PROBs and perverse sheaves I. Symmetric products
aa r X i v : . [ m a t h . C T ] F e b PROBs and perverse sheaves I. Symmetric products
Mikhail Kapranov, Vadim SchechtmanMarch 1, 2021
Abstract
Algebraic structures involving both multiplications and comultiplications (such as,e.g., bialgebras or Hopf algebras) can be encoded using PROPs (categories with PROd-ucts and Permutations) of Adams and MacLane. To encode such structures on objectsof a braided monoidal category, we need PROBs (braided analogs of PROPs). ColoredPROBs correspond to multi-sorted structures.In particular, we have a colored PROB B governing Z ě -graded bialgebras inbraided categories. As a category, B splits into blocks B n according to the grad-ing. We relate B n with the category P n of perverse sheaves on the symmetric product Sym n p C q smooth with respect to the natural stratification by multiplicities. More pre-cisely, we show that P n is equivalent to the category of functors B n Ñ Vect . This givesa natural quiver description of P n . Let k be a field. All vector spaces in this paper will be assumed k -vector spaces and allcategories will be assumed k -linear. A. PROPs and PROBs.
The concept of a PROP was introduced by Adams andMacLane [1, 23]. It allows one to axiomatize algebraic structures on a vector space (or, moregenerally, on an object A of a symmetric monoidal category) involving both multiplications A b m Ñ A and comultiplications A Ñ A b n . See [27] for a modern exposition.The term PROP is an abbreviation for “category with PROducts and Permutations”.Explicitly, a PROP is a symmetric monoidal category p P , b , q with Ob p P q “ tr m s , m P Z ` u identified with the set of non-negative integers so that the tensor operation b is, on objects,given by the addition of integers: r m s b r n s “ r m ` n s and the unit object is “ r s . In otherwords, a PROP is a strict symmetric monoidal category whose objects are tensor powers ofa single object, denoted r s .Given a symmetric monoidal category V , we can speak about algebras in V over a PROP P . Such an algebra is simply a symmetric monoidal functor F : P Ñ V . If we denote A “ F pr sq P V , then each space Hom P pr m s , r n sq is mapped into the space of mixed operations A b m Ñ A b n . 1 xample 1.1 (The PROP of Hopf algebras). A basic example of a structure involvingmultiplications and comultiplications is that of a Hopf algebra. A Hopf algebra in a sym-metric monoidal category V is an object A together with a multiplication µ : A b A Ñ A ,comultiplication c : A Ñ A b A , unit e : Ñ A and counit η : A Ñ satisfying thewell known relations. The corresponding PROP, denote it HS , can be seen as the symmet-ric monoidal category generated by the universal Hopf algebra a “ r s . That is, for anysymmetric monoidal category V , we have a bijection between:(i) Hopf algebras A in V ;(ii) Symmetric monoidal functors F : HS Ñ V ,given by A “ F p a q . At a more intuitive level, HS contains the morphisms µ : a b a Ñ a andsimilarly c, e, η as above, together with all their iterated compositions and tensor products,which are subject to the relations of a Hopf algebra “and nothing else”.Along with Hopf algebras, we can consider more general objects, namely bialgebras insymmetric monoidal categories. Thus a bialgebra A has compatible associative µ : A b A Ñ A and coassociative c : A Ñ A b A but the unit and counit are not required. As before, wehave the PROP HS ` describing bialgebras. It was introduced by Markl [26].Further, it is well known [25, 29] the concept of a Hopf algebra (or, more generally of abialgebra) can be defined in any braided, not necessarily symmetric, monoidal category V .To study such structures we need to modify the concept of a PROP to that of a PROB (withBraidings instead of Permutations). Thus, a PROB is a braided monoidal category whoseobjects are tensor powers of a single object a “ r s . Example 1.2 (The PROB of braided Hopf algebras).
As before, there is a PROB H such that Hopf algebras in a braided monoidal category V are in bijection with braidedmonoidal functors H Ñ V . Its objects are tensor powers of the universal braided Hopf algebra a “ r s P H and morphisms are iterated compositions and tensor products of the generatingmorphisms µ, c, e, η as above subject to the relations of a braided Hopf algebra. This PROBwas introduced by Habiro [13] who denoted it x H y .Despite their deceptively short definitions by universal properties, the PROP HS and thePROB H are quite non-trivial objects. In particular, their stucture as ordinary categories,i.e., some description of the spaces Hom pr m s , r n sq for all m, n , is not so easy to pin down.For the simpler PROP HS ` such a description was obtained by Pirashvili [28].The goal of this paper and the one to follow [20] is to relate versions of the PROB H to quivers describing perverse sheaves on certain configuration spaces. Let us describe theversion relevant for this paper. B. The PROP of graded bialgebras.
By a graded bialgebra in a braided monoidalcategory V we mean a bialgebra decomposed into a direct sum A “ À n ě A n so that A “ is the unit object, the multiplication and comultiplication are homogeneous and their2omponents involving A are the identities. See [17] where such objects were called primitivebialgebras. A graded bialgebra automatically has a unit, counit and antipode, see [17], Prop.2.4.11.In order to be able to speak about graded bialgebras, the existence of direct sums in V is, strictly speaking, not necessary as all the conditions can be reformulated in terms of theindividual graded components A n . Therefore we will understand a graded bialgebra A asa collection of these components: A “ p A n q n ě . The direct sum À n ě A n can be alwaysconsidered as an object of the formal direct sum completion of V but we need not requirethat it belongs to V .As before, there is a braided category B generated by the universal graded bialgebra a “ p a n q n ě , a “ . Objects of B are formal tensor products a α “ a α b ¨ ¨ ¨ b a α r associated to all the ordered partitions , i.e., sequences of positive integers α “ p α , ¨ ¨ ¨ , α r q .Morphisms are generated by the elementary formal morphisms µ p,q : a p b a q Ñ a p ` q , ∆ p,q : a p ` q Ñ a p b a q , µ p, “ µ ,p “ ∆ p, “ ∆ ,p “ Id a p , as well as the braidings, modulo the relations following from the axioms of a graded bialgebraand a braided category.The braided category B is an example of a colored PROB in that it describes algebraicstructures not on a single object but on a family of objects p A n q of a braided category. Fora discussion of colored PROPs, not PROBs see [12].Let OP be the set of all ordered partitions α as above and P Ă OP be the set of unordered (or classical) partitions , i.e., sequences α “ p α ě ¨ ¨ ¨ ě α r ą q . The a α , α P P form a system of representatives of isomorphism classes of objects of B .For α P OP let | α | “ ř α i . Let OP n Ă OP , resp. P n Ă P consist of α such that | α | “ n ,i.e., that α is an ordered resp. unordered partition of n . Let B n Ă B be the full subcategoryon objects a α with α P P n . It is not closed under the product b . The full subcategory onthe a α , α P OP n , is equivalent to B n . C. Symmetric products and perverse sheaves.
Let
Sym n p C q be the n th symmetricproduct of C , i.e., the space of monic polynomials f p x q “ x n ` a x n ´ ` ¨ ¨ ¨ ` a n , a i P C or, equivalently, the space of effective divisors z “ ř i n i z i with z i P C , n i ě , ř n i “ n .Each such divisor z has a type which is the partition t p z q P P n obtained by arranging the n i in a non-increasing order, and we denote Sym α p C q Ă Sym n p C q the subspace of z of type α .This gives an algebraic Whitney stratification of Sym n p C q which we denote S p q and call the stratification by multiplicities . The open strarum of S p q is Sym n ‰ p C q “ Sym p , ¨¨¨ , q p C q , the space of multiplicity-free divisors or, equivalently, of polynomials f p x q with non-zerodiscriminant. 3et V be any abelian category (not assumed monoidal). We can then speak about V -valued perverse sheaves (with respect to the middle perversity) on Sym n p C q which areconstructible with respect to the stratification S p q , see [17]. They form an abelian categorywhich we denote Perv p Sym n p C q ; V q . For example, if V “ Vect k is the category of k -vectorspaces, then Perv p Sym n p C q ; V q is the category of perverse sheaves of k -vector spaces in theusual sense. Here is our main result, whose proof will be given at the end of Section 4. Theorem 1.3.
We have an equivalence of categories
Perv p Sym n p C q ; V q » Fun p B n , V q . In other words, we have an elementary, or quiver description of the category of perversesheaves on
Sym n p C q . The corresponding quiver (with relations) is the category B n . Thevertices of this quiver are the objects of B n , i.e., the a α , α P P n . They are in bijection withthe strata Sym α p C q of the stratification S p q . Remark 1.4.
