TTRIPLE CLASP FORMULAS FOR C WEBS
ELIJAH BODISHA
BSTRACT . Using the light ladder basis for Kuperberg’s C webs, we derive triple clasp for-mulas for idempotents projecting to the top summand in each tensor product of fundamentalrepresentations. We then find explicit formulas for the coefficients occurring in the clasps,by computing these coefficients as local intersection forms. Our formulas provide further ev-idence for Elias’s clasp conjecture, which was given for type A webs, and suggests how togeneralize the conjecture to non-simply laced types. C ONTENTS
0. Introduction 10.1. Applications 40.2. Structure of the Paper 50.3. Acknowledgements 51. Clasps 51.1. Recollection of some notation 51.2. Neutral coefficient 71.3. Definition and basic properties 71.4. Intersection forms and triple clasp formulas 111.5. Relation to clasp conjecture 132. Deriving the recursive formulas 153. Solving the recursive formulas 28References 420. I
NTRODUCTION
Let
T L be the strict pivotal C ( q ) -linear category generated by one self dual object ofdimension − [2] q . It is well known that T L is equivalent to the category of finite dimen-sional (type I ) representations of U q ( sl ( C ) [12]. A well known application of this resultis Wenzl’s recursive formula for the idempotent in End U q ( sl ( C ) ( L ( (cid:36) ) ⊗ n ) which projects to L ( n(cid:36) ) = S n ( L ( (cid:36) )) [14]. We can express this formula using the usual graphical calculus for T L by using an n labelled oval to represent the projector. a r X i v : . [ m a t h . R T ] F e b ELIJAH BODISH (0.1) n n − n − n − n −
1+ [ n ][ n + 1] The category
T L was generalized by Kuperberg in [8] to describe the representation the-ory of U q ( g ) , when g is sl , sp , or g . The only category we consider in this article is the sp web category, which we denote by D C . We recall the definition here. Definition 0.1.
The category D C is strict pivotal Z [ q, q − ] (cid:20) (cid:21) linear category generated bytwo self dual objects, with morphisms generated by(0.2)subject to the relations(0.3) = − [6][2][3] = [6][5][3][2] = − [2] = 0 = 0 (0.4) = + 1[2] − We will refer to these relations as the circle relations, the bigon, monogon, and trigonrelation, and the H ≡ I relation.Whenever k is a field and q ∈ k so that q + q − (cid:54) = 0 , we can specialize the diagrammaticcategory D C to k . In [8] Kuperberg proves that the category of finite dimensional (type I )representations of U q ( sp ) is equivalent to the Karoubi envelope of C ( q ) ⊗ D C .In [6] Kim gives single clasp formulas for the sp projectors (and double and quadrupleclasp formulas for sl , among other things). Given a tensor product of fundamental repre-sentations so that (cid:36) occurs a times and (cid:36) occurs b times, the irreducible module L ( a, b ) occurs as a summand of this tensor product with multiplicity one. Kim’s formulas handlethe case L ( a, and L (0 , b ) . The main result of this article is a recursive triple clasp formulafor the idempotent projecting to L ( a, b ) in the case of sp . RIPLE CLASP FORMULAS FOR C WEBS 3
Theorem 0.2.
Let an oval with ( n, m ) label denote the clasp of weight ( n, m ) , then (0.5) a + 1 , b = a, b − κ ( − , a,b ) a − , b + 1 a, ba, b − κ (1 , − a,b ) a + 1 , b − a, ba, b − κ ( − , a,b ) a − , ba, ba, b where (0.6) κ ( − , a,b ) = − [ a + 1][ a ] (0.7) κ (1 , − a,b ) = [ a + 2 b + 3][2 b + 2][ a + 2 b + 2][2 b ] (0.8) κ ( − , a,b ) = − [2 a + 2 b + 4][ a + 2 b + 3][ a + 1][2 a + 2 b + 2][ a + 2 b + 2][ a ] , and (0.9) a, b + 1 = a, b − κ (0 , a,b ) a, ba, ba, b − κ ( − , a,b ) a − , b + 1 a, ba, b − κ (2 , − a,b ) a + 2 , b − a, ba, b − κ (0 , − a,b ) a, b − a, ba, b where (0.10) κ (0 , a,b ) = [ a + 2][ a + 2 b + 4][2][ a ][ a + 2 b + 2] (0.11) κ ( − , a,b ) = − [ a + 1][2 a + 2 b + 4][ a − a + 2 b + 2] (0.12) κ (2 , − a,b ) = − [2 b + 2][2 b ] (0.13) κ (0 , − a,b ) = [2 a + 2 b + 4][ a + 2 b + 3][2 b + 2][2 a + 2 b + 2][ a + 2 b + 1][2 b ] . Thanks to our construction of a double ladder basis for homomorphism spaces in D C [2], and the ideas from Elias’s work on clasps for type A webs [4], we can argue that such arecursive formula exists without knowing the κ ’s explicitly. The recursive nature of the claspformula implies recursive relations among the κ ’s. Our theorem then follows from showingthat these relations force the κ ’s to be the values specified in (0.2). ELIJAH BODISH
Applications.
Let k be a field and let q ∈ k so that q + q − (cid:54) = 0 . We write k ⊗ U Z q ( sp ) to denote the Lustig divided powers form of the quantum group and Tilt ( k ⊗ U Z q ( sp )) todenote the monoidal category of tilting modules. In [2] we proved that Tilt ( k ⊗ U Z q ( sp )) isequivalent to the Karoubi envelope of k ⊗ D C .Suppose that k = C and q = e iπ/(cid:96) for some integer (cid:96) > . There is a well known con-struction of a C -linear fusion category, as the quotient of Tilt ( k ⊗ U Z q ( sp )) by the ideal ofnegligible morphisms [1]. One may ask for a generators and relations presentation of thesequotient categories. Since we have a presentation of the category of tilting modules, it re-mains to find relations which generate the ideal of negligible morphisms, denoted N . Ingeneral the ideal of negligible morphisms is not the monoidal ideal generated by the iden-tity morphisms of all negligible objects. However, the hypothesis (cid:96) > guarantees that thenon-negligible tilting modules are such that dim Hom( T k ( λ ) , T k ( µ )) = δ λ,µ . In this case it isknown that the ideal of negligible morphisms is generated by the identity morphisms of allnegligible objects.The choice of (cid:96) gives rise to an affine Weyl group action on the weight lattice X . Extendingthis action to R ⊗ X allows us to partition the dominant weights by their relationship to al-coves in R ⊗ X . The weights in the interior of the lowest alcove in the cone − ρ + R ⊗ X + , areexactly the highest weights of the indecomposable non-negligible tilting modules. Further-more, every indecomposable negligible tilting module is a direct summand of a tensor prod-uct of some tilting module and an indecomposable tilting module with highest weight on theupper closure of the lowest alcove. Thus, the identity morphisms of all indecomposable neg-ligible tilting modules are contained in the monoidal ideal generated by the indecomposabletilting modules on the upper closure of the lowest alcove.The tilting modules on the upper closure of the lowest alcove are(0.14) T k (cid:18) k, (cid:96) − − k (cid:19) , for k = 0 , . . . , (cid:96) − when (cid:96) is odd, and(0.15) T k (cid:18) k, (cid:96) − − k (cid:19) , for k = 0 , . . . , (cid:96) − when (cid:96) is even. We may conclude that there is a monoidal functor(0.16) k ⊗ D C −→ Tilt ( k ⊗ U Z q ( sp )) / N , with kernel the monoidal ideal generated by the λ clasps for all λ on the upper closure of thelowest alcove. Furthermore, the induced functor(0.17) Kar ( k ⊗ D C / ker) −→ Tilt ( k ⊗ U Z q ( sp )) / N is an equivalence. Example 0.3. If (cid:96) = 5 , then the ideal of negligible morphisms is generated by id T k (0 , and id T k (2 , . The quotient category is equivalent to Rep ( Z / , where the sign representationcorresponds to T k (1 , . Example 0.4. If (cid:96) = 6 , then the ideal of negligible morphisms is generated by id T k (1 , and id T k (0 , . Thus, the quotient category is equivalent to Vec C . Example 0.5.
It is a pleasant exercise to use this approach (along with the diagrammaticcategory
T L ) to show that if (cid:96) = 8 , the negligible quotient of
Tilt ( k ⊗ U Z q ( sp )) is equivalent RIPLE CLASP FORMULAS FOR C WEBS 5 to Tilt ( C ( e iπ/ ) ⊗ U Z q ( sl )) / N . Note that if q = e iπ/ , then(0.18) − [6] q [2] q [3] q = − [2] q . When (cid:96) > is even, the negligible quotient of the category of tilting modules is a modulartensor category [10]. One reason this is of interest is that a modular tensor category gives riseto a dimensional TQFT, and therefore a three manifold invariant [9]. In order to computethis invariant explicitly using our graphical calculus for morphisms, it is essential to haveformulas for the non-negligible clasps. The interested reader could consult [7] to see howthis is carried out in the case of the diagrammatic category T L .With applications to modular representation theory in mind, there has been some workon writing the idempotents projecting to all indecomposable tilting modules in terms ofthe double ladder basis for SL ( F p ) [3] [13]. Such idempotents have been referred to as p -Jones-Wenzl’s. Since nobody knows the characters of tilting modules for rank two groups inpositive characteristic, this question is not appropriate in our setting. However, the tiltingcharacters are known for the quantum group at a root of unity [11]. A key first step indetermining the formulas for the p -Jones-Wenzl’s is to argue that if the characteristic p tiltingmodule is simple, then the characteristic zero clasp can be reduced modulo a maximal idealto obtain the projector in characteristic p . We are careful to point out how this works in thecase of D C (1.14), but do not explore the topic further in this article.Lastly, we mention that the analogous problem of constructing a double ladder basis andfinding the clasp formulas for g is still open, but is work in progress with Haihan Wu.0.2. Structure of the Paper.
Chapter : We recall some facts about the double ladders basisfor sp webs and deduce the triple clasp formula. Then we state the main theorem whichgives explicit formulas for the local intersection forms, which are also the coefficients in theclasp formula. Chapter : The recursive formulas for the local intersection forms are stated,and then derived via diagrammatic calculations. Chapter : Having found the recursions forlocal intersection forms and their initial conditions, we prove the main theorem by showingthe conjectured formulas satisfy the recursion.0.3. Acknowledgements.
I want to thank Ben Elias for many helpful discussions aboutidempotents. 1. C
LASPS
Recollection of some notation.
