Affine structures and a tableau model for E_6 crystals

0.31 MB·English

by  Brant Jones

AFFINE STRUCTURES AND A TABLEAU MODEL FOR E6 CRYSTALS
BRANTJONESANDANNESCHILLING
0 ABSTRACT. We provide the unique affine crystal structure for type E(1) Kirillov–Reshetikhin crystals
1 6
corresponding to themultiplesof fundamental weights sΛ ,sΛ , and sΛ for all s ≥ 1(inBourbaki’s
0 1 2 6
labelingoftheDynkinnodes,where2istheadjointnode). Ourmethodsintroduceageneralizedtableaux
2
modelforclassicalhighestweightcrystalsoftypeE andusetheorderthreeautomorphismoftheaffine

n E(1) Dynkin diagram. Inaddition, weprovide a conjecture for theaffine crystal structure of type E(1)
a 6 7
Kirillov–Reshetikhincrystalscorrespondingtotheadjointnode.
J

9
1

]
O 1. INTRODUCTION
C
A uniform description of perfect crystals of level 1 corresponding to the highest root θ was given
.
h in[BFKL06]. Ageneralizationtohigherlevelsforcertainnonexceptionaltypeswasstudiedin[Kod08].
t
a ThesecrystalsB oflevelshavethefollowingdecomposition whenremovingthezeroarrows[Cha01]:
m
s
[ (1.1) B ∼= B(kθ),
M

2 k=0
v
whereB(λ)denotesthehighestweightcrystalwithhighestweightλ.
2
4 Inthispaper, weprovidetheuniqueaffinecrystal structure fortheKirillov–Reshetikhin crystals Br,s
4 of type E(1) for the Dynkin nodes r = 1,2, and 6 in the Bourbaki labeling, where node 2 corresponds
2 6
totheadjointnode(seeFigure1). Inaddition, weprovideaconjecturefortheaffinecrystalstructurefor
.
9 (1)
typeE Kirillov–Reshetikhin crystalsoflevelscorresponding totheadjointnode.
0 7
9 Ourconstructionoftheaffinecrystalsusestheclassicaldecomposition(1.1)togetherwithapromotion
0 operator which yields the affine crystal operators. Combinatorial models of all Kirillov–Reshetikhin
:
v crystals of nonexceptional types were constructed using promotion and similarity methods in [Sch08,
i (1) (1)
X OS08, FOS09]. Perfectness was proven in [FOS08]. Affine crystals of type E and E of level 1
6 7
corresponding tominusculecoweights(r = 1,6)werestudiedbyMagyar[Mag06]usingtheLittelmann
r
a path model. Hernandez and Nakajima [HN06] gave a construction of the Kirillov–Reshetihkin crystals
Br,1 forallrfortypeE(1) andmostnodesrintypeE(1).
6 7
Fornonexceptional types,theclassicalcrystalsappearinginthedecomposition (1.1)canbedescribed
usingKashiwara–Nakashima tableaux[KN94]. Weprovideasimilarconstruction forgeneraltypes(see
Theorem2.6). ThisinvolvestheexplicitconstructionofthehighestweightcrystalsB(Λ )corresponding
i
tofundamental weightsΛ using theLenart–Postnikov [LP08]modelandthenotion ofpairwiseweakly
i
increasing columns(seeDefinition2.1).
The promotion operator for the Kirillov–Reshetikhin crystal Br,s of type E(1) for r = 1,6 is given
6
in Theorem 3.13 and for r = 2 in Theorem 3.22. Our construction and proofs exploit the notion of
Date:January19,2010.
1991MathematicsSubjectClassification. 81R50;81R10;17B37;05E99.
Keywordsandphrases. Affinecrystals,Kirillov–Reshetikhincrystals,typeE .
6
BJwaspartiallysupportedbyNSFgrant DMS-0636297. ASwaspartiallysupportedbytheNSFgrantsDMS–0501101,
DMS–0652641,andDMS–0652652.
1
2 BRANTJONESANDANNESCHILLING
compositiongraphs(Definition3.10)andthefactthatthepromotionoperatorwechoosehasorderthree.
AsshowninTheorem3.9,apromotionoperatoroforderthreeyieldsaregularcrystal. InConjecture3.26
wealsoprovideapromotionoperatorofordertwoforthecrystalsB1,s oftypeE(1). However,fororder
7
twopromotion operators theanalogueofTheorem3.9ismissing.