The monoidal structure b m,n : B m ˆ B m Ñ B m ` n corresponds, at the levelof perverse sheaves, to the functor (“comonoidal structure”) ∇ m,n : Perv p Sym m ` n p C q ; V q ÝÑ Perv p Sym m p C q ˆ Sym n p C q ; V q defined geometrically as follows. Choose two disjoint open disks D, D Ă C so that we haveopen embeddings Sym m p C q ˆ Sym n p C q i ÐÝ Sym m p D q ˆ Sym n p D q j ÝÑ Sym m ` n p C q such that the pullback functor i ˚ defines an equivalences on the categories of perverse sheaveswith respect to the natural stratifications. Then ∇ m,n “ p i ˚ q ´ ˝ j ˚ . D. The simplest example. (1) Let n “ . The category B has two objects, a b a and a , and its morphisms are generated by: R < < a b a µ , / / a , o o with R being the braiding, therefore invertible. Note that a is primitive, i.e., ∆ | a “ ∆ , ` ∆ , : a ÝÑ p a b q ‘ p b a q is the sum of two copies of the identity morphism of a . This implies that ∆ , µ , “ Id ` R ,cf. [17], §5.2. In particular, Id ´ ∆ , µ , “ ´ R is invertible. This means that Fun p B , V q isthe category of diagrams in V Ψ b / / Φ a o o such that T Ψ “ Id Ψ ´ ab is invertible (which implies that T Φ “ Id Φ ´ ba is invertible, see Eq.(1.1.6) of [17]). 42) On the other hand, Sym p C q “ C is the space of quadratic polynomails x ` a x ` a , and the only non-open stratum of S p q is the parabola given by a “ a formed bypolynomials with a double root. Factoring out by translational symmetry, we find that Perv p Sym p C q ; V q » Perv p C , V q is identified with the category of V -valued perverse sheaves on C with the only possiblesingularity at .The n “ instance of Theorem 1.3 reduces therefore to (a V -valued version of) theclassical result of [3, 10] describing Perv p C ,
0; Vect k q in terms of p Φ , Ψ q -diagrams of vectorspaces as above. E. Discussion and further plans.
Theorem 1.3 can be seen as a refinement of twoprevious results:(1) The main result of [17] which identifies the category of graded bialgebras A “ p A n q n ě in a braided monoidal V with that of factorizable systems p F n q n ě of perverse sheaves on allthe symmetric products Sym n p C q . The refinement consists in passing from such factorizablesystems to individual perverse sheaves on an individual symmetric product and in allowing V to be an arbitrary abelian (not necessarily monoidal) category.(2) The special case g “ gl n of the main result of [19] which describes perverse sheaveson any quotient W z h where h is the Cartan subalgebra of a complex reductive Lie algebra g and W is the Weyl group. If g “ gl n , then W z h “ Sym n p C q , and we get a description of Perv p Sym n p C qq . This description, however, is more cumbersome than Theorem 1.3 so thepresent refinement consists in giving a neater one-shot description.Our proof of Theorem 1.3 uses the results (1) and (2) above by assembling the categoriesconstructed in (2) into a single braided category CM and constructing a graded bialgebra inthis category so that application of (1) leads to an identification CM „ B .It is also interesting to note the similarity between the description of the PROP HS ` given by Pirashvili [28] and the description (2) above proceeding in terms of so-called mixedBruhat sheaves [19]. Both descriptions involve natural “bivariant” objects: covariant in onedirection, contravariant in the other with some base change-type relations relating the twovariances.In a sequel to this paper [20] we plan to describe the category of perverse sheaves on the Ran space
Ran p C q in terms of a category related to the PROB H governing braided Hopfalgebras.Both papers can be seen as developing the observation, going back to Lurie, that bialge-bras are Koszul dual to E -algebras and thus [6, 22] to locally constant factorization algebrason R “ C i.e., to factorizable (complexes of) sheaves on the Ran space. Informally, an E -algebra can be seen as a cochain complex E with two (homotopy) compatible (homotopy)associative multiplications. Now, Koszul duality gives an equivalence between associativedg-algebras and coalgebras. Applying it to one of the two multiplications on E , we get a5tructure consisting of (homotopy) compatible multiplication and comultiplication, i.e., ahomotopy version of a dg-bialgebra E ! . Koszul duality being a derived equivalence, for E ! to be an honest (non-dg) bialgebra, E must be a nontrivial complex. What makes our ap-proach work is a remarkable match between this type of complexes and the Cousin complexesplaying an essential role in our earlier descriptions of perverse sheaves [17, 19]. F. Outline of the paper.
Apart from the present introductory §1, the paper has threemore sections.In §2 we, first, give background material on braided categories and graded bialgebras insuch categories. In particular, we give a self-contained treatment of Deligne’s interpretationof braidings in terms of “ -dimensional tensor products” of objects labelled by points inthe plane. We recall the concept of contingency matrices and their vertical and horizontalcontractions from [18]. We further associate to a contingency matrix M “ } m ij } and a gradedbialgebra A an object A M which is the -dimensional tensor product of the components A m ij . We use the multiplication and comultiplication in A to connect the objects A M , and A N whenever M is obtained from N by a vertical or horizontal contractions and establish(Proposition 2.8) a system of relations for such connecting morphisms.In §3 we define a category CM whose objects are symbols r M s associated to contingencymatrices M and the relations of Proposition 2.8 are promoted into into a system of definingrelations for the morphisms of the category. Thus any graded bialgebra A in any braidedcategory V gives rise to a functor ξ A : CM Ñ V (Corollary 3.2) sending r M s to A M . Wefurther make CM into a braided monoidal category so that the functor ξ A above is in factbraided monoidal (Proposition 3.9).In §4 we notice that CM carries a braided bialgebra a with components r n s associated to ˆ matrices n . This allows us to connect B and CM by a braided monoidal functor whichwe show to be an eqivalence (Theorem 4.3). Finally, for each n we compare the degree n block CM n with the specialization, for g “ gl n , of the concept of mixed Bruhat sheaf whichwas introduced in [19] for description of perverse sheaves on the adjont quotient W z h of areductive Lie algebra g . This comparison yields the first in the two identifications below Perv p Sym n p C q ; V q » Fun p CM n , V q Thm. 4.3 » Fun p B n , V q thus proving Theorem 1.3. G. Acknowledgements.
The research of M.K. was supported by the World PremierInternational Research Center Initiative (WPI Initiative), MEXT, Japan.6
Graded bialgebras and contingency matrices
A. Braids, braided categories and bialgebras.
Let Br n be the braid group on n strands, with the standard generators σ , ¨ ¨ ¨ , σ n ´ and relations(2.1) σ p σ p ` σ p “ σ p ` σ p σ p ` , σ p σ q “ σ q σ p , | p ´ q | ě . Let also S n be the symmetric group with the standard generators σ , ¨ ¨ ¨ , σ n ´ subject tothe same relations as in (2.1) together with σ i “ . Thus we have the surjective morphism p : Br n Ñ S n .By a monoidal category we will mean a strictly associative monoidal category, with a strictunit object denoted by . Let V be a braided monoidal category with braiding denoted by R “ p R V,W : V b W Ñ W b V q .Given objects V , ¨ ¨ ¨ , V n of V , and an element b P Br n , we have the permutation t “ p p b q P S n , and the braiding isomorphism R b : V b ¨ ¨ ¨ b V n ÝÑ V t p q b ¨ ¨ ¨ b V t p n q . A bialgebra in V is an object A of V equipped with an associative multiplication µ : A b A Ñ A and a coassociative comultiplication ∆ : A Ñ A b A satisfying the followingcompatibility condition: ∆ is a morphism of algebras, if the multiplication on A b A isdefined using the braiding: A b A b A b A Id b R A,A b Id ÝÑ A b A b A b A µ b µ ÝÑ A b A. B. Geometric interpretation of braided categories.
Let V be a category, I be afinite set and p V i q i P I be a family of objects of V labelled by I . If V is symmetric monoidal,then we can speak about the object  i P I V i without specifying an order on I : the orderedtensor products in different orders are canonically identified with each other.If V is braided monoidal, then the notation  i P I V i does not make sense, as there is nosingle canonical identification between two given ordered products. It was pointed out byDeligne that the correct structure on I to make the product canonical is not an ordering(which of course suffices) but an embedding I ã Ñ C . In other words, once we assign toeach i P I a complex number z i P C such that z i ‰ z j for i ‰ j , there is a “well-definedobject”  i P I V i p z i q (the -dimennsional tensor product with respect to V i positioned as z i ).When the z i move, these objects unite into a local system on C I ‰ whose monodromy givesthe braiding. For convenience of the reader we recall a precise elementary construction. Definition 2.2.
Let V be a category.(a) A pseudo-object (or an object defined up to a unique isomorphism) of V is a datum V of a set K , of objects V k P V for each k P K and of isomorphisms ϕ k,k : V k Ñ V k givenfor each k, k and satisfying ϕ kk “ Id , ϕ k ,k ϕ k,k “ ϕ k,k , @ k, k , k . V k are called the determinations of the pseudo-object V , and the morphisms ϕ kk are called the transition maps of V .(b) A morphism u from a pseudo-object V “ p V k , ϕ k,k q k,k P K to a pseudo-object W “p W l , ψ l,l q l,l P L of V is a datum of morphisms u k,l : V k Ñ W l for all k P K , l P L such that u k,l “ u k ,l ϕ k,k , u k,l “ ψ l,l u k,l , @ k, k P K, l, l P L. With this definition, pseudo-objects in V form a category Ps p V q . Any actual object of V can be considered as a pseudo-object with | K | “ . This gives a functor V Ñ Ps p V q whichis easily seen to be an equivalence. In this way a pseudo-object can be seen to be “as goodas an actual object” of V .We will construct  i P I V i p z i q as a pseudo-object and start with describing its indexingset K . Definition 2.3.
