An application of TQFT to modular representation theory
AAN APPLICATION OF TQFT TO MODULARREPRESENTATION THEORY
PATRICK M. GILMER AND GREGOR MASBAUM
Abstract.
For p ≥ g ≥ p − L p ( λ ) for the symplectic group Sp(2 g, K ) where K is an algebraicallyclosed field of characteristic p . This permits explicit formulae for the dimensionand the formal character of L p ( λ ) for these highest weights. Contents
1. Introduction 12. Two lemmas 63. Results from TQFT 74. Proof of Theorems 1.1 and 1.9 125. Further Comments 17Appendix A. Proof of Lemma 3.1 18Appendix B. Some polynomial formulae for dimensions 21References 231.
Introduction
Let p be an odd prime, and K be an algebraically closed field of characteristic p . For g ≥ g, K ), thoughtof as an algebraic group of rank g . It is well-known that the classification (due toChevalley) of rational simple Sp(2 g, K )-modules is the same as in characteristic zero(see Jantzen [J, II.2]). More precisely, for every dominant weight λ there is a simplemodule L p ( λ ), and these exhaust all isomorphism classes of simple modules. Herethe set of dominant weights is the same as in characteristic zero: λ is dominantiff it is a linear combination of the fundamental weights ω i ( i = 1 , . . . , g ) withnonnegative integer coefficients.On the other hand, while the dimensions of simple Sp(2 g, C )-modules can becomputed from the Weyl character formula, it seems that explicit dimension for-mulae for the modules L p ( λ ) for p > L p ( ω i ) by reduc-ing the problem to known properties of symmetric group representations. Later, Date : April 24, 2017.2010
Mathematics Subject Classification. a r X i v : . [ m a t h . R T ] M a y PATRICK M. GILMER AND GREGOR MASBAUM
Gow [Go] gave an explicit construction of L p ( ω i ) for the last p − ω i where i ≥ g − p + 1) which allowed him to obtain a recursiveformula for their dimensions. Even later, Foulle [F] obtained a dimension formulafor all fundamental weights. As for other weights, it is known that for weights λ in the fundamental alcove the dimension of L p ( λ ) is the same as the dimension of L ( λ ) (the corresponding simple module in characteristic zero), and can thus becomputed by the Weyl character formula. But for weights outside the fundamentalalcove, no general dimension formula is known. A conjectural formula by Lusztigfor primes in a certain range was shown to hold for p >> p in the hoped-for rangeby Williamson [W].In this paper, we show that Topological Quantum Field Theory (TQFT) can givenew information about the dimensions of some of these simple modules. Specifically,we show that for every prime p ≥ g ≥
3, there is a family of p − λ , lying outside of the fundamental alcove except for oneweight in rank g = 3, for which we can express the dimension of L p ( λ ) by formulaesimilar to the Verlinde formula in TQFT. We found this family as a byproduct ofIntegral SO(3)-TQFT [G1, GM1], an integral refinement of the Witten-Reshetikhin-Turaev TQFT associated to SO(3). More precisely, we use Integral SO(3)-TQFTin what we call the ‘equal characteristic case’ which we studied in [GM5]. Thefamily of weights λ we found together with our formulae for dim L p ( λ ) is given inthe following Theorem 1.1. We can also compute the weight space decomposition of L p ( λ ) for these weights λ ; this will be given in Theorem 1.9. We follow the notationof [B, Planche III], where the fundamental weights ω i are expressed in the usualbasis { ε i } ( i = 1 , . . . , g ) of weights of the maximal torus as ω i = ε + . . . + ε i . Theorem 1.1.
Let p ≥ be prime and put d = ( p − / . For rank g ≥ , considerthe following p − dominant weights for the symplectic group Sp(2 g, K ) : λ = ( d − ω g ( Case
I)( d − c − ω g + c ω g − for ≤ c ≤ d − Case
II)( d − c − ω g + ( c − ω g − + ω g − for ≤ c ≤ d − Case
III)( d − ω g + ω g − ( Case
IV)
Put ε = 0 in Case I and II and ε = 1 in Case III and IV . Then (1) dim L p ( λ ) = 12 (cid:16) D (2 c ) g ( p ) + ( − ε δ (2 c ) g ( p ) (cid:17) where (2) D (2 c ) g ( p ) = (cid:16) p (cid:17) g − d (cid:88) j =1 (cid:18) sin πj (2 c + 1) p (cid:19) (cid:18) sin πjp (cid:19) − g (3) δ (2 c ) g ( p ) = ( − c − g p d (cid:88) j =1 (cid:18) sin πj (2 c + 1) p (cid:19) (cid:18) sin πjp (cid:19) (cid:18) cos πjp (cid:19) − g , and c is the same c used in the definition of λ , except in Case I and IV , where weput c = 0 . In Case IV in rank g = 3 , ω g − = ω should be interpreted as zero. Remark 1.2.
Formula (2) is an instance of the famous Verlinde formula in TQFT.Formula (3) appeared first in [GM5]. Note that the difference between the twoformulae is that certain sines in (2) have become cosines in (3), and the overall
N APPLICATION OF TQFT TO MODULAR REPRESENTATION THEORY 3 prefactor is different. For fixed g , both D (2 c ) g ( p ) and δ (2 c ) g ( p ) can be expressed aspolynomials in p and c . See [GM5] for more information and further references.In Appendix B, we give explicit polynomial expressions for the dimensions of our L p ( λ ) in rank g ≤ Remark 1.3.
Except for the weight λ = ( d − ω in Case IV in rank g = 3, allthe weights in the above list lie outside of the fundamental alcove. See Section 5for more concerning this. Remark 1.4.
When p = 5, the list above produces (in order) the fundamentalweights ω g , ω g − , ω g − , ω g − . These are exactly the weights considered by Gow[Go]. For p >
5, our weights are different from those of Gow. It is intriguing thatboth Gow’s and our family of weights have p − Question 1.5.
Can one find similar Verlinde-like dimension formulae for otherfamilies of dominant weights?
Remark 1.6.
In [GM4], we answered this question affirmatively for the p − e.g. [ChLe]) might pro-duce more families of weights λ where the methods of the present paper could beapplied. A difficulty here is that Integral TQFT as we need it in this paper is sofar not developed for other TQFTs. Remarks 1.7. (i) The restriction that g ≥ p − g = 1 or 2 for thoseweights λ where it makes sense ( i.e. , if no ω i with i < λ ) provided ω is interpreted as zero.(ii) Case I could be amalgamated with Case II in Theorem 1.1 by allowing c to be zero in Case II. We chose not to do this because Case I will require specialtreatment later.Throughout the paper, we assume p ≥ d = ( p − / L p ( λ ) goes as follows. For 0 ≤ c ≤ d − ε ∈ Z /
2, we construct certain simple modules which we denote by (cid:101) F p ( g, c, ε ).Note that there are p − c, ε ). The construction of the modules (cid:101) F p ( g, c, ε ) is based on results from Integral TQFT obtained in [GM5]. From theTQFT description, we shall compute the dimension and weight space decomposi-tion of (cid:101) F p ( g, c, ε ). In particular, we shall compute the highest weight occuring in (cid:101) F p ( g, c, ε ), thereby identifying (cid:101) F p ( g, c, ε ) with one of the L p ( λ ) in Theorem 1.1.Here is the construction of (cid:101) F p ( g, c, ε ). We give a description which can be readwithout any knowledge of TQFT. Consider the graph G g depicted in Figure 1which we call a lollipop tree. It has 2 g − c . The 2-valent ‘corner’ vertex to the left ofthe figure should be ignored, and the two edges meeting there are to be considereda single edge. Thus, G g has 3 g − g of which are loop edges. The edgesincident to a loop edge are called stick edges, and we refer to a loop edge togetherwith its stick edge as a lollipop.A p -color is an integer ∈ { , , . . . , p − } . A p -coloring of G g is an assignmentof p -colors to the edges of G g . A p -coloring is admissible if whenever i , j and k are PATRICK M. GILMER AND GREGOR MASBAUM
Figure 1.
Lollipop tree G g the colors of edges which meet at a vertex, then i + j + k ≡ , | i − j | ≤ k ≤ i + j , and i + j + k ≤ p − . Admissibility at the trivalent vertex of the i -th lollipop implies that the stick edgehas to receive an even color, which we denote by 2 a i , and the loop edge has toreceive a color of the form a i + b i , with b i ≥
0. We denote the colors of theremaining edges by c , c , . . . as in Figure 2, and we write an admissible p -coloringas σ = ( a , . . . , a g , b , . . . , b g , c , . . . ). a + b2 a a + b a a + b a c c g a + b gg a Figure 2.