This paper is structured as follows. In Section 2, the fundamental crystals B(Λ ) and B(Λ ) are
1 6
constructed explicitly for typeE anditisshownthatallother highest weight crystals B(λ)oftypeE
6 6
canbeconstructed from these. Similarly, B(Λ )yields allhighest weight crystals B(λ)fortypeE . In
7 7
Section2.4,ageneralizedtableauxmodelisgivenforB(λ)forgeneraltypes. Inparticular,weintroduce
thenotionofweakincrease. TheresultsareusedtoconstructtheaffinecrystalsinSection3. InSection4,
(1)
wegivesomedetailsabouttheSageimplementation oftheE ,E ,andE crystalsconstructed inthis
6 7 6
paper. Someoutlookandopenproblemsarediscussed inSection5. Appendices AandBcontaindetails
about the proofs for the construction of the affine crystals, in particular the usage of oriented matroid
theory.
Acknowledgments. We thank Daniel Bump for his interest in this work, reviewing some of our Sage
code related to E and E , and his insight into connections of B(Λ ) of type E and the Weyl group
6 7 1 6
action on 27 lines on a cubic surface. We are grateful to Jesus DeLoera and Matthias Koeppe for their
insights on oriented matroids. We thank Masato Okado for pointing [KMOY07, Theorem 6.1] out to
us, and his comments and insights on earlier drafts of this work. We thank Satoshi Naito and Mark
Shimozonofordrawingourattention tomonomialtheoryandreferences [LS86,Lit96].
Forourcomputerexplorations weusedandimplementednewfeaturesintheopen-source mathemati-
calsoftwareSage[WSea09]anditsalgebraiccombinatoricsfeaturesdevelopedbytheSage-Combinat
community [SCc09]; we are grateful to Nicolas M. Thie´ry for all his support. Figure 3 was produced
usinggraphviz,dot2tex,andpgf/tikz.
2. A TABLEAU MODEL FOR FINITE-DIMENSIONAL HIGHEST WEIGHT CRYSTALS
Inthissection,wedescribeamodelfortheclassicalhighestweightcrystalsintypeE. InSection2.1,
we introduce our notation and give the axiomatic definition of a crystal. The tensor product rule for
crystalsisreviewedinSection2.2. InSection2.3,wegiveanexplicitconstruction ofthehighestweight
crystalsassociatedtothefundamental weightsintypesE andE . InSection2.4,wegiveageneralized
6 7
tableaux model to realize all of the highest weight crystals in these types. Thegeneralized tableaux are
type-independent, and can be viewed as anextension of the Kashiwara–Nakashima tableaux [KN94]to
typeE. Forageneral introduction tocrystals wereferto[HK02].
2.1. Axiomaticdefinitionofcrystals. DenotebygaLiealgebraorsymmetrizableKac-Moodyalgebra,
P the weight lattice, I the index set for the vertices of the Dynkin diagram of g, {α ∈ P | i ∈ I}
i
the simple roots, and {α∨ ∈ P∗ | i ∈ I} the simple coroots. Let U (g) be the quantized universal
i q
envelopingalgebraofg. AU (g)-crystal[Kas95]isanonemptysetB equippedwithmapswt :B → P
q
ande ,f :B → B ∪{0}foralli ∈ I,satisfying
i i
f (b) = b′ ⇔ e (b′)= bifb,b′ ∈ B
i i
wt(f (b)) = wt(b)−α iff (b) ∈ B
i i i
hα∨,wt(b)i = ϕ (b)−ε (b).
i i i
Here,wehave
ε (b) = max{n ≥ 0 |en(b) 6= 0}
i i
ϕ (b) = max{n ≥ 0 |fn(b) 6= 0}
i i
AFFINESTRUCTURESANDATABLEAUMODELFORE6 CRYSTALS 3
for b ∈ B, and we denote hα∨,wt(b)i by wt (b). A U (g)-crystal B can be viewed as a directed edge-
i i q
coloredgraphcalledthecrystalgraphwhoseverticesaretheelementsofB,withadirectededgefromb
to b′ labeled i ∈ I, if and only if f (b) = b′. Given i ∈ I and b ∈ B, the i-string through b consists of
i
thenodes{fm(b) :0 ≤ m ≤ ϕ (b)}∪{em(b) : 0 <m ≤ε (b)}.
i i i i
Let{Λ | i ∈ I}bethefundamental weights ofg. Foreveryb ∈ B defineϕ(b) = ϕ (b)Λ and
ε(b) = i ε (b)Λ . Anelement b ∈ B iscalled highest weightife (b) = 0foralliP∈i∈II. Wiesayithat
Pi∈I i i i
B isa highest weight crystal of highest weight λ if ithas aunique highest weight element of weight λ.
Foradominantweightλ,weletB(λ)denotetheuniquehighest-weight crystalwithhighestweightλ.