Let Z Ă C be a finite subset, | Z | “ n ą . A Z -snake is a simple curve S Ă C which is a finite perturbation of R and passes through each element of Z once. Seethe center and right of Fig. 1. ‚ ‚ ‚ t t t n ¨ ¨ ¨´ ‚ ‚ ‚ S z i p z i p ` z i p ` ‚ ‚ ‚ σ p S z i p z i p ` z i p ` Figure 1: Snakes as the images of the interval p´ , q in the disk.Denote by Sn p Z q the set of isotopy classes of Z -snakes. A snake being oriented from p´8q to p`8q , each S P Sn p Z q gives an ordering i “ i p S q , ¨ ¨ ¨ , i n “ i n p S q of the set I . Itis classical that Sn p Z q is a left torsor over Br n , with σ p S being obtained from S by an “uppertwist” reversing the path between z i p p S q and z i p ` p S q , see the right of Fig. 1. Conceptually,this follows from the interpretation of Br n as the mapping class group of the n -pointed disk[7, 8]. More precisely, let D “ t| z | ď u be the closed unit disk in C with B D “ S the unitcircle and let ´ ă t ă ¨ ¨ ¨ ă t n ă be real numbers. Then Br n » π p H q , where H “ Homeo p D, Id B D , t t , ¨ ¨ ¨ , t n uq is the group of homeomorphisms of D , identical on B D and preserving t t , ¨ ¨ ¨ , t n u as a set.At the same time, let C “ C Y S be the disk compactification of C by S , the circle ofdirections at . 8 roposition 2.4. We have an identification Sn p Z q » π p M q , where M is the space ofhomeomorphisms f : D Ñ C which restrict to the standard identification B D “ S Ñ S onthe boundary and take t t , ¨ ¨ ¨ , t n u to Z . The snake corresponding to f P M is the image f pp´ , qq (flattened near ˘8 to coincide with R there). See Fig. 1. Proof:
This is equivalent to the classical encoding of braids by curve diagrams, see [7] §1.3.3,esp. Fig. 2 there.As M is a torsor over H , we have that Sn p Z q “ π p M q is a torsor over Br n “ π p H q . Definition 2.5.
Let V be a braided category, p V i q i P I , | I | “ n be a family of objects and z i P C , i P I be distinct complex numbers. Let Z “ t z i u i P I Ă C . We define a pseudo-object  i P I V i p z i q of V with the indexing set K “ Sn p Z q . For S P Sn p Z q the correspondingdetermination is defined as ˆâ i P I V i p z i q ˙ S “ V i p S q b ¨ ¨ ¨ b V i n p S q , the ordered tensor product along the snake. Given two snakes S, S P Sn p Z q with S “ bS , b P Br n , we define the transition map ϕ S,S : “ R b : V i p S q b ¨ ¨ ¨ b V i n p S q ÝÑ V i p S q b ¨ ¨ ¨ b V i n p S q to be the braiding isomorphism associated to β . C. Factorization algebra point of view on a braided category.
It is convenientto extend Definition 2.5 slightly, bringing it close to the formalism of factorization algebras[6]. By a closed disk in C we mean a subset D Ă C which is either homeomorphic to thestandard disk t| z | ď u Ă C or is a single point (a disk of radius ). Let p V i q i P I be as aboveand Z i Ă C be disjoint closed disks. We define â i P I V i p Z i q : “ â i P I V i p z i q , @ z i P Z i , i P I. Formally, we view the objects in the RHS of this definition as the stalks of a V -valued localsystem on the contractible space ś i P I Z i and we define the LHS as the object of globalsections of this local system.Alternatively, let Z “ Ť i P I Z i . Then we can speak about Z - snakes that is, simple curves S Ă C which are finite perturbations of R which intersect each Z i along a closed interval(possibly reducing to a single point). A Z -snake defines an ordering on I “ π p Z q . As before,the set Sn p Z q of isotopy classes of Z -snakes is a Br n -torsor, and we can define  i P I V i p Z i q as a pseudo-object with indexing set Sn p Z q consisting of ordered tensor products along thesnakes. 9 roposition 2.6. (1) Let V be a braided category, f : J Ñ I a surjection of finite sets and p V j q j P J a family of objects of V . Let p Z i q i P I and p Y j q j P J be two families of closed disks in C ,each consisting of disjoint disks and such that Y j Ă Z f p j q , j P J . In each such situation wehave a canonical associativity isomorphism α f : â i P I ˆ â j P f ´ p i q V j p Y j q ˙ p Z i q ÝÑ â j P J V j p Y j q . These isomorphisms satisfy the following compatibility.(2) Let K g Ñ J f Ñ I be two composable surjections of finite sets, and p V k q k P K be a familyof objects of V . Let p X k q k P K , p Y j q j P J and p Z i q i P I be three families of closed disks in C , eachconsisting of disjoint disks and such that X k Ă Y g p k q , k P K and Y j Ă Z f p j q , j P J . Foreach i P I let g i : p f g q ´ p i q Ñ f ´ p i q be the restriction of g . Then the following diagram iscommutative:  i P I ˆ  j P f ´ p i q ˆ  k P g ´ p j q V k p X k q ˙ p Y j q ˙ p Z i q α f / /  i P I α gi (cid:15) (cid:15)  j P J ˆ  k P g ´ p j q V k p X k q ˙ p Z j q α g (cid:15) (cid:15)  i P I ˆ  k Pp fg q ´ p i q V k p X k q ˙ p Z i q α fg / /  k P K V k p X k q Proof (sketch): (1) Let n “ | I | , m “ | J | , m i “ | f ´ p i q| , so m “ ř i P I m i . Let also Z “ ď i P I Z i , Y “ ď j P J Y j , Y { i “ ď j P f ´ p i q Y j , i P I. Call a p Y, Z q - snake a curve S which is both a Z -snake and a Y { i -snake for each i . A p Y, Z q -snake is also a Y -snake. Let Sn p Y, Z q be the set of isotopy classes of p Y, Z q -snakes. It is atorsor over the subgroup B in Br m which is the wreath product of the Br m i . We can viewboth the source and target of the desired α f as pseudo-objects with the indexing set Sn p Y, Z q ,after which the map α f is defined as the identity on each determination (corresponding toeach p Y, Z q -snake).To prove (2), we introduce, similarly to (1), the concept of p X, Y, Z q -snakes and theset Sn p X, Y, Z q of their isotopy classes. After this, all the arrows in the diagrams can beseen as morphisms of pseudo-objects indexed by Sn p X, Y, Z q and the statement becomesobvious. D. Contingency matrices.
We recall some constructions and terminology of [18]. Bya contingency matrix we mean a rectangular matrix M “ } m ij } j “ , ¨¨¨ ,qi “ , ¨¨¨ ,p with m ij P Z ě suchthat each row and each column contain at least one non-zero entry. We will also formallyinclude the contingency matrix H of size ˆ . The weight of M is the number Σ M “ ÿ i,j m ij P Z ą , M ‰ H , Σ H “ .
10e denote by CM the set of all contingency matrices, by CM n the set of all contingencymatrices of weight n , by CM p r, s q the set of all contingency matrices of size r ˆ s and put CM n p r, s q : “ CM n X CM p r, s q . We have the horizontal and vertical contractions B j : CM n p r, s q ÝÑ CM n p r, s ´ q , i “ , ¨ ¨ ¨ , s ´ , B i : CM n p r, s q ÝÑ CM n p r ´ , s q , j “ , ¨ ¨ ¨ .r ´ , which add up the p j ` q st and the p j ` q nd column (resp. p i ` q st and p i ` q nd row). Thesecontractions make the collection of CM n p r, s q for all r, s into an augmented bi-semisimplicialset, see [18], Prop. 1.4.For M, N P CM n we put M ď N , if M can be obtained from N by a series of horizontalcontractions B j and M ď N , if M can be obtained from N by a series of vertical contractions B i . This defines two partial orders ď , ď on CM n . Thus we have the elementary inequalities B j N ď N , resp. B i N ď N and the partial order ď , resp. ď is generated by such elementaryinequalities via transitive closure. For any M, L P CM n we write Sup p M, L q “ t O P CM n | M ď O ě P u . An elementary inequality B j N ď N is called anodyne if, for any j , among the two entries n i,j ` and n i,j ` of the p j ` q st and p j ` q nd column of N that are added in B j N , there isat least one zero. For example, B N “ ˆ ˙ ď ˆ ˙ “ N is anodyne. A general inequality M ď N is called anodyne, if there is a chain of elementaryanodyne inequalities M “ M ď ¨ ¨ ¨ ď M k “ N connecting M and N .Similarly, an elementary inequality B i N ď N is called anodyne if, for each j . among thetwo entries n i ` ,j and n i ` ,j there is at least one zero. A general inequality M ď N is calledanodyne, if there is a chain of elementary anodyne inequalities M “ M ď ¨ ¨ ¨ ď M k “ N . E. Components of a graded bialgebra associated to contingency matrices.