We follow the notation and conventions of [2].Let k be a field and let q ∈ k × such that [2] q (cid:54) = 0 . By k ⊗ D C we mean the C web categorybase changed to k . The objects in k ⊗ D C are words w in the alphabet { , } .Let U Z q ( sp ) denote Lusztig’s divided power form of the quantum group for sp . Thisalgebra has Weyl modules V Z ( λ ) for all λ ∈ X + . We will write V k ( λ ) for k ⊗ V Z ( λ ) . In thespecial case when k = C and q = 1 we will write L ( λ ) in place of V k ( λ ) . The L ( λ ) are themore familiar finite dimensional simple modules of highest weight λ for sp ( C ) .Our hypothesis that [2] q (cid:54) = 0 guarantees that the Weyl modules V k ( (cid:36) ) and V k ( (cid:36) ) areindecomposable tilting modules. The category Fund ( k ⊗ U Z q ( sp )) , is the full subcategory of Rep ( k ⊗ U Z q ( sp )) monoidally generated by these fundamental tilting modules.There is a functor eval : k ⊗ D C −→ Fund ( k ⊗ U Z q ( sp )) , such that eval (1) = V k ( (cid:36) ) and eval (2) = V k ( (cid:36) ) . The functor eval is is full, faithful, and essentially surjective. ELIJAH BODISH
Recall that the module V k (1) = V k ( (cid:36) ) has basis { v (1 , , v ( − , = F s v (1 , , v (1 , − = F t F s v (1 , , v ( − , = F s F t F s v (1 , } and the module V k (2) = V k ( (cid:36) ) has basis { v (0 , , v (2 , − = F t v (0 , , v (0 , = F s F t v (0 , , v ( − , = F (2) s F t v (0 , , v (0 , − = F t F (2) s F t v (0 , } . For w = ( w , . . . , w n ) , a word in the alphabet and , we define(1.1) v w, + ∈ V k ( w ) = V k ( w ) ⊗ . . . ⊗ V k ( w n ) by(1.2) v w, + = v w ⊗ v w ⊗ . . . ⊗ v w n where v = v (1 , and v = v (0 , . Also, for any sequence of weights ( ν , . . . , ν n ) , where ν i ∈ wt V k ( w i ) , we define(1.3) v w, ( ν ,...,ν n ) = v ν ⊗ . . . ⊗ v ν n ∈ V k ( w ) . Definition 1.1.
The subsequence basis is the set of vectors v w, ( ν ,...,ν n ) ∈ V k ( w ) , where ( ν , . . . , ν n ) runs over all weight sequences such that ν i ∈ wt V Z ( w i ) .In sections . and . of [2] we defined elementary light ladders , neutral ladders , light ladders , upside down light ladders , and double ladders . In brief, we associate an elementary light ladderdiagram in D C to each weight space in a fundamental representation: µ (cid:55)→ L µ . These dia-grams along are the building blocks of the light ladder diagrams. There is a duality D on thediagrammatic category, which takes a diagram and flips it upside down. We define upsidedown light ladders as the image, under D , of usual light ladders. The double ladder basis LL is then constructed by composing all light ladders with all upside down light ladders. Inorder make light ladder diagrams out of elementary light ladders, we also require neutraldiagrams which allow us to shuffle words in and .The following lemma appears in [2]. Lemma 1.2.
Let w and u be words in and and let N : V k ( w ) → V k ( u ) be a neutral map (i.e. theimage of a neutral ladder diagram under eval ). Then N ( v w, + ) = ξv u, + , where ξ is some invertibleelement in k . Furthermore, if ( µ , . . . , µ n ) is a sequence of weights so that µ i ∈ wt V k ( w i ) , and N ( v w, ( µ ,...,µ n ) ) has a nonzero coefficient for v u, + after being written in the subsequence basis, then v w, ( µ ,...,µ n ) = v w, + . We claim that a closer analysis shows that ξ = 1 . Since neutral maps are built from identitymaps, N , and N we just need to check that N ( v , + ) = v , + and N ( v , + ) = v , + .But from the calculations in [2] we find(1.4) N ( v , + ) = L ( − , ⊗ id( v ⊗ D ( L ( − , )( v )) = v , + and(1.5) N ( v , + ) = id ⊗ L ( − , ( D ( L ( − , )( v ) ⊗ v ) = v , + . Remark . In [2] we rescaled the generating trivalent vertex in the B spider from [8]. Explic-itly our trivalent vertex is equal to (cid:112) [2] times Kuperberg’s trivalent generator. One reasonour choice may be preferable to the original, is that using Kuperberg’s trivalent vertex theneutral maps have ξ = [2] instead of ξ = 1 . RIPLE CLASP FORMULAS FOR C WEBS 7
Definition 1.4.
Let w = ( w , . . . , w n ) with w i ∈ { , } . A sequence ( µ , . . . , µ n ) where µ i ∈ wt( L ( w i )) is a dominant weight subsequence of w if(1) µ is dominant.(2) L ( µ + . . . + µ i − + µ i ) is a summand of L ( µ + . . . + µ i − ) ⊗ L ( w i ) If we write E ( w ) for the set of all dominant weight subsequences of w , then(1.6) L ( w ) := L ( w ) ⊗ . . . ⊗ L ( w n ) ∼ = (cid:77) ( µ ,...,µ n ) ∈ E ( w ) L ( µ + . . . + µ n ) . In particular, if we denote the multiplicity of L ( λ ) as a summand of L ( w ) by [ L ( λ ) : L ( w )] and write(1.7) E ( w, λ ) := { ( µ , . . . , µ n ) ∈ E ( w ) : µ + . . . + µ n = λ } , then(1.8) [ L ( λ ) : L ( w )] = E ( w, λ ) . Neutral coefficient.
When defining the double ladder basis, we first fix, for each dom-inant weight λ , a choice of object x λ ∈ D C ) satisfying wt( x λ ) = λ . We also fix a choice ofneutral diagram from x to x λ for all objects x ∈ D C such that wt( x ) = λ .Suppose we have made such a choice. We write LL to denote the associated double ladderbasis. Let λ ∈ X + and let w and x be such that wt( w ) = λ = wt( x ) . In Hom D C ( w, x ) there isa unique double ladder diagram, I xw , which (after applying eval ) sends v w, + to v x, + (modulo”lower terms” in the subsequence basis)[2]. We call all other double ladders in Hom D C ( w, x ) strictly downward . If L is a strictly downward ladder, then we will say that D ( L ) is strictlyupward .Suppose we made another choice of x (cid:48) λ for each dominant weight λ (along with choices ofall the necessary neutral maps) and then constructed a double ladder basis LL (cid:48) . Again, thereis a unique double ladder diagram I (cid:48) xw ∈ Hom D C ( w, x ) which maps v w, + to v x, + (modulo”lower terms” in the subsequence basis). Since LL (cid:48) is a basis, we can express I xw as a linearcombination of diagrams in LL (cid:48) (1.9) I xw = c · I (cid:48) xw + (cid:88) ”lower terms” . Where ”lower terms” means a linear combination of strictly downward double ladders.Looking at how both sides of (1.9) act on v w, + we deduce that c = 1 . Definition 1.5.
Let f be an arbitrary map f : w → x . We define the neutral coefficient of f to be the coefficient of I xw when f is expressed in the basis LL . The argument given aboveensures that this coefficient is independent of any choices that are made in the double ladderalgorithm.1.3. Definition and basic properties.
Our exposition is based on [4]. The reader may findchapter eleven from [5] helpful also.
Definition 1.6.
We say that a morphism from w to x is a clasp , if it is killed by postcompositionwith any strictly downward ladder and has neutral coefficient . If wt( w ) = λ = wt( x ) , thenwe may call such a map a λ -clasp. Proposition 1.7.
If a clasp exists then it is unique, and it is also characterized as the map with neutralcoefficient which is killed by precomposition with any strictly upward ladder. The composition of aclasp with a neutral ladder is a clasp (so if any λ -clasp exists, then all λ -clasps exist), the compositionof two clasps is a clasp, and clasps are preserved by D . ELIJAH BODISH
Proof.
See [4] Proposition . (cid:3) We will depict λ -clasps as ovals labelled by λ with source w and target x .(1.10) λwx The proposition (1.7) says that the composition of a clasp with a neutral ladder is a clasp,we will refer to this as neutral absorption , depicted diagrammatically as follows.(1.11) λwx = λwx We also observe that the proposition (1.7) says the composition of two clasps is a clasp,which is what we will call clasp absorption . This is expressed diagrammatically as follows.(1.12) λwxµ = λwx Finally, note that by definition, post-composing a clasp with a non-identity elementarylight ladder results in zero. We will refer to this phenomenon as clasp orthogonality , and it canbe expressed diagrammatically as follows.(1.13) λwxL ν = 0 Remark . The λ clasps give a compatible system of idempotents [4], and therefore representsan object in the Karoubi envelope of k ⊗ D C . This object is a common summand of theobjects w so that wt( w ) = λ . RIPLE CLASP FORMULAS FOR C WEBS 9
Definition 1.9.
When the λ clasp exists, we write λ to denote the object in Kar ( k ⊗ D C ) .Given an idempotent e ∈ End k ⊗D C ( w ) we get an object ( w, e ) in Kar ( k ⊗ D C ) . Also, if ( x, f ) is another object in the Karoubi envelope we have(1.14) Hom
Kar ( k ⊗D C ) (( w, e ) , ( x, f )) = f Hom k ⊗D C ( w, x ) e. Proposition 1.10.
Suppose the λ -clasps exist. Let w and u be objects in k ⊗ D C . Fix y such that wt( y ) = λ . Suppose that y = y . . . y n and w = w . . . w m . The double ladders (1.15) LL uyw := (cid:91) χ ∈ X + ( µ ,...,µ n ,µ (cid:48) ,...,µ (cid:48) m ) ∈ E ( yw,χ )( ν ,...,ν l ) ∈ E ( u,χ ) LL u, ( ν ,...,ν l ) yw, ( µ ,...,µ n ,µ (cid:48) ,...,µ (cid:48) m ) are a basis for Hom k ⊗D C ( y ⊗ w, u ) .Consider the following subset of LL uyw : (1.16) LL uλw := { D ∈ LL u, ( ν ,...,ν l ) yw, ( µ ,...,µ n ,µ (cid:48) ,...,µ (cid:48) m ) : µ , . . . , µ n ∈ X + } After precomposing with the λ clasp (tensored with an identity map), the set LL uλw projects to a basisof Hom
Kar ( k ⊗D C ) ( λ ⊗ w, u ) .Proof. We follow the proof of Proposition . in [4]. Since clasps are orthogonal to strictlydownward ladders, if any µ i / ∈ X + , then precomposing LL u, ( ν ,...,ν l ) yw, ( µ ,...,µ n ,µ (cid:48) ,...,µ (cid:48) m ) with the λ clasp will result in zero (this follows from the inductive, tiered construction of light ladders).Thus, LL uλw projects to a spanning set after precomposing with the λ clasp.The desired result follows if we can argue that LL uλw ◦ ( λ ⊗ id w ) is linearly independent.Since the λ clasp sends v y, + to v y, + (modulo ”lower terms” in the subsequence basis) wecan repeat the argument about linear independence of double ladders from Theorem . in[2]. (cid:3) Corollary 1.11.