Itfollowsfromthegeneral theory thateveryintegrable U (g)-module decomposes asadirect sumof
q
highestweightmodules. Onthelevelofcrystals,thisimpliesthateverycrystalgraphBcorrespondingto
anintegrable module isaunion ofconnected components, andeachconnected component isthecrystal
graphofahighestweightmodule. WedenotethisbyB = B(λ)forsomesetofdominantweightsλ,
L
andwecalltheseB(λ)thecomponents ofthecrystal.
Anisomorphism of crystals isabijection Ψ : B ∪{0} → B′∪{0} such that Ψ(0) = 0, ε(Ψ(b)) =
ε(b), ϕ(Ψ(b)) = ϕ(b), f Ψ(b) = Ψ(f (b)), and Ψ(e (c)) = e Ψ(c) for all b,c ∈ B, Ψ(b),Ψ(c) ∈ B′
i i i i
wheref (b)= c.
i
Whenλisaweightinanaffinetype,wecall
e
(2.1) hλ,ci = a∨hλ,α∨i
X i i
e i∈I∪{0} e
the level of λ, where c is the canonical central element and λ = λ Λ is the affine weight. In
Pi∈I∪{0} i i
ourwork,weewilloftencomputethe0-weightλ Λ atlevele0foranodebinaclassicalcrystalfromthe
0 0
classical weightλ = λ Λ = wt(b)bysettinghλ Λ +λ,ci = 0andsolving forλ .
Pi∈I i i 0 0 0
Suppose that g is a symmetrizable Kac–Moody algebra and let U′(g) be the corresponding quantum
q
algebrawithoutderivation. ThegoalofthisworkistostudycrystalsBr,sthatcorrespondtocertainfinite
dimensional U′(g)-modules known as Kirillov–Reshetikhin modules. Here, r is a node of the Dynkin
q
diagram and s is a nonnegative integer. The existence of the crystals Br,s that we study follows from
resultsin[KKM+92],whiletheclassical decomposition ofthesecrystalsisgivenin[Cha01].
2.2. Tensorproductsofcrystals. LetB ,B ,...,B beU (g)-crystals. TheCartesian product B ×
1 2 L q 1
B × ··· × B has the structure of a U (g)-crystal using the so-called signature rule. The resulting
2 L q
crystalisdenotedB = B ⊗B ⊗···⊗B anditselements(b ,...,b )arewrittenb ⊗···⊗b where
1 2 L 1 L 1 L
b ∈ B . The reader is warned that our convention is opposite to that of Kashiwara [Kas95]. Fix i ∈ I
j j
andb = b ⊗···⊗b ∈ B. Thei-signatureofbisthewordconsistingofthesymbols+and−givenby
1 L
−···− +···+ ··· −···− +···+ .
ϕ|i(b1{)ztim}es ε|i(b1{)ztim}es ϕ|i(bL{)ztim}es εi|(bL{)ztim}es
The reduced i-signature of b is the subword of the i-signature of b, given by the repeated removal of
adjacent symbols+−(inthatorder);ithastheform
−···− +···+.
|ϕi{tizmes} |εi{tizmes}
Ifϕ = 0thenf (b) = 0;otherwise
i i
f (b ⊗···⊗b )= b ⊗···⊗b ⊗f (b )⊗···⊗b
i 1 L 1 j−1 i j L
where the rightmost symbol − in the reduced i-signature of b comes from b . Similarly, if ε = 0 then
j i
e (b)= 0;otherwise
i
e (b ⊗···⊗b )= b ⊗···⊗b ⊗e (b )⊗···⊗b
i 1 L 1 j−1 i j L
4 BRANTJONESANDANNESCHILLING
where the leftmost symbol + in the reduced i-signature ofb comes from b . Itis not hard to verify that
j
this defines the structure of a U (g)-crystal with ϕ (b) = ϕ and ε (b) = ε in the above notation, and
q i i i i
weightfunction
L
wt(b ⊗···⊗b ) = wt(b ).
1 L X j
j=1
2.3. FundamentalcrystalsfortypeE andE . LetI = {1,2,3,4,5,6} denotetheclassicalindexset
6 7
forE . WenumberthenodesoftheaffineDynkindiagram asinFigure1.
6
•0
•2 •2
•1 •3 •4 •5 •6 •0 •1 •3 •4 •5 •6 •7
(1) (1)
FIGURE 1. AffineE andE Dynkindiagrams
6 7
Classical highest-weight crystals B(λ) for E can be realized by the Lenart–Postnikov alcove path
6
modeldescribedin[LP08]. WeimplementedthismodelinSageandhaverecordedthecrystalB(Λ )in
1
Figure2. Thiscrystalhas27nodes.