Let A “ p A n q n ě be a graded bialgebra in a braided monoidal category V . Let M P CM p r, s q bea contingency matrix of size r ˆ s . We put A M “ â p p,q qPt , ¨¨¨ ,r uˆt ¨¨¨ ,s u A m p,q p p, q q , the -dimensional tensor product corresponding to A m p,q put in the position p p, q q P R , i.e., p ` q ?´ P C . We call A M the component of A associated to M (even though it is, strictlyspeaking, a tensor product of graded components).11 ‚‚‚ ‚‚‚‚ ‚‚‚‚ Lex p , q p r, qp , s q p r, s q ‚‚‚‚ ‚‚‚‚ ‚‚‚‚ Alex p , q p r, qp , s q p r, s q Figure 2: The snakes
Lex and
Alex passing through t , ¨ ¨ ¨ , r u ˆ t , ¨ ¨ ¨ , s u .By definition, A M is a pseudo-object with indexing set Sn p Z q consisting of (isotopy classesof) snakes passing through Z “ t , ¨ ¨ ¨ , r u ˆ t , ¨ ¨ ¨ , s u . Among such snakes we distinguishthe lexicographic snake Lex which reads the elements of Z one by one horizontally andthe antilexicographic snake Alex which reads them one by one vertically, see Fig. 2. Thecorresponding determinations of A M are(2.7) p A M q Lex “ p A m , b A m , b ¨ ¨ ¨ b A m r, q b ¨ ¨ ¨ b p A m ,s b A m ,s b ¨ ¨ ¨ b A m r,s q , p A M q Alex “ p A m , b A m , b ¨ ¨ ¨ A m ,s q b ¨ ¨ ¨ b p A m r, b A m r, b ¨ ¨ ¨ b A m r,s q , i.e., the ordered tensor products of the A m ij read along the columns, resp. rows of M . F. Horizontal comultiplication and vertical multiplication.
Our next goal is todefine, for any M ď N , the horizontal comultiplication map ∆ M,N : A M Ñ A N , and for any M ď N , the vertical multiplication map µ N,M : A N Ñ A M , using the comultiplication andmultiplication in A .Let first CM p r, s q Q M “ B j N ď N P CM p r, s ` q , j P t , ¨ ¨ ¨ , s ´ u be an elementary inequality, so that for each p we have: m p,j ` “ n p,j ` ` n p,j ` , m p,q “ n p,q , q ď j, m p,q “ n p,q ` , q ě j ` . Introduce closed disks Z p,q Ă C , p “ , ¨ ¨ ¨ , r , q “ , ¨ ¨ ¨ , s by: Z p,q “ $’&’% A thin ellipse encircling the points p p, j ` q and p p, j ` q , if q “ j ` the point p p, q q , if q ď j ; the point p p, q ` q , if q ě j ` . Then, on one hand, we have a canonical identification A M “ â p p,q qPt , ¨¨¨ ,r uˆt , ¨¨¨ ,s u A m p,q p p, q q ÝÑ â p p,q qPt , ¨¨¨ ,r uˆt , ¨¨¨ ,s u A m p,q p Z p,q q obtained by moving each Z p,q to the point p p, q q along the shortest (straight) path andthen contracting it to that point if needed. On the other hand, Proposition 2.6 gives anidentification A N “ â p p,q qPt , ¨¨¨ ,r uˆt , ¨¨¨ ,s u A p,q p Z p,q q , A p,q “ â p a,b qP Z p,q A n a,b “ $’&’% A n p,j ` b A n p,j ` , if q “ j ` A n p,q , if q ď j ; A n p,q ` , if q ě j ` . Using these identifications, we define a morphism ∆ M,N : A M ÝÑ A N to be the -dimensional tensor product, over p p, q q P t , ¨ ¨ ¨ , r uˆt , ¨ ¨ ¨ , s u , of the morphisms ∆ p p,q q : A m p,q Ñ A p,q given by ∆ p p,q q “ $’&’% ∆ n p,j ` ,n p,j ` : A m p,j ` ÝÑ A n p,j ` b A n p,j ` , if q “ j ` A m p,q ÝÑ A n p,q , if q ď j ;Id : A m p,q ÝÑ A n p,q ` , if q ě j ` and positioned at the Z p,q . In order to write ∆ M,N as a morphism of ordered tensor productswithout the use of the braiding, we can use the antilexicographic determinations.Similarly, let CM p r, s q Q M “ B i N ď N P CM p r ` , s q , i P t , ¨ ¨ ¨ , r ´ u be an elementary inequality, so that for each q we have m i ` ,q “ n i ` ,q ` n i ` ,q , m p,q “ n p,q , p ď i, m p,q “ n p ` ,q , p ě i ` . Introduce closed disks Z p,q Ă C , p “ , ¨ ¨ ¨ , r , q “ , ¨ ¨ ¨ , s by: Z p,q “ $’&’% A thin ellipse encircling the points p i ` , q q and p i ` , q q , if p “ i ` the point p p, q q , if p ď i ; the point p p ` , q q , if p ě i ` . Then, on one hand, we have a canonical identification A M “ â p p,q qPt , ¨¨¨ ,r uˆt , ¨¨¨ ,s u A m p,q p p, q q ÝÑ â p p,q qPt , ¨¨¨ ,r uˆt , ¨¨¨ ,s u A m p,q p Z p,q q obtained by moving each Z p,q to the point p p, q q along the shortest (straight) path andthen contracting it to that point if needed. On the other hand, Proposition 2.6 gives anidentification A N “ â p p,q qPt , ¨¨¨ ,r uˆt , ¨¨¨ ,s u A p,q p Z p,q q , where A p,q “ â p a,b qP Z p,q A n a,b “ $’&’% A n i ` ,q b A n i ` ,q , if p “ i ` A n p,q , if p ď i ; A n p ` ,q , if p ě i ` . µ N,M : A N ÝÑ A M to be the -dimensional tensor product, over p p, q q P t , ¨ ¨ ¨ , r uˆt , ¨ ¨ ¨ , s u , of the morphisms µ p p,q q : A p,q Ñ A m p,q given by µ p p,q q “ $’&’% µ n i ` ,q ,n i ` ,q : A n i ` ,q b A n i ` ,q Ñ A m i ` ,q , if p “ i ` A m p,q ÝÑ A n p,q , if p ď i ;Id : A m p,q ÝÑ A n p ` ,q , if p ě i ` and positioned at the Z p,q . In order to write µ N,M as a morphism of ordered tensor productswithout the use of the braiding, we can use the lexicographic determinations.
Proposition 2.8. (a ) Let M ď N . For all chains of elementary inequalities M “ M ď ¨ ¨ ¨ ď M k “ N , the composition ∆ M,N “ ∆ M k ´ ,M k ∆ M k ´ ,M k ´ ¨ ¨ ¨ ∆ M ,M : A M ÝÑ A N has the same value.(a ) Let M ď N . For all chains of elementary inequalities M “ M ď ¨ ¨ ¨ ď M k “ N ,the composition µ N,M “ µ M ,M µ M ,M ¨ ¨ ¨ µ M k ,M k ´ : A N ÝÑ A M has the same value.(b) The morphisms ∆ M,N , M ď N and µ N,M , M ď N thus defined satisfy the followingproperties:(b1 ) If L ď M ď N , then ∆ L,N “ ∆ M,N ∆ L,M .(b1 ) If L ď M ď N , then µ N,L “ µ M,L µ N,M .(b2) If M ě N ď L , then ∆ N,L µ M,N “ ÿ O P Sup p M,L q µ O,L ∆ M,O : A M ÝÑ A L . (b3 ) If M ď N is an anodyne inequality, then ∆ M,N is an isomorphism.(b3 ) If M ď N is an anodyne inequality, then µ N,M is an isomorphism.
Proof:
Parts (a ) and (b1 ) follow from coassociativity of the comultiplication. Parts (a )and (b1 ) follow from associativity of the multiplication. Parts (b3 ) and (b3 ) are obviousfrom the definitions. It remains to prove (b2). We do it in three steps.14 tep 1. Consider first the simplest case when L “ p l , l q is a ˆ matrix, M “ p m , m q t isa ˆ matrix and N “ p n q is a ˆ matrix so that l ` l “ m ` m “ n. In this caase
Sup p M, L q consists of ˆ contingency matrices O “ ˆ o o o o ˙ such that o ` o “ l , o ` o “ l , o ` o “ m , o ` o “ m . The claim (b2) has then the form(2.9) ∆ l ,l µ m ,m “ ÿ O P Sup p M,L q p µ o ,o b µ o ,o q ˝ p Id b R A o ,A o b Id q ˝ p ∆ o ,o b ∆ o ,o q , the appearance of the braiding in the middle coming from comparing the Alex and
Lex determinations of A O . But this equality is simply the reformulation, at the level of gradedcomponents, of the compatibility between multiplication and comultiplication in A . Cf. [17]Eq. (4.2.4). Step 2.
