aa r X i v : . [ m a t h . R T ] S e p Heisenberg algebra and a graphical calculus
Mikhail KhovanovSeptember 16, 2010
Contents H ′
335 Remarks on the Grothendieck ring of H H ′ . . . . . . . . . . . . . . . . . . . . . . . . 385.2 Grothendieck group of degenerate affine Hecke algebra . . . . . . . . . 39 Abstract
A new calculus of planar diagrams involving diagrammatics for biadjoint func-tors and degenerate affine Hecke algebras is introduced. The calculus leads to anadditive monoidal category whose Grothendieck ring contains an integral form ofthe Heisenberg algebra in infinitely many variables. We construct bases of vectorspaces of morphisms between products of generating objects in this category. Introduction
In this paper we propose a graphical calculus for a categorification of the Heisenbergalgebra. The one-variable Heisenberg algebra has generators p, q , one defining relation pq − qp = 1, and appears as the algebra of operators in the quantization of the harmonicoscillator. A fundamental role in the quantum field theory is played by its infinitely-generated analogue, the algebra with generators p i , q i , for i in some infinite set I , andrelations p i q j = q j p i + δ i,j , p i p j = p j p i , q i q j = q j q i . (1)In Section 2 we define a strict monoidal category H ′ with two generating objects Q + and Q − , and morphisms between tensor products of these objects given by linear com-binations of certain planar diagrams modulo local relations. The category is k -linearover a ground commutative ring k , and we specialize k to a field of characteristic 0.The endomorphism rings of tensor powers Q ⊗ n + and Q ⊗ n − contain the group algebra k [ S n ]of the symmetric group. The symmetrization and antisymmetrization idempotents in k [ S n ] produce objects in the Karoubi envelope H of H ′ . These objects can be viewedas symmetric and exterior powers of the generating objects Q + and Q − . Consequently,we denote them by S n + := S n ( Q + ) , Λ n + := Λ n ( Q + ) , S n − := S n ( Q − ) , Λ n − := Λ n ( Q − ) , (2)and call them the symmetric and exterior powers of Q + and Q − . When n = 0, S ∼ = S − ∼ = Λ ∼ = Λ − ∼ = , where is the identity object of the monoidal category H , ⊗ M = M for any M . Wealso set S n + = S n − = Λ n + = Λ n − = 0 if n < . Proposition 1
There are canonical isomorphisms in H S n − ⊗ Λ m + ∼ = (Λ m + ⊗ S n − ) ⊕ (Λ m − ⊗ S n − − ) ,S n − ⊗ S m − ∼ = S m − ⊗ S n − , Λ n + ⊗ Λ m + ∼ = Λ m + ⊗ Λ n + . These isomorphisms are constructed in Section 2.2. Since H is monoidal, its Grothendieckgroup K ( H ) is a ring. It has generators [ M ] over objects M of H and relations[ M ] = [ M ] + [ M ] whenever M ∼ = M ⊕ M . The multiplication is defined by[ M ][ M ] := [ M ⊗ M ]. Corollary 1
The following equalities hold in K ( H ) : [ S n − ][Λ m + ] = [Λ m + ][ S n − ] + [Λ m − ][ S n − − ] , [ S n − ][ S m − ] = [ S m − ][ S n − ] , [Λ n + ][Λ m + ] = [Λ m + ][Λ n + ] . H Z be the unital ring with generators a n , b n , n ≥ a n b m = b m a n + b m − a n − , (3) a n a m = a m a n , (4) b n b m = b m b n . (5)We simply rewrote relations in Corollary 1 using a n in place of [ S n − ] and b m instead of[Λ m + ]. Also set a = b = 1, a n = b n = 0 for n <
0, and require that the above relationshold for any n, m ∈ Z . Any product of a ’s and b ’s can be converted into a linearcombination with nonnegative integer coefficients of monomials in b ’s times monomialsin a ’s, b m b m . . . b m k a n a n . . . a n r (6)with 1 ≤ m ≤ m ≤ · · · ≤ m k , ≤ n ≤ n ≤ · · · ≤ n r . The Bergman diamondlemma [4] tells us that this set of elements is a basis of H Z viewed as a free abeliangroup. Let H = H Z ⊗ C be the C -algebra with the same generators and relations as H Z .Forming generating functions A ( t ) = 1 + a t + a t + . . . , B ( u ) = 1 + b u + b u + . . . , we can rewrite relations (3) as A ( t ) B ( u ) = B ( u ) A ( t )(1 + tu ) . Let e A ( t ) = 1 + tA ′ ( − t ) A ( − t ) , e A ( t ) = 1 + ˜ a t + ˜ a t + . . . . It is easy to check that e a , e a , . . . generate the same subalgebra of H as a , a , . . . , andthat e A ( t ) B ( u ) = B ( u ) e A ( t ) + tu − tu . Coefficients of this equation give us relations (7) below˜ a n b m = b m ˜ a n + δ n,m , (7)˜ a n ˜ a m = ˜ a m ˜ a n , (8) b n b m = b m b n . (9)Algebra H is isomorphic to the algebra generated by ˜ a n ’s, b m ’s, n, m >
0, with definingrelations (7) - (9). This allows us to identify H with the Heisenberg algebra and H Z with its integral form.Corollary 1 gives a ring homomorphism γ : H Z −→ K ( H ) (10)that takes a n to [ S n − ] and b n to [Λ n + ]. 3 heorem 1 Map γ is injective. This theorem is proved in Section 3.3.
Conjecture 1
Map γ is an isomorphism. If true, this conjecture would allow us to view the additive monoidal category H asa categorification of the integral form H Z of the Heisenberg algebra.The degenerate affine Hecke algebra, which we call degenerate AHA following a sug-gestion of Etingof, was introduced by Drinfeld [14] in the GL ( n ) case and by Lusztig [28]in the general case. Cherednik [9] classified finite-dimensional irreducible representa-tions of degenerate AHA; its centralizing properties were studied by him and Olshanskiin [10, 37]. We denote by DH n the degenerate AHA for GL ( n ), over the base field k .Under the canonical homomorphism [9, 14] from DH n to the group algebra k [ S n ] of thesymmetric group polynomial generators of DH n go to the Jucys-Murphy elements. Ok-ounkov and Vershik [35], [36] presented a detailed derivation of the basic representationtheory of the symmetric group via these elements, see also [25, Chapter 2], [8, 13]. Forsome other uses of Jucys-Murphy’s elements and degenerate AHA we refer the readerto [20, 30, 34, 41].We will prove in Section 4 that the ring of endomorphisms of the object Q ⊗ n + inour category is isomorphic to the tensor product of DH n and the polynomial algebra ininfinitely many variables. Thus, the degenerate AHA for GL ( n ) emerges naturally in ourapproach as part of a larger structure. Polynomial generators of DH n acquire graphicalinterpretation in our calculus as right-twisted curls on strands. We also describe abasis, given diagrammatically, of vector spaces of morphisms between arbitrary tensorproducts of generators Q + and Q − of H ′ .To prove our results, we construct a family of functors from H ′ to the category S ′ whose objects are compositions of induction and restriction functors between groupalgebras k [ S n ] of the symmetric group, and morphisms are natural transformationsbetween these functors. The image under these functors of the endomorphism of Q + given by the right curl diagram is the Jucys-Murphy element. The image of the coun-terclockwise circle diagram with k right curls is the k -th moment of the Jucys-Murphyelement. Products of these moments were investigated in [20, 34, 39, 41] in relationto the asymptotic representation theory of the symmetric group and free probability.It also appears that our construction should be related to the circle of ideas consid-ered by Guionnet, Jones and Shlyakhtenko [19] that intertwine planar algebras and freeprobability. In addition, one would hope for a relation between our calculus and thegeometrization of the Heisenberg algebra via Hilbert schemes by Nakajima [32], [33]and Grojnowski [18], and for a possible link with Frenkel, Jing and Wang [16].We discovered monoidal category H ′ by considering compositions of induction andrestriction functors for standard inclusions of symmetric group algebras k [ S n ] ⊂ k [ S n +1 ] . n + 1 , n + 2), anendomorphism of k [ S n +2 ] viewed as left k [ S n +2 ]-module and right k [ S n ]-module. Wedenote this natural transformation by the crossing of two upward-oriented strands. Re-lations on compositions of the crossing, cup, and cap transformations that hold forall n (universal relations) are given by equations (11)-(13). These relations togetherwith isotopies of diagrams are exactly the defining relations for the additive monoidalcategory H ′ .Cautis and Licata [7] introduced graded relatives of H ′ and H associated to finitesubgroups of SU (2), identified their Grothendieck rings with certain quantized Heisen-berg algebras, and constructed an action of their categories on derived categories ofcoherent sheaves on Hilbert schemes of points on the ALE spaces. Hom spaces inCautis-Licata monoidal categories carry a natural grading (absent in our case) withfinite-dimensional homogeneous terms and vanishing negative degree homs on certainobjects, leading to a proof that their analogue of the map γ is an isomorphism.By themselves, Heisenberg algebras are rather simple constructs. Their value isin the structures that quantum field theory builts on top of them, for instance, thestructures of vertex operator algebras. The problem posed by Igor Frenkel [15] tocategorify just the simplest vertex operator algebra remains wide open - perhaps ourpaper will serve as a small step towards this goal. Acknowledgments:
On a number of occasions Igor Frenkel emphasized to theauthor the importance of categorifying various structures related to the symmetricfunctions and Heisenberg algebras. More recently, Tony Licata asked directly if thereexists a categorification of the Heisenberg algebra, and this paper was inspired by hisquestion. The author would like to acknowledge partial support by the National ScienceFoundation via grants DMS-0706924 and DMS-0739392.
