Fusion Rings of Loop Group Representations

0.31 MB·English

by  Christopher L. Douglas

FUSION RINGS OF LOOP GROUP REPRESENTATIONS
CHRISTOPHERL.DOUGLAS
Abstract. Wecomputethefusionringsofpositiveenergyrepresentationsoftheloopgroups
ofthesimple,simplyconnected Liegroups.
1. Introduction
9 The representation theory of loop groups has become a crossroads for such diverse areas
0 as conformal field theory, operator algebras, low-dimensional topology, quantum algebra, and
0
homotopy theory. A crucial aspect of the theory is the existence of a product on the collection
2
of representations of a fixed central extension of a loop group—this product can be variously

n
described as fusion of conformal fields [14], as Connes fusion of bimodules over von Neumann
a
algebras[4,15],orasaPontryaginproductintwistedequivariantK-homology[7]. Ourpurposein
J
thispaperistocomputethefusionringstructureontherepresentationsofthecentralextensions
4
of the loop groups of simple, simply connected Lie groups.

] The Grothendieck group of positive energy representations of the level k central extension of
T
the loop group of the simple, simply connected group G is isomorphic to a free abelian group
R
on the dominant weights of G of level at most k. The fusion ring F [G] of these positive energy
k
.
h representations can be described as a quotient R[G]/Ik of the representation ring of the group
t G by the “fusion ideal” I . This ideal is generated by an infinite collection of representations
a k
m associatedtothedominantweightsofGoflevelgreaterthank. Invividcontrasttothisunwieldy
presentation,Gepner [9]conjectured that the fusionring is a globalcomplete intersection, which
[
is to say that the fusion ideal I is generated by exactly n representations, for a group G of
k
1 rank n. Despite the existence of a detailed understanding of the complexification of the fusion
v
ring [1], only recently has there been progress toward a global perspective on the integral fusion
1
ring. By an elegant Gr¨obner basis analysis, Bouwknegt and Ridout [2] showed that for any
9
3 group G the fusion ideal Ik can be generated by a collection of representations whose size grows
0 exponentiallywith the levelk—these representationsare associatedto a setof dominantweights
. that collectively encloses the level k Weyl alcove within the Weyl chamber. In the paper [5], we
1
0 useda spectralsequenceforthe twistedequivariantK-homologyofthe groupG, importedalong
9 theFreed-Hopkins-Teleman[7,8]identificationofthisK-homologywiththefusionring,toprove
0 that there exists a finite level-independent bound on the number of representations necessary to
:
v generate the fusion ideal. In this sequel, we extend the methods developed there to permit a
i complete computation:
X
r Theorem 1.1. The fusion ring Fk[G] of positive energy representations of the level k central
a extension of the loop group of the simple, simply connected Lie group G is given as follows:
Fk[SU(n+1)]=R[SU(n+1)]/h[kλ1+λn],[kλ1+λn−1],...,[kλ1+λ2],[(k+1)λ1]i
Fk[Spin(2n+1)]=R[Spin(2n+1)]/h[(k−1)λ1+λ2],[(k−1)λ1+λ3],...,[(k−1)λ1+λn−1],
[(k−1)λ1+2λn],[kλ1+λn],[kλ1+λn−1],...,[kλ1+λ3],
[kλ1+λ2]+[kλ1],[(k+1)λ1]i
Fk[Sp(n)]=R[Sp(n)]/h[kλ1+λn],[kλ1+λn−1],...,[kλ1+λ2],[(k+1)λ1]i
Fk[Spin(2n)]=R[Spin(2n)]/h[(k−1)λ1+λ2],[(k−1)λ1+λ3],...,[(k−1)λ1+λn−2],
[(k−1)λ1+λn−1+λn],[kλ1+λn],[kλ1+λn−1],[kλ1+λn−1+λn],
[kλ1+λn−2],[kλ1+λn−3],...,[kλ1+λ3],[kλ1+λ2]+[kλ1],[(k+1)λ1]i
Theauthor wassupportedbyaMillerResearchFellowship.