Next, suppose that N is a contingency matrix of arbitrary size r ˆ s and bothinequalities M ě N ď L are elementary, so N “ B j L “ B i M for some i and j .The set Sup p M, L q consists then of p r ` q ˆ p s ` q contingency matrices O such that B j O “ M , B i O “ L . The only part of O not fixed by these conditions, is the ˆ submatrix O on rows i ` , i ` and columns j ` , j ` . Therefore the situation is combinatoriallysimilar to Step 1. More precisely, Sup p M, L q is in bijection with the set of ˆ contingencymatrices O “ ˆ o i ` ,j ` o i ` ,j ` o i ` ,j ` o i ` ,j ` ˙ such that o i ` ,j ` ` o i ` ,j ` “ m i ` ,j ` , o i ` ,j ` ` o i ` ,j ` “ m i ` ,j ` ,o i ` ,j ` ` o i ` ,j ` “ l i ` ,j ` , o i ` ,j ` ` o i ` ,j ` “ l i ` ,j ` . Let λ, ρ : A m i ` ,j ` b A m i ` ,j ` ÝÑ A l i ` ,j ` b A l i ` ,j ` be the adaptations to our case of the LHS and RHS of (2.9), i.e., λ “ ∆ l i ` ,j ` ,l i ` ,j ` µ m i ` ,j ` ,m i ` ,j ` , and ρ is the sum over the O as above. Thus λ “ ρ by Step 1.We claim that the equality (b2) in our situation reduces to that in Step 1, i.e., to theequality (2.9). Indeed, let λ, ρ : A M Ñ A L be the LHS and RHS of (b2). Note that each ofthese morphisms is decomposed as a -dimensional tensor product of “elementary” morphismsof the following types: 15 The identity morphism from some A m pq to some A l p ,q for p ‰ j ` , q ‰ i ` , i ` ,the same morphism for both λ and ρ .• The multiplication A m i ` ,q b A m i ` ,q Ñ A l i ` ,q for q ‰ i ` , the same morphism forboth λ and ρ .• The comultiplication A m i ` ,q Ñ A l i ` ,q b A l i ` ,q for q ‰ i ` , j ` , the same morphismfor both λ and ρ .• The morphism λ for λ and ρ for ρ .This implies that λ “ ρ , thus establishing Step 2. Step 3.
Let now M ě N ď L be arbitrary. Let us represent both inequalities as chains ofelementary ones:(2.10) M “ M ě M ě ¨ ¨ ¨ ě M a “ N ď M a ` ď ¨ ¨ ¨ ď M a ` b “ L. Note that there is a unique chain as above with given M and L ; in particular, N with M ě N ď L is unique (if it exists, which is our assumption). We deduce the equality (b2)by applying Step 2 several times. For this, consider taxicab paths γ “ r x , x , ¨ ¨ ¨ , x a ` b s in the rectangle r , a s ˆ r , b s . Such a path consists of segments r x i ´ , x i s , i “ , ¨ ¨ ¨ , a ` b of length which can be either horizontal or vertical, with x “ p , q and x a ` b “ p a, b q , seeFig. 3. ‚‚ ‚ ‚ ‚‚ ‚ ‚‚ ‚ ‚ ‚ ‚ ‚‚‚‚ ‚ ¨ ¨ ¨ a ¨ ¨ ¨ b x x x a ` b ´ x a ` b Figure 3: A taxicab path γ in a rectangle r , a s ˆ r , b s .Given such γ “ r x , ¨ ¨ ¨ , x a ` b s , we call a γ -chain a sequence of contingency matrices L ‚ “ p L “ M, L , ¨ ¨ ¨ L a ` b “ L q such that:(1) L i ´ ě L i , if the interval r x i ´ , x i s is horizontal.(2) L i ´ ď L i , if the interval r x i ´ , x i s is vertical.16ote that the equalities in a γ -chain must be elementary. Let Ch p γ q be the set of γ -chains.For L ‚ P Ch p γ q we have the morphism T L ‚ : A M Ñ A L defined as the composition A M “ A L T L ‚ , ÝÑ A L T L ‚ , ÝÑ ¨ ¨ ¨ T L ‚ ,a ` b ÝÑ A L a ` b “ A L , where T L ‚ ,i “ µ L i ´ ,L i , if the interval r x i ´ , x i s is horizontal and T L ‚ ,i “ ∆ L i ´ ,L i , if theinterval r x i ´ , x i s is vertical. The following is straightforward. Lemma 2.11. (a) For γ “ γ min : “ rp , q , p , q , ¨ ¨ ¨ , p a, q , p a, q , ¨ ¨ ¨ , p a, b qs being the min-imal (bottom right) path, there is a unique γ -chain, namely (2.10) .(b) For γ “ γ max : “ rp , q , p , q , ¨ ¨ ¨ , p , b q , p , b q , ¨ ¨ ¨ , p a, b qs being the maximal (left top)path, the set Ch p γ q is in bijection with Sup p M, L q . More precisely, for each O P Sup p M, L q there is a unique γ -chain L ‚ such that L b “ O . The lemma implies that ∆ N,L µ M,N “ ÿ L ‚ P Ch p γ min q T L ‚ , ÿ O P Sup p M,L q µ O,L ∆ M,O “ ÿ L ‚ P Ch p γ max q T L ‚ , the sum in the RHS of the first equality consisting of one summand. So our statementreduces to the following: Lemma 2.12.
The sum ř L ‚ P Ch p γ q T L ‚ is independent on the taxicab path γ in r , a s ˆ r , b s . Proof:
It is enough to show the invariance of the sum under an elementary modification ofa path along a ˆ square which changes a horizontal-then-vertical pair of unit intervals tothe vertical-then-horizontal pair completing the square. But such invariance is a consequenceof Step 2, because the inequalities corresponding to unit intervals are elementary ones.This establishes Step 3 and Proposition 2.8 is proved.17 The category of contingency matrices as a braided monoidalcategory
A. Contingency matrices as objects of a category.
We introduce a category CM to have, as objects, formal symbols r M s for all contingency matrices M P CM . Morphismsin CM are generated by the generating morphisms δ M,N : r M s ÝÑ r N s , M ď N, δ N,M : r N s ÝÑ r M s , M ď N subject to the relations( CM ) If L ď M ď N , then δ L,N “ δ M,N δ L,M .( CM ) If L ď M ď N , then δ N,L “ δ M,L δ N,M .( CM ) If M ě N ď L , then δ N,L δ M,N “ ÿ O P Sup p M,L q δ O,L δ M,O . ( CM ) If M ď N is an anodyne inequality, then δ M,N is invertible.( CM ) If M ď N is an anodyne inequality, then δ N,M is invertible.More precisely, let CM ` be the category with the objects and generating morphisms asabove which are subject to the relations ( CM ), ( CM ) and ( CM ). Let E Ă Mor CM ` bethe set of the δ M,N , δ N,M corresponding to anodyne inequalities M ď N , M ď N . Then CM “ CM ` r E ´ s is the localization of CM ` with respect to Σ . The set E satisfies the Orecondition, as follows from the next proposition which we leave to the reader. Proposition 3.1.
Let M ě N ď L be inequalities in CM .(a) If one of these inequalities is anodyne, then Sup p M, L q consists of one element, i.e.,there exists a unique diagram of inequalities in CM O ě / / ě (cid:15) (cid:15) L ě (cid:15) (cid:15) M ě / / N. (b) Moreover, if L ě N is anodyne, then O ě M is anodyne. If M ě N is anodyne, then O ě L is anodyne. Let CM n Ă CM be the full subcategory on objects r M s , M P CM n . Since any inequality M ď N , M ď N implies equality of the weights Σ M “ Σ N , the CM n for different n aremutually orthogonal: Hom CM p CM n , CM n q “ , n ‰ n . Proposition 2.8 can be reformulated as follows.18 orollary 3.2.
Let A be a graded bialgebra in a monoidal category V . Then the correspon-dence r M s ÞÑ A M , δ M,N ÞÑ ∆ M,N , δ N,M ÞÑ µ N,M defines a functor ξ A : CM Ñ V . B. Row and column exchange isomorphisms.