Suppose the λ and χ clasps both exist. Then Hom
Kar ( k ⊗D C ) ( λ, χ ) is spanned bythe identity if λ = χ and is zero otherwise.Proof. See [4] Corollary . . (cid:3) Lemma 1.12.
The λ clasp exists in k ⊗ D C if and only if V k ( λ ) is a direct summand of V k ( x λ ) .Proof. Suppose that the λ clasp does exist. Consider the idempotent e λ ∈ End( V k ( x λ )) whichis the image under eval of the λ clasp in End k ⊗ C ( x λ ) . The map e λ projects to a direct sum-mand of V k ( x λ ) , by corollary (1.11) the summand has endomorphism ring k · id . Since thelambda clasp preserves the λ weight space, the object im( e λ ) has a non-zero lambda weightspace. An object with a one-dimensional λ weight space and a local endomorphism ringmust be the indecomposable tilting module of highest weight λ . But since the endomor-phism ring of a tilting module is k · id if and only if the indecomposable tilting module is anirreducible Weyl module, it follows that im( e λ ) ∼ = V k ( λ ) .Suppose V k ( λ ) is a summand of V k ( x λ ) (which is always the case when k = C and q = 1 ).Then there is an idempotent e λ ∈ V k ( x λ ) which projects to V k ( λ ) . We can write e λ in termsof the double ladder basis LL , and the linear combination of diagrams will be a clasp in k ⊗ D C . (cid:3) Remark . Since the finite dimensional representations of sp ( C ) are completely reducible,it follows that λ clasps exist for all λ ∈ X + . In more generality, if K is any field and q istranscendental, then λ clasps exist for all λ . Remark . We argue that if V k ( λ ) is a direct summand of V k ( x λ ) , i.e. the k -clasp exists,then the characteristic zero clasp can be used to compute the k clasp.Let A = Z [ q, q − ] (cid:20) q (cid:21) , when we say“all fields” we mean all pairs k and q ∈ k so that q + q − (cid:54) = 0 . Any quotient of A by a maximal ideal will give such a pair.From [8] we know that the set B of non-elliptic webs spans D C over A , and we knowfrom [2] that LL is linearly independent over all fields k . It follows that the coefficients ofa linear dependence among double ladders over A must all be contained in every maximalideal of A . But J ( A ) = 0 , so LL is linearly independent over A . Furthermore, we establishedin [2] that the sets B and LL both give bases of k ⊗ D C for all fields k .Fix objects w, u ∈ D C . Consider the matrix which expresses a double ladder in terms ofthe spanning set B (1.17) A uw : A LL uw −→ A B uw = Hom D C ( w, u ) . This matrix is an isomorphism over k , for all fields k , so the determinant is not contained inany maximal ideal in A . Thus, det A uw is a unit in A , and A uw is invertible over A . We mayconclude that LL spans D C over A .Let O be a complete discrete valuation ring, which is an A algebra, such that O / m = k Assume that the field K = Frac ( O ) is characteristic zero and q ∈ K is transcendental.Suppose that V k ( λ ) is a summand of V k ( x λ ) , i.e. there is a clasp e k λ ∈ End k ⊗D C ( x λ ) . Theendomorphism e k λ is an idempotent so it can be lifted to e O λ ∈ End
O⊗D C ( x λ ) . Since LL is abasis of D C over A , it follows that LL is a basis of the category O ⊗ D C . Therefore, e O λ = x id + (cid:88) LL ’s factoring through µ < λ. Since e O λ specializes to e k λ , which in turn sends v + to v + , there is some m ∈ m so that x = 1 − m .Then, since e O λ is an idempotent we have x id + (cid:88) LL (cid:48) s factoring through µ < λ = (cid:16) x id + (cid:88) LL (cid:48) s factoring through µ < λ (cid:17) . Lemma . Suppose L and L are two double ladder diagrams so that the top of L agrees with thebottom of L . If L factors through µ and L factors through µ , then when we write L L in thedouble ladder basis each double ladder factors through µ with µ ≤ min { µ , µ } .Proof. This follows from the observation that putting a light ladder with target µ on top ofan upside down light ladder with source µ and then expressing as a sum of double ladders,each term must factor through a weight less than or equal to min { µ , µ } . (cid:3) Thanks to the lemma, we deduce that x = x . Thus, − m = (1 − m ) = 1 − m + m ,which implies that m = m . Since m ∈ m , we may conclude that m = 0 .From the fact that LL is a basis over O it follows that the homomorphism spaces in O⊗D C are free and finitely generated O -modules. Thus, the O module(1.18) e O λ Hom
O⊗D C ( x λ ) e O λ is a finitely generated projective O module. Since O is local, one can use Nakayama’s lemmato show that projective and finitely generated implies free of finite rank. A consequence is RIPLE CLASP FORMULAS FOR C WEBS 11 the equality(1.19) rk O e O λ End
O⊗D C ( x λ ) e O λ = dim k e k λ End k ⊗D C ( x λ ) e k λ . We know e k λ End k ⊗D C ( x λ ) e k λ = k · e k λ , so we may deduce that e O λ Hom
O⊗D C ( x λ ) e O λ = O · e O λ .On the other hand, we know there is a characteristic zero clasp, e Kλ ∈ End K ⊗D C ( x λ ) .Using that e Kλ has neutral coefficient one and is orthogonal to all strictly lower light ladderswe may conclude that, e O λ e Kλ e O λ = e Kλ so e Kλ ∈ e O λ End K ⊗D C ( x λ ) e O λ = Ke O λ . By comparingneutral coefficients we see that e Kλ = e O λ .Over K the λ clasp exists for all λ ∈ X + , so to compute e Kλ we are free to use the recursionfrom our main theorem to expand this clasp in terms of the basis LL . The argument we justsketched implies that the coefficients, of the double ladders, in the expanded clasp actuallylie in O . So we can reduce e Kλ modulo a maximal ideal to obtain e k λ .1.4. Intersection forms and triple clasp formulas.
Let a ∈ { , } . If k = C and q = 1 , weknow that eval ( λ ⊗ a ) = L ( λ ) ⊗ L ( (cid:36) a ) decomposes as described by (1 . a ) and (1 . b ) in [2]. Definition 1.16.
Let λ ∈ X + and let µ ∈ wt( L ( ω a )) , for a ∈ { , } , so that λ + µ ∈ X + .Suppose the λ and λ + µ clasps exist. There is an elementary ladder L µ which induces a map λ ⊗ a → λ + µ . We denote this map by E λ,µ and depict it diagrammatically by(1.20) E λ,µ = L µ λ λ + µ a If the λ and λ + µ clasps exist, then from the proposition (1.10), along with the orthogonalityand absorption properties of clasps, it follows that E λ,µ is a basis for Hom
Kar k ⊗D C ( λ ⊗ a, µ ) . Definition 1.17.
The map K λ,µ := E λ,µ ◦ D E λ,µ is an endomorphism of λ + µ , and thisendomorphism space is spanned by the identity map. We define the local intersection form κ λ,µ to be the coefficient of the identity in E λ,µ ◦ D E λ,µ . (1.21) K λ,µ = λλλ + µλ + µ D ( L µ ) L µ = κ λ,µ λ + µ If the local intersection form κ λ,µ is nonzero, then κ λ,µ D E λ,µ ◦ E λ,µ will be an idempotentin End
Kar ( k ⊗D C ) ( λ ⊗ a ) which projects to λ + µ . If the local intersection form is zero, then ( λ + µ ) is not a summand of λ ⊗ a . Proposition 1.18.
Suppose that the λ clasp exists, and that the λ + µ clasps exist for all µ ∈ wt L ( (cid:36) a ) − { (cid:36) a } such that L ( λ + µ ) is a summand of L ( λ ) ⊗ L ( (cid:36) a ) . Further assume that allthe κ λ,µ are invertible in k . Then the λ + (cid:36) a clasp exists and (1.22) λ + (cid:36) a = λ ⊗ (cid:36) a − (cid:88) κ − λ,µ D E λ,µ ◦ E λ,µ , where the sum is over all µ ∈ wt L ( (cid:36) a ) − { (cid:36) a } so that L ( λ + µ ) is a summand of L ( λ ) ⊗ L ( (cid:36) a ) .Proof. See [4] Proposition . . (cid:3) Thus, to have a recursive formula for clasps, we need to compute all the κ λ,µ . The easiestcase is when µ is dominant, and we find κ λ,ω a = 1 for all λ ∈ X + and a ∈ { , } . Remark . When λ = a(cid:36) + b(cid:36) is close to the wall of the dominant Weyl chamber, i.e. a or b are“small”, there will be µ ∈ wt( L ( (cid:36) )) such that L ( λ + µ ) is not a composition factor of L ( λ ) ⊗ L ( (cid:36) ) . In this case κ − λ,µ is taken to be equal to zero. Remark . In order to simplify notation we leave off labels of clasps when the highestweight is understood. We will often leave off extra strands below (above) clasps which areon the bottom (top) of the diagram, as well as strands to the left of a diagram which has aclasp at the top or bottom. This is justified because all clasps with the same highest weightare transformed to one another by applying neutral diagrams on the top and bottom (in othercontexts this could be nontrivial to verify, but it is easy to see that any two words in and of the same weight differ by a neutral diagram). We also freely use clasp absorption (that is RIPLE CLASP FORMULAS FOR C WEBS 13 one clasp on top of another is a clasp) to simplify formulas. For example (1.21) becomes(1.23) K λ,µ = D ( L µ ) L µ = κ λ,µ and(1.24)becomes(1.25)1.5. Relation to clasp conjecture.
The main result of this paper is the calculation of all localintersection forms κ λ,µ in D C . Theorem 1.21.