Todescribeourlabelingofthenodes,observethatallofthei-stringsinB(Λ )havelength1foreach
1
i ∈ I. Therefore, the crystal admits a transitive action of the Weyl group. Also, it is straightforward
to verify that all of the nodes in B(Λ ) are determined by weight. For our work in Section 3, we also
1
computethe0-weightatlevel0ofanodebinanyclassicalcrystalfromtheclassicalweightasdescribed
inRemark3.4.
Thus,welabel thenodes ofB(Λ )byweight, whichisequivalent torecording whichi-arrows come
1
inandoutofb. Thei-arrowsintobarerecordedwithanoverlinetoindicatethattheycontributenegative
weight,whilethei-arrowsoutofbcontribute positiveweight.
UsingtheLenart–Postnikovalcovepathmodelagain,wecanverifythatB(Λ )alsohas27nodesand
6
isdualtoB(Λ )inthesensethatitscrystalgraphisobtainedfromB(Λ )byreversingallofthearrows.
1 1
Reversing the arrows requires us to label the nodes of B(Λ ) by the weight that is the negative of the
6
weight of the corresponding node in B(Λ ). Moreover, observe that B(Λ ) contains no pair of nodes
1 1
with weights µ, −µ,respectively. Hence, wecan unambiguously label any node of B(Λ )∪B(Λ )by
1 6
weight.
Itisstraightforwardtoshowusingcharactersthateveryclassicalhighest-weightrepresentationB(Λ )
i
for i ∈ I can be realized as a component of some tensor product of B(Λ ) and B(Λ ) factors. On the
1 6
level of crystals, the tensor products B(Λ )⊗k, B(Λ )⊗k and B(Λ )⊗B(Λ ) are defined for all k by
1 6 6 1
thetensorproduct ruleofSection2.2. Therefore, wecanrealizetheotherclassical fundamental crystals
B(Λ )asshowninTable1. Thereareadditional realizations forthesecrystalsobtained bydualizing.
i
(1)
The Dynkin diagram of type E is shown in Figure 1. The highest weight crystal B(Λ ) has 56
7 7
nodes and these nodes all have distinct weights (see Figure 3). Also, ϕ (b) ≤ 1 and ε (b) ≤ 1 for all
i i
i ∈{1,2,...,7}andb ∈ B(Λ ). Usingcharactercalculations, wecanshowthateveryclassicalhighest-
7
weight representation B(Λ ) appears in some tensor product of B(Λ ) factors. In Table 2, we display
i 7
realizations foralloftheclassical fundamental crystals B(Λ )intypeE .
i 7
Green [Gre07, Gre08] has another construction of the 27-dimensional crystals B(Λ ) and B(Λ )
1 6
of type E , and the 56-dimensional crystal B(Λ ) of type E in terms of full heaps, and also gives the
6 7 7
connectionofthefundamentalE crystalswiththe27linesonacubicsurface. ALittlewood-Richardson
6
rulefortypeE wasgivenin[Hos07]usingpolyhedral realizations ofcrystalbases.
6
AFFINESTRUCTURESANDATABLEAUMODELFORE6 CRYSTALS 5
1 3 4 5 6
¯01 // ¯0¯13 // ¯0¯34 // ¯0¯425 // ¯0¯526 // ¯0¯62
2 2 2
(cid:15)(cid:15) 5 (cid:15)(cid:15) 6 (cid:15)(cid:15)
¯25 // ¯2¯546 // ¯2¯64
4 4
(cid:15)(cid:15) 6 (cid:15)(cid:15) 5
¯436 // ¯4¯635 // ¯53
3 3 3
(cid:15)(cid:15) 6 (cid:15)(cid:15) 5 (cid:15)(cid:15) 4 2
¯316 // ¯3¯615 // ¯3¯514 // ¯412 // ¯210
1 1 1 1 1
(cid:15)(cid:15) 6 (cid:15)(cid:15) 5 (cid:15)(cid:15) 4 (cid:15)(cid:15) 2 (cid:15)(cid:15)
¯16 // ¯1¯65 // ¯1¯54 // ¯1¯423 // ¯1¯230
3 3
(cid:15)(cid:15) 2 (cid:15)(cid:15)
¯32 // ¯2¯340
4
(cid:15)(cid:15)
¯450
5
(cid:15)(cid:15)
¯560
6
(cid:15)(cid:15)
¯60
FIGURE 2. CrystalgraphforB(Λ1)oftypeE6
TABLE 1. Fundamentalrealizations forE6
Generator in Dimension
B(Λ ) 2¯1¯0⊗¯01 B(Λ )⊗B(Λ ) 78
2 6 1
B(Λ ) ¯0¯13⊗¯01 B(Λ )⊗2 351
3 1
B(Λ ) ¯0¯34⊗¯0¯13⊗¯01 B(Λ )⊗3 2925
4 1
B(Λ ) 5¯6¯0⊗6¯0 B(Λ )⊗2 351
5 6
2.4. Generalizedtableaux. Inthissection, wedescribehowtorealizethecrystalB(Λ +Λ +···+
i1 i2
Λ ) inside the tensor product B(Λ )⊗B(Λ )⊗···⊗B(Λ ), where the Λ are all fundamental, or
ik i1 i2 ik i
more generally dominant weights. Our arguments use only abstract crystal properties, so the results in
thissectionapplytoanyfinitetype.