Colored Lollipop tree G g A p -coloring is of type ( c, ε ) if the color 2 c is assigned to the edge incident withthe univalent vertex and if(4) c + (cid:88) a i ≡ ε (mod 2) . A p -coloring is small if the colors a i + b i of the loop edges satisfy(5) 0 ≤ a i + b i ≤ d − . Let C p ( g, c, ε ) denote the set of small admissible p -colorings of G g of type ( c, ε ).Let F p denote the finite field with p elements, and let F p ( g, c, ε ) be the F p -vectorspace with basis C p ( g, c, ε ). Theorem 1.8.
There is an irreducible representation of the finite symplectic group
Sp(2 g, F p ) on F p ( g, c, ε ) . This will be proved in Section 3.Steinberg’s restriction theorem (see e.g. [H2, 2.11]) implies that there is a uniquesimple Sp(2 g, K )-module (cid:101) F p ( g, c, ε ) characterized by the following two properties: N APPLICATION OF TQFT TO MODULAR REPRESENTATION THEORY 5 (i) The restriction of (cid:101) F p ( g, c, ε ) to the finite group Sp(2 g, F p ) is F p ( g, c, ε ) ⊗ K .(ii) (cid:101) F p ( g, c, ε ) has p -restricted highest weight.We recall that a dominant weight λ = (cid:80) gi =1 λ i ω i is p -restricted if, for each1 ≤ i ≤ g , we have 0 ≤ λ i ≤ p − (cid:101) F p ( g, c, ε ) are precisely the simplemodules L p ( λ ) listed in Theorem 1.1. Part (ii) gives the weight space decompositionand thus determines the formal character of these modules. To state the result, let (cid:102) W p ( g, c, ε ) be the multiset of weights occuring in (cid:101) F p ( g, c, ε ). (By a multiset, wemean a set with multiplicities.) Theorem 1.9. (i) The
Sp(2 g, K ) -module (cid:101) F p ( g, c, ε ) is isomorphic to L p ( λ ) wherethe highest weight λ = λ p ( g, c, ε ) is given by λ p ( g, ,
0) = ( d − ω g ( Case I) λ p ( g, c,
0) = ( d − c − ω g + c ω g − , ≤ c ≤ d − Case
II) λ p ( g, c,
1) = ( d − c − ω g + ( c − ω g − + ω g − , ≤ c ≤ d − Case
III) λ p ( g, ,
1) = ( d − ω g + ω g − ( Case
IV) (ii) We have (cid:102) W p ( g, c, ε ) = { w ( σ ) | σ ∈ C p ( g, c, ε ) } , where the weight of a coloring σ = ( a , . . . , a g , b , . . . , b g , c , . . . ) is (6) w ( σ ) = g (cid:88) i =1 ( d − − a i − b i ) ε i . Example 1.10.
In Case I, the highest weight corresponds to the p -coloring σ where all edges are colored zero. Indeed, formula (6) gives w ( σ ) = g (cid:88) i =1 ( d − ε i = ( d − ω g . In the other cases, the coloring σ is not allowed as it is not of type ( c, ε ) for( c, ε ) (cid:54) = (0 , − IV in Section 4.
Remark 1.11.
The p = 5 case of Theorem 1.9 answers affirmatively the questionraised in [GM5, p. 257 (after Theorem 8.1)] (see also [GM4, p. 83 (after Corollary3)]).The remainder of this paper is organized as follows. In Section 2, we formulatetwo results (Lemma 2.1 and Lemma 2.4) about the Sp(2 g, F p )-modules F p ( g, c, ε ).In Section 3, we review the construction of F p ( g, c, ε ) and the proof of Theorem 1.8,and then prove Lemma 2.1 and Lemma 2.4 using further arguments from TQFT. InSection 4, we prove Theorems 1.1 and 1.9. The only results from TQFT that willbe used in the proof of these two theorems are those stated in Section 2. Finally,in Section 5, we make a few further comments and discuss the rank 3 case as anexample. Acknowledgements.
We thank Henning H. Andersen for helpful discussions. Hesuggested checking our results against the Jantzen Sum Formula in the rank 3case (see Section 5) and showed us how to do it. G. M. thanks the MathematicsDepartment of Louisiana State University, Baton Rouge, the Centre for Quantum
PATRICK M. GILMER AND GREGOR MASBAUM
Geometry of Moduli Spaces, Aarhus, Denmark, and the Max Planck Institute forMathematics, Bonn, Germany, for hospitality while part of this paper was written.P. G. also thanks the Max Planck Institute for Mathematics for hospitality. Lastbut not least, we thank the referee for his insightful comments.2.
Two lemmas
We begin by fixing some notation. For k any of the rings Z , F p , or K , wetake Sp(2 g, k ) to be the subgroup of GL(2 g, k ) consisting of isometries of the skewsymmetric form given by the matrix J g = (cid:104) I g − I g (cid:105) . Let T be the maximal torusof Sp(2 g, K ) given by the diagonal matrices of Sp(2 g, K ). For 1 ≤ i ≤ g and x ∈ K ∗ , let T x,i denote the diagonal matrix with x on the i th diagonal entry, x − on the ( g + i )-th diagonal entry, and 1’s elsewhere on the diagonal. We have anisomorphism ( K ∗ ) g ≈ −→ T , ( x , . . . x g ) (cid:55)→ g (cid:89) i =1 T x i ,i . We denote by { ε i } i =1 ,...,g the standard basis of the weight lattice X ( T ) = Hom( T , K ∗ ) ≈ g (cid:77) i =1 Hom( K ∗ , K ∗ ) ≈ Z g where ε i ( T x,i ) = x and ε i ( T x,j ) = 1 for j (cid:54) = i .We also let B ( K ) denote the Borel subgroup of Sp(2 g, K ) which is the group ofblock matrices of the form (cid:20) A B A t ) − (cid:21) where A is an invertible upper triangular matrix and B satisfies AB t = BA t . Recall that F p ( g, c, ε ) is a representation of the finite symplectic group Sp(2 g, F p )on the F p -vector space with basis C p ( g, c, ε ). Let ˆ b σ denote the basis vector corre-ponding to the coloring σ ∈ C p ( g, c, ε ). Since the finite field F p is a subfield of K ,we may consider the actions of the finite maximal torus T ( F p ) = T ∩ Sp(2 g, F p ) andof the finite Borel subgroup B ( F p ) = B ( K ) ∩ Sp(2 g, F p ) on F p ( g, c, ε ). The followingLemma says that the basis vectors ˆ b σ are in some sense ‘weight vectors’ for T ( F p ). Lemma 2.1.
Each basis vector ˆ b σ is a simultaneous eigenvector for the commutingoperators { T x,i } i =1 ,...,g for x ∈ F (cid:63)p , with eigenvalues given by T x,i (ˆ b σ ) = x d − − a i − b i ˆ b σ where σ = ( a , . . . , a g , b , . . . , b g , c , . . . ) . Note that the exponents are meaningful only modulo p −
1, as F (cid:63)p is a cyclicgroup of order p −
1. We may interpret the collection of these exponents for a givenbasis vector ˆ b σ as specifying a reduced weight, by which we mean an element of X ( T ( F p )) = X ( T ) ⊗ Z / ( p − Z ≈ ( Z / ( p − Z ) g . Let W p ( g, c, ε ) be the multiset of reduced weights occuring in F p ( g, c, ε ). The fol-lowing is an immediate corollary of Lemma 2.1. Corollary 2.2.
We have W p ( g, c, ε ) = { w ( σ ) | σ ∈ C p ( g, c, ε ) } , where w ( σ ) is thereduction modulo p − of w ( σ ) as defined in (6). N APPLICATION OF TQFT TO MODULAR REPRESENTATION THEORY 7
Remark 2.3.
Note that W p ( g, c, ε ) is also the reduction modulo p − (cid:102) W p ( g, c, ε ).This is because the restriction of the Sp(2 g, K )-module (cid:101) F p ( g, c, ε ) to the finite groupSp(2 g, F p ) is F p ( g, c, ε ) ⊗ K .In Section 4, we shall see that this information is enough to determine (cid:102) W p ( g, c, ε ),except that in Case I, we will also need to use the following Lemma. Lemma 2.4.
Let σ = (0 , , . . . ) be the coloring where all edges are colored zero.Then the basis vector ˆ b σ in F p ( g, , is fixed up to scalars by the finite Borelsubgroup B ( F p ) . The proofs of Lemma 2.1 and Lemma 2.4 will be given in Section 3.In Section 4 we shall apply Lemma 2.4 through the following Corollary whoseproof we give already here.
Corollary 2.5.
