Asymptotic growth of Betti numbers of ordered configuration spaces on an elliptic curve
AASYMPTOTIC GROWTH OF BETTI NUMBERS OF ORDEREDCONFIGURATION SPACES ON AN ELLIPTIC CURVE
ROBERTO PAGARIA
Abstract.
We construct a dga to computing the cohomology of ordered con-figuration spaces on an algebraic variety with vanishing Euler characteristic. Itfollows that the k -th Betti number of Conf( C, n ) ( C is an elliptic curve) growsas a polynomial of degree exactly 2 k −
2. We also compute H k (Conf( C, n ))for k ≤ n . The ordered configuration space of n points on a smooth projective variety X isConf( X, n ) = { ( p , . . . , p n ) ∈ X n | p i (cid:54) = p j } . A central problem in the theory of configuration spaces is understand the cohomol-ogy of these topological spaces. The main tool is the Kriˇz model, introduced at thesame time by Kriˇz [Kri94] and by Totaro [Tot96], it is a differential graded algebra E ( X, n ) that codifies the rational homotopy type of Conf(
X, n ) (see Section 1).Often is useful to study all these configuration spaces Conf(
X, n ) for n ∈ N alltogether. In the case of vanishing Euler characteristic χ ( X ) = 0, we construct a fil-tration F • of the Kriˇz model E ( X, n ) and we prove that the differential is strict withrespect to this filtration (Theorem 1.15). Passing to the graded model we obtain asimpler differential graded algebra that is more feasible for computing cohomology.The mixed Hodge numbers of Conf(
X, n ) (indeed any polynomial function N → N )can be written as linear combination of binomials with coefficient in Z . We prove inProposition 1.19 that these coefficients are nonnegative integers for any algebraicvariety with χ ( X ) = 0, this is not true in a wider generality.We apply the simpler dga to configuration on an elliptic curve C . It was knownfrom [Chu12] that the k -th Betti number of Conf( C, n ) grows as polynomial ofdegree at most 2 k . We improve that result by showing that the Betti numbers growas a polynomial of degree exactly 2 k − C, n ) for n ≤ H k (Conf( C, n ); Q ) for k ≤ n ∈ N (Corollary 2.9).A recent tool to deal with sequence of homogeneous objects is the representationstability introduced by Church and Farb [CF13]. We use a variant of that: wesubstitute the category of finite sets with injections with the category F of finite The author is supported by PRIN 2017YRA3LK. a r X i v : . [ m a t h . A T ] M a y R. PAGARIA sets with all maps. The representation theory of F is developed by [Wil14], [SS17],and [Ryb18]. Ellenberg and Wiltshire-Gordon [EW15] have shown that the spacesConf(
X, n ) are an F-module if X has two linearly independent vector fields. Theirresult implies that the cohomology is an F-module, we give a short proof that theKriˇz model is an F-module. We use this structure in an essential way to define thefiltration F and to prove that is strictly compatible with the differential.1. Representation theory of the Kriˇz model
The Kriˇz model.
Let X be a smooth projective variety. For each element x ∈ H • ( X ) we denote by x i ∈ H • ( X n ) its image under the map H • ( X ) (cid:44) → H • ( X n )induced by the projection X n → X on the i -th factor.The class of the diagonal ∆ ∈ H C X ( X × X ) is the cohomological class ofthe subvariety { ( x, x ) | x ∈ X } in X × X . If we fix a graded basis { b , . . . , b k } of H • ( X ) and consider the dual basis { b ∗ , . . . , b ∗ k } with respect to the cup product,then ∆ = k (cid:88) j =1 ( − deg b ∗ j b j ⊗ b ∗ j (1)Consider also ∆ i,j ∈ H • ( X n ) as the pullback of the diagonal ∆ with respect to theprojection X n → X on the i -th and j -th factors. Definition 1.1.
Let E ( X, n ) be the differential bigraded algebra H • ( X n )[ { G i,j } i
The dga E ( X, n ) is a rational model for Conf(
X, n ). Therefore H • (Conf( X, n )) ∼ = H • ( E ( X, n ) , d). The F module structure. We will consider the spaces Conf(
X, n ) and theircohomology groups for n ∈ N all together. Consider the category of finite sets Fwhose objects are the sets [ n ] = { , , . . . , n } for all n ∈ N and with morphisms allthe maps of sets between them. Notice that the automorphism of the object [ n ]Aut F ([ n ]) ∼ = S n is the symmetric group.An F- representation over V is a functor V from the category F to the category V . A F- module V is finitely generated if there exists elements v i ∈ V ([ n i ]) for i in a finite set I such that no proper submodule of V contains all v i for i ∈ I . Amorphism τ : V → W of F- module is a natural transformation between the functors V and W . Definition 1.3.
The F-module E ( X ) is the functor defined by [ n ] (cid:55)→ E ( X, n ) onthe objects and for each map f : [ n ] → [ m ] associate the morphism f ∗ : E ( X, n ) → RDERED CONFIGURATION SPACES ON AN ELLIPTIC CURVE 3 E ( X, m ) defined by x i (cid:55)→ x f ( i ) , G i,j (cid:55)→ (cid:40) G f ( i ) ,f ( j ) if f ( i ) (cid:54) = f ( j )0 otherwiseThe submodules E p,q ( X ) are finitely generated and d : E ( X ) → E ( X ) commuteswith f ∗ for all injective maps f . Proposition 1.4.