We write ( a, b ) for a(cid:36) + b(cid:36) ∈ X + . (1.26) κ ( a,b ) , (1 , = 1 (1.27) κ ( a,b ) , (0 , = 1 (1.28) κ ( a,b ) , ( − , = − [ a + 1][ a ] (1.29) κ ( a,b ) , (2 , − = − [2 b + 2][2 b ] . (1.30) κ ( a,b ) , (0 , = [ a + 2][ a + 2 b + 4][2][ a ][ a + 2 b + 2] . (1.31) κ ( a,b ) , (1 , − = [ a + 2 b + 3][2 b + 2][ a + 2 b + 2][2 b ] . (1.32) κ ( a,b ) , ( − , = − [ a + 1][2 a + 2 b + 4][ a − a + 2 b + 2] . (1.33) κ ( a,b ) , ( − , = − [2 a + 2 b + 4][ a + 2 b + 3][ a + 1][2 a + 2 b + 2][ a + 2 b + 2][ a ] . (1.34) κ ( a,b ) , (0 , − = [2 a + 2 b + 4][ a + 2 b + 3][2 b + 2][2 a + 2 b + 2][ a + 2 b + 1][2 b ] In the following, we will reinterpret Elias’s type A clasp conjecture [4] in type C . Since theconjecture in type A only deals with κ λ,µ when µ is in the Weyl group orbit of a dominantfundamental weight, we are forced to ignore the local intersection form for the weight (0 , .Recall that the Weyl group for the C root system, which we denote simply by W , acts onthe weight lattice X by(1.35) s ( (cid:36) ) = − (cid:36) + (cid:36) and s ( (cid:36) ) = (cid:36) while(1.36) t ( (cid:36) ) = (cid:36) and t (2) = 2 (cid:36) − (cid:36) . For an arbitrary weight (cid:36) , we will denote by d (cid:36) the minimal length element in W whichwhen takes (cid:36) to a dominant weight. Thus,(1.37) d (cid:36) = 1 = d (cid:36) and(1.38) d − (cid:36) + (cid:36) = s, d (cid:36) − (cid:36) = t, d (cid:36) − (cid:36) = st, d − (cid:36) + (cid:36) = ts, d − (cid:36) = sts, and d − (cid:36) = tst. Next, for an arbitrary weight (cid:36) we define the set Φ (cid:36) = { α ∈ Φ + | d (cid:36) ( α ) ∈ Φ − } . Thus,(1.39) Φ − (cid:36) + (cid:36) = { α s } (1.40) Φ (cid:36) − (cid:36) = { α t } (1.41) Φ (cid:36) − (cid:36) = { α t , t ( α s ) } (1.42) Φ − (cid:36) + (cid:36) = { α s , s ( α t ) } RIPLE CLASP FORMULAS FOR C WEBS 15 (1.43) Φ − (cid:36) = { α s , s ( α t ) , st ( α s ) } (1.44) Φ − (cid:36) = { α t , t ( α s ) , ts ( α t ) } Let ( − , − ) be the standard inner product on X so the (cid:15) i are an orthonormal basis. Recallthat α ∨ = 2 α/ ( α, α ) , and that ρ is the sum of the fundamental weights. We define l ( α ) = 1 when α is a short root and l ( α ) = 2 when α is a long root. Corollary 1.22.
In type C , if (cid:36) is an (extremal) weight in a fundamental representation and λ ∈ X + , then (1.45) κ λ,(cid:36) = ± (cid:89) α ∈ Φ (cid:36) [( α ∨ , λ + ρ )] q l ( α ) [( α ∨ , λ + (cid:36) + ρ )] q l ( α ) Proof.
Using the formula [2 n ] v [2] v = [ n ] v it is an easy exercise to use (1.21) and our descriptionof Φ (cid:36) to check the corollary. (cid:3)
2. D
ERIVING THE RECURSIVE FORMULAS
We will compute recursive formulas for the local intersection forms κ λ,µ using the graph-ical calculus for D C . To simplify notation, we will write ( a, b ) for a(cid:36) + b(cid:36) .For certain values of ( a, b ) and fundamental weights (cid:36) , there are µ ∈ L ( (cid:36) ) so that L (( a, b )+ µ ) will not occur as a summand of L ( a, b ) ⊗ L ( (cid:36) ) . When this occurs, we set κ − a,b ) ,µ equal tozero. This results in the following initial conditions for our recursion:(2.1) κ − a,b ) , (1 , − = 0 when b = 0 , (2.2) κ − a,b ) , ( − , = κ − a,b ) , ( − , = 0 when a = 0 , (2.3) κ − a,b ) , (0 , = 0 when a = 0 , (2.4) κ − a,b ) , ( − , = 0 when a = 0 or , and(2.5) κ − a,b ) , (0 , − = κ − a,b ) , (2 , − = 0 when b = 0 . Proposition 2.1.
The κ λ,µ ’s satisfy the following relations. (2.6) κ ( a,b ) , (1 , = 1 (2.7) κ ( a,b ) , (0 , = 1 (2.8) κ ( a,b ) , ( − , = − [2] − κ − a − ,b ) , ( − , (2.9) κ ( a,b ) , (2 , − = − [4][2] − κ − a,b − , (2 , − (2.10) κ ( a,b ) , (0 , = [5][2] − κ − a − ,b ) , ( − , · κ ( a − ,b +1) , (2 , − − κ − a − ,b ) , (1 , − (2.11) κ ( a,b ) , (1 , − = [5][2] − κ − a,b − , (2 , − · κ ( a +2 ,b − , ( − , − κ − a,b − , (0 , (2.12) κ ( a,b ) , ( − , = − [6][2][3] − κ − a − ,b ) , ( − , − κ − a − ,b ) , ( − , · κ ( a − ,b +1) , (1 , − − κ − a − ,b ) , (1 , − · κ ( a,b − , ( − , (2.13) κ ( a,b ) , (0 , − = [6][5][3][2] − κ ( a,b − , (0 , − − κ ( a +2 ,b − , ( − , κ ( a,b − , (2 , − − κ ( a,b − , (0 , κ ( a,b − , (0 , − κ ( a − ,b ) , (2 , − κ ( a,b − , ( − , (2.14) κ ( a,b ) , ( − , = [5][2] · κ ( a − ,b ) , ( − , − ( − [2] − κ − a − ,b ) , ( − , ) · κ ( a − ,b ) , ( − , κ ( a − ,b ) , ( − , − κ ( a − ,b +1) , (0 , κ a − ,b ) , ( − , · κ ( a − ,b ) , ( − , Proof.
We will use the established properties of clasps to derive the recursion relations. Recallthat κ λ,µ is the coefficient of the neutral map in K λ,µ .(2.15) K λ,µ = λ λ + µλ + µ D L µ L µ Using the equation(2.16) λ = ( λ − (cid:36) ) ⊗ (cid:36) − (cid:88) ν ∈ wt L ( (cid:36) ) −{ (cid:36) } κ − λ − (cid:36),ν D E λ − (cid:36),ν ◦ E λ − (cid:36),ν to rewrite the λ clasp, and then using clasp absorption, we can rewrite K λ,µ as RIPLE CLASP FORMULAS FOR C WEBS 17 (2.17) K λ,µ = λ − (cid:36)λ + µλ + µ D L µ L µ − (cid:80) ν ∈ wt L ( (cid:36) ) κ − λ − (cid:36),ν λ − (cid:36) − νλ + µλ + µL ν D L ν D L µ L µ Having established the general pattern one follows to derive these recursive formulas, weproceed to apply it for each κ ( a,b ) ,µ .To compute κ ( a,b ) , ( − , , we resolve the ( a, b ) clasp in K ( a,b ) , ( − , as in (2.17). Since strictlylower diagrams are orthogonal to clasps, both(2.18)and(2.19)are zero. So after expanding the ( a, b ) clasp in K ( a,b ) , ( − , the only ν ∈ wt L (1) which con-tributes to the sum in (2.17) is ( − , . This means we can rewrite K ( a,b ) , ( − , as (2.20) = − κ − a − ,b ) , ( − , Using neutral ladder absorption and clasp absorption, we deduce(2.21) κ ( a,b ) , ( − , = − [2] − κ − a − ,b ) , ( − , . We proceed similarly with the remaining K λ,µ . It is useful to note that from the H ≡ I relation we have the following ”clasped” relation.(2.22) = 1[2] Similarly, using the H ≡ I relation, orthogonality to clasps, and neutral map absorption wecan also deduce the following.(2.23) = = RIPLE CLASP FORMULAS FOR C WEBS 19
For κ ( a,b ) , (2 , − , we begin by observing that from (2.22), and clasp orthogonality, the fol-lowing diagrams are zero(2.24) . Using these observations and (2.23), K ( a,b ) , (2 , − can be resolved as follows.(2.25) = − κ − a,b − , (2 , − . Then we apply H ≡ I , and cite clasp orthogonality, to find(2.26) = − . If we apply H ≡ I again, then by the monogon relation and clasp orthogonality we canrewrite the right hand side as follows.(2.27) + 1[2] − [5][2] . Finally, using the bigon relation and the clasped H ≡ I relation we can rewrite (2.27) as(2.28) − [2] · − [5][2] = − [4][2] times the clasp, and then conclude that(2.29) κ ( a,b ) , (2 , − = − [4][2] − κ − a,b − , (2 , − . To compute κ ( a,b ) , (0 , we will expand the middle clasp in K ( a,b ) , (0 , .(2.30) K ( a,b ) , (0 , = RIPLE CLASP FORMULAS FOR C WEBS 21
Since(2.31)is zero, we can rewrite K ( a,b ) , (0 , as follows.(2.32) − κ − a − ,b ) , ( − , − κ − a − ,b ) , (1 , − Observing that the second term in (2.30) is K ( a − ,b +1) , (2 , − , then using neutral map absorp-tion and clasp absorption for the third term in (2.30), we deduce that(2.33) κ ( a,b ) , (0 , = [5][2] − κ − a − ,b ) , ( − , · κ ( a − ,b +1) , (2 , − − κ − a − ,b ) , (1 , − . To compute κ ( a,b ) , (1 , − we expand the middle clasp in K ( a,b ) , (1 , − .(2.34) K ( a,b ) , (1 , − = . We begin by using H ≡ I and orthogonality, then neutral absorption, to calculate the follow-ing.(2.35) = = . Also, recall that(2.36) = 1[2]