If b is the unique highest weight node in B(λ) and c is the unique highest weight node in B(µ),
then B(λ + µ) is generated by b ⊗ c ∈ B(λ) ⊗ B(µ). Iterating this procedure provides a recursive
description ofanyhighest-weight crystalembedded inatensor product ofcrystals. Ourgoalistogivea
non-recursive description of the nodes of B(Λ +Λ +···+Λ ) for any collection of fundamental
i1 i2 ik
weightsΛ .
i
Foranorderedsetofdominantweights(µ ,µ ,...,µ )andforeachpermutationwinthesymmetric
1 2 k
groupS ,define
k
B (µ ,...,µ ) = B(µ )⊗B(µ )⊗···⊗B(µ )
w 1 k w(1) w(2) w(k)
6 BRANTJONESANDANNESCHILLING
7
7
76 5247 725 5762
6 4 7 5 2 6
65 473 52746 62
5 3 7 4 6 2
54 317 7463 264
4 1 7 3 6 4
423 17 3716 6453
3 2 7 1 6 3 5
312 23 716 3615 53
1 2 3 6 1 5 3
12 3214 615 3541
2 1 4 5 1 4
124 415 514 412
4 1 5 4 1 2
4135 561 1432 21
3 5 1 6 3 2 1
35 5136 671 32 213
5 3 6 1 7 2 3
5346 1637 71 234
4 6 3 7 1 4
426 6374 173 45
2 6 4 7 3 5
26 64275 734 56
6 2 5 7 4 6
6275 572 4725 67
5 27 2 7 5 7
5247 725 5762 7
FIGURE 3. B(Λ7)oftypeE7
AFFINESTRUCTURESANDATABLEAUMODELFORE6 CRYSTALS 7
TABLE 2. Fundamentalrealizations forE7
Generator in Dimension
B(Λ ) ¯0¯71⊗¯07 B(Λ )⊗2 133
1 7
B(Λ ) ¯12⊗¯0¯71⊗¯07 B(Λ )⊗3 912
2 7
B(Λ ) ¯0¯23⊗¯12⊗¯0¯71⊗¯07 B(Λ )⊗4 8645
3 7
B(Λ ) ¯0¯54⊗¯0¯65⊗¯0¯76⊗¯07 B(Λ )⊗4 365750
4 7
B(Λ ) ¯0¯65⊗¯0¯76⊗¯07 B(Λ )⊗3 27664
5 7
B(Λ ) ¯0¯76⊗¯07 B(Λ )⊗2 1539
6 7
B(Λ ) ¯07 B(Λ ) 56
7 7
soB (µ ,...,µ )isB(µ )⊗···⊗B(µ )wheree∈ S istheidentity.
e 1 k 1 k k
Definition2.1. Let(µ ,µ ,...,µ )bedominantweights. Then,wesaythat
1 2 k
b ⊗b ⊗···⊗b ∈ B(µ )⊗B(µ )⊗···⊗B(µ )
1 2 k 1 2 k
ispairwiseweaklyincreasing if
b ⊗b ∈ B(µ +µ )⊂ B(µ )⊗B(µ )
j j+1 j j+1 j j+1
foreach1 ≤ j < k.
Next,wefixanisomorphism ofcrystals
Φ(µ1,...,µk) :B (µ ,...,µ )→ B (µ ,...,µ )
w w 1 k e 1 k
for each w ∈ S . Observe that each choice of Φ(µ1,...,µk) corresponds to achoice for the image ofeach
k w
ofthehighest-weight nodesinB (µ ,...,µ ).
w 1 k
Letb∗denotetheuniquehighestweightnodeofthejthfactorB(µ ). Sincewearefixingthedominant
j j
weights (µ ,...,µ ), wewillsometimes drop thenotation (µ ,...,µ )from B andΦ inthe proofs
1 k 1 k w w
below.