If the Euler characteristic of X is zero, i.e. χ ( X ) = 0. Then dis a morphism of F-modules. Proof.
We verify that f ( i ) = f ( j ) implies f ∗ (d( G i,j )) = 0, the other conditions aretrivial. Let { b , . . . , b k } be a graded basis of H • ( X ) and { b ∗ , . . . , b ∗ k } be the dualbasis, we use Equation (1): f ∗ (d( G i,j )) = f ∗ (cid:32) k (cid:88) l =1 ( − deg b ∗ l ( b l ) i ( b ∗ l ) j (cid:33) = k (cid:88) l =1 ( − deg b ∗ l ( b l ) f ( i ) ( b ∗ l ) f ( i ) = k (cid:88) l =1 ( − deg b ∗ l [ X ] f ( i ) = X ) (cid:88) r =1 ( − r h r ( X )[ X ] f ( i ) = χ ( X )[ X ] f ( i ) , where h r ( X ) is the r -th Betti number of X . The hypothesis χ ( X ) = 0 completesthe proof. (cid:3) A topological interpretation of the above result can be found in [EW15].
Representation theory of F . The irreducible representations of the symmetricgroup S n are parametrize by the partitions λ (cid:96) n of the number n , we denotethem by V λ . The category V F of the F representation over V is not semisimple,however has the Jordan H¨older property as shown in [Wil14, Corollary 5.4]. The Schur projective representation of type λ (cid:96) k is the functor P λ ([ n ]) = S λ ( Q n ) , where S λ is the Schur functor (see [Wey03, Mac15] for an introductory expositionof Schur functors). The dimension dim P λ ([ n ]) is an evaluation of the Schur sym-metric polynomial, i.e. s λ (1 n ). These projective representations have the followingproperty Hom V F ( P λ , V ) ∼ = Hom S k ( V λ , V ([ k ])) , (2)where λ (cid:96) k .The simple representations are classified by Wiltshire-Gordon and they are oftwo kinds: D k for k ∈ N and C λ for λ (cid:96) k with λ >
1. The representation C λ for λ (cid:96) k is defined by C λ ([ n ]) = 0 for n < k and by C λ ([ n ]) = Ind S n S k × S n − k V λ (cid:2) n − k , R. PAGARIA where 1 n is the trivial representation of S n and the dimension is (cid:0) nk (cid:1) (cid:104) s λ , p k (cid:105) . Therepresentation D is defined by D ([0]) = Q and D ([ n ]) = 0 for all n >
0. Finally, D k is defined by D k ([ n ]) = 0 for n < k and by D k ([ n ]) = V ( n − k +1 , k − ) . Moreover D k ([ n ]) has dimension (cid:0) n − k − (cid:1) for n, k > Definition 1.5.
A finitely generated representation V is of degree k if there existsa surjection ⊕ i P λ i (cid:16) V with λ i (cid:96) n i and n i ≤ k .Since there exist surjections P λ (cid:16) C λ and P k (cid:16) D k , D k has degree k and C λ has degree | λ | . Proposition 1.6 ([Wil14, Theorem 6.39] ) . Every finitely generated representation V of degree k has a finite resolution0 → P ⊕ D → · · · → P k − ⊕ D k − → P k → V → , where P i is a direct sum of Schur projective of pure degree ≤ i and D i is directsum of some copies of D i +1 and D i +2 . Moreover, D k has an infinite projectiveresolution given by · · · → P k +2 → P k +1 → P k → D k → . Lemma 1.7.
We have the following:(1) if h (cid:54) = k + i then Ext i ( D k , D h ) = 0, and Ext i ( D k , D k + i ) = Q ,(2) let V a representation of degree k , then Ext i ( V, C λ ) = 0 if k < | λ | + i . Proof. (1) Consider the resolution P • of D k from Proposition 1.6, we compute the Extfunctor using the projective resolution Ext i ( D k , D h ) = H i (Hom( P • , D h )) . The complex Hom( P • , D h ) is almost trivial becauseHom F ( P k + i , D h ) ∼ = Hom S k + i ( V k + i , V ( k + i − h +1 , h − ) )is zero for k + i (cid:54) = h and for h = k + i is Q by the Schur Lemma.(2) Consider the resolution P • ⊕ D • of V from Proposition 1.6. It is enough toverify that Hom F ( P k − i ⊕ D k − i , C λ ) = 0, indeed Hom( D a , C λ ) = 0 becausethey are two different simple representations andHom F ( P µ , C λ ) = Hom S | µ | ( V µ ,
0) = 0 , since | µ | ≤ k − i < | λ | . (cid:3) Let V be a finitely generated representation of F of degree k . Define a canonicalfiltration F • such that F i ⊆ V is the submodule generated by V ([ j ]) for all j ≤ i . Example 1.8.