Therefore, clasp orthogonality implies the two diagrams(2.37)
RIPLE CLASP FORMULAS FOR C WEBS 23 are zero. We can rewrite K ( a,b ) , (1 , − as follows.(2.38) − κ − a,b − , (2 , − − κ − a,b − , (0 , . Identifying the second term in (2.38) as K a +2 ,b − , ( − , , we deduce that(2.39) κ ( a,b ) , (1 , − = [5][2] − κ − a,b − , (2 , − · κ ( a +2 ,b − , ( − , − κ − a,b − , (0 , . Remark . Note that at this point we could start solving these recurrence relations, as thelocal intersection forms for the weights ( − , and (2 , − are linked only to themselves intheir recursion relation. While the local intersection forms for the weights (0 , and (1 , − have recursions which link them to themselves, each other, and the local intersection formsfor the weights ( − , and (2 , − .Continuing with our derivation of recursion relations for local intersection forms, we ex-pand the middle clasp in K ( a,b ) , (0 , − (2.40) K ( a,b ) , ( − , = and apply clasp absorption to deduce(2.41) K ( a,b ) , ( − , = − κ − a − ,b ) , ( − , − κ − a − ,b ) , ( − , − κ − a − ,b ) , (1 , − By identifying the third and fourth terms on the right hand side of (2.41) as K ( a − ,b +1) , (1 , − and K ( a,b − , ( − , respectively, we find(2.42) κ ( a,b ) , ( − , = − [6][2][3] − κ − a − ,b ) , ( − , − κ − a − ,b ) , ( − , · κ ( a − ,b +1) , (1 , − − κ − a − ,b ) , (1 , − · κ ( a,b − , ( − , Remark . Again, we could stop here and solve the recurrence relations since the localintersection form for the weight ( − , is linked to itself and the weights we have computedrecursions for previously. In fact every weight in L ( ω ) appears in this recursion for the localintersection form of ( − , . We conjecture this pattern appears in general (i.e. for othertypes and/or higher ranks) when computing the local intersection form of an antidominantweight.The antidominant weight in L ( ω ) is (0 , − . The local intersection form κ ( a,b ) , (0 , − iscomputed from resolving the middle clasp in(2.43) K ( a,b ) , (0 , − = resulting in(2.44) − κ − a,b − , (0 , − − κ − a,b − , (2 , − − κ − a,b − , (0 , − κ − a,b − , ( − , Identifying the third, fourth, and fifth terms in (2.44) as K ( a +2 ,b − , ( − , , K ( a,b − , (0 , , and K ( a − ,b ) , (2 , − respectively, we may deduce(2.45) κ ( a,b ) , (0 , − = [6][5][3][2] − κ ( a,b − , (0 , − − κ ( a +2 ,b − , ( − , κ ( a,b − , (2 , − − κ ( a,b − , (0 , κ ( a,b − , (0 , − κ ( a − ,b ) , (2 , − κ ( a,b − , ( − , . The last local intersection form to resolve is K ( a,b ) , ( − , . Recall that(2.46) k ⊗ eval : k ⊗ D C −→ Fund ( k ⊗ U Z q ( sp )) is an equivalence. Also, we know that if a Weyl module k ⊗ V Z ( λ ) is simple, then we cancompute the dimension of homomorphism spaces involving that Weyl module in character-istic zero. Thus, from the calculations in [2] we see that(2.47) dim Hom k ⊗D C (( a − , b +1) , ( a, b − ⊗
2) = dim Hom sp ( C ) ( L ( a − , b +1) , L ( a, b − ⊗ L ( (cid:36) )) = 0 . RIPLE CLASP FORMULAS FOR C WEBS 25
Therefore, the diagram(2.48)is zero. When we expand the ( a, b ) clasp in K ( a,b ) , ( − , , one of the terms has (2.48) as asub-diagram and therefore is zero. So we get three terms on the right hand side.(2.49) = − κ − a − ,b ) , ( − , − κ − a − ,b ) , ( − , The first term in (2.49) simplifies to(2.50) [5][2] · κ ( a − ,b ) , ( − , . So we only need to resolve the other two terms on the right hand side of (2.49). Both termscontain the following sub-diagram.(2.51) a − , b Expanding the ( a − , b ) clasp and using clasp orthogonality and neutral absorption, we canrewrite (2.51) as follows.(2.52) a − , b a − , b = a − , b − κ − a − ,b ) , ( − , Then, from the relation (2.52), we have(2.53) a − , b = a − , b − κ − a − ,b ) , ( − , RIPLE CLASP FORMULAS FOR C WEBS 27 and(2.54) a − , b = a − , b − κ − a − ,b ) , ( − , Applying these local relations to the second and third term on the right hand side of (2.49),and simplifying diagrams using the defining relations of D C , we obtain the next two equa-tions.(2.55) = − (cid:16) [2] + κ − a − ,b ) , ( − , (cid:17) (2.56) = + κ − a − ,b ) , ( − , Note that the diagram on the right hand side of (2.55) is K ( a − ,b ) , ( − , , and, after applyingneutral absorption , the diagram on the right hand side of (2.56) is K ( a − ,b +1) , (0 , . Therefore,we can use (2.50), (2.55), and (2.56) to rewrite (2.49), deducing that(2.57) κ ( a,b ) , ( − , = [5][2] · κ ( a − ,b ) , ( − , − ( − [2] − κ − a − ,b ) , ( − , ) · κ ( a − ,b ) , ( − , κ ( a − ,b ) , ( − , − κ ( a − ,b +1) , (0 , κ a − ,b ) , ( − , · κ ( a − ,b ) , ( − , . (cid:3)
3. S
OLVING THE RECURSIVE FORMULAS
In this section we will solve the recursion relations for the local intersection forms derivedin the previous section.
Theorem 3.1.
The recursive relations (3.1) κ ( a,b ) , ( − , = − [2] − κ − a − ,b ) , ( − , (3.2) κ ( a,b ) , (2 , − = − [4][2] − κ − a,b − , (2 , − (3.3) κ ( a,b ) , (0 , = [5][2] − κ − a − ,b ) , ( − , · κ ( a − ,b +1) , (2 , − − κ − a − ,b ) , (1 , − (3.4) κ ( a,b ) , (1 , − = [5][2] − κ − a,b − , (2 , − · κ ( a +2 ,b − , ( − , − κ − a,b − , (0 , (3.5) κ ( a,b ) , ( − , = − [6][2][3] − κ − a − ,b ) , ( − , − κ − a − ,b ) , ( − , · κ ( a − ,b +1) , (1 , − − κ − a − ,b ) , (1 , − · κ ( a,b − , ( − , (3.6) κ ( a,b ) , ( − , = [5][2] · κ ( a − ,b ) , ( − , − ( − [2] − κ − a − ,b ) , ( − , ) · κ ( a − ,b ) , ( − , κ ( a − ,b ) , ( − , − κ ( a − ,b +1) , (0 , κ a − ,b ) , ( − , · κ ( a − ,b ) , ( − , RIPLE CLASP FORMULAS FOR C WEBS 29 (3.7) κ ( a,b ) , (0 , − = [6][5][3][2] − κ ( a,b − , (0 , − − κ ( a +2 ,b − , ( − , κ ( a,b − , (2 , − − κ ( a,b − , (0 , κ ( a,b − , (0 , − κ ( a − ,b ) , (2 , − κ ( a,b − , ( − , together with the initial conditions (3.8) κ − a,b ) , (1 , − = 0 when b = 0 , (3.9) κ − a,b ) , ( − , = κ − a,b ) , ( − , = 0 when a = 0 , (3.10) κ − a,b ) , (0 , = 0 when a = 0 , (3.11) κ − a,b ) , ( − , = 0 when a = 0 or , and (3.12) κ − a,b ) , (0 , − = κ − a,b ) , (2 , − = 0 when b = 0 . are uniquely solved by (3.13) κ ( a,b ) , (1 , = 1 (3.14) κ ( a,b ) , (0 , = 1 (3.15) κ ( a,b ) , ( − , = − [ a + 1][ a ] (3.16) κ ( a,b ) , (2 , − = − [2 b + 2][2 b ] . (3.17) κ ( a,b ) , (0 , = [ a + 2][ a + 2 b + 4][2][ a ][ a + 2 b + 2] . (3.18) κ ( a,b ) , (1 , − = [ a + 2 b + 3][2 b + 2][ a + 2 b + 2][2 b ] . (3.19) κ ( a,b ) , ( − , = − [ a + 1][2 a + 2 b + 4][ a − a + 2 b + 2] . (3.20) κ ( a,b ) , ( − , = − [2 a + 2 b + 4][ a + 2 b + 3][ a + 1][2 a + 2 b + 2][ a + 2 b + 2][ a ] . (3.21) κ ( a,b ) , (0 , − = [2 a + 2 b + 4][ a + 2 b + 3][2 b + 2][2 a + 2 b + 2][ a + 2 b + 1][2 b ] In our proof, we will repeatedly use the following well known quantum number identities.
Lemma 3.2.
Let m and n be integers, then (3.22) [ m ][ n ] = m − (cid:88) i =0 [ n + m − − i ] = n − (cid:88) i =0 [ n + m − − i ] . With the convention that [ − k ] = − [ k ] . An easy consequence of (3.2) is
Lemma 3.3.
Let p , q , and r be integers, then (3.23) [ p ][ q ][ r ] = (cid:88) ( i,j ) ∈ [0 ,p − × [0 ,q − [ r + q + p − − i + j )] . Proof.