The highest weight of (cid:101) F p ( g, , is congruent modulo p − to w ( σ ) = ( d − ω g .Proof. Let v be a highest weight vector in (cid:101) F p ( g, , v is fixed up to scalarsby B ( K ) (see [H1, 31.3]). Restricting to the finite symplectic group, we can view v asa vector in F p ( g, c, ε ) ⊗ K that is fixed up to scalars by B ( F p ). By [CaLu, Theorem7.1], there is a unique line fixed by B ( F p ) in F p ( g, c, ε ) ⊗ K . Since ˆ b σ is alsocontained in this line by Lemma 2.4, we conclude that v and ˆ b σ are proportional.In particular, v and ˆ b σ have the same reduced weight, which implies the result. (cid:3) Results from TQFT
In this section, we review how Integral TQFT leads to the irreducible Sp(2 g, F p )-representations F p ( g, c, ε ) of Theorem 1.8 which were the starting point for thispaper. In particular, we show how Theorem 1.8 follows from [GM5] using a result(Lemma 3.1) originally proved in [M]. We shall provide a self-contained proof ofLemma 3.1 in Appendix A. We then prove Lemma 2.1 and Lemma 2.4.Let Σ g (2 c ) denote a closed surface of genus g equipped with one marked framedpoint labelled 2 c , where c is an integer with 0 ≤ c ≤ d −
1. (Recall d = ( p − / g (2 c ) a free Z [ ζ p ]-module S p (Σ g (2 c )) of finiterank, together with a projective-linear representation of the mapping class groupof Σ g (2 c ) on this module. Here p ≥ ζ p is a primitive p -th root ofunity, and Z [ ζ p ] is the ring of cyclotomic integers. The mapping class group ofΣ g (2 c ) can be identified with Γ g, , that is, the mapping class group of Σ g, , anoriented surface of genus g with one boundary component. (Thus Γ g, is the groupof orientation-preserving diffeomorphisms of Σ g, that fix the boundary pointwise,modulo isotopies of such diffeomorphisms.) The projective-linear representationof Γ g, on S p (Σ g (2 c )) can be lifted to a linear representation of a certain centralextension of Γ g, . The representations of mapping class groups obtained in this waymay be considered as an integral refinement of the complex unitary representationscoming from Witten-Reshetikhin-Turaev TQFT associated to the Lie group SO(3).In particular, the rank of the free Z [ ζ p ]-module S p (Σ g (2 c )) is given by the Verlindeformula (2).Recall that 1 − ζ p is a prime in Z [ ζ p ], and Z [ ζ p ] / (1 − ζ p ) is the finite field F p .Thus we get a representation on the F p -vector space F p (Σ g (2 c )) = S p (Σ g (2 c )) / (1 − ζ p ) S p (Σ g (2 c )) . PATRICK M. GILMER AND GREGOR MASBAUM
It is shown in [GM3, Cor. 12.4] that this induces a linear representation of Γ g, on F p (Σ g (2 c )) ( i.e. the central extension is no longer needed). Furthermore, we provedin [GM5] that F p (Σ g (2 c )) has a composition series with (at most) two irreduciblefactors. These irreducible factors are the F p ( g, c, ε ) defined in the introduction.More precisely, we have a short sequence of Γ g, -representations(7) 0 → F p ( g, c, → F p (Σ g (2 c )) → F p ( g, c, → . It remains to show that the action of Γ g, on the irreducible factors F p ( g, c, ε )factors through an action of the finite symplectic group Sp(2 g, F p ). For g = 1, thiswas proved by explicit computation in [GM2]. For g ≥
2, we use the followinglemma whose proof is deferred to Appendix A.
Lemma 3.1.
The Torelli group I g, acts trivially on F p ( g, c, and F p ( g, c, . It follows that the action of Γ g, on the irreducible factors F p ( g, c, ε ) factorsthrough an action of the symplectic groupSp(2 g, Z ) ∼ = Γ g, / I g, . To see that this descends to an action of the finite symplectic group Sp(2 g, F p ),we invoke a result of Mennicke, who proved that for g ≥
2, the group Sp(2 g, F p )is the quotient of Sp(2 g, Z ) by the normal subgroup generated by the p -th powerof a certain transvection [Me, Satz 10]. The result follows, because transvectionslift to Dehn twists in Γ g, , and it is well-known that in SO(3)-TQFT at the prime p the p -th power of any Dehn twist acts trivially. This concludes the proof ofTheorem 1.8.For the proof of Lemma 2.1 and Lemma 2.4, we need to say more about thebasis vectors ˆ b σ associated to colorings σ . Recall the graph G g depicted in Fig-ure 1. A regular neighborhood in R of G g is a 3-dimensional handlebody H g .We identify Σ g (2 c ) with the boundary of H g , in such a way that the univalentvertex labelled 2 c in the figure meets the boundary surface in the marked point.Given this identification, there is a basis { ˜ b σ } of S p (Σ g (2 c )) called the orthogonallollipop basis (see [GM2, p. 101]). The basis vectors are indexed by colorings σ in C p ( g, c, ∪ C p ( g, c, − ζ p , we get a basis { ˆ b σ } of F p (Σ g (2 c )).Notice that as an F p -vector space, F p (Σ g (2 c )) is the direct sum of F p ( g, c,
0) and F p ( g, c, b σ where σ ∈ C p ( g, c, ε ) are a basis of F p ( g, c, ε ). Remark 3.2.
What we denote now by S p was previously denoted by S + p in [GM2],and by S in [GM5]. Similarly, in [GM5], we omitted the subscript p in F p (Σ g (2 c )).Also, we referred to ε as the parity (even or odd) of a coloring. Thus F p ( g, c,
1) wasdenoted by F odd (Σ g (2 c )) since it is spanned by the basis vectors corresponding toodd colorings ( ε = 1). The short exact sequence (7) identifies F p ( g, c,
0) with thequotient representation F p (Σ g (2 c )) /F odd (Σ g (2 c )). As a vector space, this quotientwas denoted by F even (Σ g (2 c )) in [GM5] since it is spanned by the basis vectorscorresponding to even colorings ( ε = 0). Remark 3.3.
The construction of Integral TQFT in [GM1] uses the skein-theoreticapproach to TQFT of [BHMV]. In particular, the basis vectors ˜ b σ are representedby certain skein elements (that is, linear combinations of banded links or graphs)in the handlebody H g . If a diffeomorphism f of the surface Σ g (2 c ) extends toa diffeomorphism F of the handlebody, then the projective-linear action of f on S p (Σ g (2 c )) is determined by how F acts on skein elements in the handlebody. The N APPLICATION OF TQFT TO MODULAR REPRESENTATION THEORY 9 details of this are irrelevant for our purposes, with one exception: In the case when c = 0, the basis vector ˜ b σ associated to the zero coloring σ can be representedby the empty link in the handlebody H g . In particular, it is preserved by anydiffeomorphism of the handlebody. So if f extends to a diffeomorphism of thehandlebody, then the action of f on S p (Σ g (2 c )) fixes (projectively) the basis vector˜ b σ . This will be used below in the proof of Lemma 2.4.Before giving the proof of Lemma 2.1 and Lemma 2.4, we also need to fix ourconventions for the homomorphism Γ g, (cid:16) Sp(2 g, Z ). This homomorphism comesfrom the action of the mapping class group Γ g, on the homology of the surfaceΣ g (2 c ) by isometries of the intersection form. Recall that we have identified Σ g (2 c )with the boundary of a regular neighborhood H g of the graph G g . We identify H (Σ g (2 c ); Z ) with Z g identifying the homology class of the positive meridian ofthe i th loop (oriented counterclockwise and counting from the left to right) withthe i th basis vector of Z g and denote this element by m i . Similarly we identifythe homology class of the parallel to the i th loop to be the ( g + i )th basis vector of Z g . Then the intersection pairing is described by the matrix J g = (cid:104) I g − I g (cid:105) . Proof of Lemma 2.1.