Consider the short exact sequence0 → D → P → D → , on the object [ n ], P ([ n ]) = Q n , D ([ n ]) ∼ = Q n − is the subobject of vectors whosecoordinates have zero sum, and D ([ n ]) ∼ = Q is their quotient. The sequence doesnot split and the degree of the subject D (equals to 2) is bigger then the degreeof P (equals to 1).The inclusion j : D ([ n ]) (cid:44) → P ([ n ]) has rank n −
1, but the graded map gr F j iszero. RDERED CONFIGURATION SPACES ON AN ELLIPTIC CURVE 5
Example 1.9.
In [Ryb18], minimal projective resolutions of C λ are described. Theminimal resolution of C (4) is0 → P (1 , → P (2 , ⊕ P (3) → P (4) → C (4) → . It follows from eq. (2) that dim Ext ( C (4) , C (2) ) = 1. Lemma 1.10.
Let V a representation of degree k , then the composition factors ofthe Jordan H¨older filtration are C λ for | λ | ≤ k or D i for i ≤ k + 1. Proof.
It is enough to prove the claim for V = P k a Schur projective representationof degree k . The homology functor H is defined by H ( V )([ n ]) = V /F n − V ([ n ]),this functor is right exact.Consider a Jordan H¨older filtration G • of P k . Proposition 6.32 of [Wil14] showsthat H ( G i )[ n ] = 0 for n > k +1 and that H ( G i )[ k +1] is sum of sign representationsof S k +1 . The short exact sequence H ( G i − )[ n ] → H ( G i )[ n ] → H ( G i /G i − )[ n ] → H ( G i /G i − )[ n ] = 0 for n > k + 1 and H ( G i /G i − )[ k + 1] contains onlythe sign representation. Now G i /G i − is simple and the homology of simple mod-ules of degree j is concentrate in degree j . The equalities H ( C λ )[ | λ | ] = V λ and H ( D k )[ k ] = V k imply the result. (cid:3) Theorem 1.11.
Let f : V → W be a morphism between two finitely generatedF-representations and gr F f : gr F V → gr F W be the corresponding graded map.Suppose that the composition factors of the Jordan H¨older filtration of V are dif-ferent from D i , for i ∈ N . Then:(1) gr F V is semisimple and the addenda of F n /F n − are of the type C λ for λ (cid:96) n ,(2) rk( f ) = rk(gr f ), i.e the map f is strict with respect the filtration F • . Proof.
Consider the composition factors of F n V /F n − V have degree at least n because F n V /F n − V ([ n − n by Lemma 1.10 using the hypothesisthat D n +1 does not appear. Therefore the composition factors are C λ for λ (cid:96) n .Since Ext ( C λ , C µ ) = 0 for λ, µ (cid:96) n , an inductive reasoning proves that F n /F n − is semisimple.Suppose that f ( y ) = x with y ∈ F n V \ F n − V and x ∈ F n − W , we show that x ∈ f ( F n − V ). Let ( x ) and ( y ) be the subrepresentation generated by x and y , f induces a surjective morphism f : ( y ) (cid:30) ( y ) ∩ F n − V → ( x ) (cid:30) ( x ) ∩ f ( F n − V ) . As noticed above H (( y ) / ( y ) ∩ F n − V ) is concentrate in degree n and does notcontain the sign representation, instead H (( x ) / ( x ) ∩ f ( F n − V ))[ n ] contains onlysome copies of the sign representation. Thus, H ( f ) is zero and surjective because f does. This implies H (( x ) / ( x ) ∩ f ( F n − V )) = 0, so ( x ) / ( x ) ∩ f ( F n − V ) = 0 andhence x ∈ f ( F n − V ). (cid:3) Representation theory of E ( X ) . From now on, we assume that χ ( X ) = 0. LetF( k, n ) be the set of all maps from [ k ] to [ n ]. The increasing filtration F • E ( X )defined in the previous section can be described as F k E ( X ) = (cid:88) f ∈ F( k,n ) Im f ∗ . R. PAGARIA
The module F k E ( X )[ n ] is, by definition, the submodule of E ( X, n ) generated byall monomials in G i,j and x i (for i, j ≤ n and x ∈ H • ( X )) with at most k differentindices. Example 1.12.
Let x ∈ H • ( X ) be any element. We have G , ∈ F E ( X ) \ F E ( X ), x − x ∈ F E ( X ) and x x ∈ F E ( X ) \ F E ( X ).We fix a graded basis { b i } i of H • ( X ) and assume b = 1 ∈ H ( X ). A labelledpartition λ of n is λ (cid:96) n whose blocks are decorate with elements of the fixedbasis. We define q ( λ ) = n − l ( λ ) the difference between n and the number of blocks l ( λ ), and p ( λ ) = (cid:80) l ( λ ) i =1 deg( λ i ) the sum of the cohomological degrees of all thelabels. Let L ( λ ) < S n be a subgroup generated by disjoint cycles of length λ i for i = 1 , . . . , l ( λ ), L ( λ ) is isomorphic to × l ( λ ) i =1 C λ i . Let N ( λ ) < S n be the subgroup ofthe stabilizer of L ( λ ) that permutes the cycles with the same labels, it is isomorphicto × j S n j for some n j . Finally, define Z ( λ ) < S n as the subgroup generated by L ( λ ) and N ( λ ), i.e. the semidirect product L ( λ ) (cid:111) N ( λ ).We denote the sign representation of S i by (cid:15) i and a faithful representation ofthe cyclic group C i by ϕ i . Let ϕ λ , α λ , ξ λ be the one dimensional representations of L ( λ ) , N ( λ ) , Z ( λ ) defined by: ϕ λ = (cid:15) n | L ( λ ) ⊗ ( (cid:2) l ( λ ) i =1 ϕ λ i ) ,α λ = (cid:2) j (cid:15) ⊗ m j j ,ξ λ = ϕ λ (cid:2) α λ , where m j = λ i + deg( λ i ) + 1 where λ i is any block permuted by S n j . Example 1.13.