Fix a commutative ring k and consider the following additive k -linear monoidal category H ′ generated by two objects Q + and Q − . An object of H ′ is a finite direct sum of tensorproducts Q ǫ ⊗ · · · ⊗ Q ǫ m , denoted Q ǫ , where ǫ = ǫ . . . ǫ m are finite sequences of signs.Thus, Q ǫǫ ′ ∼ = Q ǫ ⊗ Q ǫ ′ for sequences ǫ , ǫ ′ and their concatenation ǫǫ ′ . The unit objectcorresponds to the empty sequence: = Q ∅ . The space of homomorphisms Hom H ′ ( Q ǫ , Q ǫ ′ ) for sequences ǫ, ǫ ′ is the k -modulegenerated by suitable planar diagrams, modulo local relations. The diagrams consistof oriented compact one-manifolds immersed into the plane strip R × [0 , = = PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB (11) = = PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB (12) = 0= 1 PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB (13)We require that the endpoints of the one-manifold are located at { , . . . , m } × { } and { , . . . , k } × { } and call these the lower and upper endpoints, respectively, where m and k are lengths of sequences ǫ and ǫ ′ . Moreover, orientation of the one-manifold atthe endpoints must match the signs in the sequences ǫ and ǫ ′ . For instance, the diagram + ++ PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB is a morphism from Q ++ −− to Q − + . A diagram without endpoints is an endomorphismof . Composition of morphisms is given by concatenating the diagrams. The sequenceof n pluses is denoted + n , the sequence of n minuses − n .We have the Heisenberg relation Q − + ∼ = Q + − ⊕ . Q + − Q − + Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB (14)The four arrows are given by four morphisms, two of which are crossings and twoare U-turns. The condition that these maps describe a decomposition of Q − + as thedirect sum of objects Q − + and is equivalent to relations (11) and (13), modulo thecondition that an isotopy of a diagram does not change the morphism. The lattercondition is equivalent to the biadjointness of functors of tensoring with Q + and Q − ,with the biadjointness transformations given by the four U -turnsPSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB (see Section 3.1 for details).Moving the lower endpoints of a diagram up via a multiple cups diagram leads tocanonical isomorphismsHom H ′ ( Q ǫ , Q ǫ ′ ) ∼ = Hom H ′ ( , Q ǫǫ ′ ) ∼ = Hom H ′ ( , Q ǫ ′ ǫ ) , (15)related to the biadjointness of tensoring with Q ǫ and Q ǫ . Here ǫ is the sequence ǫ with the order and all signs reversed. Biadjointness natural transformations satisfy thecyclicity condition [1, 2, 11, 27], which follows at once from the definition of H ′ .The two relations in (11) allow simplification of a double crossing for oppositelyoriented intervals. The first relation in (13) says that a counterclockwise oriented circleequals one. Thus, an innermost counterclockwise circle can be erased from the diagramwithout changing the value of the diagram viewed as an element of the hom spacebetween the functors.There are two possible types of curls on strands: a left curlPSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB and a right curlPSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB . The second relation in (13) says that a diagram that contains a left curl subdiagram iszero.Defining local relations in H ′ imply the following relations7 + PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB , = PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB .The second relation, jointly with the original ones, implies that the triple intersectionmove holds for any orientation of the 3 strands: = PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB Furthermore, right curls can be moved across intersection points, modulo simplerdiagrams: = += +
PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB It will be convenient to denote a right curl by a dot on a strand, and k -th power ofa right curl by a dot with k next to it: (cid:0)(cid:0)(cid:1)(cid:1) : = (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) : = k k dots PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB The above relations can be rewritten as (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) = + (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) = +
PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB Together with the earlier ones, these relations show that there is a homomorphismfrom the degenerate affine Hecke algebra DH n with coefficients in k to the k -algebra ofendomorphisms of the object Q + n . The permutation generator s i of DH n goes to thepermutation diagram of the i -th and i + 1-st strands, and the polynomial generator x i goes to the dot on the i -th strand: 8 i i+1 ni x i (cid:0)(cid:0)(cid:1)(cid:1) i n PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB Note thatPSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB = 0 . Indeed, the figure eight diagram, for any orientation,contains both left and right curls, and, therefore, equals to zero in our calculus.A strand with k dots can be closed into either a clockwise-oriented or a counterclockwise-oriented circle with k dots. Denote these circles by c k and ˜ c k , respectively: c k := (cid:0)(cid:0)(cid:1)(cid:1) k PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB , ˜ c k := (cid:0)(cid:0)(cid:1)(cid:1) k PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB . Counterclockwise circles can be expressed as linear combinations of products ofclockwise circles. For the first few values of k , these are˜ c = 1 , ˜ c = 0 , ˜ c = c , ˜ c = c , ˜ c = c + c , ˜ c = c + 2 c c . These equations are obtained by expanding each dot into a left curl and then operat-ing on the resulting diagram via the rules of the graphical calculus. A counterclockwisecircle with one dot expands into the figure eight diagram, which is 0. For anotherexample, ˜ c = = = PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB = c . Proposition 2
For k > we have ˜ c k +1 = k − X a =0 ˜ c a c k − − a . (16) Proof is the following computation: 9 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) a (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) k−1−a = Σ a=0k−1 (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) k (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) kk+1 = = (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) a (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) Σ a=0k−1 k−1−a (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) k = + PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB In the second equality, we converted a dot to a right curl, and, in the third equality,moved k dots through a crossing. The first term on the second line equals 0, since itcontains a left curl. (cid:3) Iterating this formula, one obtains an expession for ˜ c k as a polynomial function of c m , m ≤ k −
2. Vice versa, each c m can be written as a polynomial in ˜ c k , k ≤ m + 2.Let t be a formal variable and write c ( t ) = ∞ X i =0 c i t i , ˜ c ( t ) = ∞ X i =0 ˜ c i t i . Formula (16) turns into t c ( t )˜ c ( t ) = ˜ c ( t ) − , so that˜ c ( t ) = 11 − t c ( t ) . The following identities, called bubble moves by analogy with [27], hold. (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) k (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) k (cid:0)(cid:0)(cid:1)(cid:1) k−b−2 b(k+1) k = + PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) k (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) k (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) b k−b−2 = PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB
10 closed diagram D defines an endomorphism of the object of H ′ . Using the localmoves, such diagram D can be converted into a linear combinations of crossinglessdiagrams that consist of nested dotted circles. Furthermore, bubble moves can beused to split apart nested circles. Lastly, convert counterclockwise circles into linearcombinations of products of clockwise circles. Therefore, a closed diagram can bewritten as a linear combination of products of dotted clockwise circles. We see thatthe endomorphism algebra End H ′ ( ) is a quotient of the polynomial algebra Π := k [ c , c , c , . . . ] in countably many variables via the map ψ : Π = k [ c , c , c , . . . ] −→ End H ′ ( ) (17)that takes c k to the clockwise circle with k dots (we allowed ourselves the liberty ofusing c k to denote both a formal variable and its image in the endomorphism algebra). Proposition 3
