1 0
1122 0 −1 1 1 0 −1
1221 −1 1 0 0 −1 1
2121 1 0 −1 −1 1 0
2211 0 −1 1 1 0 −1
2112 −1 1 0 0 −1 1.
Table1:Spechtmatrixϕ(1122,1212)
WewilldescribenowtheactionofS ontherowsofthisSpechtmatrix.Consider
n
theactionofσ =(123)ontherowwordw .Thisactionchangestheorderofrows
1
to2112,1212,2211,1221,2121and1122.Whateffectdoesithaveonthecolumns
of the Specht matrix? Let us look, for example, at the first column, labelled by
1122.Withthenewroworderthiscolumnbecomesc=(−1,1,0,−1,1,0),which
is the original column for r =2112. If we consider also the action of σ on the
2
rearrangements of w , we see that the rearrangement r = 2112 of w becomes
2 2 2
σ·2112=1122,whichisthelabelofthefirstcolumn.Thepermutationσ actsthis
wayontheSpechtmodule.
Wehavethefollowingfundamentalfactaboutthisrepresentation.
Theorem1.19.TheSpechtmoduleV(w ,w )isanirreduciblerepresentationofS .
1 2 n
Proof. BytheproofofCorollary1.14,weseethathavingauniquefreeorbitgives
aunique-up-to-scalingmapϕ:V(w )⊗ε →V(w )whoseimageisirreducible.It
2 1
remainsonlytoshowthattheSpechtmatrixprovidesanexplicitchoiceforϕ.This
wasaccomplishedinProposition1.16,whichshowsthatYprovidesamap
ε →V(w )⊗V(w )∨,
1 2
wherewehaveusedthefactthattheactionofS onV(w )andV(w )∨ arecanoni-
n 2 2
callyequal. (cid:116)(cid:117)
Therepresentationdoesnotactuallydependonthewordsbutonlyontheparti-
tiondiagram,sowemakethefollowingdefinition:
Definition1.20.Ifλ isapartitionofn,thentheSpechtmoduleV(λ)istheSpecht
moduleV(w ,w )foranychoiceofcomplementaryw andw suchthatthediagram
1 2 1 2
10 JohnD.Wiltshire-Gordon,AlexanderWoo,andMagdalenaZajaczkowska
showing w and w are complementary is D(λ). We call w the row word and w
1 2 1 2
thecolumnword.
Forexample,thematrixinExample1.18istheSpechtmatrixV(2,2).
Remark1.21.Since every entry of the Specht matrix is a 0, 1, or −1, the Specht
module can be similarly defined over any field. However, over a field of positive
characteristic, Maschke’s Theorem does not hold. Nevertheless, over any field of
characteristic other than 2, the statements above show that the Specht module is
indecomposable,meaningthatV(w ,w )cannotbewrittenasthedirectsumoftwo
1 2
subrepresentations.
Notethat,ifw andw arecomplementarywithassociateddiagramD,thenw
1 2 2
andw arealsocomplementary,withanassociateddiagramwhichisthetranspose
1
ofD.Itwillbeusefultohaveadefinitiondescribingthisrelationship.
Definition1.22.Given a partition λ, the conjugate partition λ∗ is the partition
whosediagramD(λ∗)isthetransposeofthediagramD(λ).Formally,wehave
λ∗=#{k|λ ≥i}.
i k
For example, if λ =(4,3,1,1), then λ∗ =(4,2,2,1). This natural combinatorial
constructionhasrepresentation-theoreticmeaning,asthenextresultshows.
Theorem1.23.WehaveanisomorphismofS representationsV(λ∗)(cid:39)V(λ)∨⊗ε.
n
Proof. ObservethattransposingtheSpechtmatrixφ(w ,w )givestheSpechtma-
1 2
trix φ(w ,w ), which is the Specht matrix for the conjugate partition. After trans-
2 1
posing,however,thesymmetricgroupactsonφ(w ,w )byrearrangingthecolumn
2 1
wordw .ThisisnotthecorrectactionofthesymmetricgroupontheSpechtmatrix.
1
However,theactionisoffonlybyasignandadualbecause,bytheidentity
Y(w ,σw )=(−1)σ·Y(σ−1w ,w ),
2 1 2 1
thesymmetricgroupactsontherowwordw byσ−1 andpicksupasignwithodd
2
permutations. (cid:116)(cid:117)
Remark1.24.We have not needed it, but it is actually the case that Specht mod-
ulesareself-dualinthesensethatthereisanabstractisomorphismV(λ)∨(cid:39)V(λ).
Choosing a basis from the columns of the Specht matrix, it would be possible to
write matrices for the action of the symmetric group. Evidently, these matrices
would contain only real numbers—in fact, only rational numbers—and so their
traces would be real as well. It follows that the character of a Specht module is
real,andsoitsdual,whosecharacterisgivenbycomplexconjugation,isthesame.
Withthisfactinmind,Theorem1.23givesthatV(λ∗)∼=V(λ)⊗ε.
We remark briefly on the relationship between the construction above and the
moreusualpresentationfound,forexample,inJamesandKerber[10,Chapter7.1]