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W-ALGEBRAS FROM HEISENBERG CATEGORIES SABINCAUTIS,AAROND.LAUDA,ANTHONYM.LICATA,ANDJOSHUASUSSAN 5 1 0 Abstract. Thetrace(orzerothHochschildhomology)ofKhovanov’sHeisenbergcategoryisidenti- 2 fiedwithaquotientofthealgebraW1+∞. ThisinducesanactionofW1+∞ onsymmetricfunctions. n a J 3 Contents ] 1. Introduction 1 A 2. The Heisenberg category H 3 Q 3. The algebra W 6 1+∞ . h 4. Trace of H 8 t References 30 a m [ 1 1. Introduction v The algebraSym ofsymmetric functions in infinitely many variablesis importantin classicalrepre- 9 8 sentation theory in part because it describes the characters of representations of symmetric groups in 5 characteristic 0. More precisely, there are isomorphisms 0 ∞ ∞ .0 Sym∼= Z(C[Sn])∼= K0(C[Sn]−mod), 1 n=0 n=0 M M 0 where Z(C[S ]) is the center of the group algebra, and K (C[S ]−mod) is the Grothendieck group 5 n 0 n of the category of finite dimensional representations. Since Z(C[S ]) is isomorphic to the center of 1 n : the category C[Sn]−mod, the rightmost isomorphism says that two basic decategorifications of the v categories C[S ]−mod, the Grothendieck group and the center, are isomorphic to each other. i n X It is fruitful to consider a structure of interest for symmetric functions and then try to interpret it r in the language of symmetric groups. For example, Gessinger realized the Hopf algebra structure on a Sym by considering the maps on characters induced by induction and restriction [G]. Closely related tothis Hopfalgebrastructureisthe factthatSymis the underlyingvectorspaceofthe canonicalFock spacerepresentationofthe Heisenbergalgebrah, where generatorsofhactonthe Grothendieckgroup as the maps induced by induction and restriction between different symmetric groups. The algebra of symmetric functions Sym thus becomes the canonical level one Fock space module, which we denote by V , for the Heisenberg algebra. 1,0 From this point of view, it is also desirable to understand how to categorify the algebra h itself (rather than just the representation V ) and seek a monoidal category which acts directly on 1,0 ∞ C[S ]−mod. One definition of such a monoidal category, which is the central object of interest n=0 n in the current paper, is due to Khovanov [K]. His Heisenberg category H, which uses a graphical L calculus of planar diagrams in the definition of the morphism spaces, has a Grothendieck groupwhich contains the Heisenberg algebra h.1 1Conjecturally,thereisanalgebraisomorphismK0(H)∼=h. 1 2 SABINCAUTIS,AAROND.LAUDA,ANTHONYM.LICATA,ANDJOSHUASUSSAN The canonical level one module V of the Heisenberg algebra can be equipped with the action 1,0 of even larger infinite dimensional algebras. Much of this symmetry, including the actions of both the Heisenberg and Virasoro algebras, is unified in the action of the W-algebra W on V . The 1+∞ 1,0 Heisenberg and Virasoro actions on V have already been interpreted in the language of symmetric 1,0 groupcharacters: the Heisenberg actionis essentiallydeterminedby the Hopfalgebrastructure onthe polynomial algebra, and is thus implicit in [G] (see also [W] and references therein for closely related constructions using more general finite groups). A Virasoro algebra action on ∞ Z(C[S ]) was n=0 n described explicitly by Frenkel and Wang in [FW]. The natural question which remains is to extend L this description to obtain a character-theoretic description of the action of the entire W action on 1+∞ V . Betterstill wouldbe to obtainacharacter-theoreticconstructionofthe algebraW itself, and 1,0 1+∞ then as a consequence of such a construction to recover a construction of the canonical module V . 