Map ψ is an isomorphism. This proposition will be proved in Section 4.The endomorphism algebra of Q + m is spanned by all diagrams that have m upperand m lower endpoints and such that at each endpoint the strand is oriented upward.A homomorphism from the degenerate affine Hecke algebra DH m to End H ′ ( Q + m ) wasdescribed earlier. Placing a closed diagram to the right of a diagram representing anelement of DH m gives a homomorphism ψ m : DH m ⊗ Π −→ End H ′ ( Q + m ) . (18)It is easy to see that ψ m is surjective, by taking a diagram representing an element onthe right hand side, and inductively simplifying it to a linear combination of diagramsthat come from a standard basis of the left hand side. Namely, any diagram representingan endomorphism of Q + m is a combination of diagrams that consist of a permutation σ ∈ S m , some number (possibly zero) of dots on each strand above the permutationdiagram, and a monomial in dotted clockwise circles to the right: (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB We write these basis elements as x a . . . x a m m · σ · c b c b . . . c b k k . In the above example,the element is x x x · (1324) · c c c . Surjectivity of ψ m can be strengthened to the following result.11 roposition 4 ψ m is an isomorphism. Injectivity of ψ m is proved in Section 4.We now describe a spanning set in the k -module Hom H ′ ( Q ǫ , Q ǫ ′ ) for any sequences ǫ, ǫ ′ . Let k be the total number of pluses in these two sequences. This hom space isnontrivial only if the total number of minuses in these two sequences is k as well, whichwe’ll assume from now on to be the case. The spanning set, denoted B ( ǫ, ǫ ′ ), is obtainedby forming all possible oriented matchings of these two points via k oriented segments inthe plane strip R × [0 , ǫ and ǫ ′ are written at the bottomand top of the strip, the segments are embedded in the strip, and their orientations atthe endpoints match corresponding elements of ǫ and ǫ ′ . Each two segments intersect atmost once, and no triple intersections are allowed, see an example below for ǫ = − − +and ǫ ′ = + − − − +. + ++ − −− −− PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB Select an interval disjoint from intersections near the out endpoint of each intervaland put any number (perhaps 0) of dots on it. In the rightmost region of the diagram,draw some number of clockwise oriented disjoint nonnested circles with no dots, somenumber of such circles with one dot, two dots, etc., with finitely-many circles in total.The resulting set of diagrams B ( ǫ, ǫ ′ ) is parametrized by k ! possible matchings of the2 k oriented endpoints, by a sequence of k nonnegative integers describing the numberof dots on each interval, and by a finite sequence of nonnegative integers listing thenumber of clockwise oriented circles with no dots, one dot, and so on. An example ofa diagram in B ( − − + , + − − − +) is depicted below. PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB
12n this picture, we put no dots, one dot, three dots, and four dots on the four arcsof the matching, added one bubble with no dots, two with two dots and one with fivedots.It is rather straightforward to check that B ( ǫ, ǫ ′ ) is a spanning set of the k -vectorspace Hom H ′ ( Q ǫ , Q ǫ ′ ). Proposition 5
For any sign sequences ǫ, ǫ ′ the set B ( ǫ, ǫ ′ ) constitutes a basis of the k -vector space Hom H ′ ( Q ǫ , Q ǫ ′ ) . Thus, we claim that the set B ( ǫ, ǫ ′ ) is also linearly independent. This propositionholds as well for k being any commutative ring rather than a field, with B ( ǫ, ǫ ′ ) beinga basis of the free k -module Hom H ′ ( Q ǫ , Q ǫ ′ ) . Notice that Proposition 4 is a special case of this proposition, for ǫ = ǫ ′ = + m .Proposition 5 follows from proposition 4, functor isomorphisms Q − + ∼ = Q + − ⊕ Id andarguments similar to the ones in [24, Section 2.2]. First, canonical isomorphisms (15)take the set B ( ǫ, ǫ ′ ) to B ( ∅ , ǫǫ ′ ) and B ( ∅ , ǫ ′ ǫ ), respectively, and it is then enough to showthat B ( ∅ , ǫ ) is linearly independent for any sequence ǫ with k pluses and k minuses.Proposition 4 implies linear independence for k = 0 ,
1, and for the sequence + k − k for any k . Assume that ǫ = ǫ − + ǫ for some sequences ǫ , ǫ . Assume by induction on k and by induction on the lexicographic order among length 2 k sequences that the sets B ( ∅ , ǫ ǫ ) and B ( ∅ , ǫ + − ǫ ) are linearly independent in their respective hom spaces.Two upper arrows in the diagram (14) lead to a canonical decomposition Q ǫ − + ǫ ∼ = Q ǫ + − ǫ ⊕ Q ǫ ǫ . Under this isomorphism sets B ( ∅ , ǫ ǫ ) and B ( ∅ , ǫ + − ǫ ) get mapped to two subsetsof Hom H ′ ( , Q ǫ − + ǫ ). Denote by B the union of these two subsets. It is easy to seethat linear independence of B ( ∅ , ǫ − + ǫ ) is equivalent to linear independence of B ,which we know by induction. Proposition 5 follows. (cid:3) By the thickness of a diagram in B ( ǫ, ǫ ′ ) we call the number of arcs connecting lowerand upper endpoints. The diagram depicted earlier has thickness one. For each k and ǫ , the subset of diagrams of thickness at most k is a 2-sided ideal in the endomorphismring of Q ǫ , since thickness cannot increase upon composition. For ǫ = + n − m and k = n + m − J n,m . It is spanned by diagramswith at least one arc connecting a pair of upper endpoints (and, necessarily, at leastone arc connecting a pair of lower endpoints). It is easy to see that the quotient ofthe endomorphism ring of Q + n − m by this ideal is naturally isomorphic to the tensorproduct DH n ⊗ DH opm ⊗ Π, and the short exact sequence0 −→ J n,m −→ End H ′ ( Q + n − m ) −→ DH n ⊗ DH opm ⊗ Π −→ DH opm ∼ = DH m .13e now list some obvious symmetries of H ′ . The map that assigns ( − w ( D ) D toa diagram D , where w ( D ) is the number of crossings plus the number of dots of D ,extends to an involutive autoequivalence ξ of H ′ . We have ξ = Id (equality and notjust isomorphism). Autoequivalence ξ exchanges S n + with Λ n + and S n − with Λ n − .Denote by ξ the symmetry of category H ′ given on diagrams by reflecting about thex-axis and reversing orientation. This symmetry is an involutive monoidal contravariantautoequivalence of H ′ .Denote by ξ the symmetry of category H ′ given on diagrams by reflecting aboutthe y-axis and reversing orientation. This symmetry is an involutive antimonoidalautoequivalence of H ′ . Being antimonoidal means that it reverses the order of elementsin the tensor product: ξ ( M ⊗ N ) = ξ ( N ) ⊗ ξ ( M ) . Symmetries ξ , ξ , ξ pairwise commute and generate an action of ( Z / . The two relations in (12) tell us that upward oriented crossings satisfy the symmetricgroup relations and give us a canonical homomorphism k [ S n ] −→ End H ′ ( Q + n )from the group algebra of the symmetric group to the endomorphism ring of the n -th tensor power of Q + . Turning the diagrams by 180 degrees, we obtain a canonicalhomomorphism k [ S n ] −→ End H ′ ( Q − n ) . Assume that k is a field of characteristic 0. Then we can use symmetrizers and an-tisymmetrizers, and, more generally, Young symmetrizers, to produce idempotents inEnd H ′ ( Q + n ). At this point it is convenient to introduce the Karoubi envelope of H ′ , thecategory H whose objects are pairs ( P, e ), where P is an object of H ′ and e : P −→ P is an idempotent endomorphism, e = e . Morphisms from ( P, e ) to ( P ′ , e ′ ) are maps f : P −→ P ′ in H ′ such that e ′ f e = f . It is immediate that H is a k -linear additivemonoidal category.To the complete symmetrizer e ( n ) ∈ k [ S n ] , e ( n ) = 1 n ! X σ ∈ S n σ we assign the object S n + := ( Q + n , e ( n )) in H . Following Cvitanovi´c [12], which containsdiagrammatics for Young symmetrizers and antisymmetrizers, we depict S n + as a whitebox labelled n . The inclusion morphism S n + −→ Q + n is depicted by a white boxwith n upward oriented lines emanating from the top. The projection Q + n −→ S n + isdepicted by a white box with n upward oriented lines at the bottom. The composition Q + n −→ S n + −→ Q + n is depicted likewise. 14 PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB To the complete antisymmetrizer e ′ ( n ) ∈ k [ S n ] , e ′ ( n ) = 1 n ! X σ ∈ S n sign( σ ) σ we assign the object Λ n + := ( Q + n , e ′ ( n )) in H and depict it and related inclusions andprojections to and from Q + n by black boxes with up arrows n PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB Define the objects S − n := ( Q − n , e ( n )) and Λ − n := ( Q − n , e ′ ( n )) as the subobjects of Q − n associated to the symmetrizer e ( n ) and the antisymmetrizer e ′ ( n ) idempotents,respectively, under the canonical homomorphism k [ S n ] −→ End H ′ ( Q − n ). We draw S − n and Λ − n as white, respectively black, boxes, but with the lines at the boxes orienteddownward.We plan to develop the graphical calculus of these diagrams elsewhere. Part of thecalculus that deals with the lines oriented only upwards (or only downwards) is thegraphical calculus of symmetrizers and antisymmetrizers in the symmetric group, andcan be found in [12]. The latter calculus implies the second and third isomorphismsfrom Proposition 1 in the introduction. For instance, the second isomorphism is realizedby the diagram n mnm PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB The first family of isomorphisms in Proposition 1 is realized by the maps15 mn−1m−1n mn−1m−1 n nmmmm nn
PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB A straightforward manipulation of diagrams shows that α β = Id , α β = 0 , α β = 1 mn Id , α β = 0 . Let β ′ = mnβ . Then the mapsΛ m + ⊗ S n − α ←− −→ β S n − ⊗ Λ m + α −→ ←− β ′ Λ m − ⊗ S n − − satisfy α β = Id , α β ′ = 0 , α β ′ = Id , α β = 0 , β α + β ′ α = Id . The last equality follows from a direct diagrammatic manipulation as well. Thus, thereis an isomorphism S n − ⊗ Λ m + ∼ = (cid:0) Λ m + ⊗ S n − (cid:1) ⊕ (cid:0) Λ m − ⊗ S n − − (cid:1) , concluding the proof of Proposition 1 and Corollary 1.There is a natural bijection between partitions λ of n and (isomorphism classesof) irreducible representations of k [ S n ]. To each partition λ = ( λ , . . . , λ k ) with | λ | =16 + · · · + λ k = n there corresponds the unique common irreducible summand L λ of the representation induced from the trivial representation of parabolic subgroup S λ = S λ × · · · × S λ k ⊂ S n and the representation induced from the sign representationof parabolic subgroup S λ ∗ = S λ ∗ × · · · × S λ ∗ m , where λ ∗ is the dual partition. Let e λ ∈ k [ S n ] be the Young idempotent, so that e λ = e λ and L λ ∼ = k [ S n ] e λ . We denote by Q + ,λ := ( Q + n , e λ ) the object of H which is the direct summand of Q + n corresponding to the idempotent e λ , where we view the latter as an idempotent in theendomorphism ring via the standard homomorphism k [ S n ] −→ End H ( Q + n ). Likewise,let Q − ,λ := ( Q − n , e λ ) be the corresponding direct summand of Q − n , where we view e λ as an endomorphism of the latter object. In particular, S n + = Q + , ( n ) , Λ n + = Q + , (1 n ) , S n − = Q − , ( n ) , Λ n − = Q − , (1 n ) . The Grothendieck ring K ( H ) is an abelian group with generators–symbols [ M ], overall objects M of H and defining relations [ M ] = [ M ] + [ M ] whenever M ∼ = M ⊕ M .Monoidal structure on H descends to an associative multiplication on K ( H ), with [ ]being the identity for multiplication. Hence, K ( H ) is an associative unital ring.Recall the ring H Z from the introduction. We can now define homomorphism γ : H Z −→ K ( H ) discussed there: γ ( a n ) = [ Q − , ( n ) ] = [ S n − ] , γ ( b m ) = [ Q + , (1 m ) ] = [Λ m + ] . If we identify the subring of H Z generated by the a n ’s with the ring of symmetricfunctions Sym so that a n corresponds to n -th complete symmetric function h n , then γ will take the Schur function associated to partition λ to [ Q + ,λ ]. This function is oftendenoted s λ ; for us it is convenient to call it a λ , so that a ( n ) = a n .Similarly, we identify the subring generated by the b m ’s with Sym by taking b m tothe m -th elementary symmetric function e m . Denote by b λ the polynomial in b m ’s thatcorresponds to the Schur function s λ under this identification. In particular, b (1 m ) = b m .We have γ ( a λ ) = [ Q − ,λ ] , γ ( b λ ) = [ Q + ,λ ∗ ] . Littlewood-Richardson coefficients r νλ,µ that appear in decompositions of the productof Schur functions a λ a µ = X ν r νλ,µ a ν ,b λ b µ = X ν r νλ,µ b ν , also appear in the isomorphism formulas in H : Q + ,λ ⊗ Q + ,µ ∼ = ⊕ ν ( Q + ,ν ) r νλ,µ ,Q − ,λ ⊗ Q − ,µ ∼ = ⊕ ν ( Q − ,ν ) r νλ,µ . Q + ,λ ][ Q + ,µ ] = X ν r νλ,µ [ Q + ,ν ] , [ Q − ,λ ][ Q − ,µ ] = X ν r νλ,µ [ Q − ,ν ] . The ring H Z has a basis { b µ a λ } λ,µ over all partitions λ, µ . Consequently, elements[ Q + ,µ ][ Q − ,λ ] over all λ, µ span the subring γ ( H Z ) of K ( H ). Remark:
The symmetries ξ , ξ , ξ of H ′ extend to self-equivalences of category H ,also denoted ξ , ξ , ξ . On objects Q + ,λ ⊗ Q − ,µ they act as follows: ξ ( Q + ,µ ⊗ Q − ,λ ) = Q + ,µ ∗ ⊗ Q − ,λ ∗ ,ξ ( Q + ,µ ⊗ Q − ,λ ) = Q + ,µ ⊗ Q − ,λ ,ξ ( Q + ,µ ⊗ Q − ,λ ) = Q + ,λ ⊗ Q − ,µ . These self-equivalences induce involutions [ ξ ] , [ ξ ] and antiinvolution [ ξ ] on K ( H ).The involution of H Z corresponding to [ ξ ] is the identity. We don’t know whether [ ξ ]is the identity involution on the entire K ( H ); this would follow from Conjecture 1. Recall [29] that a functor L : A −→ B between categories A and B is left adjoint to afunctor R : B −→ A whenever there are natural transformations α : LR ⇒ Id B , β : Id A ⇒ RL (20)that satisfy the relations( α ◦ Id L )(Id L ◦ β ) = Id L , (Id R ◦ α )( β ◦ Id R ) = Id R . (21)Assume that L is both left and right adjoint to R , and the second adjunction maps α : RL ⇒ Id A , β : Id B ⇒ LR (22)are fixed as well. They satisfy( α ◦ Id R )(Id R ◦ β ) = Id R , (Id L ◦ α )( β ◦ Id L ) = Id L . (23)Out of α, α, β, β one can construct more general natural transformations betweencompositions of functors L and R by placing the basic four transformation in variouslocations in the composition of functors, and then composing several such transforma-tions. It’s convenient to draw these compositions via planar diagrams, with transfor-mations (20), (22) depicted as U-turns 18 L LR αβ R LL R
PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β A AAA BBB B
General diagrams are built out of U-turns and vertical lines, the latter denotingidentity natural transformations of R and L . For instance,PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β A ABB B is the following natural transformation from
RLR to R ( α ◦ Id)(Id ◦ α ◦ Id ⊗ )(Id ◦ β ◦ Id ⊗ )( β ◦ Id) α The four biadjointness equations (21), (23), which can drawn as = == =
PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β A AA A AAA A B BB B BBB B are equivalent to the condition that the lines and circles can be isotoped without chang-ing the natural transformation associated to the diagram. This was observed in [11, 31].The graphical calculus of biadjoints can be further enhanced. Assume given a col-lection of categories and a collection of functors between them such that each functorhas a biadjoint, which is also in the collection, and the biadjointness transformationsare fixed. Natural transformations generated by the biadjointness ones can be drawn19ia diagrams on the plane strip R × [0 , z in the center of a category A (i.e. z is the endomorphismof the identity functor Id A ) can be shown as freely floating in a region labelled A . Twosuch central elements can freely move past each other.Given two functors F, G : A −→ B in the collection, a natural transformation a : F −→ G can be depicted as a labelled dot on a line separating a segment labelled F from a segment labelled G . (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) FG a PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β A B
When a is dragged through a U -turn, it can change in two possible ways, dependingon the type of a U -turn, into natural transformations a ∗ , ∗ a : G ′ −→ F ′ , where F ′ , G ′ are the biadjoints of F and G . If ∗ a = a ∗ , we say that the biadjointness data is cyclic .For more on the cyclicity condition see [1, 2, 11, 27].Biadjoint functors appear throughout categorification and TQFTs, see discussionsin [21, Section 5], [22] and references therein, also [6]. We fix a commutative ring k and denote by k G the group algebra of finite group G with coefficients in k . We denote by ( G ) the group algebra k G , viewed as a ( k G, k G )-bimodule. If H is a subgroup of G and we view k G as a ( k G, k H )-bimodule via theleft action of k G and the right action of k H , we denote it by ( G H ) . When viewing k G as a k H -bimodule, denote it by ( H G H ), etc. Similar shortcut notation is adopted fortensor products of bimodules. For instance, ( G H G ) denotes k G -bimodule k G ⊗ k H k G ,while ( H G H G ) denotes the same space, but viewed as a ( k H, k G )-bimodule.Start with the 2-category BF in whose objects are finite groups G , morphisms from G to H are ( k H, k G )-bimodules, and 2-morphisms are bimodule homomorphisms. Con-sider the 2-subcategory IRF in ′ of BF in with the same objects as
BF in, while mor-phisms are finite direct sums of tensor products of bimodules ( G H ) and ( H G ) corre-sponding to the induction and restriction functors between categories of H and G -modules. Thus, a 1-morphism from G to G ′ is a finite direct sum of bimodules isomor-phic to ( G n H n − G n − H n − . . . H G H G )where G ′ = G n ⊃ H n − ⊂ G n − ⊃ · · · ⊂ G ⊃ H ⊂ G = G is a zigzag of inclusionsbetween finite groups. The 2-morphisms in IRF in ′ are bimodule homomorphisms.20lternatively, we can think of 1-morphisms in IRF in ′ as given by compositions ofinduction and restriction functors between categories of (left) G -modules, over finitegroups G .In this section, we develop basics of a graphical calculus for studying 2-morphismsin IRF in ′ . In general, given a unital inclusion of rings B ⊂ A , the induction functorInd : B − mod −→ A − mod that takes M to A ⊗ B M is left adjoint to the restrictionfunctor. An inclusion ι : H ⊂ G of finite groups produces an inclusion k H ⊂ k G ofgroup algebras, with the induction functorInd GH : k H − mod −→ k G − modbeing both left and right adjoint (i.e. biadjoint) to the restriction functorRes HG : k G − mod −→ k H − mod . The biadjointness endomorphisms are given by the following four bimodule maps1) : ( G H G ) −→ ( G ) , x ⊗ y xy, x, y ∈ ( G ) ,
2) : ( H ) −→ ( H G G G H ) , x x ⊗ ⊗ x, x ∈ ( H ) ,