Recalling the base cases(3.24) κ − ,b ) , ( − , = 0 = κ − a, , (2 , − , it is an easy calculation with quantum numbers to verify that(3.25) κ ( a,b ) , ( − , = − [ a + 1][ a ] and(3.26) κ ( a,b ) , (2 , − = − [2( b + 1)][2 b ] . Plugging these formulas into the next two recurrence relations we obtain the followingsimplified relations(3.27) κ ( a,b ) , (0 , = [5][2] − [ a − b + 2)][ a ][2( b + 1)] − κ − a − ,b ) , (1 , − (3.28) κ ( a,b ) , (1 , − = [5][2] − [ a + 3][2( b − a + 2][2 b ] − · κ − a,b − , (0 , . We claim that(3.29) κ ( a,b ) , (0 , = [ a + 2][ a + 2 b + 4][2][ a ][ a + 2 b + 2] and(3.30) κ ( a,b ) , (1 , − = [ a + 2 b + 3][2 b + 2][ a + 2 b + 2][2 b ] . Let us suppose inductively that the conjectured formulas (3.29) and (3.30) hold for all ( a, b ) with a + b < n . If a + b = n , we have(3.31) κ ( a,b ) , (0 , = [5][2] − [ a − b + 4][ a ][2 b + 2] − [ a + 2 b + 1][2 b ][ a + 2 b + 2][2 b + 2] and(3.32) κ ( a,b ) , (1 , − = [5][2] − [ a + 3][2 b − a + 2][2 b ] − [ a ][ a + 2 b ][2][ a + 2][ a + 2 b + 2] . We claim that the right hand side of these equalities simplify to the conjectured closed formformula, which after clearing denominators is equivalent to the equalities(3.33) [ a +2][2 b +2][ a +2 b +4] = [5][ a ][2 b +2][ a +2 b +2] − [2][ a − b +4][ a +2 b +2] − [2][ a ][2 b ][ a +2 b +1] and(3.34) [2][ a +2][2 b +2][ a +2 b +3] = [5][ a +2][2 b ][ a +2 b +2] − [2][ a +3][2 b − a +2 b +2] − [ a ][2 b ][ a +2 b ] . Let us first focus on proving the first equality, initially rearranging terms so there are nosubtractions in the equation(3.35) [5][ a ][2 b +2][ a +2 b +2] = [ a +2][2 b +2][ a +2 b +4]+[2][ a − b +4][ a +2 b +2]+[2][ a ][2 b ][ a +2 b +1] . RIPLE CLASP FORMULAS FOR C WEBS 31
We expand the products into the sums to find the following equivalent form of the desiredequality a − (cid:88) i =0 2 b +1 (cid:88) j =0 [5][2 a + 4 b + 2 − i − j ] = a +1 (cid:88) i =0 2 b +1 (cid:88) j =0 [2 a + 4 b + 6 − i − j ] + a − (cid:88) i =0 2 b +3 (cid:88) j =0 [2][2 a + 4 b + 3 − i − j ]+ a − (cid:88) i =0 2 b − (cid:88) j =0 [2][2 a + 4 b − − i − j ] . (3.36)Reindexing the sums we find a − (cid:88) i =0 2 b +1 (cid:88) j =0 [5][2 + 2( i + j )] = a +1 (cid:88) i =0 2 b +1 (cid:88) j =0 [2 + 2( i + j )] + a − (cid:88) i =0 2 b +3 (cid:88) j =0 [2][1 + 2( i + j )]+ a − (cid:88) i =0 2 b − (cid:88) j =0 [2][3 + 2( i + j )] . (3.37)The left hand side of (3.37) can be expanded as(3.38) (cid:88) ( i,j ) ∈ [0 ,a − × [0 , b +1] [6 + 2( i + j )] + [4 + 2( i + j )] + [2 + 2( i + j )] + [2( i + j )] + [ − i + j )] . Next, we split up the sum (3.38), obtaining three separate sums(3.39) (cid:88) [0 ,a − × [0 , b +1] [6+2( i + j )]+ (cid:88) { }× [0 , b +1] [4+2( i + j )]+ (cid:88) { }× [1 , b +1] [2+2( i + j )]+ (cid:88) (0 , [2( i + j )] , (3.40) (cid:88) [1 ,a − × [0 , b +1] [4+2( i + j )]+ (cid:88) [1 ,a − × [0 , b +1] [2+2( i + j )]+ (cid:88) [1 ,a − × [0 , [2( i + j )]+ (cid:88) [1 ,a − × [0 , [ − i + j )] , and(3.41) (cid:88) [0 ,a − × [2 , b +1] [2( i + j )] + (cid:88) [0 ,a − × [2 , b +1] [ − i + j )] . Reindexing, the three sums (3.39), (3.40), and (3.41) become(3.42) (cid:88) [2 ,a +1] × [0 , b +1] [2 + 2( i + j )] + (cid:88) { }× [0 , b +1] [2 + 2( i + j )] + (cid:88) { }× [1 , b +1] [2 + 2( i + j )] + [2] , (3.43) (cid:88) [0 ,a − × [2 , b +3] [2+2( i + j )]+ (cid:88) [0 ,a − × [0 , [2+2( i + j )]+ (cid:88) [0 ,a − × [2 , b +3] [2( i + j )]+ (cid:88) [0 ,a − × [0 , [2( i + j )] , and(3.44) (cid:88) [0 ,a − × [0 , b − [4 + 2( i + j )] + (cid:88) [0 ,a − × [0 , b − [2 + 2( i + j )] . Finally, we observe that the three sums (3.42), (3.43), and (3.44) simplify to(3.45) (cid:88) [0 ,a +1] × [0 , b +1] [2 + 2( i + j )] , (3.46) (cid:88) [0 ,a − × [0 , b +3] [2][ a + 2( i + j )] , and(3.47) (cid:88) [0 ,a − × [0 , b − [2][3 + 2( i + j )] respectively, proving (3.37).To prove that(3.48) [2][ a + 2][2 b + 2][ a + 2 b + 3] = [5][ a + 2][2 b ][ a + 2 b + 2] − [2][ a + 3][2 b − a + 2 b + 2] − [ a ][2 b ][ a + 2 b ] we will show that (cid:88) [0 ,a +1] × [0 , b − [5][2 a + 4 b + 2 − i + j )] = (cid:88) [0 ,a +1] × [0 , b +1] [2][2 a + 4 b + 5 − i + j )]+ (cid:88) [0 ,a +2] × [0 , b − [2][2 a + 4 b + 1 − i + j )]+ (cid:88) [0 ,a − × [0 , b − [2 a + 4 b − − i + j )] . (3.49)After reindexing, (3.49) becomes (cid:88) [0 ,a +1] × [0 , b − [5][2 + 2( i + j )] = (cid:88) [0 ,a +1] × [0 , b +1] [2][1 + 2( i + j )]+ (cid:88) [0 ,a +2] × [0 , b − [2][3 + 2( i + j )]+ (cid:88) [0 ,a − × [0 , b − [2 + 2( i + j )] . (3.50)Expanding the left hand side we have(3.51) (cid:88) [0 ,a +1] × [0 , b − [6 + 2( i + j )] + [4 + 2( i + j )] + [2 + 2( i + j )] + [2( i + j )] + [ − i + j )] which we can regroup into three summands as (cid:88) [0 ,a +1] × [0 , b − [6 + 2( i + j )] + (cid:88) [0 ,a +1] × [0 , [2 + 2( i + j )] + (cid:88) (0 , [2( i + j )]+ (cid:88) [0 ,a +1] × [0 , b − [4 + 2( i + j )] + (cid:88) [0 ,a +1] × [0 , [2( i + j )] , (3.52)(3.53) (cid:88) [0 ,a +1] × [2 , b − [2+2( i + j )]+ (cid:88) { }× [2 , b − [ − i + j )]+ (cid:88) [0 ,a +1] × [2 , b − [2( i + j )]+ (cid:88) { }× [2 , b − [ − i + j )] , and(3.54) (cid:88) [2 ,a +1] × [0 , b − [ − i + j )] . RIPLE CLASP FORMULAS FOR C WEBS 33
Next, we reindex the sums (3.52), (3.53), and (3.54) obtaining (cid:88) [0 ,a +1] × [2 , b +1] [2 + 2( i + j )] + (cid:88) [0 ,a +1] × [0 , [2 + 2( i + j )] + [2]+ (cid:88) [0 ,a +1] × [2 , b +1] [2( i + j )] + (cid:88) [0 ,a − × [0 , [2( i + j )] , (3.55)(3.56) (cid:88) [1 ,a +2] × [0 , b − [4+2( i + j )]+ (cid:88) { }× [0 , b − [4+2( i + j )]+ (cid:88) [1 ,a +2] × [0 , b − [2+2( i + j )]+ (cid:88) { }× [0 , b − [2+2( i + j )] , and(3.57) (cid:88) [0 ,a − × [0 , b − [2 + 2( i + j )] . Observing that the three summands (3.55), (3.56), and (3.57) simplify to be(3.58) (cid:88) [0 ,a +1] × [0 , b +1] [2][1 + 2( i + j )] , (3.59) (cid:88) [0 ,a +2] × [0 , b − [2][3 + 2( i + j )] , and(3.60) (cid:88) [0 ,a − × [0 , b − [2 + 2( i + j )] respectively, we may conclude that the equality (3.50) holds. Remark . Showing the base case for (3.29) requires the same work to show the inductivestep. This is because if b = 0 , then κ − a − ,b ) , (1 , − = 0 by definition. So our base case would bewhen b = 0 , and we would have to verify that (3.29) satisfies(3.61) κ ( a,b ) , (0 , = [5][2] − [ a − b + 2)][ a ][2( b + 1)] − On the other hand, in our induction step we assume(3.62) κ ( a − ,b ) , (1 , − = [ a + 2 b + 2][2 b + 2][ a + 2 b + 1][2 b ] and to prove (3.29) we are reduced to proving that(3.63) [ a + 2][ a + 2 b + 4][2][ a ][ a + 2 b + 2] = [5][2] − [ a − b + 4][ a ][2 b + 2] − [ a + 2 b + 1][2 b ][ a + 2 b + 2][2 b + 2] . But when b = 0 , this reduces to(3.64) κ ( a,b ) , (0 , = [5][2] − [ a − b + 2)][ a ][2( b + 1)] . which is the inductive step in the proof of our base case. The same remarks apply mutatismutandis to (3.30), as well as to all the base cases below, so we will not argue each base caseseparately, only justifying the inductive step which can be interpreted as proving the basecase. Having solved the recurrences for the local intersection forms corresponding to the weight’s (1 , − and ( − , , we may simplify the recurrence(3.65) κ ( a,b ) , ( − , = − [6][2][3] − κ − a − ,b ) , ( − , − κ − a − ,b ) , ( − , · κ ( a − ,b +1) , (1 , − − κ − a − ,b ) , (1 , − · κ ( a,b − , ( − , to(3.66) κ ( a,b ) , ( − , = − [6][2][3] + [ a − a + 2 b + 3][2 b + 4][ a ][ a + 2 b + 2][2 b + 2] + [ a + 2 b + 1][2 b ][ a + 1][ a + 2 b + 2][2 b + 2][ a ] − κ − a − ,b ) , ( − , . We will show that(3.67) κ ( a,b ) , ( − , = − [2 a + 2 b + 4][ a + 2 b + 3][ a + 1][2 a + 2 b + 2][ a + 2 b + 2][ a ] . Suppose this holds for ( a, b ) with a + b < n and assume that a + b = n , then we have(3.68) κ ( a,b ) , ( − , = − [6][2][3] + [ a − a + 2 b + 3][2 b + 4][ a ][ a + 2 b + 2][2 b + 2] + [ a + 2 b + 1][2 b ][ a + 1][ a + 2 b + 2][2 b + 2][ a ] + [2 a + 2 b ][ a + 2 b + 1][ a − a + 2 b + 2][ a + 2 b + 2][ a ] , and so we are tasked to prove that − [2 a + 2 b + 4][ a + 2 b + 3][ a + 1][2 a + 2 b + 2][ a + 2 b + 2][ a ] = − [6][2][3] + [ a − a + 2 b + 3][2 b + 4][ a ][ a + 2 b + 2][2 b + 2] + [ a + 2 b + 1][2 b ][ a + 1][ a + 2 b + 2][2 b + 2][ a ]+ [2 a + 2 b ][ a + 2 b + 1][ a − a + 2 b + 2][ a + 2 b + 2][ a ] . (3.69)Clearing denominators and rearranging the terms we see that this is equivalent to provingthat [6][2][ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b + 2] = [3]([ a + 1][2 b + 2][ a + 2 b + 3][2 a + 2 b + 4]+ [ a − b + 4][ a + 2 b + 3][2 a + 2 b + 2]+ [ a + 1][2 b ][ a + 2 b + 1][2 a + 2 b + 2]+ [ a − b + 2][ a + 2 b + 1][2 a + 2 b ]) . (3.70)Using that [6][2] / [3] = [5] − [1] we can rewrite (3.70) as [5][ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b + 2] = [ a + 1][2 b + 2][ a + 2 b + 3][2 a + 2 b + 4]+ [ a − b + 4][ a + 2 b + 3][2 a + 2 b + 2]+ [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b + 2]+ [ a + 1][2 b ][ a + 2 b + 1][2 a + 2 b + 2]+ [ a − b + 2][ a + 2 b + 1][2 a + 2 b ] . (3.71) RIPLE CLASP FORMULAS FOR C WEBS 35
We expand the left hand side of (3.71) [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b + 6] + [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b + 4]+[ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b + 2] + [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b ]+ [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b − a + 1][2 b + 2][ a + 2 b + 3][2 a + 2 b + 4]+ [ a − b + 4][ a + 2 b + 3][2 a + 2 b + 2]+ [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b + 2]+ [ a + 1][2 b ][ a + 2 b + 1][2 a + 2 b + 2]+ [ a − b + 2][ a + 2 b + 1][2 a + 2 b ] , (3.72)which immediately simplifies to [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b + 6] + [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b + 4]++ [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b ]+ [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b − a + 1][2 b + 2][ a + 2 b + 3][2 a + 2 b + 4]+ [ a − b + 4][ a + 2 b + 3][2 a + 2 b + 2]+ [ a + 1][2 b ][ a + 2 b + 1][2 a + 2 b + 2]+ [ a − b + 2][ a + 2 b + 1][2 a + 2 b ] . (3.73) Lemma 3.5.