First, let us prove Lemma 2.1 in the special case when g = 1.Then F p (1 , c,
1) is zero and F p (1 , c,
0) = F (Σ (2 c )) has dimension d − c , as thegraph G is just a single lollipop, with stick color 2 a equal to 2 c , so that only thecolor a + b of the loop edge may vary, and there are d − a = d − c possibilitiesfor b . The representation of SL(2 , F p ) = Sp(2 , F p ) on F p (1 , c,
0) is shown in [GM2, §
5] to be isomorphic to the standard representation of SL(2 , F p ) on homogeneouspolynomials of degree d − c − X and Y . Explicitly, thisrepresentation is given by: (cid:20) a bc d (cid:21) X d − c − − b Y b = ( a X + c Y ) d − c − − b ( b X + d Y ) b . Note that (cid:20) x x − (cid:21) X d − c − − b Y b = x d − − c − b X d − c − − b Y b . Thus X d − c − − b Y b , which is the b -th element in the monomial basis for the poly-nomials, is an eigenvector for (cid:2) x x − (cid:3) with eigenvalue x d − − c − b . One can checkthat the intertwiner Φ [GM2, §
5] defining the isomorphism sends X d − c − − b Y b toa multiple of ˆ b σ for σ the coloring which is 2 c on the stick edge and c + b on theloop edge of G . Thus Lemma 2.1 holds when g = 1.The general case is now proved as follows. The torus T ( F p ) is contained in thesubgroup of Sp(2 g, F p ) isomorphic to a product of g copies of SL(2 , F p ) = Sp(2 , F p )arising from the g copies of SL(2 , Z ) = Sp(2 , Z ) in Sp(2 g, Z ) corresponding to eachloop of G g . Specifically, the element T x,i defined in Section 2 lies in the i -th copy.Similarly, the mapping class group Γ g, contains a subgroup isomorphic to a productof g copies of Γ , . The i -th copy of Γ , is generated by the Dehn twist about themeridian and the Dehn twist about the parallel to the i th loop of G g . Since thehomomorphism Γ g, (cid:16) Sp(2 g, F p ) is surjective, we can lift T x,i (non-uniquely) toa mapping class, say φ , which we may assume to lie in the i -th copy of Γ , . Thiscopy of Γ , is the mapping class group of the one-holed torus which is cut offfrom Σ g (2 c ) by the simple closed curve γ on Σ g (2 c ) which is a meridian of the i -th stick edge of the graph G g . Using the integral modular functor properties of [GM1,Section 11], we have an injective linear map(8) d − (cid:77) a i =0 S p (Σ (2 a i )) ⊗ S p (Σ g − (2 a i , c )) −→ S p (Σ g (2 c ))given by gluing along γ . Here, Σ g − (2 a i , c ) stands for a genus g − a i and 2 c , respectively. The module S p (Σ g − (2 a i , c ))is again a free Z [ ζ p ]-lattice by [GM1, Theorem 4.1]. The image of the gluing map(8) is a free sublattice of S p (Σ g (2 c )) of full rank. When tensored with the quotientfield of Z [ ζ p ], the map (8) becomes an isomorphism familiar in TQFTs defined overa field under the name of ‘factorization along a separating curve’. Over the ring Z [ ζ p ] the gluing map (8) is, however, not surjective in general.On the sublattice of S p (Σ g (2 c )) given by the image of the map (8), the mappingclass φ preserves the direct sum decomposition and in each summand, φ acts onlyon the first tensor factor S p (Σ (2 a i )). When reduced modulo 1 − ζ p , the actioninduced by φ on F p (Σ (2 a i )) is as described in the genus one case. In particular,the lollipop with stick color 2 a i and loop color a i + b i indexes an eigenvector forthe induced action of φ on F p (Σ (2 a i )) with eigenvalue x d − − a i − b i . If the map(8) were an isomorphism of Z [ ζ p ]-modules, this would prove the lemma by familiarTQFT arguments, since it would then also induce an isomorphism when reducedmodulo 1 − ζ p .Although (8) is not an isomorphism of Z [ ζ p ]-modules, we are saved by thefollowing fact (see [GM1, Theorem 11.1]). Pick a basis { b ( a i ) ν } of the lattice S p (Σ g − (2 a i , c )) associated to a lollipop tree as in [GM1, Theorem 4.1]. Thenthe image under the map (8) of the direct summand S p (Σ (2 a i )) ⊗ b ( a i ) ν of theL.H.S. of (8) is a certain power of 1 − ζ p times a direct summand of the R.H.S. of(8), that is, of S p (Σ g (2 c )). (The power of 1 − ζ p may depend on the summand.)Thus the action of φ on this direct summand, and hence the action of T x,i on thereduction modulo 1 − ζ p of this direct summand, can be computed from the ac-tion of φ on S p (Σ (2 a i )) ⊗ b ( a i ) ν (cid:39) S p (Σ (2 a i )). Since this action is given by thegenus one case, where the lemma is already proved, it follows that for a coloring σ = ( a , . . . , a g , b , . . . , b g , c , . . . ), the basis vector ˆ b σ is an eigenvector of T x,i witheigenvalue x d − − a i − b i . This completes the proof. (cid:3) Remark 3.4.
As a word of caution, we mention that a basis { b ( a i ) ν } of the lattice S p (Σ g − (2 a i , c )) as needed in the proof above cannot be obtained from coloringsof the graph obtained from G g by cutting G g at the mid-point of the stick edge ofthe i -th lollipop and removing the connected component containing the loop edge ofthe lollipop, as one would do when working with TQFTs defined over a field. Thisis because the remaining graph would not be a lollipop tree. See [GM1, Section 10].We now prepare the way for the proof of Lemma 2.4. Let L be the span of thehomology classes of the meridians m , m , . . . , m g in H (Σ g ) = Z g . Note that L is a lagrangian subspace with respect to the form J g . Let L ( Z ) be the subgroup ofSp(2 g, Z ) consisting of the matrices which preserve L . One has that L ( Z ) is the setof matrices of the form (cid:104) A B A t ) − (cid:105) where A ∈ GL( g, Z ), and B satisfies AB t = BA t . We call this subgroup the lagrangian subgroup.
N APPLICATION OF TQFT TO MODULAR REPRESENTATION THEORY 11
Let Γ g be the mapping class group of the closed surface Σ g of genus g , viewedas the boundary of the handlebody H g . Note that L is the kernel of the map H (Σ g ) → H ( H g ). If f ∈ Γ g , and f extends to a diffeomorphism F : H g → H g then f ∗ ∈ L ( Z ). We have a converse: Proposition 3.5. If f ∈ L ( Z ) , then f is induced by an element of Γ g which extendsto a diffeomorphism F : H g → H g .Proof. Consider the special case when f ∈ L ( Z ) has the form (cid:104) I g B I g (cid:105) . It followsthat B = B t . Consider the building blocks (cid:104) I g E ( i,j )0 I g (cid:105) where E ( i, i ) has zero entrieseverywhere except for the ( i, i ) location where it has a 1, and E ( i, j ) (for i (cid:54) = j ) haszero entries everywhere except for the ( i, j ) location and the ( j, i ) location where ithas 1’s. These E ( i, j ) are realized by Dehn twists along m i in the case i = j , andalong a curve representing m i + m j when i (cid:54) = j . These curves may be chosen sothat they bound disks in H g . Thus these Dehn twists extend over H g . Products ofsuch Dehn twists realize any symmetric matrix B . See [GL, p 312-313].We can reduce the general case to the above case, using another special case: f ∈ L ( Z ) has the form [ A D ]. We note that this is the case when A = ( D t ) − , where D ∈ GL( g, Z ). An elementary matrix in SL( g, Z ) can be realized, as D , bysliding one 1-handle in H g over another. Permuting two handles realizes, as D , atransposition matrix. Any D ∈ GL( g, Z ) is a product of elementary matrices andperhaps a transposition matrix. Thus (cid:104) ( D t ) − D (cid:105) for any D ∈ GL( g, Z ) can berealized by a diffeomorphism which extends over H g . (cid:3) We let U ( F p ) denote the unipotent radical of the finite Borel subgroup B ( F p ). Proposition 3.6.
The image of L ( Z ) under the quotient map π : Sp(2 g, Z ) (cid:16) Sp(2 g, F p ) contains U ( F p ) .Proof. We have that U ( F p ) is the group of block matrices over F p of the form (cid:104) V B V t ) − (cid:105) where V is an invertible upper triangular matrix with 1’s on the diagonaland B satisfies V B t = BV t . Each such matrix may be factored (cid:104) I g BV t I g (cid:105) (cid:104) V
00 ( V t ) − (cid:105) , and BV t will equal its transpose. As above, we note that (cid:104) I g BV t I g (cid:105) can be writtenas a product of (cid:104) I g E ( i,j )0 I g (cid:105) matrices. Thus any matrix of the form (cid:104) I g BV t I g (cid:105) haslifts under the quotient map π : Sp(2 g, Z ) (cid:16) Sp(2 g, F p ) that lie in L ( Z ). Also anymatrix of the form (cid:104) V
00 ( V t ) − (cid:105) has such a lift. It follows that any element of U ( F p )lifts to an element of L ( Z ). (cid:3) Proof of Lemma 2.4.
Recall that σ = (0 , , . . . ) denotes the coloring where alledges are colored zero. We are to show that the basis vector ˆ b σ in F p ( g, ,
0) isfixed up to scalars by B ( F p ). It will suffice to show that ˆ b σ is fixed by U ( F p ). Notethat ˆ b σ is the reduction modulo 1 − ζ p of the basis vector ˜ b σ of S p (Σ g (0)) whichis represented by the empty skein, and is thus fixed by any element of Γ g whichextends to a diffeomorphism of H g , as observed in Remark 3.3. By Proposition3.6, any element of U ( F p ) lifts to an element of L ( Z ), and by Proposition 3.5, anyelement of L ( Z ) is induced by an element of Γ g which extends to a diffeomorphismof H g . This implies the result. (cid:3) Proof of Theorems 1.1 and 1.9
Theorem 1.1 follows easily from Theorem 1.9 and the dimension formulae of[GM5, p. 229], where we computed the cardinality of the sets C p ( g, c, ε ) in thevarious cases in terms of the Verlinde formula (2) and its cousin (3).In the proof of Theorem 1.9, we shall need one more result from modular repre-sentation theory. Recall that the set of simple positive roots for the symplectic Liealgebra consists of α = ε − ε , . . . , α g − = ε g − − ε g , and α g = 2 ε g [B, PlancheIII]. Lemma 4.1.