3) : ( H G G G H ) ∼ = ( H G H ) −→ ( H ) , g g if g ∈ H, g , if g ∈ G \ H. We denote this projection map by p H : ( H G H ) −→ ( H ), p H ( g ) = g if g ∈ H and p H ( g ) = 0 if g ∈ G \ H , extended by k -linearity. Clearly, p H is a map of k H -bimodules.4) Let G = F mi =1 Hg i be a decomposition of G into left H -cosets, so that m = [ G : H ]is the index of H in G . Notice that the element m X i =1 g − i ⊗ g i ∈ ( G H G )does not depend on the choice of coset representatives { g i } mi =1 of H in G : if g ′ i = h i g i then m X i =1 g ′− i ⊗ g ′ i = m X i =1 g − i h − i ⊗ h i g i = m X i =1 g − i ⊗ g i , since the tensor product is over k [ H ], and h i can be moved through the tensor productsign. Define bimodule map ( G ) −→ ( G H G ) (24)by the condition that 1 m X i =1 g − i ⊗ g i , so that g m X i =1 g − i ⊗ g i g = m X i =1 gg − i ⊗ g i . g i g = h i g i ′ for some h i ∈ H and i ′ ∈ { , , . . . , m } . Theassigment i i ′ is a bijection of { , , . . . , m } . We have m X i =1 g − i ⊗ g i g = m X i =1 g − i ⊗ h i g i ′ = m X i =1 g − i h i ⊗ g i ′ = m X i ′ =1 gg − i ′ ⊗ g i ′ = m X i =1 gg − i ⊗ g i . Combining with the earlier remark, we see that (24) is a bimodule map which does notdepend on the choices of H -coset representatives g i .We associate to these four bimodule maps the following four pictures HG G HG GGH HGH H
PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB Thus,
HG G
PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB , denoted α GH , is the map( G H G ) −→ ( G ) , x ⊗ y xy, x, y ∈ ( G ) . GH H
PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB , denoted β GH , is the map( H ) −→ ( H G G G H ) ∼ = ( H G H ) , x x ⊗ ⊗ x, x ∈ ( H ) . GH H
PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB , denoted α HG , is the map p H described earlier,( H G G G H ) ∼ = ( H G H ) −→ ( H ) , x p H ( x ) , x ∈ ( G ) . HG G
PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB , denoted β HG , is the k -linear map (24)( G ) −→ ( G H G ) , g m X i =1 g − i ⊗ g i g, g ∈ G. Theorem 2
These four bimodule maps turn induction and restriction functors
Ind GH and Res HG into a cyclic biadjoint pair.Proof: First, we check that the adjointness equations (21) and (23) hold for thesemaps. The bimodule map ( G H ) −→ ( G H G H ) −→ ( G H ) corresponding to the left handside of the first equation in (21) is given by g g ⊗ g g , hence the map isthe identity: 22 G H G H
PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB The bimodule map ( H G ) −→ ( H G H G ) −→ ( H G ) for the left hand side of the secondequation in (21) is given by g ⊗ g g = g , and the map is the identity: GH = GH PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB The bimodule map ( H G ) −→ ( H G H G ) −→ ( H G ) for the left hand side of the firstequation in (23) is given by g m X i =1 gg − i ⊗ g i m X i =1 p H ( gg − i ) g i . Notice that p H ( gg − i ) = 0 iff g / ∈ Hg i , and p H ( hg i g − i ) = h . Therefore, g g underthe map, and the first equation in (23) holds: GH = H G
PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB The bimodule map ( G H ) −→ ( G H G H ) −→ ( G H ) for the left hand side of the secondequation in (23) is given by g m X i =1 g − i ⊗ g i g m X i =1 g − i p ( g i g ) = g by a similar computation, so that = G HHG
PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB ,and the four bimodule maps above determine biadjointness morphisms for inductionand restriction functors between H and G .23onsider the k -algebra k G H := { a ∈ k G | ha = ah ∀ h ∈ H } of H -invariants in k G with respect to the conjugation action. This algebra is canonicallyisomorphic to the endomorphism ring of the bimodule ( H G ), and, therefore, to theendomorphism ring of the functor Res HG , via the map that assigns to a ∈ k G H theendomorphism ′ a ( x ) := ax , where x ∈ ( H G ). Likewise, the opposite algebra of k G H iscanonically isomorphic to the endomorphism ring of the bimodule ( G H ) and, therefore,to that of the functor Ind GH , via the map that assigns to a ∈ k G H the endomorphism a ′ ( x ) := xa , where x ∈ ( G H ).Thus, to a ∈ k G H we assign endomorphisms a ′ and ′ a of Ind GH and Res HG and depictthem by (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) G H a (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)
H a G
PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB Lemma 1
For any a ∈ k G H there are equalities of bimodule homomorphisms (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) aG H (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) G H a = PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB α GH ( a ′ ◦ Id) = α GH (Id ◦ ′ a ) , (25) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) a G H (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) aG H = PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB ( ′ a ◦ Id) β GH = (Id ◦ a ′ ) β GH , (26) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) aGH (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) a GH = PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB α HG ( ′ a ◦ Id) = α HG (Id ◦ a ′ ) , (27) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) G H a (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)
G H a = PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB ( a ′ ◦ Id) β HG = (Id ◦ ′ a ) β HG . (28)The left hand side of the first equality is a map ( G H G ) −→ ( G ) given by g ⊗ g g a ⊗ g g ag . The right hand side is g ⊗ g g ⊗ ag g ag , and theequality is obvious.The second equality follows from an equally trivial computation.The third equality is the equation p H ( ga ) = p H ( ag ) for g ∈ G and a ∈ k G H . Itsuffices to check it when k = Z and a = P h ∈ H hkh − for some k ∈ G . The equationbecomes X h ∈ H p H ( ghkh − ) = X h ∈ H p H ( hkh − g ) (29)24he left hand side equals X h ∈ H p H ( ghk ) h − = X h − g − u = k p H ( u ) p H ( h − ) = X hg − u = k p H ( u ) p H ( h )where in the first equality we set u = ghk , the sum being over all u, h ∈ G with h − g − u = k . For the second equality, we converted h to h − .The right hand side of (29) equals X h ∈ H hp H ( kh − g ) = X ug − h = k p H ( h ) p H ( u ) , where we set u = kh − g and the sum is over all u, h ∈ G . Interchanging h and u , wesee that (29) holds.For the last of the four equations, it suffices to check that the image of 1 ∈ G is thesame under these two bimodule homomorphisms. For the one on the left,1 m X i =1 g − i ⊗ g i m X i =1 g − i ⊗ ag i . For the one on the right,1 m X i =1 g − i ⊗ g i m X i =1 g − i a ⊗ g i . Again, we can assume k = Z and a = P h ∈ H hkh − for some k ∈ G . The equationbecomes X i,h g − i ⊗ hkh − g i = X i,h g − i hkh − ⊗ g i , (30)summing over 1 ≤ i ≤ m and h ∈ H . We have X i,h g − i ⊗ hkh − g i = X i,h g − i h ⊗ kh − g i = X u ∈ G u ⊗ ku − , where, in the first equality, h is moved to the left (the tensor product is over k H ), andin the second equality u = g − i h runs over all elements of G as i changes from 1 to m and h runs over all elements of H . Likewise, X i,h g − i hkh − ⊗ g i = X i,h g − i hk ⊗ h − g i = X u ∈ G uk ⊗ u − . Equation (30) and Lemma 1 follow.Since the box labelled by a ∈ k G H can be dragged through any U -turn, we see thatthe biadjointness maps have the cyclic property – dragging the box labelled a all theway along a circle brings us back to the original diagram. This concludes the proof ofTheorem 2. (cid:3) There are obvious simplification relations25
H H = [ G : H ] = G H G
PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB The first relation says that a counterclockwise bubble with G inside and H outsidecan be erased. The second relation allows to remove a clockwise bubble at the cost ofmultiplying the diagram by the index of H in G .For each inclusion of finite groups H ⊂ K ⊂ G there is a canonical isomorphismbetween induction functors Ind GH ∼ = Ind GK ◦ Ind KH which corresponds to a canonicalisomorphism of bimodules ( G ) H ∼ = ( G ) K ( K ) H . Likewise, canonical isomorphism be-tween restrictions Res HG ∼ = Res HK ◦ Res KG is given by a natural isomorphism of bimodules H ( G ) ∼ = H ( K ) K ( G ). We draw these isomorphisms via trivalent diagrams G HK GH KG HK GH K
PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB Since the isomorphisms are mutually-inverse, we have, for the first two isomor-phisms,
G HKKKG H = KG H G H = PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB Mutual inversion of the other two isomorphisms can be similarly depicted. Thesedefinitions are compatible with isotopies–identities
G HK G HK G HK = =
PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB hold (likewise for the other pair of isomorphisms). These identities imply that variousdefinitions of trivalent vertices in other positions relative to the y -coordinate are all thesame. For instance, if we define G KH G KH := PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB then 26 KH G KH = PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB The associativity relation for the inclusions of four groups L ⊂ H ⊂ K ⊂ G has theform G HK L G K HL = PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB The induction and restriction functors for H and G depend on the inclusion ι : H ֒ → G . Conjugating the inclusion by an element g ∈ G, so that ι ′ ( h ) = ghg − , ι ′ : H ֒ → G ,leads to induction and restriction functors isomorphic to the original ones, via bimodulemaps ( G ) ι ( H ) −→ ( G ) ι ′ ( H ) , f f g − , f ∈ G, ι ( H ) ( G ) −→ ι ′ ( H ) ( G ) , f gf, f ∈ G. We depict these conjugation isomorphisms via a mark on a line with g next to it: g HG PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB The Mackey induction-restriction theorem says that, given subgroups