Two row vectors p m , ¨ ¨ ¨ , m r q and p n , ¨ ¨ ¨ , n r q of the same size r will be called disjoint , if, for each i “ , ¨ ¨ ¨ , r , at least one ofthe two numbers m i , n i is equal to , i.e., m i n i “ . Similarly for column vectors.Let M “ } m ij } P CM p r, s q be a contingency matrix of size r ˆ s . We denote by M i “ p m i , ¨ ¨ ¨ , m is q , M j “ p m j , ¨ ¨ ¨ , m rj q t , i “ , ¨ ¨ ¨ , r, j “ , ¨ ¨ ¨ , s, the i th row and the j th column of M . For i “ , ¨ ¨ ¨ , r ´ let σ i,i ` M be the matrix obtainedfrom M by interchanging the i th and p i ` q st rows. For j “ , ¨ ¨ ¨ , s ´ let σ j,j ` M be thematrix obtained from M by interchanging the j th and p j ` q st columns.Recall that the vertical contraction B i ´ , i “ , ¨ ¨ ¨ , r ´ , adds together the i th and p i ` q st rows of a contingency matrix. Suppose that our M P CM p r, s q is such that M i and M i ` are disjoint. Then the inequalities M ě B i M “ B i ´ p σ i,i ` M q ď σ i,i ` M are anodyne, and we define the row exchange isomorphism ε i,i ` “ p δ σ i,i ` M, B i ´ M q ´ ˝ δ M, B i ´ M : r M s ÝÑ r σ i,i ` M s in the category CM .Similarly, suppose that M j and M j ` are disjoint. Then we have anodyne inequalities M ě B j ´ M “ B j ´ p σ j,j ` M q ď σ j,j ` M, and we define the column exchange isomorphism in CM ε j,j ` “ δ B j ´ M,σ j,j ` M ˝ p δ B j ´ M,M q ´ : r M s ÝÑ r σ j,j ` M s . Proposition 3.3. (a) Let M P CM p r, s q and i “ , ¨ ¨ ¨ , r ´ be such that M i , M i ` , M i ` are mutually disjoint. For any permutation τ “ p τ p q , τ p q , τ p qq of t , , u let M τ be thematrix obtained from M by permuting the i th, p i ` q st and p i ` q nd rows of M according to τ , e.g., M p q “ M , M p q “ σ i,i ` M etc. Then the hexagon of row exchange isomorphisms r M p q s ε i ` ,i ` / / r M p q s ε i,i ` ) ) ❙❙❙❙❙❙ r M p qs ε i,i ` ❦❦❦❦❦❦ ε i ` ,i ` ) ) ❙❙❙❙❙❙ r M p qs r M p q s ε i,i ` / / r M p q s ε i ` ,i ` ❦❦❦❦❦❦ is commutative (braid relation).(b) A similar braid relation for column exchange isomorphisms in the case when M j , M j ` and M j ` are mutually disjoint. roof: We show (a), since (b) is similar. By construction, each arrow in the hexagon is thecomposition of two isomorphisms going through an intermediate object: one isomorphism isof the form δ corresponding to an anodyne inequality ě , the other an inverse of a δ ofthis kind. Let us restore these intermediate objects and draw the corresponding morphisms δ (without inverting them). We get a diagram with vertices. Let also M “ B i ´ B i ´ M “ B i ´ B i M be the p r ´ qˆ s matrix obtained by summing all three rows, the i th, the p i ` q st and p i ` q nd,of M . Then we have an anodyne inequality M ď N , where N is any of the matricescorresponding to the vertices of the extended diagram above. Therefore, putting theobject r M s inside that diagram, we decompose it into triangles which commute becauseof the relation ( CM ) (transitivity of the maps δ ). In this way we get a diagram whoseshape is the barycentric subdivision of the original hexagon (considered as a -dimensionalcell complex) and which consists of commuting triangles. This impllies the commutativityof the ( -dimensional boundary of the) hexagon, which is the claim. Proposition 3.4. (a) Let M P CM p r, s q and i “ , ¨ ¨ ¨ , r ´ have the following property:any vector from the set t M i , M i ` u and any vector from the set M i , M i ` u are disjoint.For any permutation ˜ a “ p τ p q , τ p q , τ p q , τ p qq of t , , , u let M τ be the matrix obtainedfrom M by permuting the i th, p i ` q st, p i ` q nd and p i ` q rd rows according to τ , e.g., M p q “ σ i ` ,i ` M . Then the diagram of row exchange isomorphisms r M p q s ε i ` ,i ` * * ❱❱❱❱❱❱❱❱ r M p q s ε i ` ,i ` / / r M p q s ε i,i ` ❤❤❤❤❤❤❤❤ ε i ` ,i ` * * ❱❱❱❱❱❱❱❱ r M p q s ε i ` ,i ` / / r M p q sr M p q s ε i,i ` ❤❤❤❤❤❤❤❤ commutes.(b) A similar statement for column exchange isomorphisms in the case when any vectorfrom t M j , M j ` u and any vector from t M j ` , M j ` u are disjoint. Proof:
We prove (a), since (b) is similar. It suffices to prove the commutativity of thecentral diamond. The argument is similar to that of Proposition 3.3. That is, we expandthe diamond (a -gon) to an -gon by restoring the intermediate objects and drawing the δ -morphisms without inverting anything. Let Ă M “ B i B i ´ M p q be the matrix obtained bysumming the i th and p i ` q st rows and separately summing the p i ` q nd and p i ` q rd rowsof M p q . Then we have an anodyne inequality Ă M ď N where N is any of the matricesfrom the -gon above. So putting r Ă M s inside the -gon, we fill the -gon by commutativetriangles which impllies that the original diamond commutes as well. C. The monoidal structure on CM . We make CM into a monoidal category byputting, on the level of objects, r M s b r N s “ r M ‘ N s , M ‘ N “ ˆ M N ˙ .
20n the level of morphisms, if M ď M and N ď N , then M ‘ N ď M ‘ N , and weput δ M ,M b δ N ,N “ δ M ‘ N ,M ‘ N . Similarly, if M ď M and N ď N , then M ‘ N ď M ‘ N , and we put δ M ,M b δ N ,N “ δ M ‘ N ,M ‘ N . Further, let M ď N and M ď N . Then we have the diagram of inequalities N ‘ N ě / / ě (cid:15) (cid:15) N ‘ M ě (cid:15) (cid:15) M ‘ N ě / / M ‘ M and Sup p M ‘ N , N ‘ M q “ t N ‘ N u consists of one element. Therefore δ M ‘ M ,N ‘ M δ M ‘ N ,M ‘ M “ δ N ‘ N ,N ‘ M δ M ‘ N ,N ‘ N and we define δ M ,N b δ N ,M to be equal to this common value.Similarly, let M ď N and M ď N . We have the diagram of inequalities N ‘ N ě / / ě (cid:15) (cid:15) M ‘ N ě (cid:15) (cid:15) N ‘ M ě / / M ‘ M and put δ N ,M b δ M ,N : “ δ M ‘ M ,M ‘ N δ N ‘ M ,M ‘ M “ δ N ‘ N ,M ‘ N δ N ‘ M ,N ‘ N , the second inequality following from Sup p N ‘ M , M ‘ N q “ t N ‘ N u . Proposition 3.5.
The above data on objects and generating morphisms define a monoidalstructure b on CM with unit object “ rHs . Proof:
By construction, the operation b is strictly associative on objects. What remains toprove is that b extends to a functor in each argument, i.e., that our definitions are compatiblewith the relations in CM . For this, we proceed as follows.First, our definitions imply that for any two generating morphisms f : r M s Ñ r M s , g : r N s Ñ r N s we have f b g “ p f b Id r N s qp Id r M s b g q “ p Id r M s b g qp f b Id r N s q . So it suffices to show that for any
M, N P CM the operations p´ b Id r N s q , p Id r M s b´q ongenerating morphisms preserve the relations in CM . We consider p´ b Id r N s q , the case of p Id r M s b´q being similar. 21or ( CM ), ( CM ) such preservation is obvious. For ( CM ) it follows from the identi-fication Sup p L, M q ÝÑ
Sup p L ‘ N, M ‘ N q , O ÞÑ O ‘ N. For ( CM ), ( CM ) it follows from the following obvious fact: if M ď M is anodyne, then M ‘ N ď M ‘ N is anodyne also, and similarly for M ď M . D. Braiding on CM . Let M P CM p p, q q and N P CM p r, s q . We define the braidingisomorphism R r M s , r N s : r M sbr N s “ r M ‘ N s ÝÑ r N ‘ M s “ r N sbr M s by mimicking the standard Eckmann-Hilton procedure in topology (“jeu de taquin” provingthe commutativity of π ). More precisely, we define R r M s , r N s as the composition r M ‘ N s “ „ˆ M N ˙ R M,N ÝÑ „ˆ NM ˙ R M,N ÝÑ „ˆ N M ˙ “ r N ‘ M s , where:• R M,N is the composition of pr row exchange isomorphisms moving r rows of p N q pastthe p rows of p M q . This can be done in several ways but Proposition 3.4(a) mpliesthat all of them lead to the same result, which is denoted R M,N .• R M,N is the composition of qs column exchange isomorphisms moving s columns of ˆ N ˙ past q columns of ˆ M ˙ . Again, this can be done in several ways but Proposition3.4(b) mplies that all of them lead to the same result, which is denoted R M,N . Proposition 3.6.