1
h[3,ℓ−1],[1,ℓ],[0,ℓ+1]+[0,ℓ]i if k even
8
Fk[G2]=R[G2]/<
h[0,hk+1i],[3,hk−1i]+[3,hk−3i],[2,hk−1i]i if k odd
:
h[ℓ,0,0,0]+[ℓ+1,0,0,0],[ℓ−2,1,0,0]+[ℓ,1,0,0],[ℓ−3,0,2,0]+[ℓ,0,2,0],
8 [ℓ−2,0,1,1]+[ℓ,0,1,1],[ℓ−1,0,1,0]+[ℓ,0,1,0],[ℓ−3,1,0,2]+[ℓ−1,1,0,2],
Fk[F4]=R[F4]/>>>>>>>>>>>>>>>>>>>>>>>>>>< h[[[[[[ℓℓℓℓhℓk−−−−−−42313,,,,,310201i,,,,,,111021,,,,,,022212,]]]]]0,++++[]ℓ+[[[[−ℓℓℓℓ[−−−−h2k,2143−0,,,,,102012,,,,i,1102,1,,,,1]2221,i]]]]0i,,,,,f[[[[0ℓℓℓℓk]−−−−,e[1321hv,,,,ke0211−n,,,,00105,,,,i2010,]]]]0,+++,[2ℓ,[[[−ℓℓℓ0,−−]20+,23,1[0,,h,21,k0,,2−,01]2,,,1][01iℓ,]],−[,,ℓ0[[−2,ℓℓ2,−−11,,310,0,,0],10,,11,,,02]1,,+]30,]][[ℓ++ℓ−,0[[1ℓℓ,,−−01,22,10,,]10,,,,102],,,30]],,
[hk−3i,0,1,1]+[hk−1i,0,1,1],[hk−5i,1,0,2]+[hk−3i,1,0,2],
>>>>>>>>>>>>>>>>>>>>>>>>>>: [[[[[[[hhhhhhhkkkkkkk−−−+−+−1373115iiiiiii,,,,,,,1110001,,,,,,,1101000,,,,,,,2110203]]]]]]]+++++,,[[[[[[[hhhhhhhkkkkkkk−−−−−−−3357531iiiiiii,,,,,,,1011100,,,,,,,0211010,,,,,,,1021100]]]]]]],,,,,,,[[[[[hh[[hhhhhkkkkkkk−−−+−++3711151iiiiiii,,,,,,,0220010,,,,,,,1000200,,,,,,,2221100]]]]]]],,++++,[[hh[[[[[hkkhhhhkkkkk−−−−−−−551ii9175i,,iiii,21,,,,0,,2001,01,,,,1,,0020,01,,,,0]]2110i],]]]],,,,,if k odd
Fk[E6]=R[E6]/h[k−1,1,0,0,0,0],[k−2,0,0,1,0,0],[k−3,0,1,0,1,0],[k−1,0,0,0,1,0],
[k−2,0,1,0,0,1],[k−3,0,1,0,1,1],[k−1,0,1,0,0,0],[k−2,0,0,1,0,1],
[k−2,0,1,0,1,0],[k,0,0,0,0,1],[k−2,1,1,0,0,1],[k−1,0,0,1,0,0],
[k−1,0,1,0,0,1],[k−1,1,0,0,1,0],[k−1,1,0,0,1,1],[k,0,0,0,1,0],
[k,1,0,0,0,1],[k,1,0,0,1,0],[k,0,1,0,0,0],[k+1,0,0,0,0,0],
[k−1,1,0,0,0,1]+[1,0,0,0,0,0]·ν,[k−3,0,1,1,0,1]−[1,1,0,0,0,0]·ν,
[k−2,1,0,1,0,1]−[1,0,0,0,0,1]·ν,[k−1,0,0,1,0,1]−[1,0,0,0,0,0]·ν,
[k,1,0,0,0,0]+ν,[k,0,0,1,0,0]−νi
Fk[E7]=R[E7]/h[1,0,0,0,0,0,k−1],[0,0,1,0,0,0,k−2],[0,0,0,1,0,0,k−3],[0,1,0,0,1,0,k−4],
[0,0,0,0,1,0,k−2],[0,1,0,0,0,1,k−3],[0,1,0,0,1,1,k−5],[0,1,0,0,0,0,k−1],
[0,0,0,1,0,1,k−4],[0,1,0,0,1,0,k−3],[0,0,1,0,0,1,k−3],[0,1,0,1,0,1,k−5],
[0,0,0,1,0,0,k−2],[0,0,0,0,0,1,k−1],[1,1,0,0,0,1,k−3],[0,0,1,1,0,1,k−5],
[0,1,0,0,0,1,k−2],[1,0,0,1,0,1,k−4],[0,0,0,1,0,1,k−3],[1,0,1,0,1,1,k−5],
[0,0,1,0,0,0,k−1],[0,0,1,0,1,1,k−4],[1,0,0,0,1,0,k−2],[1,0,0,0,1,1,k−3],
[1,0,1,0,1,0,k−3],[0,0,0,0,1,0,k−1],[1,0,1,0,1,1,k−4],[1,0,0,1,0,0,k−2],
[0,0,1,0,1,0,k−2],[1,1,0,0,0,0,k−1],[1,0,0,1,1,0,k−3],[1,1,0,0,0,1,k−2],
[0,0,0,1,0,0,k−1],[1,1,0,0,1,0,k−2],[1,0,0,0,0,1,k−1],[1,1,0,1,0,0,k−2],
[1,0,0,0,1,0,k−1],[0,1,1,0,0,0,k−1],[0,1,0,0,0,0,k],[1,1,1,0,0,0,k−1],
[1,1,0,0,0,0,k],[0,1,1,0,0,0,k],[0,0,0,0,1,0,k],[0,0,0,0,0,1,k],[0,0,0,0,0,0,k+1],
[1,0,0,0,0,1,k−2]+[1,0,0,0,0,0,0]·µ,[0,1,1,0,0,1,k−4]−[1,1,0,0,0,0,0]·µ,
[0,0,1,0,1,0,k−3]−[0,0,1,0,0,0,0]·µ,[1,0,1,0,0,1,k−3]−[1,0,0,0,0,0,1]·µ,
[0,0,1,0,0,1,k−2]−[1,0,0,0,0,0,0]·µ,[1,0,0,1,0,1,k−3]+[1,0,0,0,0,0,1]·µ,
[1,0,0,0,0,0,k]+µ,[1,0,0,1,0,0,k−1]+[0,0,0,0,0,0,1]·µ,
[0,0,1,0,0,0,k]−µ,[0,0,0,1,0,0,k]+µi
The fusion ring of E at even and odd levels is given in Table 1 and Table 2 in Section 4.3.