H, K of afinite group G , there is an isomorphismRes KG ◦ Ind GH ∼ = ⊕ i ∈ I Ind KK ∩ g i Hg − i ◦ Res K ∩ g i Hg − i H , (31)where the sum is over representatives g i of ( K, H )-cosets of G , G = ⊔ i ∈ I Kg i H. Let L i = K ∩ g i Hg − i . Diagrams L i HGK L i K HG
PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB K, H )-bimodule maps α i : ( K ) L i ′ ( H ) −→ K ( G ) H , β i : K ( G ) H −→ ( K ) L i ′ ( H ) . Here ( K ) L i ′ ( H ) is k [ K ] ⊗ k [ L i ] k [ H ], with x ∈ L i acting on H by right multiplication by g − i xg i . Proposition 6
The maps P i ∈ I α i and P i ∈ I β i are mutually-inverse isomorphisms ofbimodules ⊕ i ∈ I ( K ) L i ′ ( H ) and K ( G ) H . This proposition is a pictorial restatement of the Mackey theorem. Proof is left tothe reader and amounts to checking the following relations L i GG HKGK H = PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB L i L i GK HL i K H = PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB L i L j GK H = 0
PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβ g j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB We now specialize the earlier construction to the case of the symmetric group S n ,viewed as the permutation group of { , , . . . , n } , and induction/restriction functorsfor inclusions S n ⊂ S n +1 , where S n is identified with the stabilizer of n + 1 in S n +1 .Notations for bimodules will be further simplified, so that, for instance, n ( n + 1) n − stands for k [ S n +1 ], viewed as a ( k [ S n ] , k [ S n − ])-bimodule for the standard inclusions S n ⊂ S n +1 ⊃ S n − , and n ( n + 1) n ( n + 2) stands for k [ S n +1 ] ⊗ k [ S n ] k [ S n +2 ], viewedas a ( k [ S n ] , k [ S n +2 ])-bimodule. The regions of the strip R × [0 ,
1] are now labelled bynonnegative integers n . An upward oriented line separating regions labelled n and n + 128 n+1 PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB denotes the identity endomorphism of the induction functorInd n +1 n : k [ S n ] − mod −→ k [ S n +1 ] − mod. This is the functor of tensoring with the bimodule ( n + 1) n .A downward oriented line separating regions n + 1 and n n+1n PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB denotes the identity endomorphism of the restriction functorRes nn +1 : k [ S n +1 ] − mod −→ k [ S n ] − mod . The bimodule for this functor is n ( n + 1).The four U -turns are given by the following bimodule maps: n n+1 PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB ( n + 1) n ( n + 1) −→ ( n + 1) ,g ⊗ h gh, g, h ∈ S n +1 , (32) nn+1 PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB ( n ) −→ n ( n + 1) n , g g, g ∈ S n (33) nn+1 PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB p n : n ( n + 1) n −→ ( n ) , p n ( g ) = ( g if g ∈ S n , n n+1 PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB q n : ( n + 1) −→ ( n + 1) n ( n + 1) , (35)where the bimodule map q n is determined by the condition q n (1) = n +1 X i =1 s i s i +1 . . . s n ⊗ s n . . . s i +1 s i , s i = ( i, i + 1) . Notice that { s n . . . s s , s n . . . s s , . . . , s n s n − , s n , } are n + 1 coset representatives of S n ⊂ S n +1 , and q n ( g ) = n +1 X i =1 gs i s i +1 . . . s n ⊗ s n . . . s i +1 s i = n +1 X i =1 s i s i +1 . . . s n ⊗ s n . . . s i +1 s i g, g ∈ S n +1 . p n and q n are the second adjointness maps for the group inclusion S n ⊂ S n +1 . For an arbitrary inclusion of finite groups H ⊂ G these maps were describedin the previous subsection.Denote by an upward-pointing crossing n PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB the endomorphism of ( n + 2) n givenby right multiplication by s n +1 , so that g gs n +1 , g ∈ S n +2 .Denote by a downward-pointing crossing n PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB the endomorphism of n ( n + 2) givenby left multiplication by s n +1 , so that g s n +1 g, g ∈ S n +2 .Denote by a right-pointing crossing n PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB the bimodule endomorphism ( n ) n − ( n ) −→ n ( n + 1) n that takes g ⊗ h for g, h ∈ S n to gs n h ∈ S n +1 .Denote by a left-pointing crossing n PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB the bimodule endomorphism n ( n +1) n −→ ( n ) n − ( n ) that takes g ∈ S n ⊂ S n +1 to 0 and gs n h for g, h ∈ S n to g ⊗ h ∈ ( n ) n − ( n ).These four definitions-notations are compatible with the isotopies of diagrams inthe plane strip–there are equalities of bimodule endomorphisms = n nn n = = n nn n = PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB They can be checked by direct computations.30 roposition 7
The following relations hold for any n ∈ Z . = n nn = 0= n n = n nn nn = n = 1 PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB (36)Here and further we follow the convention that a diagram equal 0 if it has a regionlabelled by a negative number. Notice that the top two relations come from the relationsin the symmetric groups: s n +1 = 1 and s n +1 s n +2 s n +1 = s n +2 s n +1 s n +2 and follow at oncefrom the definition of the bimodule homomorphism associated to an upward-pointingcrossing. The four remaining relations encode the bimodule decomposition n ( n + 1) n ∼ = ( n ) n − ( n ) ⊕ ( n )giving an isomorphism Res nn +1 ◦ Ind n +1 n ∼ = Ind nn − ◦ Res n − n ⊕ Idof endofunctors in the category k [ S n ] − mod. This is a special case of the Mackeydecomposition theorem which was given a diagrammatic interpretation in an earliersection. Proposition 7 relations are identical to the ones in the definition of the category H ′ , see Section 2.1.Let S ′ be the category whose objects are compositions of induction and restrictionfunctors between symmetric groups for standard embeddings S k ⊂ S k +1 . The mor-phisms are natural transformations of functors (again, we work over a field k of charac-teristic 0). The category S ′ is the sum of categories S ′ k over k ≥
0; in the latter the firstinduction or restriction starts from S k . For instance, Ind k +2 k +1 ◦ Ind k +1 k ◦ Res kk +1 ◦ Ind k +1 k is an object of S ′ k . Morphisms in S ′ k are natural transformation of functors and can beidentified with homomorphisms of associated bimodules.Thus, for each k ≥
0, there is a functor F ′ k : H ′ −→ S ′ k that takes Q ǫ to thecorresponding composition of induction and restriction functors. For instance, F ′ k ( Q ++ − + ) = Ind k +2 k +1 ◦ Ind k +1 k ◦ Res kk +1 ◦ Ind k +1 k . If, for some m , the last m terms of ǫ contain at least k +1 more minuses than pluses, then F ′ k ( Q ǫ ) = 0. On morphisms F ′ k is defined as follows. It takes a diagram representing a31orphism in H ′ , labels the rightmost region of the diagram by k , and views the diagramas a natural transformation between compositions of induction and restriction functors.The functor F ′ k is not monoidal, since S ′ k does not have a monoidal structure matchingthat of H ′ .Let S , respectively S k , be the Karoubi envelope of S ′ , respectively S ′ k . Functor F ′ k induces a functor on Karoubi envelopes F k : H −→ S k . We summarize relevantcategories and functors below. S = Kar( S ′ ) , S k = Kar( S ′ k ) , F ′ k : H ′ −→ S ′ k , F k : H −→ S k . S ′ ⊕ k ≥ S ′ k Kar y y
Kar
S ⊕ k ≥ S k H ′ F ′ k −−−→ S ′ k Kar y y
Kar H F k −−−→ S k Functor F k induces a homomorphism of abelian groups[ F k ] : K ( H ) −→ K ( S k ) . (37)Notice that K ( H ) is a ring, while K ( S k ) is only an abelian group.An object of S k is a direct summand of a finite sum of composition of induc-tion and restriction functors that start with the category of k [ S k ]-modules, thus ittakes any finite-dimensional k [ S k ]-module to a module over ⊕ m ≥ k [ S m ]. Descending toGrothendieck groups, we obtain a homomorphism θ k : K ( S k ) −→ Hom Z ( K ( k [ S k ]) , ⊕ m ≥ K ( k [ S m ])) . From here until the end of this paper we assume that c har ( k ) = 0. Consider thecomposite homomorphism θ k [ F k ] : K ( H ) −→ Hom Z ( K ( k [ S k ]) , ⊕ m ≥ K ( k [ S m ])) . This homomorphism takes [ Q + ,µ ] to a map that assigns to [ M ] ∈ K ( k [ S k ]) , for a k [ S k ]-module M , the symbol [Ind S | µ | + k S | µ | × S k ( L µ ⊗ M )] of induced module over k [ S | µ | + k ]. In otherwords, tensor M with L µ , producing a module over k [ S | µ | ] × S k , induce to k [ S | µ | + k ],then pass to the Grothendieck group.Likewise, θ k [ F k ] takes [ Q − ,λ ] to the zero map if | λ | > k and, if k ≥ | λ | , to the mapthat assigns to [ M ] as above the symbol of the moduleHom k [ S | λ | ] ( L λ , M ) ∈ k [ S k −| λ | ] − mod32n other words, restrict M to being a module over the group algebra of S | λ | × S k −| λ | ⊂ S k ,and form homs from the simple module L λ over S | λ | . The result is a representation ofthe symmetric group S k −| λ | . Now consider composition θ k [ F k ] γ : H Z −→ Hom Z ( K ( k [ S k ]) , ⊕ m ≥ K ( k [ S m ])) . We claim that the sum of these maps, over all k ≥