The isomorphisms R r M s , r N s make CM into a braided monoidal category. Proof:
We first show that the R r M s , r N s are natural in each variable. Naturality in the firstvariable means that for any morphism ϕ : r M s Ñ r L s and any object r N s in CM the diagram(the naturality square) r M s b r N s ϕ br N s (cid:15) (cid:15) R r M s , r N s / / r N s b r M s r N rb ϕ (cid:15) (cid:15) r L s b r N s R r L s , r N s / / r N s b r L s is commutative. To show this, it suffices to assume that ϕ is one of the elementary generatingmorphisms, i.e., we are in either of the two cases:(i) ϕ “ δ ML , where M “ B j L is obtained from L by a horizontal contraction (adding twoadjacent columns);(ii) ϕ “ δ ML , where L “ B i M is obtained from M by a vertical contraction (adding twoadjacent rows). 22onsider the case (i). The naturality square whose commutativity we need to prove, decom-poses into two:(3.7) „ˆ M N ˙ R MN / / ϕ br N s (cid:15) (cid:15) „ˆ NM ˙ R MN / / ψ (cid:15) (cid:15) „ˆ N M ˙ r N sb ϕ (cid:15) (cid:15) „ˆ L N ˙ R LN / / „ˆ NL ˙ R LN / / „ˆ N L ˙ , where ψ is induced by the inequality ˆ NM ˙ ď ˆ NL ˙ . Note that the other two verticalarrows are, by construction, also induced by the corresponding inequalities. We prove thecommutativity of each of the two squares separately.Left square: The morphisms R MN and R LN are compositions of row exchange isomorphisms,i.e., of δ -isomorphisms induced by anodyne vertical contractions and of the inverses ofsuch isomorphisms. These isomorphisms go through intermediate objects corresponding tomatrices obtained from ˆ M N ˙ and ˆ L N ˙ by some number of row exchanges and then,possibly, summation of two disjoint adjacent rows. Let us restore these intermediate objectsand the δ -isomorphisms connecting them, without inverting these isomorphisms. In thisway we replace the square by a diagram of the form(3.8) „ˆ M N ˙ ϕ br N s δ (cid:15) (cid:15) δ / / ‚ δ (cid:15) (cid:15) ✤✤✤✤✤ ‚ δ o o δ (cid:15) (cid:15) ✤✤✤✤✤ δ / / ‚ δ (cid:15) (cid:15) ✤✤✤✤✤ ¨ ¨ ¨ δ o o δ / / ‚ δ (cid:15) (cid:15) ✤✤✤✤✤ „ˆ NM ˙ δ o o ψδ (cid:15) (cid:15) „ˆ L N ˙ δ / / ‚ ‚ δ o o δ / / ‚ ¨ ¨ ¨ δ o o δ / / ‚ „ˆ NL ˙ δ o o where the horizontal maps are δ -isomorphisms. Note further that we have morphisms be-tween the corresponding intermediate objects indicated by the dotted vertical arrows. Theycorrespond to the horizontal contractions of the intermediate matrices. We obtain a ladderdiagram consisting of many squares, with horizontal maps being δ -isomorphisms and verti-cal maps being δ -morphsms. We claim that each of these squares is commutative. Indeed,such a square corresponds to a square of inequalities of the type discussed in Proposition3.1: two of the inequalities of the same type (in our case, ď ) are anodyne, In this situ-ation Proposition 3.1 and the relation ( CM ) give that the square is commutative, as thesum in ( CM ) consists of one summand. This implies that the boundary of the entire di-agram, formed by inverting the isomorphisms oriented Ð , i.e., the left square in (3.7), iscommutative.Right square: The morphisms R MN and R LN are compositions of column exchange isomor-phisms, i.e., of δ -isomorphisms induced by anodyne horizontal contractions and of the in-23erses of such isomorphisms. Restoring the itnermedaite objects involved in these isomor-phisms, we obtain a diagram somewhat similar to (3.8): „ˆ NM ˙ ψ δ (cid:15) (cid:15) ‚ δ o o δ / / ‚ ‚ δ o o δ / / ¨ ¨ ¨ ‚ δ o o δ / / „ˆ N M ˙ r N sb ϕδ (cid:15) (cid:15) „ˆ NL ˙ ‚ δ o o δ / / ‚ ‚ δ o o δ / / ¨ ¨ ¨ ‚ δ o o δ / / „ˆ N L ˙ . This diagram consists entirely of δ -morphisms. Further, unlike (3.8), the bottom row here islonger than the top one, since M “ B j L has one fewer column than L , being obtained from L by adding the p j ` q st and p j ` q nd columns. Let us denote these columns for short by l “ L j ` and l “ L j ` .Now, some objects in the bottom row can be assigned “matches” in the top one, fromwhich they receive δ -maps which we add to the diagram as vertical arrows. These objectscorrespond to matrices which contain the columns ˆ l ˙ and ˆ l ˙ situated next to each other,and the corresponding matching matrix in the top row is obtained by adding these columns.In this way we get several vertical arrows which decompose our diagram into fragments oftwo types.A fragment of the first type is a square obtained when two vertical arrows are positionednext to each other. Each such square is commutative by transitivity of δ -maps.A fragment of the second type is obtained when a column of N , denote it n , is movedpast l and l . Such a fragment has the form (we do not depict any other columns that areunchanged throughout the procedure): „ˆ nl ` l ˙ δ (cid:15) (cid:15) „ˆ nl ` l ˙ δ o o δ y y s s s s s δ (cid:15) (cid:15) ✤✤✤ δ % % ❑❑❑❑❑ δ / / „ˆ n l ` l ˙ δ (cid:15) (cid:15) „ˆ nl l ˙ „ˆ nl l ˙ δ o o δ / / „ˆ n l l ˙ „ˆ n l l ˙ δ o o δ / / „ˆ n l l ˙ To show that this fragment becomes commutative after inverting the arrows oriented Ð , wedecompose it by the dotted arrows (which are likewise δ -morphisms) into two -gons andtwo triangles. Each of them is commutative by transitivity of δ -morphisms.This proves that the right square in (3.7) is commutative in the situation of Case (i)above, i.e,. under the assumption that ϕ “ δ ML , where M “ B j L . In this way we show thenaturality of the R r M s , r N s in the first variable in Case (i).Naturality in the first argument in Case (ii) when ϕ “ δ ML , L “ B i M , is analyzed com-pletely analogously except the roles of the left and right squares in (3.7) will be interchanged.Further, the naturality in the second argument is also completely analogous. This provesthat the R r M s , r N s are natural in both arguments.24o prove that R is a braiding, it remains to show the commutativity of the braidingtriangles [2, 14]. These triangles are of two classes. The triangles of the first class have theform r M s b r M s b r N s R r M sbr M , r N s (cid:15) (cid:15) r M sb R r M , r N s / / r M s b r N s b r M s R r M s , r N s br M s s s ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ r N s b r M s b r M s for any three objects r M s , r M s , r N s . The triangles of the second class are similarly associatedto any r M s , r N s , r N s and express two ways of passing from r M s b r N s b r N s to r N s b r N s br M s . The commutativity of such triangles follows straightforwardly from Propositions 3.3(braid relation for row or column exhanges) and 3.4. Proposition 3.6 is proved.We now notice the following refinement of Corollary 3.2. Proposition 3.9.
In the situation of Corollary 3.2, the functor ξ A : CM Ñ V is braidedmonoidal. Proof:
We first construct isomorphisms ξ A pr M sq b ξ A pr N sq “ A M b A N ϕ
M,N ÝÑ A M ‘ N “ ξ A pr M s b r N sq . Suppose M is of size p ˆ q and N is of size r ˆ s . By definition, the component A M ,being a -dimensional tensor product, is a pseudo-objecr and as such, is given in termsof determinations corresponding to snakes. In particular (2.7), the determination p A M q Lex corresponding to the
Lex snake, is the ordered tensor product of the A m ij along the columnsof M . Similarly, p A M q Alex , the determination corresponding to the
Alex snake, is the orderedtensor product of the A m ij along the rows of M . They are idenfitied by the braiding R b M ,where b M P Br pq is the braid (depending only on p and q ) connecting Lex and
Alex snakes for M (in fact, for any p ˆ q matrix). Similarly for p A N q Lex and p A N q Alex and R b for b N P Br rs .Now note that reading M ‘ N along the columns (and ignoring the ’s in the off-diagonalblocks) is the same as first reading M along the columns and then reading N in the sameway. This gives an isomorphism ϕ Lex
M,N : p A M q Lex b p A N q Lex
Ñ p A M ‘ N q Lex . Similarly, reading M ‘ N along the rows (and ignoring the s as above) is the same as first reading M and thenreading N in this way. This gives an isomorphism ϕ Alex
M,N : p A M q Lex b p A N q Alex
Ñ p A M ‘ N q Alex .We claim that ϕ Lex
M,N and ϕ Alex
M,N give the same morphism of pseudo-objects ϕ M,N : A M b A N Ñ A M ‘ N . Indeed, consider the juxtaposition (direct sum) homomorphism ‘ : Br pq ˆ Br rs Ñ Br pq ` rs . The
Lex and
Alex determinations of A M ‘ N are related by the braiding R b M ‘ N , where b M ‘ N P Br pq ` rs is the braid relating the Lex and
Alex snakes for block-diagonal matrices p p ` r q ˆ q ` s q . We notice that b M ‘ N “ b M ‘ b N and therefore we have a commutative square p A M q Lex b p A N q Lex R bM b R bN (cid:15) (cid:15) ϕ Lex
M,N / / p A M ‘ N q Lex R bM ‘ N (cid:15) (cid:15) p A M q Alex b p A N q Alex ϕ Lex
M,N / / p A M ‘ N q Alex which implies that the resulting morphism of pseudo-objects is the same for both
Lex and
Alex determinations. This defines ϕ M,N .Next, we show that the ϕ M,N are natural in r M s and r N s . It suffice to check the naturalityon generating morphisms δ or δ for M or N . Naturality for δ (horizontal contractions,adding some adjacent columns) is immediate in the Alex determination. Indeed, whenreading the A m ij along the rows, the action of δ , i.e., horizontal comultiplication, will respectthe order of the product, i.e., will produce new tensor factors in positions which are adjacentwith respect to the order. Similarly, naturality for δ is immediate in the Lex determination,reading the A m ij along the columns.This naturality makes ξ A into a monoidal functor. It remains to show that ξ A is in facta braided monoidal functor, i.e., preserves the braiding. This verification is straightforwardand left to the reader. Proposition 3.9 is proved.26 The category of contingency matrices and the PROB B of graded bialgebras A. The graded bialgebra a in CM . We now define a graded bialgebra a “ p a n q n ě in CM with components a n “ r n s (the object corresponding to the ˆ contingency matrix p n q ) for n ą and a “ “ rHs . The multiplication and comultiplication are given by µ m,n : r m s b r n s “ „ˆ m n ˙ p δ q ´ ÝÑ „ˆ mn ˙ δ ÝÑ r m ` n s , ∆ m,n : r m ` n s δ ÝÑ rp m, n qs p δ q ´ ÝÑ „ˆ m n ˙ “ r m s b r n s . Proposition 4.1.