1,0 Our main aim in this paper is to give such a construction, and for this aim it is important to consider Khovanov’s Heisenberg category H, rather than just the representation categories of all symmetric groups. In this categorical context, the notion of character is captured by the trace Tr(H), or zeroth Hochschild homology. Let I ⊂W the two-sided ideal generated by C −1 and w for the central 1+∞ 0,0 elements C,w ∈W (see Section 3). Our main theorem is the following. 0,0 1+∞ Theorem 1. There is an isomorphism Tr(H) ∼= W /I. The action of Tr(H) on the center Z(H) 1+∞ is irreducible and faithful, and is thus identified with the canonical level one representation V . 1,0 The theorem thus gives a graphical construction of both W /I itself, and of its canonical repre- 1+∞ sentation. Asacorollary,anycategoricalrepresentationofKhovanov’sHeisenbergcategoryHgivesrise to a representation of W /I.2 Thus, the categorical representation of H on ∞ C[S ]−mod also 1+∞ n=0 n immediatelygivesrisetoanactionofW on ∞ Z(C[S ]). Suchanactionhasalsorecentlybeen 1+∞ n=0 n L given in independent work of Shan, Varagnolo, and Vasserot in [SVV]. A closely related categorical L actionofH onthe categoryof polynomialfunctors [HY]allows us to realizea W -algebraactionin 1+∞ this context as well. In order to prove that Tr(H) is a quotient of W we follow the proof in [SV] of the isomorphism 1+∞ between a limit of spherical degenerate double affine Hecke algebras and a quotient of W . A 1+∞ completesetofgeneratorsandrelationsofthelimitofsphericaldegeneratedoubleaffineHeckealgebras wasgivenin[AS]. Composingtheisomorphismin[SV]withtheisomorphismfromTheorem1,wegeta complete generators and relations description of Tr(H). Crucial in this comparison is the relationship between H and the degenerate affine Hecke algebra DH . Roughly speaking, DH appears in the n n morphismspacesofintheupperhalfofH. AsavectorspaceTr(DH )wasdeterminedbySolleveld[S] n and this calculation plays a role in our computation of Tr(H). From the point of view of Theorem 1, it is perhaps equally reasonable to refer to H as a categorification of the W-algebra; the difference between this statement and the statement that H categorifies theHeisenbergalgebraisthatwehavemadeadifferentchoiceinwhichdecategorificationprocedureto invert. It is interesting that the two natural decategorifications of the category H – the Grothendieck group and the trace – are quite different from one another. This is in contrast to the situation for the category C[S ]−mod, for example, where the Grothendieck group and the trace are isomorphic. Our n analysisshowsthatthetracedecategoricationcanhavestructuremissingfromtheGrothendieckgroup decategorification,sincethealgebraW containstheHeisenbergalgebraandalotmore. Inaddition 1+∞ to being able to directly identify Tr(H) with W /I, in the trace decategorification we are also able 1+∞ to find natural graphical realizations of the standard Heisenberg algebra generators. By contrast, in the Grothendieckdecategorificationthe standardHeisenberggeneratorsdo notadmitanobviouscate- goricallift and therefore an alternative presentationfor the Heisenberg algebrais required. We