8
2
In the above equations for the classical groups, the expression [λ]∈R[G] refers to the irreducible
representation of G with highest weight λ. For the group Spin(2n+1), the rank n is assumed to
be at least 3, and for the group Spin(2n), the rank n is assumed to be at least 4. In the equa-
tions for the exceptional groups, the expression [a ,a ,...,a ] ∈ R[G] refers to the irreducible
1 2 n
representation of G with highest weight a λ +a λ +···+a λ . Throughout λ are the funda-
1 1 2 2 n n i
mental weights of G in the Bourbaki ordering. The abbreviations ℓ and hmi refer respectively to
k and m; the abbreviations ν and µ refer respectively to the representations [k,0,0,0,0,0] and
2 2
[0,0,0,0,0,0,k].
For the fusion rings of the exceptional groups, we have implicitly assumed that the level
k is large enough that the entries in the expressions [a ,a ,...,a ] are all at least −1; the
1 2 n
representation [a ,a ,...,a ] is taken to be zero if some a is exactly −1. Thus for F , k > 5;
1 2 n i 4
for E , k > 1; for E , k > 3; and for E , k > 19. For smaller levels, the fusion ideal is
6 7 8
obtained by inducing the errant weights back into the Weyl chamber. Thanks to the various
available perspectives on representations of loop groups, Theorem 1.1 could be rephrased as a
computationofthetwistedG-equivariantK-homologyofthegroupG,orasacomputationofthe
fusion ring of the Wess-Zumino-Witten rational conformal field theory over the group manifold
G.
The above descriptions of the fusion rings for type A and type C were known previously [2,
3, 6, 9, 10] and a proof appears in [2]. By exploiting Jacobi-Trudy determinantal identities,
Bouwknegt and Ridout [2] also produced an explicit description of the fusion ideal in type B
whose complexity grows only linearly with the level. In both type B and type D, Boysal and
Kumar [3] wrote down a finite collection of representations which they conjecture generate the
fusion ideal; they also describe an intriguing conjecture relating the fusion ideals for these types
to a simple radical ideal.
The paper is organized as follows. In Section 2, we recall the computational methods devel-
oped in the preceding paper [5], particularly those concerning the resolution of the fusion ring
comingfromthe twistedMayer-Vietorisspectralsequence. We then proveourmainlemma,that
the fusion ideal is the induction image of the kernel of an affine induction map to the twisted
representation module of the centralizer of any one affine vertex of the Weyl alcove; for brevity
we refer to this kernel as the “fusion kernel”. Section 3 establishes a series of techniques for
computing the fusion kernel. These techniques rely on specialized bases, called affine Steinberg
bases, for twisted representation modules; we describe these bases and discuss their relationship
toclassicalSteinbergbasesforrepresentationrings. InSection3.1,wecompute the fusionkernel
whenthereisanaffineSteinbergbasisthatisclosedwithrespecttoinductiontothechosenaffine
vertex of the alcove. In Section 3.2, we compute the fusion kernel when the the affine vertex of
the alcove is central. In Section 3.3, we compute the fusion kernel when the edge of the Weyl
alcove connecting the origin to the affine vertex is orthogonal to the affine wall of the alcove.
Section 4 harnesses the results of the preceding two sections to prove Theorem 1.1.
Acknowledgments. We thank Eckhard Meinrenken and Andr´e Henriques for illuminating corre-
spondence, and Arun Ram and Lauren Williams for helpful pointers to the literature.
2. Affine centralizers control the fusion ideal
In this section we recall the resolution of the fusion ring F [G] in terms of the twisted repre-
k
sentationmodules ofcentralizersubgroupsofG, andnote the abstractpresentationofthe fusion
ring resulting from this resolution. We then prove our main lemma, that the fusion ideal is the
induction image of the kernel of affine induction to any one affine vertex of the Weyl alcove.
Thoughwewillbrieflyreviewthenecessarynotationandconcepts,weareworkingthroughout
in the context established in our preceding paper [5]; the reader may want to refer there for
background material and for more extensive exposition. Fix a simple, simply connected Lie
3
group G, with Lie algebra g; unless indicated otherwise, all of the following items refer to this
Lie algebra.