The morphisms µ m,n , ∆ m,n make a into a graded bialgebra in CM . Proof:
We first prove associativity. For this, we must compare two morphisms r m s b r n s br p s Ñ r m ` n ` p s corresponding to two bracketing of the triple product. There morphismsare the compositions of the upper and lower paths in the boundary of the following diagram,the paths obtained by inverting the δ -isomorphisms: »–¨˝ m n p ˛‚fifl δ { { ✈✈✈✈✈✈✈✈✈ δ / / „ˆ m ` n p ˙ „ˆ m ` np ˙ δ o o δ " " ❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊ »–¨˝ m n
00 0 p ˛‚fifl »–¨˝ mnp ˛‚fifl δ O O ✤✤✤ δ (cid:15) (cid:15) ✤✤✤ δ ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ δ ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ r m ` n ` p s »–¨˝ m n p ˛‚fifl δ c c ❍❍❍❍❍❍❍❍❍ δ / / „ˆ m n ` p ˙ „ˆ mn ` p ˙ . δ o o δ < < ②②②②②②②②②②②②②②②② To prove that these two paths have the same composition, we decompose the diagram intofour -gons by the dotted arrows as shown and notice that each of these -gons is commu-tative.Indeed, the leftmost -gon commutes by transitivity of δ -morphisms. The rightmost -gon commutes by transitivity of δ -morphisms. The remaining two -gons commute bythe relation ( CM ) since the sum in that relation consists of one summand by Proposition3.1.This proves associativity of µ . The proof of coassociativity of ∆ is similar.Finally, we prove compatibility of ∆ and µ . This is expressed by Eq. (2.9), we we assumethat we are in the situation of (2.9). The composition ∆ l ,l µ m ,m is, in our case, given by27he border (top horizontal followed by the right vertical) path in the following diagram: „ˆ m m ˙ p δ q ´ β / / „ˆ m m ˙ δ ψ O (cid:15) (cid:15) δ ϕ / / r n s δ ψ (cid:15) (cid:15) „ˆ o o o o ˙ ϕ O δ / / rp l , l qs p δ q ´ α (cid:15) (cid:15) „ˆ l l ˙ Here the matrix O “ ˆ o o o o ˙ is a (so far arbitrary) element of Sup p M, L q . We denotedfor short by ψ, ψ O , ϕ, ϕ O the δ and δ -morphisms in the square and by α and β the invertedisomorphisms at the end and the beginning of the border path. By the relation ( CM ) wehave ψϕ “ ÿ O P Sup p M,L q ϕ O ψ O . So it suffices to show that for each O P Sup p M, L q we have(4.2) αϕ O ψ O β “ p µ o ,o b µ o ,o q ˝ p Id b R A o ,A o b Id q ˝ p ∆ o ,o b ∆ o ,o q . so that the summands in the RHS of (2.9) match those in ( CM ). We represent the twosides of (4.2) by the upper and lower path in the boundary of the following diagram (moreprecisely, the paths, going from left to right, are obtained by inverting the isomorphismsoriented the other way): „ˆ m m ˙ δ v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠ δ ψO / / „ˆ o o o o ˙ κ w w ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ρ & & ▲▲▲▲▲▲▲▲▲▲▲▲▲▲ δ ϕO / / rp l , l qs „ˆ m m ˙ δ (cid:15) (cid:15) „ˆ l l ˙ δ (cid:15) (cid:15) δ f f ◆◆◆◆◆◆◆◆◆◆◆◆ „ˆ o o o o ˙ »——–¨˚˚˝ o o o o ˛‹‹‚fiffiffifl δ z z ✈✈✈✈✈✈✈✈✈ »——–¨˚˚˝ o o o
00 0 0 o ˛‹‹‚fiffiffifl δ f f ▼▼▼▼▼▼▼▼▼▼▼▼▼ Id b R b Id / / »——–¨˚˚˝ o o o
00 0 0 o ˛‹‹‚fiffiffifl To show the equality of the compositions of these paths, we decompose the diagram intotwo -gons and a pentagon by the dotted arrows κ and ρ , where:28 κ is the composition „ˆ o o o o ˙ δ ÝÑ „ˆ o o o o ˙ Column exchange ÝÑ „ˆ o o o o ˙ , so it composed entirely of δ -morphisms and their inverses.• ρ is the composition „ˆ o o o o ˙ p δ q ´ ÝÑ »——–¨˚˚˝ o o o o o ˛‹‹‚fiffiffifl Row exchange ÝÑ »——–¨˚˚˝ o o o o ˛‹‹‚fiffiffifl , so it is composed entirely of δ -morphisms and their inverses. The left -gon in the decom-posed diagram is commutative by transitivity of δ -morphisms. The right -gon is commu-tative by transitivity of δ -morphsms. Finally, the pentagon consists entirely of anodyne δ - or δ -isomorphisms and their inverses which move the o ij around in the plane. We canview them as moving points in the plane. After we go around the pentagon, we return tothe same position. Moreover, the braid on strands representing this move, is trivial. Thistriviality of the braid implies the commutativity of the pentagon. We leave further detailsto the reader. Proposition 4.1 is proved. B. The category CM and the PROB B . Recall the PROB B governing gradedbialgebras, see §1 B. Theorem 4.3.
We have an equivalence of braided monoidal categories ξ : CM Ñ B . Inparticular, for any n ą we have an equivalence of ordinary (non-monoidal) categories ξ n : CM n Ñ B n . Proof:
Recall that B has a graded bialgebra a . By definition, the components a n aregenerating objects for B , i.e., any other object is isomorphic to a tensor product of severalof the a n . Similarly, the objects r n s associated to ˆ matrices, are generating objects for CM . Indeed, any object r M s associated to any r ˆ s contingency matrix M “ } m ij } , isisomorphic to the tensor product (in any order) of the individual r m ij s , the latter productbeing represented by the diagonal rs ˆ rs matrix with entries m ij in the corresponding order.This can be easily seen by moving the m ij around in the matrix by using anodyne δ - and δ -isomoprhisms and their inverses.Next, the graded bialgebra a , Corollary 3.2 and Proposition 3.9 give a braided monoidalfunctor ξ “ ξ a : CM ÝÑ B , r M s ÞÑ a M . We prove that ξ is an equivalence. For this, we use the graded bialgebra a in CM constructedin Proposition 4.1. As a P B is the universal graded bialgebra, we get a braided monoidalfunctor F “ F a : B ÝÑ CM , a n ÞÑ a n “ r n s .
29e claim that the functors ξ and F are quasi-inverse to each other. Indeed, look at thecomposition F ξ : CM Ñ CM , a braided monoidal functor. It takes any generating object r n s to itself. Therefore F ξ is isomorphic to Id . Similarly, look at ξF : B Ñ B . It is abraided monoidal functor which takes any generating object a n to itself. Therefore ξF isisomorphic to Id . C. Proof of Theorem 1.3.
Because of Theorem 4.3, Theorem 1.3 can be reformulatedas follows.
Reformulation 4.4.
For any abelian category V we have an equivalence of categories Perv p Sym n p C q ; V q » Fun p CM , V q . This statement is a consequence (particular case) of the main result of [19] (Theorem 2.6)which describes perverse sheaves on W z h where h is the Cartan subalgebra of a reductivecomplex Lie algebra g and W is the Weyl group of g . More precisely, [19] deals with Vect k -valued perverse sheaves, but extension to perverse sheaves with values in an arbitraryabelian category V is trivial. Our case corresponds to g “ gl n , when h “ C n and W “ S n ,so W z h “ Sym n p C q . The description of [19] is in terms of mixed Bruhat sheaves (Definition2.1 there) which are certain diagrams with objects labelled by the set Ξ “ Ξ g “ ğ I,J Ă ∆ sim W zp W { W I ˆ W { W j q where ∆ sim is the set of simple roots of g and W I , I Ă ∆ sim is the subgroup in W generatedby the simple reflections s α , α P I . For g “ gl n the set Ξ g is identified with CM n , see [18],and the axioms of a mixed Bruhat sheaves become identical to the relations in the category CM . This finishes the proof. 30 eferences [1] J. F. Adams. Infinite Loop Spaces. Princeton Univ. Press. 1978.[2] B. Bakalov, A. Kirillov, Jr. Lectures on Tensor Categories and Modular Functors. Amer.Math. Soc. Publ. 2000.[3] A. Beilinson. How to glue perverse sheaves. In: K -theory, arithmetic and geometry (Moscow,1984), Lecture Notes in Math. , Springer-Verlag, 1987, 42 - 51.[4] A. Beilinson, V. Drinfeld. Chiral Algebras. AMS Coll. Publ. , 2004.[5] R. Bezrukavnikov, M. Finkelberg, V. Schechtman. Factorizable Sheaves and Quantum Groups, Lecture Notes in Math. „ gaitsgde/GL/FS.pdf>(2008)[10] A. Galligo, M. Granger, Ph. Maisonobe. D-modules et faisceaux pervers dont le supportsingulier est un croisement normal. Ann. Inst. Fourier (Grenoble) (1985), 1-48.[11] M. Granger, Ph. Maisonobe. Faisceaux pervers relativement à un point de rebroussement. C.R. Acad. Sci. Paris. Sér. I (1984), 567-570.[12] P. Hackney, M. Robertson. On the category of props.
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V.S.: Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne,31062 Toulouse, France. Email: [email protected]@math.ups-tlse.fr