Consider an elliptic curve C = ( S ) and the basis of H • ( C ) givenby 1 , x, y, xy . Let λ = (4 , , , , , , (cid:96)
19 with labels ( xy, xy, xy, , x, x, x ). Wehave l ( λ ) = 7, q ( λ ) = 12, p ( λ ) = 9. The associated groups are L ( λ ) ∼ = ( C ) × generated by (1 , , , , (5 , , , , (9 , , , , (13 , , , N ( λ ) ∼ = S × S generated by (1 , , , , , , , , , , Z ( λ ) = ( C (cid:111) S ) × C × S < S . The representations are ϕ λ = ϕ (cid:2) ϕ (cid:2) ϕ (cid:2) ϕ (because (cid:15) | C ( λ ) = 1), α = (cid:15) (cid:2) (cid:15) (because 4 + 2 + 1 and 1 + 1 + 1 are odd) and ξ λ = ( ϕ (cid:111) (cid:15) ) (cid:2) ϕ (cid:2) (cid:15) .A decomposition of E ( X, n ) into S n representations is provided in [AAB14]. Theorem 1.14.
Let X be a smooth projective variety. The Kriˇz model decomposesas E p,q ( X, n ) ∼ = (cid:77) q ( λ )= qp ( λ )= p Ind S n Z ( λ ) ξ λ . For each representation V = (cid:76) λ V n λ λ of S n , we define the F -representation C V = (cid:76) λ C n λ λ . For any labelled partition λ let f ( λ ) the number of blocks of size1 labelled with 1 ∈ H • ( X ). Theorem 1.15.
Let X be a smooth projective algebraic variety with χ ( X ) = 0.Then gr • F H • , • ( E ( X ) , d) = H • , • (gr • F E ( X ) , gr • F d)and for q > E p,q ( X ) has associated graded:gr rF E p,q ( X ) = (cid:77) λ C Ind S rZ ( λ ) ξ λ , RDERED CONFIGURATION SPACES ON AN ELLIPTIC CURVE 7 where the sum is taken over all labelled partitions λ (cid:96) r with p ( λ ) = p , q ( λ ) = q ,and f ( λ ) = 0. Proof.
We first prove that the sign representation (cid:15) n does not appear in E p,q ( X, n )for q >
0. We have (cid:104) (cid:15) n , E p,q ( X, n ) (cid:105) S n = (cid:88) λ (cid:104) (cid:15) n , Ind S n Z ( λ ) ξ λ (cid:105) S n = (cid:88) λ (cid:104) (cid:15) n | Z ( λ ) , ξ λ (cid:105) Z ( λ ) = (cid:88) λ (cid:104) (cid:15) n | L ( λ ) , ϕ λ (cid:105) L ( λ ) (cid:104) (cid:15) n | N ( λ ) , α λ (cid:105) N ( λ ) = (cid:88) λ (cid:104) L ( λ ) , (cid:2) (cid:107) λ (cid:107) i =1 ϕ λ i (cid:105) L ( λ ) (cid:104) (cid:15) n | N ( λ ) , α λ (cid:105) N ( λ ) = (cid:88) λ (cid:104) (cid:15) n | N ( λ ) , α λ (cid:105) N ( λ ) (cid:107) λ (cid:107) (cid:89) i =1 (cid:104) λ i , ϕ λ i (cid:105) C λi = 0 , because (cid:104) λ i , ϕ λ i (cid:105) C λi (cid:54) = 0 if and only if 1 λ i = ϕ λ i if and only if λ i = 1. The signrepresentation appears only in E p, ( X, n ).It follows that D i cannot be a composition factor of E p,q ( X ) for q >
0, so byTheorem 1.11 we obtain that d is strict.The second part of the statement follows from the fact that the submoduleInd S n Z ( λ ) ξ λ is contained in F r E p,q ( X )([ n ]) if and only if f ( λ ) ≥ n − r . Therefore, F r E p,q ( X ) (cid:30) F r − E p,q ( X )([ r ]) = (cid:77) λ s.t. f ( λ )=0 Ind S n Z ( λ ) ξ λ , and since gr rF E p,q ( X ) is semisimple, we obtain the claimed equality. (cid:3) Example 1.16.
In the case X = C , we have d( G i,j ) = ( x i − x j )( y i − y j ) andgr F d( G i,j ) = − x i y j − x j y i . Moreover gr F d( x i G i,j ) = x i x j y j − x i x j y i and noticethat the product of x i and G i,j in gr • F E ( C ) is zero. Remark 1.17.