Let n, m ∈ Z , if we interpret [ − k ] as − [ k ] then (3.74) [ n ][ m ] = [ n − m + 1] + [ m − n + 1] and (3.75) [ n ][ m ] = [ n − m + 2] + [2][ m − n + 2] . By repeatedly applying lemma (3.5) we can rewrite each summand on the right hand sideof (3.73) as follows [ a + 1][2 b + 2][ a + 2 b + 3][2 a + 2 b + 4] = [ a + 1][2 b + 2][ a + 2 b + 3][2 a + 2 b + 6]+ [ a + 1][2 b + 2][2][ a + 3]= [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b + 6]+ [2 b + 1][2 b + 2][2 a + 2 b + 6]+ [2][ a + 1][2 b + 2][ a + 3] , (3.76) [ a − b + 4][ a + 2 b + 3][2 a + 2 b + 2] = [ a − b + 2][ a + 2 b + 3][2 a + 2 b + 4]+ [2][2 a ][ a − a + 2 b + 3]= [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b + 4]+ [ − b − b + 2][2 a + 2 b + 4]+ [2][2 a ][ a − a + 2 b + 3] , (3.77) [ a + 1][2 b ][ a + 2 b + 1][2 a + 2 b + 2] = [ a + 1][2 b + 2][ a + 2 b + 1][2 a + 2 b ]+ [2][ − a ][ a + 1][ a + 2 b + 1]= [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b ]+ [2 b + 1][2 b + 2][2 a + 2 b ]+ [2][ − a ][ a + 2][ a + 2 b + 1] , (3.78)and [ a − b + 2][ a + 2 b + 1][2 a + 2 b ] = [ a − b + 2][ a + 2 b + 3][2 a + 2 b − − a + 3][ a − b + 2]= [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b − − b − b + 2][2 a + 2 b − − a + 3][ a − b + 2] . (3.79)Substituting these equations into (3.73) we obtain [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b + 6] + [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b + 4]++ [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b ]+ [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b − a ][2 b + 2][ a + 2 b + 2][2 a + 2 b + 6]+ [2 b + 1][2 b + 2][2 a + 2 b + 6]+ [2][ a + 1][2 b + 2][ a + 3]+ [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b + 4]+ [ − b − b + 2][2 a + 2 b + 4]+ [2][2 a ][ a − a + 2 b + 3]+ [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b ]+ [2 b + 1][2 b + 2][2 a + 2 b ]+ [2][ − a ][ a + 2][ a + 2 b + 1]+ [ a ][2 b + 2][ a + 2 b + 2][2 a + 2 b − − b − b + 2][2 a + 2 b − − a + 3][ a − b + 2] (3.80)which is equivalent to [2][ a − a ][ a + 2 b + 3] + [2 b + 1][2 b + 2][2 a + 2 b + 6]+ [2][ a + 1][2 b + 2][ a + 3]+ [2 b + 1][2 b + 2][2 a + 2 b ]= [2][2 a ][ a + 1][ a + 2 b + 1]+ [2 b + 3][2 b + 2][2 a + 2 b + 4]+ [2 b + 3][2 b + 2][2 a + 2 b − a − a − b + 2] . (3.81) RIPLE CLASP FORMULAS FOR C WEBS 37
Again, we apply lemma (3.5) to rewrite the first summands of each side of (3.81) [2][ a − a ][ a + 2 b + 3] = [2][ a ][2 a ][ a + 2 b + 2]+ [2][2 a ][ − b − (3.82)and [2][2 a ][ a + 1][ a + 2 b + 1] = [2][2 a ][ a ][ a + 2 b + 2]+ [2 b + 1][2][2 a ] . (3.83)Substituting these new expressions into (3.81) and rearranging we have [2 b + 1][2 b + 2][2 a + 2 b + 6] + [2][ a + 1][2 b + 2][ a + 3]+ [2 b + 1][2 b + 2][2 a + 2 b ]= [2][2 a ][2][2 b + 2]+ [2][2 a ][ a + 1][ a + 2 b + 1]+ [2 b + 3][2 b + 2][2 a + 2 b + 4]+ [2 b + 3][2 b + 2][2 a + 2 b − a − a − b + 2] . (3.84)We can then simplify (3.84) by dividing by [2 b + 2] , obtaining the equivalent equality [2 b + 1][2 a + 2 b + 6] + [2][ a + 1][ a + 3] + [2 b + 1][2 a + 2 b ] = [2][2][2 a ] + [2 b + 3][2 a + 2 b + 4]+ [2 b + 3][2 a + 2 b −
2] + [2][ a − a − . (3.85)We will apply lemma (3.5) one last time to some summands on each side of (3.85)(3.86) [2 b + 3][2 a + 2 b −
2] = [2 b + 1][2 a + 2 b ] + [2][2 a − and(3.87) [2 b + 3][2 a + 2 b + 4] = [2 b + 1][2 a + 2 b + 6] + [2][2 a + 3] . Substituting these formulas into (3.85) and simplifying, we end up with the following(3.88) [ a + 1][ a + 3] = [2 a + 3] + [2][2 a ] + [2 a −
3] + [ a − a − or equivalently,(3.89) [ a + 1][ a + 3] − [ a − a −
1] = [2 a + 3] + [2 a + 1] + [2 a −
1] + [2 a − . The validity of (3.89) is easy to see by expanding the left hand side into two sums(3.90) [ a + 1][ a + 3] = [2 a + 3] + [2 a + 1] + . . . + [3] and(3.91) [ a − a −
1] = [2 a −
5] + [2 a −
7] + . . . + [3] . The penultimate recurrence can now be simplified to the following κ ( a,b ) , ( − , = [5][2] · κ ( a − ,b ) , ( − , − κ a − ,b ) , ( − , κ ( a − ,b ) , ( − , − κ ( a − ,b +1) , (0 , κ a − ,b ) , ( − , · κ ( a − ,b ) , ( − , = − [5][ a ][2][ a −
1] + [ a ][ a ][2 a + 2 b ][ a + 2 b + 1][ a − a − a − a + 2 b + 2][ a + 2 b + 2][ a ] + [ a ][ a + 2 b + 4][ a − a − a − a − a + 2 b + 2][ a − a − a ]= − [5][ a ][2][ a −
1] + [ a ][2 a + 2 b ][ a + 2 b + 1][ a − a + 2 b + 2][ a + 2 b + 2] + [ a + 2 b + 4][ a − a + 2 b + 2][ a − . (3.92)Interestingly, the local intersection form for the weight ( − , is given by a recursive formulawhich does not involve previous local intersection forms for the weight ( − , . We claimthat the right hand side simplifies to yield the formula(3.93) κ ( a,b ) , ( − , = − [ a + 1][2 a + 2 b + 4][ a − a + 2 b + 2] . Similar to the above we can clear denominators and rearrange terms to obtain the equation [5][ a ][ a + 2 b + 2][2 a + 2 b + 2] = [2][ a + 1][ a + 2 b + 2][2 a + 2 b + 4] + [2][ a ][ a + 2 b + 1][2 a + 2 b ]+ [ a − a + 2 b + 4][2 a + 2 b + 2] . (3.94)The left hand side of (3.94) expands into the sum(3.95) (cid:88) [0 ,a − × [0 ,a +2 b +1] [5][4 a + 4 b + 2 − i + j )] which we may reindex and write as(3.96) (cid:88) [0 ,a − × [0 ,a +2 b +1] [5][2 + 2( i + j )] We then further expand (3.96) and rearrange into three summands (cid:88) [0 ,a − × [0 ,a +2 b +1] [6 + 2( i + j )] + (cid:88) { }× [0 ,a +2 b +1] [2 + 2( i + j )] + (cid:88) [0 ,a − × [0 ,a +2 b +1] [4 + 2( i + j )]+ (cid:88) { }× [0 ,a +2 b +1] [2 + 2( i + j )] + (cid:88) (1 , [2( i + j )] , (3.97)(3.98) (cid:88) [0 ,a − × [1 ,a +2 b +1] [2( i + j )] + (cid:88) [0 ,a − × [1 ,a +2 b +1] [ − i + j )] , and(3.99) (cid:88) [2 ,a − × [0 ,a +2 b +1] [2 + 2( i + j )] + (cid:88) [2 ,a − ×{ } [2( i + j )] + (cid:88) [0 ,a − ×{ } [ − i + j )] . RIPLE CLASP FORMULAS FOR C WEBS 39