0, is injective. Let y = X λ,µ y λ,µ b µ a λ , y λ,µ ∈ Z be an arbitrary nonzero element of H Z . We have γ ( y ) = X λ,µ y λ,µ [ Q + ,µ ∗ ][ Q − ,λ ] . When | λ | = k , the map θ k [ F k ] γ ( a λ ) = θ k [ F k ]([ Q − ,λ ])takes [ L ν ] to 0 if | ν | = k and ν = λ . The same map takes [ L λ ] to [ L ∅ ], the symbol ofthe simple module over k [ S ] = k .Choose k such that y λ,µ = 0 for some λ with | λ | = k and some µ , while y λ,µ = 0 forall µ whenever | λ | < k . Also choose ν with | ν | = k and y ν,µ = 0 for some µ . The map θ k [ F k ] γ ( y ) takes [ L ν ] to X µ y ν,µ [ L µ ∗ ] = 0 . Therefore, θ k [ F k ] γ ( y ) is a nonzero map, and γ ( y ) = 0. This concludes the proof that γ is injective (Theorem 1). H ′ In this section we will prove Propositions 3 and 4. Consider the right curl with therightmost region labelled n (also recall the shorthand of denoting this curl by a dot).This curl can be realized as the composition of a cup with a crossing with a cap. n n = PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB n + 1) n takes 1 to J n := n X i =1 s i . . . s n − s n s n − . . . s i = (1 , n + 1) + (2 , n + 1) + · · · + ( n, n + 1) . This endomorphism of ( n + 1) n is the right multiplication by J n : g gJ n , g ∈ S n +1 . Notice that J n is the Jucys-Murphy element, ubiquitous in the representation theoryof the symmetric group. Our diagrammatics realizes it via the right curl and interpretsthe endomorphism of multiplication by J n as the composition of three natural trans-formations, two of which (cup and cap) come from biadjointness of the induction andrestriction functors.Commutativity of Jucys-Murphy elements now acquires a graphical interpretationas isotopies of curls (or dots) on upward-oriented disjoint strands past each other. = PSfrag replacements Q + − Q − + ιι ′ g i g − i P i ∈ I αβg j if i = j S n − ⊗ Λ m + Λ m + ⊗ S n − Λ m − ⊗ S n − − − P k − b =0 ( k − b − α α β β AB For each n ≥ F ′ n , applied to and its endomorphisms, produces a ho-momorphism from End H ′ ( ) to Z ( k [ S n ]), the center of the group ring of the symmetricgroup. Composing with the homomorphism ψ , we obtain the homomorphism ψ ,n : Π ∼ = k [ c , c , . . . ] −→ Z ( k [ S n ])that takes c k to c k,n = n X i =1 s i . . . s n − J kn − s n − . . . s i (38)= n X i =1 ( i, i + 1 , . . . , n ) (cid:0) (1 , n ) + · · · + ( n − , n ) (cid:1) k ( n, n − , . . . , i ) . (39)We’d like to show that the union of ψ ,n over all n ,Π −→ ⊕ n ≥ Z ( k [ S n ]) ,
34s injective. For the first few values of k we have c ,n = n,c ,n = 2 X ≤ i
0, the lower order terms is a sumover permutations of disturbance less than k + 1, where we define the disturbance of apermutation σ ∈ S n as the number of elements moved by σ , dist ( σ ) = |{ i | ≤ i ≤ n, σ ( i ) = i }| . Notice that c k,n is the sum of conjugates of J kn − and contains terms of disturbance atmost k + 1, for k > har ( k ) = 0, the coefficient of the sum of ( k + 1)-cycles in (40) is nonzero.Consider an increasing filtration Z = k ⊂ Z ⊂ Z ⊂ · · · ⊂ Z n = Z ( k [ S n ])by having Z i be the span over conjugacy classes that consist of permutations of distur-bance at most i. We make k [ c , c , . . . ] graded by deg( c ) = 0 and deg( c k ) = k + 1 for k >
0, and then consider associated increasing filtration on k [ c , c , . . . ], where the i -thfiltered terms is the sum of graded terms of degree at most i . The homomorphism ψ ,n respects the two filtrations.To show asymptotic faithfulness of ψ ,n as n −→ ∞ we examine induced homomor-phism of adjoint graded rings. Assume that there is a universal relation in End H ′ ( ) X I a I c I = 0which holds for all sufficiently large n . Here I = ( a , a , . . . , a i ) is a finite sequence ofnon-negative integers, a I ∈ k , and c I = c a c a . . . c a i i is the corresponding monomial in k [ c , c , . . . ]. The sum is over finitely many sequences I . This implies relations X I a I c I,n = 0for all n , where c I,n = c a ,n c a ,n . . . c a i i,n . Choose any term from the sum corresponding to I = ( a , a , . . . , a i ) with the highest possible degree monomial c I . The only terms in35he sum P I a I c I,n that contribute to the conjugacy class of type ( a + 1 , . . . , a i + 1) cancome from sequences I ′ = ( a, a , . . . , a i ) that differ from I only in the first term. Since,for such I ′ , c I ′ ,n = n a c a ,n . . . c a i i,n the relation P I a I c I,n = 0 implies P a a I ′ n a = 0, which, in turn, leads to a I ′ = 0 forall I ′ as above, since the only polynomial with infinitely many positive integers n asroots is the zero polynomial. This contradiction implies asymptotic faithfulness of ψ ,n .Therefore, ψ is injective, which concludes the proof of Proposition 3 when c har k = 0.Take k = Q , then pass to the subring Z of Q . Our arguments imply that ψ is anisomorphism over Z , and, hence, over any commutative ring k , including any field.We next prove injectivity of ψ m (Proposition 4), see equation (18), again first in thecharacteristic zero case. The algebra DH m ⊗ Π has a basis of elements x a . . . x a m m · σ · c b c b . . . c b k k over permutations σ ∈ S m and a i , b j ∈ Z + , see discussion before Proposi-tion 4.Applying functor F ′ n to Q + m and its endomorphism ring gives us a homomorphismEnd H ′ ( Q + m ) −→ End(( n + m ) n )to the endomorphism ring of the ( k [ S n + m ] , k [ S n ])-bimodule k [ S n + m ], which we alsodenote ( n + m ) n . The composite homomorphism ψ m,n : DH m ⊗ Π −→ End(( n + m ) n )takes elements of DH m ⊗ Π to endomorphisms given by right multiplication by suitableelements of k [ S n + m ]. Namely, ψ m,n ( σ ), for a permutation σ ∈ S m ⊂ DH m , is the rightmultiplication by σ , where we define the latter as the translate of σ by n : σ ( i + n ) = σ ( i ) + n, ≤ i ≤ m, σ ( i ) = i, ≤ i ≤ n.ψ m,n ( x i ), where x i the the diagram of m vertical lines with the dot (right curl) onthe i -th strand from the left, is the endomorphism of right multiplication by J n + m − i ,and ψ m,n ( c k ) is the right multiplication by c k,n . The map ψ m,n is described by thecorresponding homomorphism ψ ′ m,n from DH m ⊗ Π to the opposite of the group algebra k [ S n + m ] op ⊃ End(( n + m ) n ), with ψ ′ m,n ( x i ) = J n + m − i , ψ ′ m,n ( c k ) = c k,n , etc. We need totake the opposite algebra since the ring of endomorphism of a ring A viewed as a left A -module is naturally isomorphic to the opposite of A : End A ( A A, A A ) ∼ = A op . Define m -disturbance of a permutation σ ∈ S n + m as the number of integers between1 and n that are moved by σ : dist m ( σ ) = |{ i | ≤ i ≤ n, σ ( i ) = i }| . Notice that dist m ( στ ) ≤ dist m ( σ ) + dist m ( τ ). On the group algebra k [ S n + m ] we canintroduce an increasing filtration k [ S m ] = Z m ⊂ Z m ⊂ · · · ⊂ Z mn = k [ S n + m ]36here Z mk is spanned by permutations of disturbance at most k .We turn DH m ⊗ Π into a filtered algebra by setting deg( c ) = 0 , deg( c k ) = k + 1if k >
0, deg( x i ) = 1 and deg( σ ) = 0 and then making the k -th term in the increasingfiltration be spanned by the basis elements of total degree at most k .Homomorphism ψ ′ m,n is that of filtered algebras, and we can pass to the homomor-phism of associated graded algebras. To show asymptotic faithfulness of ψ ′ m,n we fix m and will be taking n large compared to m . Assume that there exists a relation X d σ, a , b x a . . . x a m m · σ · c b c b . . . c b r r = 0 (41)in End H ′ ( Q + m ) for some d σ, a , b ∈ k \ { } , with a = ( a , . . . , a m ) , b = ( b , . . . , b r ), thesum over finitely many triples ( σ, a , b ). Let x ( σ, a , b ) = x a . . . x a m m · σ · c b c b . . . c b r r denote the elements of our basis of DH m ⊗ Π. The element ψ ′ m,n ( x ( σ, a , b )) ∈ k [ S n + m ]belongs to the k -th term of the filtration of the latter, where k = a + · · · + a m + 2 b + 3 b + · · · + ( r + 1) b r , but not to the ( k − x ( σ, a , b ) in the sum select only thosewith the maximal possible k (denote such k by k ). It’s enough to show that, as we sumover only these terms, the image of P ψ ′ m,n ( x ( σ, a , b )) in the associated graded ring of k [ S n + m ] relative to the above filtration is nonzero. In other words, we need to showthat coefficients of permutations of disturbance k are not all zero for some sufficientlylarge n in the expansion of ψ m,n applied to the LHS of (41). This is obtained by lookingat the structure of these permutations. They are disjoint unions of cycles, with someof the cycles containing one or more elements of the set P = { n + 1 , . . . , n + m } . Therelative positions of elements of P in the cycles, lengths of portions of the cycles betweenelements of P , together with the number of cycles of each length without elements of P uniquely determine σ , a , . . . , a m and c , . . . , c r that can contribute to the coefficientof such permutation. Coefficients at different powers of c are taken care in the sameway as in the m = 0 case. Linear independence of our spanning set of End H ′ ( Q + m )and Proposition 4 follow when c har k = 0. Same argument as in the m = 0 case thenimplies that ψ m is an isomorphism over any commutative ring k .The formula (40) implies that the natural homomorphism End H ′ ( ) −→ Z ( k [ S n ])from the endomorphisms of the identity object of H ′ to the center of the group al-gebra is surjective when the field k has characteristic 0. Combining with the resultof Cherednik [10] and Olshanskii [37], [8, Theorem 3.2.6] that the centralizer algebraof k [ S n ] in k [ S n + m ] is generated by DH m and the center of k [ S n ], we obtain that thehomomorphism DH m ⊗ Π −→ End(( n + m ) n ) introduced above is surjective whenc har k = 0. 