should 2ThequotientofW1+∞ weseehereisnotexactlythesameasthequotients ofW1+∞ thathaveappearedinclosely relatedliterature. Forexample,in[SV],thequotientofW1+∞ whichappearshasw0,0= 21 insteadofw0,0=0. W-ALGEBRAS FROM HEISENBERG CATEGORIES 3 also mention that a completely different categorification problem – to construct a monoidal category whose Grothendieck group is the algebra W – is open, though the Theorem 1 suggests that such a 1+∞ category might be related to Khovanov’s category H. An analogof Theorem1 (andof this difference between the Grothendieckgroupandthe trace)also arisesinthecontextofquantumgroupsassociatedtotypeADEKac-Moodyalgebras. Associatedtoan arbitrarysymmetrizable Kac-Moodyalgebrasg, a 2-categoryU(g) was defined in [KL] whose the split Grothendieck group is isomorphic to the integral form of Lusztig’s indepotented form of the quantum groupU (g). However,the trace of the 2-categoryU(g) is then identified with the idempotented form q of quantum current algebra U (g[t]) [SVV, BHLZ, BHLW]. q 1.1. Further directions. More generally, one can associate a Heisenberg category H to any finite F dimensional Frobenius algebra F. This is the categorical analogue of the fact that one can associate a Heisenberg algebra h to any Z-lattice L. If one takes the simplest Frobenius algebra, namely the L one dimensional algebra C, then one recovers H, and for this reason we first study the trace of H. In future work we plan to compute the traces of these more general Heisenberg categories, and thus associate a W-algebra W to any Frobenius algebra F. Of particular interest are the Heisenberg F categoriesstudiedin[CL1]andassociatedtozig-zagalgebras(thesecategorifytheHeisenbergalgebras associated to quantum lattices of type A,D,E). Via categorical vertex operators [CL2, C] these Heisenberg categories were related to categorified quantum groups. So one might expect their trace to be related to [BGHL, BHLZ] or to W-algebras associated to quantum groups [FR]. Finally, one may also consider the trace on the categorification of twisted Heisenberg algebras such as the twisted version of Khovanov’s category from [CS]. In this particular case the trace should be related to the twisted W algebra from [KWY]. 1+∞ In the recent work [MS], the algebra W appears in relation to the skein module of the torus. It 1+∞ could also be interesting to relate directly the appearance of W in Heisenberg categorification to 1+∞ this skein module or to other invariants in quantum topology. 1.2. Outline. In Section 2 Khovanov’s Heisenberg algebra h and its associated category H are intro- duced andits basic properties arereviewed. InSection 3 the algebraW is defined, andsome of its 1+∞ important features are recalled. In Section 4 the trace of H is determined in terms of W . 1+∞ Acknowledgements. The authors would like to thank Peter Bouwknegt, Mikhail Khovanov,Olivier Schiffmann, Davesh Maulik, and Weiqiang Wang for helpful conversations. S.C. was supported by an NSERC discoverygrantand the JohnTempleton Foundation. A.D.L. was partially supportedby NSF grantDMS-1255334and by the John Templeton Foundation. A.M.L. was supported by an Australian Research Council Discovery Early Career fellowship. J.S. was supported by NSF grant DMS-1407394 and PSC-CUNY Award 67144-0045. 