Let p > and suppose λ is a p -restricted dominant weight for Sp(2 g, K ) . Let Π( λ ) be the set (without multiplicities) of weights occuring in thesimple module L p ( λ ) . (i) If µ ∈ Π( λ ) is such that µ + α i (cid:54)∈ Π( λ ) for all i = 1 , . . . , g , then µ = λ . (ii) If λ = (cid:80) η i ω i and η i > for some i = 1 , . . . , g , then λ − α i ∈ Π( λ ) .Proof. This is true for the sets of weights of simple Sp(2 g, C )-modules. By a resultof Premet [P] (see also the discussion in [H2, § λ ) is the same when workingover K or C as long as λ is p -restricted and the characteristic p >
2. The resultfollows. (cid:3)
Let us now prove Theorem 1.9. Recall that we must determine (cid:102) W p ( g, c, ε ) (=the multiset of weights occuring in (cid:101) F p ( g, c, ε )), and we must determine which of theweights in (cid:102) W p ( g, c, ε ) is the highest weight, which we denote by λ p ( g, c, ε ). As thedetails of this are somewhat involved, let us first outline the strategy of the proof.The proof proceeds in four steps, as follows. Step 1.
The first step is to compute (cid:102) W p ( g, c, ε ) modulo p −
1. As observedin Remark 2.3, we already know the answer: it is the multiset W p ( g, c, ε ) whichwas determined in Corollary 2.2. Recall that the elements of W p ( g, c, ε ) are re-duced weights, and that every coloring in C p ( g, c, ε ) determines a reduced weightin W p ( g, c, ε ). Step 2.
The second step is to identify λ p ( g, c, ε ), that is, the reduced weight in W p ( g, c, ε ) which is the reduction modulo p − λ p ( g, c, ε ) is the reduced weight associatedto a coloring illustrated in Figure 3.This will be proved by showing that these colorings satisfy the hypothesis of thefollowing Lemma. We let α i denote the reduction modulo p − α i . Lemma 4.2.
Let σ be a coloring in C p ( g, c, ε ) and let w ( σ ) be its associated reducedweight. If w ( σ ) + α i / ∈ W p ( g, c, ε ) for all i = 1 , . . . , g , then w ( σ ) = λ p ( g, c, ε ) .Proof. Let µ be any lift of w ( σ ) to (cid:102) W p ( g, c, ε ). The hypothesis implies that for all i = 1 , . . . , g , the weight µ + α i does not occur in (cid:102) W p ( g, c, ε ). By the construction ofthe module (cid:101) F p ( g, c, ε ), we know that its highest weight is p -restricted, so that wecan apply Lemma 4.1(i). It follows that µ is the highest weight, and so w ( σ ) is thereduction modulo p − (cid:3) Step 3.
Once λ p ( g, c, ε ) is known, the third step will be to determine λ p ( g, c, ε ).Write λ p ( g, c, ε ) = (cid:80) η i ω i with 0 ≤ η i ≤ p − λ p ( g, c, ε ) is p -restricted). The N APPLICATION OF TQFT TO MODULAR REPRESENTATION THEORY 13 , c12 2c 2c ,
11 12 2 2
Figure 3.
The colorings associated to the highest weights in CaseII, Case III, and Case IV. The leftmost part of G g is not drawn asit is colored zero. For the same reason the rightmost edge is notdrawn in Case IV as c is zero. These graphs could be guessed bytaking the smallest coloring which is rightmost on the graph andhas the given type.coefficient η i is determined by its reduction η i modulo p − η i (cid:54) = 0. Butif η i = 0, then η i can be either 0 or p −
1. We shall show that η i = 0 whenever η i = 0 in all our cases by means of the following Lemma. Lemma 4.3.
Let λ p ( g, c, ε ) = (cid:80) η i ω i be the highest weight in (cid:102) W p ( g, c, ε ) . If λ p ( g, c, ε ) − α i / ∈ W p ( g, c, ε ) , then η i = 0 .Proof. This follows immediately from Lemma 4.1(ii). (cid:3)
Step 4.
Once Step 3 is completed, it only remains to prove that (cid:102) W p ( g, c, ε ) is asclaimed in the theorem. This is now easy. Recall the notation d = ( p − /
2. Wesimply note that all weights in (cid:102) W p ( g, c, ε ) must lie in [1 − d, d − g as they mustlie in the convex hull of the orbit under the Weyl group of the highest weight, andin each case this highest weight has been shown in Step 3 to lie in [0 , d − g . Butno two distinct integer points in [1 − d, d − g agree modulo p − d in eachcoordinate. Thus (cid:102) W p ( g, c, ε ) is determined by its reduction modulo p −
1, and weare done.In the rest of this section, we shall now carry out Steps 2 and 3 in the variouscases. Having done this, the proof will be complete. To simplify notation, we shalldenote the highest weight λ p ( g, c, ε ) simply by λ . Also, from now on when we saycoloring, we mean a small admissible p -coloring.Recall that a coloring σ = ( a , . . . , a g , b , . . . , b g , c , . . . ) assigns the color 2 a i tothe i th stick edge and the color a i + b i to the i th loop edge (see Figure 2). Recallalso that a i ≥ b i ≥ a i + b i ≤ d −
1, and thecoefficient of ε i in the weight w ( σ ) is d − − a i − b i (see (6)).Before we begin with the cases, we state two lemmas. Both are an easy conse-quence of Corollary 2.2 and the smallness condition. Recall 2 d = p − . Lemma 4.4. If n i ≡ d (mod 2 d ) for some ≤ i ≤ g , then (cid:80) gi =1 n i ε i / ∈ W p ( g, c, ε ) , Lemma 4.5.
Suppose σ = ( a , . . . , a g , b , . . . , b g , c , . . . ) , and w ( σ ) = (cid:80) gi =1 n i ε i .If n i ≡ d − d ) for some ≤ i ≤ g , then a i = b i = 0 . Case I . Recall σ is the coloring which is zero on every edge. Step 2 was alreadytaken in Corollary 2.5 and so we know that λ = w ( σ ) = ( d − ω g . In Step 3,we must show that λ = ( d − ω g . By Lemma 4.3, it is enough to show that λ − α j / ∈ W p ( g, ,
0) for 1 ≤ j ≤ g −
1, which follows easily from Lemma 4.4. Thiscompletes the proof in Case I.
Remark 4.6.
One cannot accomplish Step 2 in Case I in the same way as we dobelow in Cases II, III, and IV, as σ does not satisfy the hypotheses of Lemma 4.2.This is because w ( σ ) + α g = ( d − ω g − + ( d + 1) ε g = w ( σ (cid:48) ) ∈ W p ( g, , σ (cid:48) denotes the coloring with all a i ’s, b i ’s and c i ’s zero except for b g = d − . Case II . Let σ be the coloring on the left of Figure 3, then w ( σ ) = ( d − g − (cid:88) i =1 ε i + ( d − c − ε g = ( d − c − ω g + c ω g − . We shall show that λ = w ( σ ) = ( d − c − ω g + c ω g − . As explained above, Step 2 in the proof is based on Lemma 4.2. We mustshow that w ( σ ) + α i / ∈ W p ( g, c,
0) for all 1 ≤ i ≤ g . For i (cid:54) = g this follows fromLemma 4.4. For i = g , it is proved by contradiction, as follows. Assume that w ( σ ) + α g ∈ W p ( g, c, σ (cid:48) = ( a (cid:48) , . . . , b (cid:48) , . . . , c (cid:48) , . . . ) oftype ( c,
0) with w ( σ (cid:48) ) = w ( σ ) + α g = ( d − g − (cid:88) i =1 ε i + ( d − c + 1) ε g . We will see such a coloring is impossible. By Lemma 4.5, a (cid:48) i = b (cid:48) i = 0 for i (cid:54) = g . Inother words, σ (cid:48) must color all but the rightmost lollipop by zero. By admissibility,it follows that a (cid:48) g = c . On the other hand, since the coefficient of ε g in w ( σ (cid:48) ) is d − − a (cid:48) g − b (cid:48) g (mod 2 d ), we have(9) d − c + 1 ≡ d − − a (cid:48) g − b (cid:48) g = d − − c − b (cid:48) g (mod 2 d ) , so b (cid:48) g ≡ − d ) , so b (cid:48) g = d − a (cid:48) g = 0 again bysmallness (see (5)). This contradicts a (cid:48) g = c >
0. Thus σ (cid:48) does not exist and hence w ( σ ) + α g / ∈ W p ( g, c, λ = w ( σ ) = ( d − c − ω g + c ω g − .Next, Step 3 in the proof is based on Lemma 4.3. We must show that λ − α i / ∈ W p ( g, c,
0) whenever the coefficient of ω i in λ is zero. For 1 ≤ j ≤ g −
2, thisfollows from Lemma 4.4. Then if 1 ≤ c ≤ d −
2, Step 3 is already complete, as thecoefficient of ω g and of ω g − in λ is non-zero. But if c = d −
1, the coefficient of ω g in λ is zero and we therefore also need to show that λ − α g / ∈ W p ( g, d − , λ − α g ∈ W p ( g, d − , σ (cid:48)(cid:48) of type ( d − ,
0) with(10) w ( σ (cid:48)(cid:48) ) = λ − α g = ( d − g − (cid:88) i =1 ε i − ε g . As before when we ruled out the coloring σ (cid:48) in Step 2, it follows from Lemma 4.5that σ (cid:48)(cid:48) must color all but the rightmost lollipop by zero. By admissibility, it followsthat a (cid:48)(cid:48) g = c = d − b (cid:48)(cid:48) g = 0 by smallness. But then the coefficient of ε g in w ( σ (cid:48)(cid:48) ) must be d − − a (cid:48)(cid:48) g − b (cid:48)(cid:48) g = 0 (mod 2 d ), which contradicts (10) where thiscoefficient is − d ). This contradiction shows that σ (cid:48)(cid:48) does not exist. Thiscompletes the proof in Case II. N APPLICATION OF TQFT TO MODULAR REPRESENTATION THEORY 15
In Case III and IV, we will also need the following Lemma which says that oddcolorings must assign nonzero colors to at least two lollipop sticks, and to at leastthree lollipop sticks if c = 0. Lemma 4.7.