D =affine Dynkin diagram
{α }n =simple affine roots
i i=0
{λ }n =fundamental weights, generating the lattice Λ
i i=1 W
A=Weyl alcove
w =Weyl reflection corresponding to α
i i
w =reflection in the plane −kα0 +kerh ,for h the coroot of α
k·α0 2 α0 α0 0
h{w |i∈S}i if 0∈/ S ⊂D,
Wk = i
S (h{wi|i∈S\0}∪{wk·α0}i if 0∈S ⊂D
F =face of A fixed by W1
S S
Z =centralizer of F
S S
ρ =half sum of the positive roots of Z
S S
Rk[ZS]=Z[ΛW]WSk, the twisted representation module of ZS
For a weight λ and a reflection group U in the affine Weyl group, let AU denote the anti-
λ
symmetrization of λ with respect to U. The twisted Mayer-Vietoris spectral sequence for the
twistedequivariantK-homologyof the groupG, reinterpretedin terms of twistedrepresentation
modules, has the following form:
Proposition 2.1. There is a complex of R[G]-modules R [Z ] whose differential
S⊂D,|S|=n−p k S
has components dS,T :R [Z ]→R [Z ], for T =S∪j , given by twisted induction:
k S k T sL
Wk Wk
A S A T
dS,T µ+ρS =(−1)s µ+ρT
 W0   W0 
(cid:18) AρSS (cid:19) AρTT
This complex is acyclic except in degree zero,where it hashomology the level k fusion ring F [G].
k
This proposition is discussed in Section 2.3 of [5] and is closely related to a resolution described
by Meinrenken [11]—the original identification of the homology of the complex with the fusion
ring is due to Freed, Hopkins, and Teleman [7]. Investigatingthe cokernelof the first differential
in this resolution produces a presentation of the fusion ring in terms of induction maps:
Lemma 2.2. The level k fusion ring F [G] is isomorphic to the quotient of the representation
k
ring R[G] by the ideal hdc01,b0(kerdc01,b1),dc02,b0(kerdc02,b2),...,d0cn,b0(kerd0cn,nb)i.
Here S denotes the complement of the collectionS in the affine Dynkin diagramD. This lemma
c b
is establishedas partofthe proofofTheorem4.3 of[5]: firstobservethatthe differentiald0ij,ij :
Rk[Z0cbij]→Rk[Zibj] is surjective; the fusion ring is therefore the cokernelof the sum ⊕ni=1(d0bi,b0−
bb bb
d0i,i); second observe that the differential d0i,i is also surjective; the lemma follows.
Thenextlemmashowsthattheabovepresentationisredundant—indeed,theinductionofthe
affine induction kernel for any one affine vertex of the Weyl alcove spans the fusion ideal:
Main Lemma 2.3. For any node i of the Dynkin diagram of G, the level k fusion ring is given
by
b b bb
F [G]=R[G]/hd0i,0(kerd0i,i)i
k
4
Henceforth we will only be concerned with a single chosen affine vertex of the Weyl alcove.
We therefore introduce new and more specialized notation before proceeding to the proof of the
lemma.
• o,v,e denoterespectivelythe originvertexofthe alcoveA,the affinevertexofthe alcove
associatedtothesimplerootα ,andtheedgeofthealcoveconnectingthosetwovertices.
v
cb
• de,o :R[Ze]→R[G] is the induction map d0v,0.
• de,v :R[Ze]→Rk[Zv] is the twisted induction map d0cv,vb.
• Ak isthelevelkalcove,boundedbythefixedplanesofthereflections{wk·α0,w1,...,wn}.
• Wv is the reflection group hwk·α0,w1,...,wv,...,wni.
• We is the reflection group hw1,...,wv,...,wni.
• W−ρ is the group generated by the reflecticons λ7→wi(λ+ρ)−ρ for i∈D\α0.
• W−vρv is the group generated by thecreflections λ 7→ wi(λ+ρv)−ρv for v 6= i ∈ D\α0
and by the reflection λ7→w (λ+ρ )−ρ .
• W−k+ρh∨ is the reflection groukp·αg0eneratved byvthe group W−ρ and the groups W−vρv for all
v ∈D\α ;theinteriorweightsofthefundamentalchamberforthisgrouparetheweights
0
in the closed level k alcove A .
k
• C,Cv,Ce denoterespectivelytheuniqueWeylchambersforthereflectiongroupsW,Wv,
and We containing A .
k
• [λ] ∈ R[G] and [λ]e ∈ R[Ze] denote the irreducible representations with highest weight
λ, for respectively λ∈C and λ∈C .
e
• [λ]v ∈ Rk[Zv] denotes the twisted irreducible representation with highest weight λ, for
λ∈C .
v
b b bb
Proof. The ideal DK :=de,o(kerde,v)=d0i,0(kerd0i,i) is certainly contained in the ideal
hdc01,b0(kerdc01,b1),dc02,b0(kerdc02,b2),...,d0cn,b0(kerd0cn,nb)i
By Lemma 2.2, that latter ideal is the fusion ideal I [G]. The fusion ideal is generated by the
k
following collection of representations:
[λ]+(−1)aλ+1[a ·λ]|λ∈C\(A ∪S ) ∪{[λ]|λ∈C∩S }
λ k k k
See for instance t(cid:8)he papers [9] and [2]. Here S is the co(cid:9)llection of weights on the (−ρ)-shifted
k
(k+h∨)-scale Stiefel diagram walls, that is the weights with nontrivial isotropy in Wk+h∨. The
−ρ
transformationa ∈Wk+h∨ is the unique element such that a ·λ∈A .
λ −ρ λ k
It suffices, therefore, to check that
(1) [λ]+(−1)aλ+1[aλ·λ]∈DK for λ∈C\(Ak∪Sk), and
(2) [λ]∈DK for λ∈C∩S .
k
Weestablishbothbyinductiononthelengthofthevectorλ+ρ. Foranyweightλ,letv ∈Wv
λ −ρv
denoteanelementsuchthatv ·λ∈C ,ifoneexists,andsimilarlyletc ∈W denoteanelement
λ v λ −ρ
such that c ·λ∈C, if one exists.