37 Remarks on the Grothendieck ring of H H ′ For a sequence ǫ of pluses and minuses denote by h ǫ i the difference between the numberof pluses and minuses in ǫ (the weight of ǫ ). Then Hom H ′ ( Q ǫ , Q ǫ ′ ) = 0 if and only if h ǫ i 6 = h ǫ ′ i . The ”if” part of this observation implies that categories H ′ and H , viewedas additive categories, decompose into the direct sum of subcategories H ′ = ⊕ ℓ ∈ Z H ′ ℓ , H = ⊕ ℓ ∈ Z H ℓ , where H ′ ℓ is a full subcategory of H ′ which contains objects Q ǫ over all sequences ofweight ℓ , and H ℓ is the Karoubi envelope of H ′ ℓ .This direct sum decomposition induces a grading on the Grothendieck ring K ( H ) = ⊕ ℓ ∈ Z K ( H ℓ )The Heisenberg algebra H and its integral form H Z are graded by deg( a n ) = n = − deg( b n ), and the homomorphism γ : H Z −→ K ( H ) is that of graded rings.We can redefine the Grothendieck groups K ( H ) and K ( H ℓ ) via idempontentedrings. For a sequence ǫ let End( ǫ ) := End H ( Q ǫ )denote the endomorphism algebra of Q ǫ . Likewise, denoteHom( ǫ, ǫ ′ ) := Hom H ( Q ǫ , Q ǫ ′ ) . To the category H we can assign the idempotented ring of all homomorphisms betweenvarious tensor products of generating objects Q + and Q − : R := ⊕ ǫ,ǫ ′ Hom( ǫ, ǫ ′ ) , the sum over all sequences ǫ, ǫ ′ . Ring R is nonunital, but has a family of idempotents1 ǫ = 1 ∈ End( ǫ ) . Right projective R -modules 1 ǫ R correspond to objects Q ǫ , in the sense thatHom R (1 ǫ R, ǫ ′ R ) = Hom( ǫ, ǫ ′ ) = Hom H ′ ( Q ǫ , Q ǫ ′ ) , (42)and the Grothendieck group of finitely-generated projective right R -modules is canon-ically isomorphic to the Grothendieck group of H : K ( R ) ∼ = K ( H ) . This isomorphism takes [1 ǫ R ] to [ Q ǫ ]. Usually we use K ( A ) to denote the Grothendieckgroup of finitely-generated projective left, not right, A -modules. Here, because of (42),38e use right R -modules in the definition of K ( R ). Alternatively, we could use left R op -modules, or even left R -modules after fixing an isomorphism R ∼ = R op (involution ξ induces one such isomorphism). We have R = ⊕ ℓ ∈ Z R ℓ , R ℓ := ⊕ h ǫ i = h ǫ ′ i = ℓ Hom( ǫ, ǫ ′ ) . Assume from now on that ℓ ≥ ξ to reverse ℓ ). Given a sequence ǫ with n + ℓ pluses and n minuses, the object Q ǫ of H ℓ decomposes into the direct sum of objects Q + k + ℓ − k with0 ≤ k ≤ n with some multiplicities. Hence, R ℓ is Morita equivalent to the idempotentedring R ℓ = ∞ ⊕ k,k ′ =0 Hom(+ k + ℓ − k , + k ′ + ℓ − k ′ ) , and the inclusion R ℓ ⊂ R ℓ induces an isomorphism of Grothendieck groups K ( R ℓ ) ∼ = K ( R ℓ ) . Let R ℓ,m = m ⊕ k,k ′ =0 Hom(+ k + ℓ − k , + k ′ + ℓ − k ′ ) . The ring R ℓ is the union of rings in the increasing chain R ℓ, ⊂ R ℓ, ⊂ . . . . Formationof Grothendieck group commutes with direct limits, implying that K ( R ℓ ) is the directlimit of K ( R ℓ,m ) as m goes to infinity. Thus, there is an isomorphism K ( H ℓ ) ∼ = lim m →∞ K ( R ℓ,m ) . For each k between 0 and m natural inclusion of rings End(+ k + ℓ − k ) ⊂ R ℓ,m inducesa homomorphism of groups K (End(+ k + ℓ − k )) −→ K ( R ℓ,m ). Conjecture 1 would followfrom the following two conjectures: Conjecture 1.1.
The standard inclusion k [ S n × S m ] ⊂ End(+ n − m ) induces an isomor-phism of Grothendieck groups of these two rings. Conjecture 1.2.
Ring inclusion m ⊕ k =0 End(+ k + ℓ − k ) ⊂ R ℓ,m induces an isomorphism on Grothendieck groups.We don’t know how to prove either statement, but will present now some weakevidence in favor of Conjecture 1.1. Here we prove Conjecture 1.1 in the case m = 0 ( n = 0 case follows by symmetry).By Proposition 4, the endomorphism ring of the object Q + n of H is isomorphic to the39ensor product of the degenerate affine Hecke algebra DH n and the polynomial algebraΠ: End(+ n ) ∼ = DH n ⊗ Π . The inclusion k [ S n ] ֒ → DH n is split, via the homomorphism τ n : DH n −→ k [ S n ] whichtakes generators s i of DH i to transpositions ( i, i +1) and generators x i to Jucys-Murphyelements. The split inclusion induces a split short exact sequence of two rings and anideal 0 −→ ker( τ n ) −→ DH n −→ k [ S n ] −→ , which, in turn, induces a split short exact sequence of K -groups0 −→ K (ker( τ n )) −→ K ( DH n ) −→ K ( k [ S n ]) −→ , see [40, 42]. Introduce an increasing filtration0 = Z − DH n ⊂ Z DH n ⊂ Z DH n ⊂ . . . on DH n , where Z k DH n is spanned by elements of the form x a . . . x a n n σ over all σ ∈ S n and a + · · · + a n ≤ k . Then Z k DH n × Z m DH n ⊂ Z k + m DH n and Z DH n = k [ S n ]. Let B = gr DH n with respect to this filtration. B is a graded algebra isomorphic to thecross-product of the polynomial algebra in n generators with the group algebra of thesymmetric group, B ∼ = k [ x , . . . , x n ] ∗ k [ S n ].Algebra B is Koszul, with the Koszul dual algebra isomorphic to the cross-productof the exterior algebra on n generators with the group algebra of the symmetric group(recall that char( k ) = 0). Hence, B has finite Tor dimension and, in particular, Z DH n = k [ S n ] has Tor dimension n as a right B -module. B has Tor dimension 0as a right Z DH n -module, since k [ S n ] is semisimple. We are in a position to invokeQuillen’s theorem [38, Theorem 7, page 112], [5]. Theorem. L et A be a ring equipped with an increasing filtration Z k A , and suchthat Z A is regular. Suppose that B = gr( A ) has finite Tor dimension as a right Z A module and that Z A has finite Tor dimension as a right B module. Then the inclusion Z A ⊂ A induces an isomorphism K i ( Z A ) ∼ = K i ( A ).In our case A = DH n . The regularity of Z DH n is obvious due to it being semisimple(a regular ring is a noetherian ring such that every left module has finite projectivedimension). Applying the theorem in i = 0 case we obtain Proposition 8
The inclusion k [ S n ] ⊂ DH n and the surjection DH n −→ k [ S n ] inducemutually-inverse isomorphisms K ( k [ S n ]) ∼ = K ( DH n ) . Same argument shows that inclusion k [ S n ] ⊂ DH n ⊗ k [ c , . . . , c r ]40nduces an isomorphism of Grothendieck groups K ( k [ S n ]) ∼ = K ( DH n ⊗ k [ c , . . . , c r ]) . Formation of Grothendieck groups commutes with taking direct limit of rings [40, Sec-tion 1.2], and Π is the limit of k [ c , . . . , c r ] as r → ∞ . We obtain an isomorphism K ( k [ S n ]) ∼ = K (End(+ n ))proving Conjecture 1.1 when m = 0. (cid:3) Induction and restriction functors for inclusions k [ S n ] ⊗ k [ S m ] ⊂ k [ S n + m ] DH n ⊗ DH m ⊂ DH n + m induce “multiplication” and “comultiplication” maps on Grothendieck groups that turn ⊕ n ≥ K ( k [ S n ]) and ⊕ n ≥ K ( DH n )into graded birings, see Geissinger [17] for the symmetric group, Zelevinsky [43] forsemisimple generalizations, and Bergeron and Li [3], Khovanov and Lauda [23] fornonsemisimple ones. We write birings rather than bialgebras , since these K groupsare abelian groups rather than vector spaces over some field. Isomorphisms in theabove proposition are compatible with multiplication and comultiplication, and inducea biring isomorphism ⊕ n ≥ K ( DH n ) ∼ = ⊕ n ≥ K ( k [ S n ]) . The second biring can be canonically identified [17, 43] with an integral form Sym ofthe ring of symmetric functions in infinitely many variables.What can we say about K of End(+ n − m ) when n, m >
0? Recall (end of Sec-tion 2.1) that End(+ n − m ) contains 2-sided ideal J n,m spanned by basis elements ofthickness less than n + m , which fits into short split exact sequence0 −→ J n,m −→ End(+ n − m ) −→ DH n,m −→ , (43)where DH n,m := DH n ⊗ DH m ⊗ Πis the tensor product of two degenerate AHA with the polynomial algebra in infinitelymany generators. The earlier argument via Quillen’s theorem shows that the inclusion k [ S n ] ⊗ k [ S m ] −→ DH n ⊗ DH m ⊗ k [ c , c , . . . , c r ]41nduces an isomorphisms of K -groups K ( k [ S n ] ⊗ k [ S m ]) ∼ = K ( DH n ⊗ DH m ⊗ k [ c , c , . . . , c r ]) . Passing to the limit r → ∞ , the inclusion k [ S n ] ⊗ k [ S m ] −→ DH n,m induces an isomorphism of Grothendieck groups K ( k [ S n ] ⊗ k [ S m ]) ∼ = K ( DH n,m ) . (44)The split short exact sequence (43) gives rise to a split short exact sequence0 −→ K ( J n,m ) −→ K (End(+ n − m )) −→ K ( DH n,m ) −→ K -group of a 2-sided ideal see [40, 42]) and canonical decom-position K (End(+ n − m )) ∼ = K ( DH n,m ) ⊕ K ( J n,m ) . Conjecture 1.1 is equivalent to the vanishing of K ( J n,m ) for all n, m . References [1] B. Bartlett, The geometry of unitary 2-representations of finite groups and their2-characters, 2008, math.QA/0807.1329.[2] B. Bartlett, On unitary 2-representations of finite groups and topological quantumfield theory,
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Mikhail Khovanov, Department of Mathematics, Columbia University, New York, NY 10027 email: [email protected]: [email protected]