Suppose σ = ( a , . . . , a g , b , . . . , b g , c , . . . ) is a small admissible p -coloring of type ( c, . (i) There are at least two distinct i ∈ { , . . . , g } with a i > . (ii) If moreover c = 0 , then there are at least three distinct i ∈ { , . . . , g } with a i > .Proof. (i) If only one of the a i is non-zero, say a i , then admissibility implies that a i = c and so condition (4) in the definition of colorings of type ( c,
1) is violated.(ii) If moreover c = 0, and only two of the a i are non-zero, say a i and a i , thenadmissibility implies that a i = a i and so condition (4) is again violated. (cid:3) Case
III . Let σ be the coloring in the middle of Figure 3, then w ( σ ) = ( d − g − (cid:88) i =1 ε i +( d − ε g − +( d − c − ε g = ( d − c − ω g +( c − ω g − + ω g − . We shall show that λ = w ( σ ) = ( d − c − ω g + ( c − ω g − + ω g − . In Step 2, we must show that w ( σ ) + α i / ∈ W p ( g, c,
1) for all 1 ≤ i ≤ g . For i ≤ g − i ∈ { g − , g } it is proved by contradiction,as follows.(Case i = g − w ( σ ) + α g − ∈ W p ( g, c, σ (cid:48) of type ( c,
1) with w ( σ (cid:48) ) = w ( σ ) + α g − = ( d − g − (cid:88) i =1 ε i + ( d − c − ε g . Note σ (cid:48) must color all but the rightmost lollipop with zeros by Lemma 4.5, and so σ (cid:48) contradicts Lemma 4.7(i).(Case i = g .) Assume that w ( σ ) + α g ∈ W p ( g, c, σ (cid:48)(cid:48) oftype ( c,
1) with w ( σ (cid:48)(cid:48) ) = w ( σ ) + α g = ( d − g − (cid:88) i =1 ε i + ( d − ε g − + ( d − c + 1) ε g . By Lemma 4.5 we have a (cid:48)(cid:48) i = 0 for 1 ≤ i ≤ g −
2. By admissibility, it follows thatthe three colors meeting at the trivalent vertex at the bottom of the rightmost stickare 2 a (cid:48)(cid:48) g − , a (cid:48)(cid:48) g , and 2 c . Computing the coefficient of ε g − , we have d − ≡ d − − a (cid:48)(cid:48) g − − b (cid:48)(cid:48) g − (mod 2 d )or a (cid:48)(cid:48) g − + 2 b (cid:48)(cid:48) g − ≡ d ). By smallness, it follows that a (cid:48)(cid:48) g − = 1. This impliestwo things. First, since ε = 1 ( i.e., the coloring is odd), it follows that a (cid:48)(cid:48) g ≡ c (mod 2). Second, by the triangle inequality in the admissibility condition at thetrivalent vertex at the bottom of the rightmost stick, it follows that c − ≤ a (cid:48)(cid:48) g ≤ c + 1. One concludes that a (cid:48)(cid:48) g = c . The rest of the proof is now the same as inCase II Step 2. Computing the coefficient of ε g exactly as was done there (see (9)),we deduce b (cid:48)(cid:48) g ≡ − d ), hence b (cid:48)(cid:48) g = d − a (cid:48)(cid:48) g = 0 by smallness. Thiscontradicts a (cid:48)(cid:48) g = c > Thus Step 2 is complete, and we now know that λ = w ( σ ) = ( d − c − ω g + ( c − ω g − + ω g − . For Step 3, we must show that λ − α i / ∈ W p ( g, c,
0) whenever the coefficient of ω i in λ is zero. For 1 ≤ i ≤ g −
3, this follows from Lemma 4.4. Then if 2 ≤ c ≤ d − ω g , ω g − , and ω g − in λ is non-zero.But if c = 1, then we also need to show that λ − α g − / ∈ W p ( g, , c = d − λ − α g / ∈ W p ( g, d − , d = 2 (which corresponds to p = 5)we have both c = 1 and c = d −
1, so that we need to use both arguments.(Case c = 1.) Assume for a contradiction that λ − α g − ∈ W p ( g, , σ (cid:48)(cid:48)(cid:48) of type (1 ,
1) with w ( σ (cid:48)(cid:48)(cid:48) ) = λ − α g − = ( d − g − (cid:88) i =1 ε i + ( d − ε g − + ( d − ε g . By Lemma 4.5 σ (cid:48)(cid:48)(cid:48) must color all but the ( g − λ − α g − / ∈ W p ( g, , c = d − λ − α g ∈ W p ( g, d − , σ of type ( d − ,
1) with(11) w (˜ σ ) = λ − α g = ( d − g − (cid:88) i =1 ε i + ( d − ε g − − ε g . By the exact same reasoning as when showing that a (cid:48)(cid:48) g = c for the coloring σ (cid:48)(cid:48) in Step 2 (Case i = g ), we have ˜ a g = c . Hence ˜ a g = d − c = d −
1) and so ˜ b g = 0 by smallness. But then the coefficient of ε g in w (˜ σ )must be d − − ˜ a g − b g = 0 (mod 2 d ) which contradicts (11). This shows that λ − α g / ∈ W p ( g, d − , Case IV . Let σ be the coloring on the right of Figure 3, then w ( σ ) = ( d − g − (cid:88) i =1 ε i + ( d − g (cid:88) i = g − ε i = ( d − ω g + ω g − . We shall show that λ = w ( σ ) = ( d − ω g + ω g − . In Step 2, we must show that w ( σ ) + α i / ∈ W p ( g, c,
0) for all 1 ≤ i ≤ g . For i ≤ g − i = g , this follows from Lemma 4.4. For i = g −
2, it is provedby contradiction, as follows. Assume that w ( σ ) + α g − ∈ W p ( g, , σ (cid:48) of type (0 ,
1) with w ( σ (cid:48) ) = w ( σ ) + α g − = ( d − g − (cid:88) i =1 ε i + ( d − ε g − + ( d − ε g . Note that σ (cid:48) must color all but two of the lollipops with zeros by Lemma 4.5 andso σ (cid:48) contradicts Lemma 4.7(ii). This shows that w ( σ ) + α g − / ∈ W p ( g, , N APPLICATION OF TQFT TO MODULAR REPRESENTATION THEORY 17
For i = g −
1, the proof is similar: We have that w ( σ ) + α g − = ( d − g − (cid:88) i =1 ε i + ( d − ε g − + ( d − ε g − + ( d − ε g and the two lollipop argument shows that w ( σ ) + α g − / ∈ W p ( g, , λ = w ( σ ) = ( d − ω g + ω g − . Concerning Step 3, we have that λ − α i / ∈ W p ( g, ,
1) for i ≤ g − i = g − g − d = 2 (hence p = 5) in which case we also need to show that λ − α g / ∈ W ( g, , Further Comments
We elaborate on Remark 1.3. Let C denote the closure of the fundamentalalcove. (See e.g. [H2, 3.5].) A dominant weight λ lies in C iff (cid:104) λ + ρ, β ∨ (cid:105) ≤ p , where ρ is sum of the fundamental weights, and β is the highest short root (thus β ∨ is the highest root of the dual root system). By [B, Planche II], we have β ∨ = α ∨ + 2 α ∨ + . . . + 2 α ∨ g . Using (cid:104) ω i , α ∨ j (cid:105) = δ ij , one can check that all the weights λ arising in Theorem 1.1lie outside of C ( i.e., one has (cid:104) λ + ρ, β ∨ (cid:105) > p ) except for the weight λ = ( d − ω in rank g = 3 (for which (cid:104) λ + ρ, β ∨ (cid:105) = p ). In fact, as soon as the rank g ≥
5, oneuniformly has (cid:104) λ + ρ, β ∨ (cid:105) = p + 2 g − g .On the other hand, Theorem 1.1 also holds in rank g = 2 for those weights whereit makes sense ( cf. Remark 1.7(i)). It turns out that those weights λ in rank g = 2all lie in C . Hence the dimension of L p ( λ ) is given by the Weyl character formula,as it is a well-known consequence of the linkage principle that for dominant weights λ in C , the simple module L p ( λ ) is isomorphic to the Weyl module ∆ p ( λ ) (see e.g. [H2, 3.6]). We have checked that indeed for g = 2 our dimension formulae (seeAppendix B.2) agree with the Weyl character formula.A further consistency check is possible in rank g = 3. In this case, although ourweights (with one exception) lie outside of C , the distance to C is not too big(one has (cid:104) λ + ρ, β ∨ (cid:105) ≤ p + 2) and one can use the Jantzen Sum Formula to computethe formal character of L p ( λ ). (See [J, II.8] and references therein. See also thesummary in [H2, 3.9].) Here is the answer in the case ε = 0 . We have λ = λ p (3 , c,