λ
For case (1), we have λ∈C\(Ak∪Sk). Note that [λ]e+(−1)vλ+1[vλ·λ]e ∈kerde,v. It follows
that
[λ]+(−1)c(vλ·λ)+vλ+1[c(vλ·λ)·vλ·λ]∈DK
BecauseλisinthechamberC butoflevelstrictlylargerthank+1,thevectorv ·λ+ρisstrictly
λ
shorter than the vector λ+ρ; the vector c ·v ·λ+ρ is the same length as v ·λ+ρ. By
(vλ·λ) λ λ
induction
[c(vλ·λ)·vλ·λ]+(−1)a(c(vλ·λ)·vλ·λ)+1[a(c(vλ·λ)·vλ·λ)·c(vλ·λ)·vλ·λ]∈DK
5
Of course, the weight a ·c ·v ·λ just is a ·λ. Adding the above two displayed
(c(vλ·λ)·vλ·λ) (vλ·λ) λ λ
elements of the ideal DK (after multiplying the second element by (−1)c(vλ·λ)+vλ) gives
[λ]+(−1)c(vλ·λ)+vλ+a(c(vλ·λ)·vλ·λ)+1[aλ·λ]∈DK
Finally observe that the sum of orders |c | + |v | + |a | and the order |a | are
(vλ·λ) λ (c(vλ·λ)·vλ·λ) λ
congruent modulo 2. The whole induction begins with the situation λ ∈ A ; here a is the
k λ
identity and so evidently [λ]+(−1)aλ+1[aλ·λ]∈DK.
Case (2), where λ ∈ C ∩ S , is analogous. If v does not exist, then [λ] ∈ kerde,v and
k λ e
so [λ] ∈ DK. If vλ exists but c(vλ·λ) does not, then again [λ]e + (−1)vλ+1[vλ·λ]e ∈ kerde,v
which implies [λ] ∈ DK. Those two cases begin the induction. In general we find as before
[λ]+(−1)c(vλ·λ)+vλ+1[c(vλ·λ)·vλ·λ]∈DK. Again, provided λ is not level k+1 (in which case vλ
doesnotexist),thevectorc ·v ·λ+ρisshorterthanλ+ρ. Byinduction,[c ·v ·λ]∈DK,
(vλ·λ) λ (vλ·λ) λ
and so [λ]∈DK as desired. (cid:3)
3. Affine Steinberg bases and the fusion kernel
Lemma 2.3 shows that the fusion ideal is the induction image of the kernel of the affine
induction map de,v. In this section we establish techniques for computing that “fusion kernel”
kerde,v. The map de,v :R[Z ]→R [Z ] is an R[G]-module map; we begin by discussing highest
e k v
weightR[G]-modulebasesforR[Z ]thatdeterminebyrestrictionR[G]-modulebasesforR [Z ]—
e k v
these “level k affine Steinberg bases for (R[Z ],R [Z ])” will facilitate an analysis of the fusion
e k v
kernel. In Section 3.1, we provide a complete description of the kernel in case there exists an
affineSteinberg basisthat is closedunder the inductionmapde,v; suchabasis existsfor a vertex
of the alcove of each of the classical groups and a vertex of the alcove of G . In Section 3.2,
2
we analyze the kernel when the vertex v of the alcove is central, that is Z = G; this holds in
v
particularforavertexofthealcoveofE andavertexofthealcoveofE . FinallyinSection3.3,
6 7
we specify the kernel when the edge e is orthogonal to the affine wall of the Weyl alcove; this is
the case for a vertex of the alcove of F and a vertex of the alcove of E .
4 8
MostofourdiscussionutilizesaffineSteinbergbasesfortherepresentationmodulesassociated
to the centralizers Z and Z . More generally, the notion is as follows.
e v
Definition 3.1. Let S ⊂ T be two subsets of the affine Dynkin diagram of the simple, simply
connected Lie group G; as before denote by Z ⊂Z the centralizers of the corresponding faces
S T
of the Weyl alcove. Choose compatible Weyl chambers C ⊂ C for the twisted representation
T S
modules R [Z ] and R [Z ] such that both chambers contain the level k Weyl alcove A . Two
k T k S k
collectionsofweightsB ⊂Λ andB ⊂Λ ,withB =B ∩C ,formalevelk affineSteinberg
S W T W T S T
basis (B ,B )forthepair(Z ,Z )ifthecollectionoftwistedirreduciblerepresentations{[λ] ∈
S T S T S
R [Z ]|λ∈B } is an R[G]-module basis for R [Z ], and if the collection of twisted irreducible
k S S k S
representations {[λ] ∈R [Z ]|λ∈B } is an R[G]-module basis for R [Z ].
T k T T k T
Because,given the chosen Weyl chambers, the collection B is determined by the collection B ,
T S
we sometimes refer to B by itself as the affine Steinberg basis for the pair (Z ,Z ).