Let M be a even-dimensional smooth manifold with χ ( M ) = 0.The analogous result of Theorem 1.15 holds for the Leray spectral sequence forthe inclusion Conf( M, n ) (cid:44) → M n . Indeed the collection of spectral sequences foreach n ∈ N is an F-module and the filtration F is strictly compatible with thedifferentials d i (for all i ∈ N ) because the sign representation can appears only inthe 0-th row (i.e. q = 0). Behaviour of mixed Hodge numbers.
Recall that every polynomial P ( t ) in Q [ t ] such that P ( n ) ∈ N for all n ∈ N can be written uniquely as P ( t ) = deg P (cid:88) i =0 a i (cid:18) ti (cid:19) , for some a i ∈ Z . The value a i not need to be positive. Example 1.18.
Consider dim H (Conf( S , n ); Q ) for n >
0, it is known that ispolynomial in n and it value is ( n − n − . However a = − H (Conf( S , n ); Q ) = ( n − n − (cid:18) n (cid:19) − (cid:18) n (cid:19) + (cid:18) n (cid:19) . R. PAGARIA
Indeed H (Conf( S )) ∼ = D as F-module (included in position (0 ,
2) of the Lerayspectral sequence).We collect the information about dim H p,q ( E ( X, n ) , d) in a polynomial P p,q ( n ) := dim H p,q ( E ( X, n ) , d) = p +2 q (cid:88) i =0 a p,qi (cid:18) ni (cid:19) , for some unique integers a p,qi . Proposition 1.19.
For q >
0, the coefficients a p,qi are positive and coincide with a p,qi = dim H p,q (cid:16) E ( X, i ) (cid:30) F i − E ( X, i ) , d (cid:17) . Proof.
We use Theorem 1.15 and the fact that gr iF E p,q ( X ) has not compositionfactors of type D k to obtain:dim H p,q ( E ( X, n ) , d) = (cid:88) i ≤ p +2 q dim gr iF H p,q ( E ( X, n ))= (cid:88) i ≤ p +2 q dim H p,q (gr iF E ( X, n ))= (cid:88) i ≤ p +2 q dim H p,q (cid:16) Ind S n S i × S n − i E ( X, i ) (cid:30) F i − E ( X, i ) (cid:17) = (cid:88) i ≤ p +2 q dim Ind S n S i × S n − i H p,q (cid:16) E ( X, i ) (cid:30) F i − E ( X, i ) (cid:17) = (cid:88) i ≤ p +2 q (cid:18) ni (cid:19) dim H p,q (cid:16) E ( X, i ) (cid:30) F i − E ( X, i ) (cid:17) Since N is infinite we obtain the corresponding equality between polynomials. (cid:3) The elliptic case
Let C be an elliptic curve, topologically C = ( S ) , the cohomology H • ( C ; Q )is the exterior algebra on two generator x and y such that x (cid:94) y = [ C ]. Theconstruction of the previous section is compatible with the action of SL ( Q ).We recall two results from [Pag18, Lemma 1.6, Theorem 3.9]. Proposition 2.1.
The cohomology H k (Conf( C, n )) vanishes for k > n + 1 and so a p,qi = 0 for p + q > i + 1 or p + 2 q < i . Moreover the mixed Hodge numbers for q = 0 are given by: P p, ( t ) = ( p + 1) (cid:18) tp (cid:19) + ( p − (cid:18) tp − (cid:19) . Indeed, an easy computation shows that H p, (Conf( C )) = ( P p (cid:2) V p ) ⊕ ( P p − (cid:2) V p − ) , as representation of F × SL ( Q ). Remark 2.2.
The multiplicity of the representation C ( k ) for k > • F H • (Conf( C )) can be deduced from [Pag19].For the sake of notation, we denote Ind S n + m S n × S m V (cid:2) W by V · W . RDERED CONFIGURATION SPACES ON AN ELLIPTIC CURVE 9
Lemma 2.3.
The following decomposition of S p +2 q × SL ( Q )-representationsholds: gr p +2 qF E p,q ( C, p + 2 q ) ∼ = (cid:98) p (cid:99) (cid:77) a =0 (cid:16) Ind S n S (cid:111) S q V (cid:2) q (2) ⊗ V (1 q ) (cid:17) · V (2 a , k ) (cid:2) V k , where p = 2 a + k . Proof.