0) = c ω + ( d − − c ) ω . One finds that L p ( λ ) is equal to the Weyl module ∆ p ( λ ) if c ∈ { , } , but for c ≥ → ∆ p ( µ ) → ∆ p ( λ ) → L p ( λ ) → µ = λ − ω = ( c − ω + ( d − − c ) ω . In particular dim L p ( λ ) can be computed from the Weyl character formula asdim L p ( λ ) = dim ∆ p ( λ ) − dim ∆ p ( µ )and we have checked that our dimension formulae (see Appendix B.3) agree withthis.In rank g ≥
4, we have not attempted to compute L p ( λ ) with the Jantzen SumFormula. Note that it is easy to see that L p ( λ ) can only very rarely be equal tothe Weyl module ∆ p ( λ ) for a weight λ that arises in Theorem 1.1, because of thefollowing observations (the first two of which imply the third). • By [GM5, Corollary 2.10], dim L p ( λ p ( g, c, ε )), with g ≥ c , and ε heldfixed and viewed as a function of p is polynomial of degree 3 g − • By the Weyl character formula [FH, 24.20], dim ∆ p ( λ p ( g, c, ε )), with g ≥ c , and ε held fixed and viewed as a function of p is polynomial of degree g ( g + 1) / • For each g ≥ c , and ε , there is a integer N ( g, c, ε ) such that for all p ≥ N ( g, c, ε ), dim L p ( λ p ( g, c, ε )) < dim ∆ p ( λ p ( g, c, ε )). Appendix A. Proof of Lemma 3.1
In this appendix, we assume some familiarity with Integral TQFT, in particularwith the results of [GM2, §
3] and [GM5]. See Remark 3.2 above for the correspon-dence between our present notations and those in [GM2] and [GM5]. It is shownin [GM5, Cor. 2.5] that every mapping class f ∈ Γ g, is represented on F p (Σ g (2 c ))by a matrix of the form (cid:18) (cid:63) (cid:63) (cid:63) (cid:19) with respect to the direct sum decomposition (as vector spaces) F p (Σ g (2 c )) = F p ( g, c, ⊕ F p ( g, c, . (Here, the top left (cid:63) stands for an element of End F p ( F p ( g, c, (cid:63) s.) Lemma 3.1 is equivalent to the following Lemma A.1.
Any f in the Torelli group I g, is represented by a matrix of theform (cid:18) (cid:63) (cid:19) To prove this, let us review how the coefficients of this matrix can be computed.Throughout this appendix we put h = 1 − ζ p . Recall that Z [ ζ p ] / ( h ) = F p . We usethe basis { ˆ b σ } of F p (Σ g (2 c )); it is the reduction modulo h of the orthogonal lollipopbasis { ˜ b σ } of S p (Σ g (2 c )) constructed in [GM2]. The basis { ˜ b σ } is orthogonal withrespect to the Hopf pairing (( , )) defined in [GM2, § { ˜ b (cid:63)σ } satisfying((˜ b σ , ˜ b (cid:63)σ (cid:48) )) = δ σ,σ (cid:48) . Here ˜ b (cid:63)σ lies in the dual lattice S (cid:93)p (Σ g (2 c )) ⊂ S p (Σ g (2 c )) ⊗ Q ( ζ p ). We will use that(12) ˜ b (cid:63)σ ∼ ˜ b (cid:93)σ , N APPLICATION OF TQFT TO MODULAR REPRESENTATION THEORY 19 where ˜ b (cid:93)σ is defined in [GM2, Cor. 3.4] to be a certain power of h times ˜ b σ , and ∼ means equality up to multiplication by a unit in Z [ ζ p ]. (The power of h dependson σ .)The Hopf pairing is a symmetric Z [ ζ p ]-valued form on S p (Σ g (2 c )) which dependson a choice of Heegaard splitting of S . Given x, y ∈ S p (Σ g (2 c )), their Hopf pairingis computed skein-theoretically as follows. Think of x and y as represented by skeinelements in the handlebody H g . Pick a complementary handlebody H (cid:48) g such that H g ∪ Σ g H (cid:48) g = S . Then(13) (( x, y )) = (cid:104) x ∪ r ( y ) (cid:105) , where the right hand side of (13) is the Kauffman bracket evaluation (at a certainsquare root of ζ p ) of the skein element in S obtained as the union of x in H g and r ( y ) in H (cid:48) g where r is a certain identification of H g with H (cid:48) g (see [GM2, §
3] for amore precise definition).Given now a mapping class f , the ( σ, σ (cid:48) ) matrix coefficient of f acting on S p (Σ g (2 c )) is computed skein-theoretically as the evaluation(14) (cid:104) ˜ b σ ∪ s ∪ r (˜ b (cid:63)σ (cid:48) ) (cid:105) in S = H g ∪ Σ g (Σ g × I) ∪ Σ g H (cid:48) g , where s is a certain skein element in Σ g (2 c ) × I obtained from f in the usual way:there is a banded link L in Σ g, × I so that surgery on L gives the mapping cylinderof f ; one then obtains s by replacing each component of L by a certain skein element ω p (see [BHMV, p. 898]) and placing the resulting skein element in Σ g, × I intoΣ g (2 c ) × I in the standard way. Here L and s are not uniquely determined by f ,but it is shown in [BHMV] that this procedure is well-defined, and gives the correctanswer. (More precisely, it gives the correct answer up to multiplication by a globalprojective factor which is a power of ζ p . Here, we can safely ignore this projectiveambiguity as we are eventually interested in the matrix of f modulo h = 1 − ζ p only.)We are now ready to prove Lemma A.1. The main idea is that if f lies inthe Torelli group I g, , then L and hence s are very special, because the mappingcylinder of f can be obtained by Y -surgery on Σ g, × I [Ha, MM, HM]. The notionof Y -surgery goes back to Matveev [Mat] (who called it Borromean surgery), andthen Goussarov [Gou] and Habiro [Ha] (who called it clasper surgery). We referthe reader to § Y -surgery and alsofor more references to the original papers.The result we need is stated in [HM, Prop. 5.5] and can be formulated as follows.There is a certain 6-component banded link Y in a genus 3 handlebody with thefollowing property. For every f ∈ I g, , there exists an embedding of a finite disjointunion of, say, n copies of the pair ( H , Y ) into Σ g, × I, giving rise to a 6 n -componentbanded link L in Σ g, × I such that the mapping cylinder of f is obtained by surgeryon this banded link L .Cabling each component of Y by ω p gives rise to a skein element Y p in H , andthe skein element s appearing in our computation of matrix coefficients (see (14))is obtained by placing n copies of Y p into Σ g (2 c ) × I. We can view Y p as an element of the free Z [ ζ p ]-module S p (Σ ). The orthogonallollipop basis { ˜ b σ } of S p (Σ ) is indexed by colorings of the form σ = ( a , a , a , b , b , b ).As before, let σ denote the zero coloring. The following lemma is the key to prov-ing Lemma A.1, as it shows that all but two coefficients of Y p in the { ˜ b σ } basis aredivisible by h . Lemma A.2. Y p − ˜ b σ = α ˜ b (1 , , , , , (mod h ) for some α ∈ Z [ ζ p ] .Proof of Lemma A.1 from Lemma A.2. We have that ˜ b σ is represented by theempty link and ˜ b (1 , , , , , is h − times the elementary tripod (see [GM1, Fig. 2 onp. 824]). Thus s = n (cid:88) k =0 h − k s k , where s k is the disjoint union of k elementary tripods embedded in Σ g (2 c ) × I. Thecontribution of h − k s k to the ( σ, σ (cid:48) ) matrix coefficient of f is(15) h − k (cid:104) ˜ b σ ∪ s k ∪ r (˜ b (cid:63)σ (cid:48) ) (cid:105) . For k = 0 this is δ σ,σ (cid:48) , as s is the empty link. For k >
0, a straightforwardapplication of the lollipop lemma [GM1, Thm. 7.1] shows that if σ and σ (cid:48) have thesame parity, then (15) is divisible by h (since a tripod has 3 elementary lollipops).This proves Lemma A.1. (cid:3) Proof of Lemma A.2.