S S T
Though we expect that for any S ⊂ T ⊂ D, there exists a level k affine Steinberg basis
for the pair (Z ,Z ), this remains an open problem. Observe that Steinberg’s construction of
S T
bases for representation rings of full rank subgroups of Lie groups [13] depends only on working
with reflection subgroups of the Weyl group. If the twisted representation module R [Z ] is
k T
in fact untwisted, in the sense that it is isomorphic to R[Z ] as an R[Z ]-module, then the
T T
corresponding Weyl group Wk ⊂ Waff is translation conjugate to a subgroup of the ordinary
T
Weylgroup. Steinberg’sargumentsthenapplymutatismutandistoconstructanaffineSteinberg
basis:
Proposition 3.2. Suppose the representation module R [Z ] is untwisted, and therefore R [Z ]
k T k S
is also untwisted. In this case, there exists a level k affine Steinberg basis for the pair (Z ,Z ).
S T
6
More concretely, a specific affine Steinberg basis can be described as follows. Let Fk denote
T
the affine subspace of the dual torus t∗ fixed by the reflection group Wk; this plane is spanned
T
by the face of the level k alcove A associated to the collection T ⊂ D. If the representation
k
module R [Z ] is untwisted, then the plane Fk contains some element ξ of the weight lattice
k T T
of G—see [5] for a more detailed discussion of the circumstances under which representation
modules are twisted or untwisted. Choose Weyl chambers C ⊂ C ⊂ C for R[G], R[Z ], and
T S T
R[Z ] such that the shifted affine chambers C +ξ and C +ξ contain the level k Weyl alcove
S T S
A . If B¯ is the ordinary Steinberg basis for R[Z ] with respect to the pair of chambers C ⊂C ,
k S S
then {λ+ξ|λ∈B¯} is an affine Steinberg basis for the pair (Z ,Z ).
S T
3.1. InductionclosedSteinbergbases. Thefusionkernelhasaparticularlysimpleformwhen
the collection of weights of the level k affine Steinberg basis B for the pair (Z ,Z ) is closed
e e v
under the induction map de,v. Let B denote the intersection B ∩C .
v e v
Proposition3.3. Suppose(B ,B )isalevelkaffineSteinbergbasisforthecentralizers(Z ,Z ).
e v e v
For any weight λ, let v ∈Wv be an affine Weyl element such that v ·λ∈C , if one exists; if
λ −ρv λ v
no such v exists, we say the weight λ is v-singular, and denote by S the collection of v-singular
λ v
weights in B . If for every weight λ∈B \B either λ∈S or v ·λ∈B , then the fusion kernel
e e v v λ v
is
kerde,v = [λ] +(−1)vλ+1[v ·λ] |λ∈B \(B ∪S ) ∪ [λ] |λ∈S
e λ e e v v e v
Proof. Evidently,thel(cid:10)is(cid:8)tedelements{[λ]e+(−1)vλ+1[vλ·λ]e|λ∈Be(cid:9)\(B(cid:8)v∪Sv)}and{(cid:9)[(cid:11)λ]e|λ∈Sv}
are in the kernel. Given a representationr ∈R[Z ] in the kernel of de,v, write
e
r = r [λ] + r [λ]
λ e λ e
! !
λX∈Sv λ∈XBe\Sv
Here for any weight λ the coefficient r is in R[G]. Because de,vr=0, we have
λ
(−1)vλr [v ·λ] =0
λ λ v
λ∈XBe\Sv
For each µ ∈ B , let O denote the collection of weights λ ∈ B such that v ·λ = µ. For any
v µ e λ
such µ, it follows that
(−1)vλr =0
λ
λX∈Oµ
Rewriting this coefficient equation as
r = (−1)vλ+1r
µ λ
λ∈XOµ\µ
we find that
r = r [λ] + r [λ]
λ e λ e
! !
λX∈Sv λ∈XBe\Sv
= r [λ] + r [λ] +(−1)vλ+1[v ·λ]
λ e λ e λ e
! !
λX∈Sv λ∈Be\X(Sv∪Bv) (cid:0) (cid:1)
as desired. (cid:3)
7
3.2. Central Weyl alcove vertices. Often the affine Steinberg basis (B ,B ) for the central-
e v
izers (Z ,Z ) has the property that B is a small collection of weights, and so there is little
e v v
hope that affine induction takes all the weights of B into B , as considered in the last section.
e v
However, if B has exactly one weight, that is if the centralizer Z is the whole group G, then
v v
again the kernel of induction to Z has a particularly clean description.
v
Proposition3.4. Supposethevertexv oftheWeylalcove, correspondingtothesimplerootα ,is
i
central in G, and (B ,B ) is a level k affine Steinberg basis for the pair (Z ,Z ). The collection
e v e v
B is the single weight at that vertex of the level k alcove, namely kλi, where h∨ is the dual
v h∨ i
i
Coxeter label of the root α ; we abbreviate that weight by θ. In this case the fusion kernel is
i
kerde,v = [λ] +(−1)vλ+1[w(v ·λ−θ)]·[θ] |λ∈B \(B ∪S ) ∪ [λ] |λ∈S
e λ e e v v e v
Here Sv is again(cid:10)t(cid:8)he v-singular weights in Be, and w ∈W is the Weyl e(cid:9)lem(cid:8)ent such that(cid:9)w(cid:11)(Cv−
θ)=C, that is the Weyl element aligning the Weyl chambers of the vertex and the origin of the
alcove.