Definition2.2. Letw beapermutation thatfixes{1,2,...,j}. WesaythatΦ(µ1,...,µk) isalazyisomor-
w
phismiftheimageofeveryhighest weightnodeoftheform
b ⊗b ⊗···⊗b ⊗b∗ ⊗···⊗b∗
1 2 j j+1 k
underΦ(µ1,...,µk) isequalto
w
b ⊗b ⊗···⊗b ⊗b∗ ⊗···⊗b∗ .
1 2 j w−1(j+1) w−1(k)
WewanttochooseourisomorphismsΦ(µ1,...,µk) tobelazy,butwewillseeinthecourseoftheproofs
w
thatourresultsdonototherwisedependuponthechoiceofΦ(µ1,...,µk).
w
Definition 2.3. Let T be any subset of S , and {Φ(µ1,...,µk)} be a collection of lazy isomorphisms.
k w w∈T
WedefineI(µ1,...,µk)(T)tobe
Φ(µ1,...,µk)({pairwiseweaklyincreasing nodesofB (µ ,...,µ )}) ⊂ B (µ ,...,µ ).
\ w w 1 k e 1 k
w∈T
Proposition 2.4. Let T be any subset of Sk. Then, whenever b ∈ I(µ1,...,µk)(T) we have ei(b),fi(b) ∈
I(µ1,...,µk)(T).
8 BRANTJONESANDANNESCHILLING
Proof. Wefirstclaimthatthecrystal operators e andf preserve thepairwiseweaklyincreasing condi-
i i
tioninanytensorproduct ofhighestweightcrystals. Let
b = b ⊗b ⊗···⊗b
1 2 k
beapairwiseweaklyincreasing nodeinB = B(µ )⊗···⊗B(µ ).
1 k
Weneedtoshowthate (b)ispairwiseweaklyincreasing. Supposethate actsonthej-thtensorfactor
i i
inb,thatis,e (b) =b ⊗···⊗e (b )⊗···⊗b . Henceitsufficestoshowthatb ⊗e (b )∈ B(µ +µ )
i 1 i j k j−1 i j j−1 j
and e (b ) ⊗ b ∈ B(µ + µ ). Since e acts on b in b, in the tensor product rule the leftmost
i j j+1 j j+1 i j
unbracketed+isassociatedtob . Thismeansthatany+fromb mustbebracketedwitha−fromb .
j j−1 j
Butthen e (b ⊗b ) = b ⊗e (b ) ∈ B(µ +µ ). Similarly, since e acts on b , not all +inb
i j−1 j j−1 i j j−1 j i j j
arebracketedwith−inb ⊗···⊗b . Buttherefore, alsonotall+inb arebracketed with−inb
j+1 k j j+1
andhencee (b ⊗b ) = e (b )⊗b ∈ B(µ +µ ). Thearguments forf areanalogous.
i j j+1 i j j+1 j j+1 i
Next,supposethatb ∈ I(µ1,...,µk)(T) ⊂ Be. Then,forallw ∈ Sk wehaveΦ−w1(b)ispairwiseweakly
increasing in B . By the argument above, we then have that e (Φ−1(b)) is pairwise weakly increasing
w i w
in B . Since Φ is an isomorphism, it commutes with e , so Φ−1(e (b)) is pairwise weakly increasing
w w i w i
inBw forallw ∈ Sk. Hence,ei(b) ∈ I(µ1,...,µk)(T). Thearguments forfi areanalogous. (cid:3)
Corollary 2.5. For any subset T of Sk, we have that I(µ1,...,µk)(T) is a direct sum of highest weight
crystals B(λ)forsomecollection ofweightsλ.
λ
L
Proof. Proposition 2.4implies that whenever b ∈ I(µ1,...,µk)(T), the entire connected component ofthe
crystalgraphcontaining bisinI(µ1,...,µk)(T). (cid:3)
Theorem2.6. Fixasequence (µ ,...,µ )ofdominantweights. Then,
1 k
I(µ1,...,µk)(S ) ∼= B(µ +µ +...+µ ).
k 1 2 k
Proof. Let b∗ be the unique highest weight node of B with highest weight µ for each j = 1,...,k.
j j j
Thenb∗ = b∗1⊗b∗2⊗···⊗b∗k generates B(µ1+...+µk)andthisnodeliesinI(µ1,...,µk)(Sk).
Suppose there exists another highest weight node in I(µ1,...,µk)(Sk). Then, at least one of the factors
b must have ε (b ) > 0 for some i. Choose j to be the rightmost factor having ε (b ) > 0 for some
j i j i j
i ∈I. Thenfixsomechoiceofisuchthatε (b )> 0. Ourhighestweightnodehastheform
i j
b = b ⊗···⊗b ⊗b∗ ⊗···⊗b∗.