The 6-component banded link Y in H can be described asfollows (see [HM, Fig. 5]). We can number the components of Y as Y , Y , Y , Y (cid:48) , Y (cid:48) , Y (cid:48) such that the following holds. The components ( Y , Y , Y ) lie in a ball B ⊂ H and form zero-framed Borromean rings. For i = 1 , ,
3, the component Y (cid:48) i is azero-framed unknot going once around the i th ‘hole’ of H ; moreover Y (cid:48) i is linkedexactly once with Y i , and unlinked with the two other Y j s.Recall that Y p is obtained by cabling each of these 6 components by ω p . For thelemma, we need to compute the coefficients of the basis vectors ˜ b ( a ,a ,a ,b ,b ,b ) in Y p . (In fact, except for the zero coloring, we only need these coefficients modulo h .) Such a coefficient is given by the Hopf pairing(( Y p , ˜ b (cid:63) ( a ,a ,a ,b ,b ,b ) )) . As explained above, this is computed as the Kauffman bracket of a certain skeinelement in S . Using now the handle slide property of ω p (which is at the basis of theskein-theoretic construction of TQFT [BHMV]), we can compute this by performingsurgery on Y , which gives back S , but transforms the standard embedding of H (cid:48) in S into an embedding ϕ of H (cid:48) in S where the three handles of ϕ ( H (cid:48) ) are linkedas in the Borromean rings. Thus the coefficient of ˜ b ( a ,a ,a ,b ,b ,b ) in Y p is theKauffman bracket evaluation(16) (cid:104) ϕ ( r (˜ b (cid:63) ( a ,a ,a ,b ,b ,b ) )) (cid:105) . For the zero coloring, the basis vector ˜ b (cid:63)σ is represented by the empty link, andso (16) evaluates to 1, as asserted. The following Lemma A.3 shows that (16) isdivisible by h as soon as max( b i ) > a i ) >
1. Thus it only remains tocompute (16) for the colorings with max( a i ) = 1 and all b j = 0. Assume all b j = 0.For a = a = a = 1, there is nothing to prove, while if one of the a i is zero, then N APPLICATION OF TQFT TO MODULAR REPRESENTATION THEORY 21 (16) evaluates to zero (as the two remaining handles are unlinked). This proves thelemma. (cid:3)
For r ∈ Q , we define its floor (cid:98) r (cid:99) to be the largest integer ≤ r , and its roof (cid:100) r (cid:101) to be the smallest integer ≥ r . Lemma A.3.
We have that (16) is divisible by h (cid:98) ( E + E − E ) / (cid:99) where E i = a i + 2 b i ( i = 1 , , ), and w. l. o. g. we may assume E ≥ E ≥ E .Proof. Put B ( a ,a ,a ,b ,b ,b ) = h (cid:100) ( E + E + E ) / (cid:101) ˜ b (cid:63) ( a ,a ,a ,b ,b ,b ) . Then B ( a ,a ,a ,b ,b ,b ) is represented by a linear combination of skein elements in H with coefficients in Z [ ζ p ]. (This follows from (12) and the definition of { ˜ b (cid:93)σ } in[GM2, § β de-note the result of evaluating (16) with B ( a ,a ,a ,b ,b ,b ) in place of ˜ b (cid:63) ( a ,a ,a ,b ,b ,b ) .Since the three lollipops are linked as in the Borromean rings, we can pull the firsttwo of them apart and consider the third as just some skein element in the comple-ment of the first two. In other words, by cutting B ( a ,a ,a ,b ,b ,b ) at the midpointof the edge labelled 2 a , we can write β as the genus two Hopf pairing of two skeinelements, say s and s (cid:48) , in a genus 2 handlebody with one banded point labelled 2 a .Moreover, one of these two skein elements, say s (namely the one containing thefirst two lollipops), is a power of h times the basis element ˜ b (cid:63) ( a ,a ,b ,b , a ) of theorthogonal lollipop basis of S p (Σ (2 a )). Rewriting s (cid:48) also in this basis and usingthe formulae in [GM2, §
3] for the Hopf pairing in the orthogonal lollipop basis, onefinds that β is divisible by h E + E . (Here it is important to observe that both s and s (cid:48) are skein elements without denominators.) Thus (16) is divisible by h −(cid:100) ( E + E + E ) / (cid:101) + E + E = h (cid:98) ( E + E − E ) / (cid:99) . This completes the proof. (cid:3)
Remark A.4.
Other applications of Lemma A.3 include a skein-theoretic con-struction of Ohtsuki’s power series invariant for integral homology 3-spheres, anda Torelli group representation inducing this invariant [M].
Appendix B. Some polynomial formulae for dimensions
In [GM5, Prop. 7.7, Prop. 7.8] we gave residue formulae for D (2 c ) g ( p ) and δ (2 c ) g ( p )valid for g ≥
1. Using Equation (1), one can then express dim L p ( λ ), for the λ thatarise in Theorem 1.1, for specified g as polynomials in p and c using mathematicalsoftware. Below we write down these polynomials in rank g = 2, 3, and 4. Theseformulae hold for p ≥ ≤ c ≤ d − d = ( p − /
2. The firstpolynomial in each rank is the second polynomial with c set to zero. Similarly thefourth polynomial in each rank (except rank 2) is the third polynomial with c setto zero. As noted in Section 5, our formulae in rank g = 2 and also the first one inrank g = 3 agree with Weyl’s character formula. The second formula in rank g = 3agrees with Weyl’s character formula for c = 1, but not for c > B.2.
Rank g = 2 . dim L p (cid:0) ( d − ω (cid:1) = 124 ( p − p ( p + 1)dim L p (cid:0) ( d − c − ω + c ω (cid:1) = 124 ( c + 1)( p + 1)( p − c − p − c )dim L p (cid:0) ( d − c − ω + ( c − ω (cid:1) = 124 c ( p − p − c − p − c − Rank g = 3 . dim L p (cid:0) ( d − ω (cid:1) = 12880 ( p − p ( p + 1) ( p + 2)( p + 3)dim L p (cid:0) ( d − c − ω + c ω (cid:1) = 12880 ( p − c − (cid:16) p (2 c + 1) + p (4 c + 4 c + 7)+ p ( − c − c + 28 c + 17) + p (6 c + 12 c − c − c + 17)+6 p ( − c − c + c + 1) + 6 c ( c + 2 c − c − (cid:17) dim L p (cid:0) ( d − c − ω + ( c − ω + ω (cid:1) = 12880 ( p − p + 1)( p − c − (cid:16) p (2 c + 1)+ p (4 c + 4 c −
5) + 6 p ( − c − c + c + 1) + 6 c ( c + 2 c − c − (cid:17) dim L p (cid:0) ( d − ω (cid:1) = 12880 ( p − p − p − p ( p + 1)B.4. Rank g = 4 . dim L p (cid:0) ( d − ω (cid:1) = 1120960 ( p − p ( p + 1) (cid:0) p + 37 p + 142 p + 36 (cid:1) dim L p (cid:0) ( d − c − ω + c ω (cid:1) = 1120960 ( p + 1)( p − c − (cid:16) p (2 c + 1) + 2 p c (2 c + 1)+ p ( − c − c + 18 c + 37) + 2 p c ( − c − c + 22 c + 38)+ p (18 c + 57 c − c − c + 22 c + 142) − p c ( c + 6 c + c − c − c + 24)+3 p (2 c + 12 c + 5 c − c − c + 32 c + 12) − c ( c + 3 c − c − c + 4 c + 12) (cid:17) dim L p (cid:0) ( d − c − ω + ( c − ω + ω (cid:1) = 1120960 ( p − p − c − (cid:16) p (2 c + 1)+ p (4 c + 6 c + 2) + p ( − c − c + 26 c − − p (6 c + 15 c − c − c + 13)+ p (18 c + 33 c − c − c + 164 c + 23) − p ( c − c − c + 33 c + 16 c − p ( − c + 75 c + 30 c − c − c + 36) − c ( c + 3 c − c − c + 4 c + 12) (cid:17) dim L p (cid:0) ( d − ω + ω (cid:1) = 1120960 ( p − p − p − p ( p + 1) ( p + 2)( p + 3) N APPLICATION OF TQFT TO MODULAR REPRESENTATION THEORY 23
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Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803,USA
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