Proof. First check that for λ∈B \(B ∪S ), we have
e v v
de,v [λ] +(−1)vλ+1[w(v ·λ−θ)]·[θ] =(−1)vλ[v ·λ] +(−1)vλ+1[w(v ·λ−θ)]·[θ] =0
e λ e λ v λ v
Second(cid:0)given r ∈kerde,v, write (cid:1)
r = r [λ] +r [θ] + r [λ]
λ e θ e λ e
! !
λX∈Sv λ∈Be\X(Bv∪Sv)
Inducing we have
0=r [θ] + (−1)vλr [v ·λ] = r + (−1)vλr [w(v ·λ−θ)] [θ]
θ v λ λ v θ λ λ v
!
λ∈Be\X(Bv∪Sv) λ∈Be\X(Bv∪Sv)
The coefficient of [θ] in this last expression is therefore 0, and we conclude that
v
r = r [λ] + r [λ] +(−1)vλ+1[w(v ·λ−θ)]·[θ]
λ e λ e λ e
! !
λX∈Sv λ∈Be\X(Bv∪Sv) (cid:0) (cid:1)
as desired. (cid:3)
3.3. Kernels of orthogonal induction. For every simple, simply connected Lie group except
SU(n),n>2,thereis exactlyone edgeofthe Weyl alcoveperpendicularto the affine wallofthe
alcove. The kernel of induction from that edge to the corresponding affine vertex of the alcove
has a concrete description in terms of a weight basis for the representation ring R[Z ] of the
e
centralizerofthatedge. We begin,though, withanintermediatedescriptionofthis fusionkernel
in terms of a generating set for a quotient of R[Z ] by irreducible representations associated to
e
weights nonsingular for affine induction.
Lemma 3.5. Assume that the edge e of the Weyl alcove is perpendicular to the affine wall, and
has affine vertex v. Let B be a collection of weights in the chamber C such that {[λ] |λ∈B }
e e e e
is an R[G]-module basis for R[Z ]. Let S denote the collection of all weights in C of level k+1,
e e
and let R[S] ⊂ R[Z ] denote the Z-module span of {[λ] |λ ∈ S}. The module R[S] is also
e e
the quotient q of R[Z ] by the Z-submodule generated by {[λ] |λ ∈ C \S}. For ǫ ∈ R[G] and
e e e
η ∈ R[S], the operation ǫ·η := q(ǫη) ∈ R[S] does not define an R[G]-action, but it does define
an R[G]-action on the quotient R[S]/(2). Suppose B ⊂S is a collection of weights such that the
s
quotients {[λ] /(2)|λ∈B } generate R[S]/(2) as an R[G]-module. Then the fusion kernel is
e s
kerde,v = [λ] |λ∈B ∪ [λ] |λ∈B ∩S ∪ [λ] +[w λ] |λ∈B \(B ∩S)
e s e e e (k+1)·α0 e e e
Here w(k+1)·α(cid:10)0(cid:8)is reflection(cid:9)in t(cid:8)he affine wall of t(cid:9)he l(cid:8)evel k+1 alcove. (cid:9)(cid:11)
8
Proof. The listed elements are certainly in the kernelof de,v. For a weightλ∈C , the induction
v
is de,v([λ] )=[λ] ; for singular weights, the induction is zero; for a weight λ∈C \(C ∪S), the
e v e v
induction is de,v([λ] )=−[w λ] . The collection
e (k+1)·α0 v
[λ] |λ∈S ∪ [λ] +[w λ] |λ∈C \(C ∪S)
e e (k+1)·α0 e e v
is therefore a Z-basis(cid:8)for the ker(cid:9)nel o(cid:8)f de,v. It suffices to check that each o(cid:9)f these basis elements
is in the R[G]-span of the collection
[λ] |λ∈B ∪ [λ] |λ∈S ∪ [λ] +[w λ] |λ∈B \S
e s e v e (k+1)·α0 e e v
Here we have w(cid:8)ritten S for(cid:9)the(cid:8)v-singular b(cid:9)asis(cid:8)weights B ∩S. (cid:9)
v e
For µ∈C \(C ∪S), write
e v
[µ] = r [λ] + r [λ]
e λ e λ e
! !
λX∈Sv λ∈XBe\Sv
Next observe that
[µ] +[w µ] = r [λ] + r [λ] + r [λ] + r [w λ]
e (k+1)·α0 e λ e λ e λ e λ (k+1)·α0 e
! ! ! !
λX∈Sv λ∈XBe\Sv λX∈Sv λ∈XBe\Sv
= 2r [λ] + r [λ] +[w λ]
λ e λ e (k+1)·α0 e
! !
λX∈Sv λ∈XBe\Sv (cid:0) (cid:1)
as desired.