1 j j+1 k
Inparticular,j < ksinceanyrightmostfactorofahighestweighttensorproductmustbehighestweight.
Since b is highest weight, we have that all + entries for factor b are canceled by − entries lying to
j
therightinthei-signature forthetensor product rule. Suppose thatbj′ istheleftmost factorforwhicha
− cancels a + from b in the i-signature. Let w be the permutation that interchanges factors j +1 and
j
j′. Then, byour choice of Φ wehave that Φ−1(b) is obtained from b just by interchanging the factors
w w
b∗ andb∗.
j+1 j′
Hence,wehavethatΦ−1(b)inB hasanadjacent+/−paironfactorsj,j+1. Sincethispairispart
w w
ofapairwiseweaklyincreasingelement,theremustexistasequenceofei′ operationsthatbringsbj⊗b∗j′
to b∗j ⊗ b∗j′. However, ei′ can only operate on the first tensor factor in this pair because b∗j′ is already
highestweight. Moreover,wehavethatε ofthefirstfactorandϕ ofthesecondfactorarebothpositive.
i i
This remains true regardless of how we apply ei′ operations where i 6= i′ by [Ste03, Axiom (P4)]. We
can potentially apply the e operation max{ε (b )−ϕ (b∗),0} times, but since ϕ (b∗) > 0, we have
i i j i j′ i j′
thatε ofthefirstfactorwillalwaysremainpositive. Hence,wecanneverreachb∗⊗b∗,acontradiction.
i j j′
Thus,b∗ istheuniquehighest weightnodeofI(µ1,...,µk)(Sk). (cid:3)
AFFINESTRUCTURESANDATABLEAUMODELFORE6 CRYSTALS 9
Remark 2.7. The condition that there is a unique highest weight element that we used in the proof
of Theorem 2.6 is equivalent to the hypothesis of [KN94, Proposition 2.2.1] from which the desired
conclusion alsofollows.
Remark 2.8. Observe that only a finite constant amount of data is ever required to check the pairwise
weaklyincreasing condition, regardlessofhowlargethenumberoftensorfactorskis. Theorem2.6and
its refinements willallow usto formulate arguments that apply toall highest-weight crystals simultane-
ously.
When we are considering a specific highest-weight crystal, it may be computationally easier to gen-
erate B(µ + ··· + µ ) by simply applying f operations to the highest-weight node in all possible
1 k i
ways.
WewillsaythatanynodeofI(µ1,...,µk)(Sk)isweaklyincreasing. ItturnsoutthatwecanoftentakeT
tobemuchsmaller than Sn bystarting withT = {e}and adding permutations toT until I(µ1,...,µk)(T)
contains a unique highest weight node. In particular, the next result shows that we can take T = {e}
whenweareconsidering alinearcombination oftwodistinctfundamental weights.
Lemma 2.9. Let Λ and Λ be distinct fundamental weights, and k ,k ∈ Z with k = k + k .
i1 i2 1 2 ≥0 1 2
Then,thenodesof
B(k Λ +k Λ ) ⊂ B(Λ )⊗k1 ⊗B(Λ )⊗k2
1 i1 2 i2 i1 i2
arepreciselythepairwiseweaklyincreasingtensorproductsb1⊗b2⊗···⊗bkofB(Λi1)⊗k1⊗B(Λi2)⊗k2.
Proof. We order the fundamental weights as (Λ ,...,Λ ,Λ ,...,Λ ) and apply the same argument
i1 i1 i2 i2
asintheproofofTheorem2.6toseethatanyhighestweightnodeinI(Λi1,...,Λi1,Λi2,...,Λi2)({e})mustbe
oftheform
b ⊗···⊗b ⊗b∗ ⊗b∗ ⊗···⊗b∗.
1 k1−1 k1 k1+1 k
Inthiscase,itisnevernecessary toapplyΦ toreorderthefactors becauseallofthefactors totheright
w
offactork mustbethesame.
1
Next,weletj = k −1. Wehavethatb = b∗ andweworkbydownwardinductiontoarguethat
1 j+1 j+1
b mustbeb∗. Thisfollowsbecauseduetothepairwiseweakincreasingconditionthereexistsasequence
j j
ofe that takes b ⊗b∗ to b∗ ⊗b∗ . Thehighest weight node of thefundamental crystal B(Λ )has
i j j+1 j j+1 i1
a unique i -arrow. If b 6= b∗ then we could never traverse this edge because in the i -signature any +
1 j j 1
wouldbecanceledbya−fromb∗ . Hence,b = b∗,andtheinduction continues.
j+1 j j
Thus,thereisauniquehighest-weight nodeinI(Λi1,...,Λi1,Λi2,...,Λi2)({e}). (cid:3)
Allofthecrystals inourworkhave classical decompositions thathave been givenbyChari[Cha01].