Reindexing the sums (3.97), (3.98), and (3.99) as (cid:88) [1 ,a ] × [0 ,a +2 b +1] [4 + 2( i + j )] + (cid:88) { }× [0 ,a +2 b +1] [4 + 2( i + j )] + (cid:88) [1 ,a ] × [0 ,a +2 b +1] [2 + 2( i + j )]+ (cid:88) { }× [0 ,a +2 b +1] [2 + 2( i + j )] + [2] , (3.100)(3.101) (cid:88) [0 ,a − × [0 ,a +2 b ] [2 + 2( i + j )] + (cid:88) [0 ,a − × [0 ,a +2 b ] [2( i + j )] , and(3.102) (cid:88) [0 ,a − × [2 ,a +2 b +3] [2 + 2( i + j )] + (cid:88) [0 ,a − ×{ } [2 + 2( i + j )] + (cid:88) [0 ,a − ×{ } [2 + 2( i + j )] . we can can simplify the sums (3.100), (3.101), and (3.102) to be(3.103) (cid:88) [0 ,a ] × [0 ,a +2 b +1] [2][3 + 2( i + j )] = [2][ a + 1][ a + 2 b + 2][2 a + 2 b + 4] , (3.104) (cid:88) [0 ,a − × [0 ,a +2 b ] [2][1 + 2( i + j )] = [2][ a ][ a + 2 b + 1][2 a + 2 b ] , and(3.105) (cid:88) [0 ,a − × [0 ,a +2 b +3] [2 + 2( i + j )] = [ a − a + 2 b + 4][2 a + 2 b + 2] respectively. This establishes the desired equality.The last local intersection form is given by the recurrence relation(3.106) κ ( a,b ) , (0 , − = [6][5][3][2] − κ ( a,b − , (0 , − − κ ( a +2 ,b − , ( − , κ ( a,b − , (2 , − − κ ( a,b − , (0 , κ ( a,b − , (0 , − κ ( a − ,b ) , (2 , − κ ( a,b − , ( − , . Using the previously established formulas for the local intersection forms and that [6][5] / [3][2] − / [2] we can write (3.106) as(3.107) κ ( a,b ) , (0 , − = [8][2] − [ a + 3][2 a + 2 b + 4][2 b − a + 1][2 a + 2 b + 2][2 b ] − [ a − a + 2 b ][2 b + 2][ a + 1][2 a + 2 b + 2][2 b ] − κ ( a,b − , (0 , − . We claim that(3.108) κ ( a,b ) , (0 , − = [2 a + 2 b + 4][ a + 2 b + 3][2 b + 2][2 a + 2 b + 2][ a + 2 b + 1][2 b ] which as above amounts showing that this formula can be obtained by manipulating theright hand side of(3.109) κ ( a,b ) , (0 , − = [8][2] − [ a + 3][2 a + 2 b + 4][2 b − a + 1][2 a + 2 b + 2][2 b ] − [ a − a + 2 b ][2 b + 2][ a + 1][2 a + 2 b + 2][2 b ] − [2 a + 2 b ][ a + 2 b − b − a + 2 b + 2][ a + 2 b + 1][2 b ] . Again, we clear denominators and rearrange terms in the desired equality to obtain the fol-lowing equivalent equality of quantum numbers [8][ a + 1][2 b ][ a + 2 b + 1][2 a + 2 b + 2] = [2][ a + 1][2 b + 2][ a + 2 b + 3][2 a + 2 b + 4]+ [2][ a + 3][2 b − a + 2 b + 1][2 a + 2 b + 4]+ [2][ a − b + 2][ a + 2 b + 1][2 a + 2 b ]+ [2][ a + 1][[2 b − a + 2 b − a + 2 b ] . (3.110)The left hand side expands as(3.111) [ a + 1][2 b ][ a + 2 b + 1] (cid:88) i =0 [2 a + 2 b + 7 − i ] and the right hand side can be considered (after multiplying the [2 a + 2 b + i ] terms by the [2] term) as eight summands. We will apply the [ n ][ m ] = [ n − m + 2] + . . . lemma to each ofthese eight summands as follows [ a + 1][2 b + 2][ a + 2 b + 3][2 a + 2 b + 5] = [ a + 1][2 b + 2][ a + 2 b + 1][2 a + 2 b + 2]+ [2][ a + 1][2 b + 2][ a + 3]= [ a + 1][2 b ][ a + 2 b + 1][2 a + 2 b + 9]+ [2][ a + 1][ a + 2 b + 1][2 a + 7]+ [2][ a + 1][2 b + 2][ a + 4] , (3.112) [ a + 1][2 b + 2][ a + 2 b + 3][2 a + 2 b + 3] = [ a + 1][2 b + 2][ a + 2 b + 1][2 a + 2 b + 5]+ [2][ a + 1][2 b + 2][ a + 2]= [ a + 1][2 b ][ a + 2 b + 1][2 a + 2 b + 7]+ [2][ a + 1][ a + 2 b + 1][2 a + 5]+ [2][ a + 1][2 b + 2][ a + 2] , (3.113) [ a + 3][2 b − a + 2 b + 1][2 a + 2 b + 5] = [ a + 3][2 b ][ a + 2 b + 1][2 a + 2 b + 3]+ [2][ a + 3][ a + 2 b + 1][ − a −
5= [ a + 1][2 b ][ a + 2 b = 1][2 a + 2 b + 5]+ [2][2 b ][ a + 2 b + 1][ a + 2 b + 2]+ [2][ a + 3][ a + 2 b + 1][ − a − , (3.114) [ a + 3][2 b − a + 2 b + 1][2 a + 2 b + 3] = [ a + 3][2 b ][ a + 2 b + 1][2 a + 2 b + 1]+ [2][ a + 3][ a + 2 b + 1][ − a − a + 1][2 b ][ a + 2 b + 1][2 a + 2 b + 3]+ [2][2 b ][ a + 2 b + 1][ a + 2 b ]+ [2][ a + 3][ a + 2 b + 1][ − a − , (3.115) [ a − b + 2][ a + 2 b + 1][2 a + 2 b + 1] = [ a − b ][ a + 2 b + 1][2 a + 2 b + 3]+ [2][ a − a + 2 b + 1][2 a + 1]= [ a + 1][2 b ][ a + 2 b + 1][2 a + 2 b + 1]+ [2][2 b ][ a + 2 b + 1][ − a − b − a − a + 2 b + 1][2 a + 1] , (3.116) RIPLE CLASP FORMULAS FOR C WEBS 41 [ a − b + 2][ a + 2 b + 1][2 a + 2 b −
1] = [ a − b ][ a + 2 b + 1][2 a + 2 b + 1]+ [2][ a − a + 2 b + 1][2 a − a + 1][2 b ][ a + 2 b + 1][2 a + 2 b − b ][ a + 2 b + 1][ − a − b ]+ [2][ a − a + 2 b + 1][2 a − , (3.117) [ a + 1][2 b − a + 2 b − a + 2 b + 1] = [ a + 1][2 b − a + 2 b + 1][2 a + 2 b − a + 1][2 b − − a ]= [ a + 1][2 b ][ a + 2 b + 1][2 a + 2 b − a + 1][ a + 2 b + 1][ − a + 1]+ [2][ a + 1][2 b − − a ] , (3.118)and [ a + 1][2 b − a + 2 b − a + 2 b −
1] = [ a + 1][2 b − a + 2 b + 1][2 a + 2 b − a + 1][2 b − − a + 2]= [ a + 1][2 b ][ a + 2 b + 1][2 a + 2 b − a + 1][ a + 2 b + 1][2 a + 2 b − a + 1][ a + 2 b + 1][ − a + 3]+ [2][ a + 1][2 b − − a + 2] . (3.119)Comparing the left hand and right hand side of the desired equality, we can now cancel alleight of the terms of the form [ a + 1][2 b ][ a + 2 b + 1][2 a + 2 b + i ] (this is the entire left handside). Then we divide both sides by [2] (the left hand side is zero so this is fine), and rearrangeterms so that the equality between the (new) left hand side [ a + 1][ a + 2 b + 1][2 a + 7] + [ a + 1][2 b + 2][ a + 4] + [ a + 1][ a + 2 b + 1][2 a + 5] + [ a + 1][2 b + 2][ a + 2]+ [2 b ][ a + 2 b + 1][ a + 2 b + 2] + [2 b ][ a + 2 b + 1][ a + 2 b ] + [ a − a + 2 b + 1][2 a + 1]+ [ a − a + 2 b + 1][2 a − (3.120)and the (new) right hand side [ a + 3][ a + 2 b + 1][2 a + 5] + [ a + 3][ a + 2 b + 1][2 a + 3] + [2 b ][ a + 2 b + 1][ a + 2 b + 2] + [2 b ][ a + 2 b + 1][ a + 2 b ]+ [ a + 1][ a + 2 b + 1][2 a −
1] + [ a + 1][2 b − a ] + [ a + 1][ a + 2 b + 1][2 a − a + 1][2 b − a − . (3.121)is equivalent to our original desired equality (3.110). Using that ([ n + 1] + [ n − / [2] = [ n ] we can rewrite the left hand side (3.120) as(3.122) [ a + 1][ a + 2 b + 1][2 a + 6] + [ a + 1][2 b + 2][ a + 3] + [2 b ][ a + 2 b + 1][ a + 2 b + 1] + [ a − a + 2 b + 1][2 a ] and rewrite the right hand side (3.121) as(3.123) [ a +3][ a +2 b +1][2 a +4]+[2 b ][ a +2 b +1][ a +2 b +1]+[ a +1][ a +2 b +1][2 a − a +1][2 b − a − . Then, we cancel the common term of, [2 b ][ a + 2 b + 1] , and use the identities(3.124) [ a − a + 2 b + 1][2 a ] = [ a + 1][ a + 2 b + 1][2 a − − [2][ a + 2 b + 1][ a − and(3.125) [ a + 3][ a + 2 b + 1][2 a + 4] = [ a + 1][ a + 2 b + 1][2 a + 6] + [2][ a + 2 b + 1][ a + 3] to reduce the work to showing the following equality [ a + 1][2 b + 2][ a + 3] + [ a + 1][ a + 2 b + 1][2 a −
2] = [2][ a + 2 b + 1][ a −
1] + [2][ a + 2 b + 1][ a + 3]+[ a + 1][ a + 2 b + 1][2 a −
2] + [ a + 1][2 b − a − . (3.126)Using the identity(3.127) [4][2] [ a + 1] = ([3] − [1])[ a + 1] = [ a + 3] + [ a − we can rewrite (3.126) as [ a + 1][2 b + 2][ a + 3] + [ a + 1][ a + 2 b + 1][2 a −
2] = [4][ a + 2 b + 1][ a + 1]+[ a + 1][ a + 2 b + 1][2 a −
2] + [ a + 1][2 b − a − . (3.128)which simplifies to(3.129) [2 b + 2][ a + 3] = [4][ a + 2 b + 1] + [2 b − a − . To verify (3.129), observe that(3.130) [2 b + 2][ a + 3] = a +2 (cid:88) i =0 [ a + 2 b + 4 − i ] = [4][ a + 2 b + 1] + a − (cid:88) i =0 [ a + 2 b − − i ] and(3.131) [4][ a + 2 b + 1] + a − (cid:88) i =0 [ a + 2 b − − i ] = [4][ a + 2 b + 1] + [2 b − a − . (cid:3) R EFERENCES [1] Henning Haahr Andersen. Tensor products of quantized tilting modules.
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