From Theorem 1.15, we havegr p +2 qF E p,q ( C, p + 2 q ) ∼ = p (cid:77) b =0 (cid:16) Ind S q C (cid:111) S q V (cid:2) q (2) ⊗ V (1 q ) (cid:17) · V (1 p − b ) · V (1 b ) because the labelled partitions λ (cid:96) p + 2 q with p ( λ ) = p , q ( λ ) = q and f ( λ ) = 0 are(2 q , p − b , b ) for some b with blocks labelled respectively by 1, x , and y . Moreoverfor such λ we have ϕ λ = V (cid:2) q (2) and α λ = V (1 q ) (cid:2) V (1 p − b ) (cid:2) V (1 b ) . By definition themaximal torus of SL ( Q ) acts with weight p − b on the b -th addendum, sogr p +2 qF E p,q ( C, p + 2 q ) ∼ = (cid:98) p (cid:99) (cid:77) a =0 W · (cid:16) V (1 a + k ) · V (1 a ) (cid:9) V (1 a + k +1 ) · V (1 a − ) (cid:17) (cid:2) V k , where k = p − a and W = Ind S q C (cid:111) S q V (cid:2) q (2) ⊗ V (1 q ) . The representation V (1 a + k ) · V (1 a ) (cid:9) V (1 a + k +1 ) · V (1 a − ) has dimension (cid:0) pa (cid:1) − (cid:0) pa − (cid:1) . Using the Littlewood-Richardson rule(see [Ful97]), we observe that the representation V (2 a , k ) appears in V (1 a + k ) · V (1 a ) butnot in V (1 a + k +1 ) · V (1 a − ) . The hook formula shows that dim V (2 a , k ) = p !( k +1) a ! a + k +1! = k +1 a + k +1 (cid:0) pa (cid:1) and an easy computation show thatdim V (2 a , k ) = dim (cid:0) V (1 a + k ) · V (1 a ) (cid:9) V (1 a + k +1 ) · V (1 a − ) (cid:1) . Since the first representation is contained in the second one, we complete the proof. (cid:3)
Oyster partitions.
We need the Frobenius notation for partitions: for any se-quences a > a > · · · > a k > b > b > · · · > b k > a , . . . , a k | b , . . . , b k ) be the partition of (cid:80) ki =1 ( a i + b i −
1) such that the i -th row has length a i + i − i -th column has length b i + i − i ≤ k Let Q ( n ) be the setof all the partitions of n of the form ( a , . . . , a k | a − , . . . , a k − S n S (cid:111) S q V (cid:2) q (2) ⊗ V (1 q ) ∼ = (cid:77) λ ∈ Q (2 q ) V λ . Definition 2.4. A k - core partition of 2 q + k is any partition of the form ( λ , . . . , λ k )such that ( λ − , . . . , λ k −
1) is in Q (2 q ). A ( k, a )- shell partition is any partitionof the form ( b + 3 , . . . , b a + 3 | b , . . . , b a ) with b a > k . A ( k, a )- oyster partition is a partition ( c , . . . , c a + k | d , . . . , d a + k ) such that ( c , . . . , c a | d , . . . , d a ) is a( k, a )-shell partition and ( c a +1 , . . . , c a + k | d a +1 , . . . , d a + k ) is a k -core partition. Example 2.5.
Let k = 2 and consider the 2-core partition (4 ,
4) obtained from(3 , | , ∈ Q (6) and the (2 , , , the core give the following oyster partition λ = (6 , , V λ (cid:2) V is an addendum of V (4 , , · V (2 , , (cid:2) V ⊂ (cid:16) Ind S S (cid:111) S V (cid:2) ⊗ V (1 ) (cid:17) · V (2 , , (cid:2) V ⊂ gr F E , ( C, . Indeed, 6 = 3 + 2 + 1 and the partition (4 , ,
4) is in Frobenius notation (4 , , | , , V λ in V (4 , , · V (2 , , is one and correspond to the skew semistandar Young tableaux of shape(6 , , / (4 , ,
4) of content (2 , ,
1) shown above.
Lemma 2.6.
The module gr p +2 qF H p,q (Conf( C )) contains (cid:98) p (cid:99) (cid:77) a =0 (cid:77) λ ( k,a )-oyster C V λ (cid:2) V k , where k = p − a and the sum is taken over all ( k, a )-oyster partitions of p + 2 q . Proof.
It is enough to prove that for every ( k, a )-oyster partition λ (cid:96) p +2 q the repre-sentation V λ (cid:2) V k appears in gr p +2 qF E p,q ( C, p +2 q ) but not in gr p +2 qF E p +2 ,q − ( C, p +2 q ) neither in gr p +2 qF E p − ,q +1 ( C, p + 2 q ).For λ = ( λ , . . . , λ r ) consider the partition µ = ( µ , . . . , µ r ) such that µ i = λ i − i ≤ a , µ i = λ i − a < i ≤ a + k , and µ i = λ i for a + k < i . Byconstruction µ belongs to Q (2 q ) and so V µ appears in V (1 q ) [ V ]. Applying theLittlewood-Richardson rule to V µ · V (2 a , k ) , we obtain that V λ has multiplicity onein V µ · V (2 a , k ) . We have proven V λ (cid:2) V k ⊂ gr p +2 qF E p,q ( C ).In order to show that gr p +2 qF E p − ,q +1 ( C, p + 2 q ) does not contain V λ (cid:2) V k isequivalent to show that V λ is not contained in V η · V (2 a − , k ) for all η ∈ Q (2 q +2). Suppose by contradiction that there exist η ∈ Q (2 q + 2) and a Littlewood-Richardson skew tableau of shape λ/η and content (2 a − , k ). The i -th column of λ has length λ i − i ≤ a and the one of µ has length µ i − i . Since µ is contained in λ then µ i ≤ λ i − i ≤ a . In particular λ/η has at least 2 a boxes in the first a rows. From the reverse lattice word property of λ/η the number j cannot appear in the first j − λ/η has at most 2 a − a rows. We have obtained a contradiction.Suppose that V λ is contained in V η · V (2 a +1 , k ) for some η ∈ Q (2 q − η ∈ Q (2 q −
2) and the existence of a Littlewood-Richardson skewtableau of shape λ/η and content (2 a +1 , k ) imply that η i = λ i − i ≤ a . Theinequality η i ≤ λ i − η are shorter than the correspondingones of λ , the other inequality η i ≥ λ i − λ/η hasat most 2 i elements in the first i -th rows for all i . Therefore λ/η has at most a + k nonempty rows, but each Littlewood-Richardson skew tableau of content (2 a +1 , k )must have at least a + k + 1 nonempty rows. This is contrary to the hypothesisthat V λ is contained in V η · V (2 a +1 , k ) for some η ∈ Q (2 q − (cid:3) RDERED CONFIGURATION SPACES ON AN ELLIPTIC CURVE 11
Corollary 2.7.