These crystals satisfy the requirement of Lemma 2.9 that at most two fundamental weights appear. On
theotherhand,Example2.10showsthatnoordering ofthefactorsinB(Λ )⊗B(Λ )⊗B(Λ )intype
2 1 6
E admitsananalogous weaklyincreasing condition thatisdefinedusingonlypairwisecomparisons.
6
Example2.10. ObservethateachofthefollowingnodesintypeE isacounterexampletothecondition
6
required in [KN94, Proposition 2.2.1]. Each of the given nodes is highest weight, and pairwise weakly
increasing, butnoneofthenodescorrespond tothehighestweightnodeofB(Λ +Λ +Λ ).
1 6 2
10 BRANTJONESANDANNESCHILLING
(3¯1¯6⊗1)⊗u ⊗u ∈ B(Λ )⊗B(Λ )⊗B(Λ )
1 6 2 1 6
(5¯3⊗¯13)⊗u ⊗u ∈ B(Λ )⊗B(Λ )⊗B(Λ )
6 1 2 6 1
¯25⊗u ⊗u ∈ B(Λ )⊗B(Λ )⊗B(Λ )
6 2 1 6 2
¯62⊗u ⊗u ∈ B(Λ )⊗B(Λ )⊗B(Λ )
2 6 1 2 6
¯23⊗u ⊗u ∈ B(Λ )⊗B(Λ )⊗B(Λ )
1 2 6 1 2
2¯1⊗u ⊗u ∈ B(Λ )⊗B(Λ )⊗B(Λ )
2 1 6 2 1
Here, u is the highest weight node of B(Λ ). Hence, it is not possible to obtain a pairwise weakly
i i
increasing condition thatcharacterizes thenodesofB(Λ +Λ +Λ ).
1 6 2
Remark 2.11. In standard monomial theory [LS86], the condition that a tensor product of basis ele-
ments lies in B(λ+µ) can also be formulated as a comparison of the lift of these elements in Bruhat
order[Lit96]. Forseveraltensorfactors, oneneedstocomparesimultaneous lifts.
WenowrestricttotypeE . Lemma2.9impliesthatwehaveanon-recursivedescriptionofallB(kΛ )
6 i
determined by the finite information in B(2Λ ). In the case of particular fundamental representations,
i
wecanbemorespecificabouthowtotestfortheweaklyincreasing condition.
Proposition 2.12. Wehavethatb ⊗b ∈ B(2Λ ) ⊂ B(Λ )⊗2 ifandonlyifb canbereached fromb
1 2 1 1 2 1
byasequence off operations inB(Λ ).
i 1
Proof. Thisisafinitecomputation onB(2Λ ). (cid:3)
1
The crystal graph for B(Λ ) of Figure 2 can be viewed as a poset. Then Proposition 2.12 implies in
1
particular thatincomparable pairsinB(Λ )arenotweaklyincreasing.
1
There are 78 nodes in B(Λ ). We construct B(Λ ) as the highest weight crystal graph generated by
2 2
2¯1¯0⊗¯01inside B(Λ )⊗B(Λ ). Notethat weonly needtousethe nodes inthe“tophalf” ofFigure 2
6 1
andtheirduals. Thereare2430nodesinB(2Λ ).
2
Proposition 2.13. Wehavethat
(b ⊗c )⊗(b ⊗c ) ∈ B(2Λ ) ⊂ (B(Λ )⊗B(Λ ))⊗2
1 1 2 2 2 6 1
ifandonlyif
(1) b can be reached from b by f operations in B(Λ ), and c can be reached from c by f
2 1 i 6 2 1 i
operations inB(Λ ),and
1
(2) Wheneverc isdualtob ,wehavethatthereisapathoff operationsfrom(b ⊗c )to(b ⊗c )
1 2 i 1 1 2 2
oflengthatleast1(soinparticular, theelementsarenotequal)inB(Λ ).
2
Proof. Thisisafinitecomputation onB(2Λ ). (cid:3)
2
3. AFFINE STRUCTURE
(1)
In this section, we study the affine crystals of type E . We introduce the method of promotion
6
to obtain a combinatorial affine crystal structure in Section 3.1 and the notion of composition graphs
in Section 3.2. It is shown in Theorem 3.9 that order three twisted isomorphisms yield regular affine
crystals. Thisisusedtoconstruct Br,s oftype E(1) fortheminuscule nodes r = 1,6inSection3.3and
6
theadjointnoder = 2inSection3.4. InSection3.5wepresentconjectures forB1,s oftypeE(1).
7