2. The Heisenberg category H 2.1. The Heisenberg algebra and Fock space. Let h be the associative algebra over C with generators h for n∈Z−{0} with relations [h ,h ]=mδ . n m n m,−n Another wayto presenth is as follows. Set p(n) =q(n) =0 for n<0. For n≥0 define p(n) and q(n) by h h p(n)zn =exp( −nzn) q(n)zn =exp( nzn). n n n≥0 n≥1 n≥0 n≥1 X X X X In these generators the relations are: p(n)p(m) =p(m)p(n) q(n)q(m) =q(m)q(n) 4 SABINCAUTIS,AAROND.LAUDA,ANTHONYM.LICATA,ANDJOSHUASUSSAN q(n)p(m) = p(m−k)q(n−k). k≥0 X The Heisenberg algebra h has a natural representation F, known as the Fock space. Let h+ ⊂ h denote the subalgebra generated by the q(n) for n ≥ 0. Let triv denote the trivial representation of 0 h+, where all q(n) (n>0) act by zero. Then F :=Indh (triv ) h+ 0 iscalledthe Fockspacerepresentationofh. Itinherits a gradingF =⊕ F(m)bydeclaringtriv m∈ 0 to have degree zero, p(n) degree n and q(n) degree −n. Z N 2.2. HeisenbergcategoryH. In[K]KhovanovintroducedacategoricalframeworkfortheHeisenberg algebra h. This framework consists of a monoidal category H which is the Karoubi envelope of a monoidal category H′ whose definition we now sketch (see [K] for more details). The category H′ is generated by objects P and Q. These can be denoted by an upward pointing strandandadownwardpointingstrand. Monoidalcompositionofsuchobjectsisthengivenbysideways concatenation of diagrams. The space of morphisms between products of P’s and Q’s is a C-algebra described by certain string diagrams with relations. By convention, composition of morphisms is done vertically from the bottom and going up. The morphisms are generated by crossings, caps and cups as shown below (2.1) Thus, for instance, the left crossing is a map in End(PP) while the right cap is a map PQ → id. These morphisms satisfy the following relations = = (2.2) = − = (2.3) =1. = 0. (2.4) Moreover, two morphisms which differ by planar isotopies are equal. Relation (2.2) implies that there is a map C[S ] →End(Pn). Since H is assumed to be idempotent complete this means that we n also get objects P(λ), for any partition λ ⊢ n, associated with the corresponding minimal idempotent W-ALGEBRAS FROM HEISENBERG CATEGORIES 5 e ∈ C[S ]. Likewise we also have Q(λ) for any λ ⊢ n. We will denote by (m) and (1m) the unique λ n one-part and m-part partitions of m. Let H≥ and H≤ be the full subcategories of H generated by P and Q respectively. Let H0 be the full subcategory of of H generated by the identity object Id. Theorem 2 ([K]). Inside H we have the following relations (1) P(λ) and P(µ) commute for any partitions λ,µ, (2) Q(λ) and Q(µ) commute for any partitions λ,µ, (3) Q(n)P(m) ∼= P(m−k)Q(n−k) and Q(1n)P(1m) ∼= P(1m−k)Q(1n−k), k≥0 k≥0 (4) Q(n)P(1m) ∼=P(1m)Q(n)⊕P(1m−1)Q(n−1) and Q(1n)P(m) ∼=P(m)Q(1n)⊕P(m−1)Q(1n−1). L L Thus, at the level of Grothendieck groups we have a map h → K (H). This map is known to be 0 injective but it is not known if it is surjective. 2.3. The degenerate affine Hecke algebra. Inside H consider the element X ∈ End(Pn), acting i on the ith factor P, as illustrated in the left hand side of (2.5). In [K] this element was studied and it was encoded diagrammatically by a solid dot, as shown = (2.5) In [K] it was shown that these X ’s together with the symmetric group C[S ] ⊂ End(Pn) generate i n a copy of the degenerate affine Hecke algebra. In particular, using equation (2.2)– (2.4) the equations (2.6) − = = − follow. More precisely, denote a crossing of the ith and (i+1)st strands by T . i Proposition 3. [K] We have the following