We have the following lower bounds for the bi-graded Betti num-bers: gr q +2 F H ,q ( E ( C ) , d) ⊇ C ( q +3 | q ) (cid:2) V , gr F H , ( E ( C ) , d) = C (6 | (cid:2) V ⊕ C (4 , | , (cid:2) V , gr F H , ( E ( C ) , d) = C (5 | (cid:2) V ⊕ C (4 , | , (cid:2) V , gr F H , ( E ( C ) , d) = C (4 | (cid:2) V ⊕ C (3 | (cid:2) V , gr F H , ( E ( C ) , d) = C (3 | (cid:2) V . In particular dim gr q +2 F H ,q ( E ( C, n ) , d) ≥ (cid:0) q +1 q − (cid:1)(cid:0) n q +2 (cid:1) . Proof.
The first inclusion follows from Lemma 2.6 and the fact that ( q + 3 | q ) is an(0 , q + 2 for q >
0. The following oyster partitions (withempty shells): 12 12 12 1imply that the right hand sides are contained in the left hand sides. The othercontainment follows from a dimensional argument: the dimensions of the left handsides are computed in Tables 7, 8 and 10 and eq. (11) and coincide with the dimen-sion of the representations on the right. Finally, since dim V ( q +3 , q − ) = (cid:0) q +1 q − (cid:1) , itfollows that dim C V ( q +3 , q − ([ n ]) (cid:2) V = (cid:0) q +1 q − (cid:1)(cid:0) n q +2 (cid:1) . (cid:3) Upper bounds for Betti numbers.
We denote by E p,q ( C ) k the subspace in E p,q ( C ) of highest vectors for SL ( Q ) of weight k and similar for gr F E p,q ( C ) k andfor H p,q ( E ( C ) , d) k . Therefore E p,q ( C ) ∼ = (cid:98) p (cid:99) (cid:77) a =0 E p,q ( C ) p − a (cid:2) V p − a . In order to give upper bounds for Betti numbers, we need the following result.
Lemma 2.8.
The following cohomology groups are zero: H ,q ( E ( C ) , d) = 0 q > , (3)gr q +1 F H ,q ( E ( C ) , d) = 0 q > , (4)gr qF H ,q ( E ( C ) , d) = 0 q > , (5)gr q +2 F H ,q ( E ( C ) , d) = 0 q > . (6) Proof.
Equation (3) is proven in [AAB14, Proposition 1.2].For eq. (4) we proceed by induction, the base case follows from the entry (1 , G • the filtration of the complex D r = ⊕ q gr rF E r − q,q ( C ) for fixed r defined by G = 0, G = D r and G q = (cid:104) x α, y α | α ∈ E r − − q,q ( C, { , . . . , r } ) (cid:105) . The complex G is isomorphic to two copies of gr r − F E r − − q,q ( C ) and the quotientcomplex G /G is identified with 2 q copies of gr r − F E r − q,q − ( C ) (one for each G i ).For r = 2 q + 1 we have:dim gr q +1 F H ,q ( E ( C ) , d) ≤ dim H q ( G ) + dim H q ( G /G ) = 2 dim gr qF H ,q ( E ( C ) , d) + 2 q dim gr q − F H ,q − ( E ( C ) , d) . The first addendum is zero by eq. (3) and the second one by inductive step.For eq. (5) we proceed by induction, the base case follows from the entry (1 , G (cid:48) • of D (cid:48) r = ⊕ q gr rF E r − q +1 ,q ( C ) defined by G (cid:48) = 0, G (cid:48) = D (cid:48) r , G (cid:48) = (cid:104) x α, y α, x y β | α, β w/o index 1 (cid:105) ,G (cid:48) = G (cid:48) + (cid:104) G ,i α, G ,i x β, G ,i y β, G ,i x y γ | α, β, γ w/o indices 1 , i (cid:105) i =2 ,...,r . For r = 2 q , we have H q ( G (cid:48) ) ∼ = gr q − F H ,q ( E ( C )) ⊕ y gr q − F H ,q ( E ( C )) = 0 (7)by eq. (3). Similarly, H q ( G (cid:48) /G (cid:48) ) is equal togr q − F H ,q − ( E ( C )) ⊕ q − ⊕ gr q − F H ,q − ( E ( C )) ⊕ q − = 0 (8)by inductive step and by eq. (3). The top cohomology vanishes: H q ( G (cid:48) /G (cid:48) ) ∼ = (cid:77)
Corollary 2.9.