Forµ∈S,byassumptionthereexistrepresentationsr ∈R[G]suchthat[µ] − r [λ] =
0 in R[S]/(2). Thus, there exist integers t ∈Z such thaλt e λ∈Bs λ e
λ
P
[µ] − r [λ] − 2t [λ] =0
e λ e λ e
λX∈Bs λX∈S
in R[S]. Because the representations r ∈ R[G] are w -symmetric, there furthermore exist
λ 0·α0
integers s ∈Z such that in R[Z ] we have
λ e
[µ] − r [λ] − s [λ] +[w λ] − 2t [λ] =0
e λ e λ e (k+1)·α0 e λ e
λX∈Bs λ∈CeX\(Cv∪S) (cid:0) (cid:1) λX∈S
The first sum is in the required R[G]-span. By the previous paragraph, the second sum is also
in that span. For the third sum, for each λ∈S write
[λ] = r [ν] + r [ν]
e ν e ν e
νX∈Sv ν∈XBe\Sv
with r ∈R[G], from which we have
ν
2[λ] =[λ] +[w λ] = 2r [ν] + r [ν] +[w ν]
e e (k+1)·α0 e ν e ν e (k+1)·α0 e
νX∈Sv ν∈XBe\Sv (cid:0) (cid:1)
as needed. (cid:3)
Next we observe that the basis B for the singular weight module R[S]/(2), as in the above
s
lemma,isdeterminedbythe basisB forR[Z ]. Tosimplify the statement,weassumethe group
e e
Gis notSU(2)orSp(n)—this restrictionensures thatthe fundamentalweightonthe edgee has
level 2.
9
Lemma 3.6. Suppose the edge e of the Weyl alcove is perpendicular to the affine wall, and has
affine vertex v corresponding to the simple root α . Assume the group G is not SU(2) or Sp(n).
i
Let B be a collection of weights in the chamber C such that {[λ] |λ ∈ B } is an R[G]-module
e e e e
basis for R[Z ]. Then for B as follows, the collection {[λ] /(2)|λ ∈ B } generates, over R[G],
e s e s
the modulo 2 singular weight module R[S]/(2):
k+1−Lv(λ)
B = λ+ λ |λ∈B ,Lv(λ)=k+1(mod2)
s i e
2
n o
Here Lv(λ) is the level of the weight λ.
Proof. Consider the series of maps
R[Z ]→R[Z ] →R[S]→R[S]/(2)
e e S-cong
Here R[Z ] is the quotient of R[Z ] by the Z-submodule generated by {[λ] | Lv(λ) 6=
e S-cong e e
k+1(mod2)},thatisthe quotientbyallrepresentationswhoselevelisnotcongruenttothe sin-
gularlevel. ThemapR[Z ] →R[S]takesarepresentation[λ] ,forLv(λ)=k+1(mod2),to
e S-cong e
therepresentation[λ+k+1−Lv(λ)λ ] . ThoughneitherR[Z ] norR[S]isanR[G]-module,the
2 i e e S-cong
compositeR[Z ]→R[S]/(2)isasurjectiveR[G]-modulemap. Therepresentations{[λ] |λ∈B }
e e e
form an R[G]-basis for R[Z ]; this collection of representations maps to the collection
e
k+1−Lv(λ)
[λ+ λ ] ∈R[S]/(2)|λ∈B ,Lv(λ)=k+1(mod2)
i e e
2
which thereforne generates the quotient module R[S]/(2). o (cid:3)
The cases SU(2) and Sp(n) present no difficulty—one need only replace the factor λ by 2λ
i i
throughout.
Altogether, then, the basis B determines the fusion kernel:
e
Corollary 3.7. If the edge e of the Weyl alcove, with affine vertex v corresponding to the simple
root α , is perpendicular to the affine wall, and the group G is not SU(2) or Sp(n), then the
i
fusion kernel is given by
k+1−Lv(λ)
kerde,v = [λ+ λ ] |λ∈B ,Lv(λ)=k+1(mod2)
i e e
2
(cid:10)(cid:8)∪ [λ] |λ∈B ∩S (cid:9)
e e
∪(cid:8)[λ]e+[w(k+1)·α0(cid:9)λ]e|λ∈Be\(Be∩S)
As before Be is a collection(cid:8)of weights in the chamber Ce whose co(cid:9)r(cid:11)responding irreducible repre-
sentations form an R[G]-basis for R[Z ], and S is the set of level k+1 weights of C .
e e
4. Computation of the fusion ideals
Section 2 showedthat the fusion ideal is the image de,o(kerde,v). Here, de,o :R[Z ]→R[G] is
e
inductionfromthecentralizerofanedgeoftheWeylalcove,andde,v :R[Z ]→R [Z ]istwisted
e k v
inductionfromthatedgetothetwistedrepresentationmoduleofthe correspondingvertexofthe
alcove. Section 3 computed the kernel kerde,v in terms of appropriate affine Steinberg bases for
the centralizers Z . In this section, we describe the necessary Steinberg bases and compute the
e
induction images of the fusion kernels, for all simple, simply connected groups.
We begin by recalling a convenient form of Steinberg’s original construction of bases for rep-
resentation modules:
Proposition 4.1 ([13]). Suppose E ⊂ F are two full rank subgroups of the simple, simply
connected group G. Let R ⊂R ⊂R be the root systems for the three groups, with Weyl groups
E F
WE, WF, and W. Denote by R[E] = Z[ΛW]WE, R[F] = Z[ΛW]WF, and R[G] = Z[ΛW]W the
three representation rings. Choose a set of positive roots R+, and let D(R+) be the simple roots
10