The Betti numbers of Conf(
C, n ) are: b = 1 ,b = 2 n,b = 2 (cid:18) n (cid:19) + 3 (cid:18) n (cid:19) + n,b = 14 (cid:18) n (cid:19) + 8 (cid:18) n (cid:19) + 2 (cid:18) n (cid:19) ,b = 32 (cid:18) n (cid:19) + 74 (cid:18) n (cid:19) + 32 (cid:18) n (cid:19) + 5 (cid:18) n (cid:19) ,b = 63 (cid:18) n (cid:19) + 427 (cid:18) n (cid:19) + 490 (cid:18) n (cid:19) + 154 (cid:18) n (cid:19) + 18 (cid:18) n (cid:19) , RDERED CONFIGURATION SPACES ON AN ELLIPTIC CURVE 13 b k = c k (cid:18) n k − (cid:19) + o ( n k − ) , where c k ≥ (cid:0) k − k − (cid:1) . Proof.
The case b i for i ≤ C, n ) and of Conf(
C, n ) /C differ by a factor (1 + t ) and the case q = 0 follows from Proposition 2.1. The case b need the vanishingresults of Lemma 2.8 and eq. (11).For general k , we have b k ( n ) = (cid:88) p + q = k p +2 q (cid:88) i = p + q − (cid:18) ni (cid:19) dim H p,q (cid:16) E ( C, i ) (cid:30) F i − E ( C, i ) (cid:17) . Eq. (3) and (4) ensure that the polynomial has degree at most 2 k −
2. Eq. (5) and(6) implies that b k ( n ) = (cid:18) n k − (cid:19) dim H ,k − (cid:16) E ( C, k − (cid:30) F k − E ( C, k − (cid:17) + o ( n k − ) . Finally, Corollary 2.7 implies the desired result. (cid:3)
Conjecture 2.10.
We claim that b k = (cid:18) k − k − (cid:19)(cid:18) n k − (cid:19) + o ( n k − ) . Remark 2.11.
Let S be a connected orientable surface of finite type, the stablerange for H k (Conf( S, n ); Q ) in the sense of [CF13] is n ≥ k as proven in [Chu12,Theorem 1]. The above discussion implies that for k > S = C is n ≥ k − Appendix A. Small cases
The elliptic curve C acts on Conf( C, n ) by p · ( p , . . . , p n ) = ( p + p, . . . , p n + p )where + is the group operation on the elliptic curve C . This action is compatiblewith the structure of F-module and the fibration C → Conf(
C, n ) → Conf(
C, n ) (cid:30) C has a non-canonical section s : Conf( C, n ) (cid:30) C → Conf(
C, n ). This induces an iso-morphism H • (Conf( C, n )) ∼ = H • ( C ) ⊗ H • (Conf( C, n ) /C ) (10)as rings, but not as F-modules because the section cannot be chosen in an equi-variant way. We used a Python3 to compute the cohomology of Conf( C, n ) /C for n ≤
7. The code is available at . and the bigraded Betti numbers arepresented in Tables 1 to 6.Since gr rF H p,q (Conf( C )) is semisimple for q >
0, we have a decomposition anal-ogous to eq. (10):gr • F H • ,q (Conf( C, n )) ∼ = gr • F H • ( C ) ⊗ gr • F H • ,q (Conf( C, n ) /C )Tables 7 to 11 report the numbers a p,qi for q > • F H p,q (Conf( C, n ) /C ). These entries are computed from the corresponding onesof Tables 1 to 6 as convolution with binomial coefficients.
01 2
Table 1.
The dimension of the cohomology H p,q (Conf( C, /C ).00 21 4 3 Table 2.
The dimension of the cohomology H p,q (Conf( C, /C ).00 40 8 101 6 9 4 Table 3.
The dimension of the cohomology H p,q (Conf( C, /C ).00 120 20 380 20 50 241 8 18 16 5 Table 4.
The dimension of the cohomology H p,q (Conf( C, /C ).As an example we consider dim H , (Conf( C, n ) /C ), the values in tables Tables 1to 6 are 0 , , , , , (cid:18) n (cid:19) + 32 (cid:18) n (cid:19) at n = 2 , , , , ,
7. The coefficient of this polynomial are reported in the corre-sponding entries (2 ,
2) of Tables 7 to 11. (Notice that in those tables the 0-th rowis omitted.)The values of Tables 1 and 2 corresponding to the cases n = 2 , H , (Conf( C, /C ; Q ) = 7063 , (11)and so a , = 63. Acknowledgement.
I would thank Gian Marco Pezzoli for the useful discussionsand John Wiltshire-Gordon for notifying me the reference [Ryb18].
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The coefficients a p,q of the F-module H p,q (Conf( C ) /C )( q >
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Roberto Pagaria
Universit`a di Bologna, Dipartimento di Matematica , Piazza di Porta San Donato 5 - 40126Bologna, Italy.
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