relations inside End (Pn): H T X =X T +1 i i i+1 i X T =T X +1 i i i i+1 X X =X X i j j i T2 =1 i T T =T T if |i−j|>1 i j j i T T T =T T T . i i+1 i i+1 i i+1 The algebra generated by T for i = 1,...,n−1 and X for i = 1,...,n satisfying the relations in i i Proposition 3 is the degenerate affine Hecke algebra DH . n We now define bubbles which are also endomorphisms of Pn. cn = n cn = n (2.7) e 6 SABINCAUTIS,AAROND.LAUDA,ANTHONYM.LICATA,ANDJOSHUASUSSAN Proposition 4. [K, Proposition 2] For n>0 n−1 c = c c . n+1 i n−1−i i=0 X Proposition 5. [K, Proposition 4] Theree is an isomeorphism of algebras End(Pn)∼=DHn⊗C[c0,c1,...]. In particular, when n=0 we have End(id)∼=C[c ,c ,...]. 0 1 Let J be the ideal of End (PmQn) generated by diagrams which contain at least one arc con- m,n H′ necting a pair of upper points. Proposition 6. [K, Equation (19)] There exists a short exact sequence 0→J →End (PmQn)→DH ⊗DHop⊗C[c ,c ,...]→0. m,n H′ m n 0 1 Furthermore, this sequence splits. 3. The algebra W 1+∞ InthissectionwedefinetheW-algebraofinterestandlistseveralimportantproperties. Thisalgebra is defined to be the universal enveloping algebra of a central extension of differential operators on C∗. TheW-algebra,W comesequippedwitha -gradingandacompatible -filtration. Wewillalso 1+∞ ≥0 Z Z defineafaithfulirreduciblerepresentationofaquotientofW . Bothofthesefeatureswillbecrucial 1+∞ in proving that the W-algebra is related to the trace of H. This section follows closely the exposition in [SV, Appendix F]. Let W be the C-associativealgebrageneratedby w for l ∈ and k ∈ andC with relations 1+∞ l,k Z N that w and C are central, and 0,0 (3.1) w =tlDk (l,k)6=(0,0) l,k exp(−lα)−exp(−kβ) (3.2) [tlexp(αD),tkexp(βD)]=(exp(kα)−exp(lβ))tl+kexp(αD+βD)+δ C l,−k 1−exp(α+β) where t is the parameter on C∗ and D =t∂ . t Note that for l∈Z−{0} the set {w } generates a Heisenberg subalgebra because l,0 (3.3) [w ,w ]=lδ C. l,0 k,0 l,−k We will need the following relations which are direct consequences of (3.2). k3−k (3.4) [w ,w ]=(k−l)w + Cδ l,1 k,1 l+k,1 l,−k 6 a b a b (3.5) [w ,w ]= w − (−1)s w +δ (−1)b+1C −1,a 1,b 0,a+b−r 0,a+b−s a,0 r s r=1(cid:18) (cid:19) s=1 (cid:18) (cid:19) X X k(k−1) (3.6) [w ,w ]=kw −δ C l,1 k,0 l+k,0 l,−k 2 (3.7) [w ,w ]=−2lw −l2w . l,0 0,2 l,1 l,0 We define Virasoro elements 1 L¯ =−w − (l+1)w . l l,1 l,0 2 Then it is easy to check that (3.2) gives the Virasoro relations l3−l [L¯ ,L¯ ]=(l−k)L¯ +δ C. l k l+k l,−k 12 W-ALGEBRAS FROM HEISENBERG CATEGORIES 7 Let W> ,W0 ,W< be subalgebras of W generated as follows: 1+∞ 1+∞ 1+∞ 1+∞ W> :=Chw ;l≥1,k≥0i 1+∞ l,k W0 :=ChC,w ;l≥0}i 1+∞ 0,l W< :=Chw ;l ≥1,k ≥0i. 1+∞ −l,k This algebra has a -grading called the rank grading where w is in degree l and C is in degree l,k 0. The algebra is alsoZ -filtered where w is in degree ≤ k. Let Wω [r,≤ k] denote the set of Z≥0 l,k 1+∞ elements in -degree r and -degree k where ω ∈{<,>,0,∅}. ≥0 Z Z Denote the associated graded algebra of W with respect to the -filtration by 1+∞ ≥0 Z W =gr(W ). 1+∞ 1+∞ Since the -grading on W is compatible with the -filtering, the associated graded W is 1+∞ ≥0 1+∞ Z f Z ( × )-graded. LetWω [r,k]denotethesubspaceofWω inbidegree(r,k). Defineagenerating Z Z≥0 1+∞ 1+∞ series for the graded dimension of Wω by f 1+∞ f f PWfω f(t,q)= dimW1ω+∞[r,k]trqk. 1+∞ rX∈Zk∈XZ≥0 f Proposition 7. The graded dimensions of W> and W< are given by: 1+∞ 1+∞ 1 1 PWf1>+∞ = 1−tfrqk, PWff1<+∞ = 1−trqk. r>0k≥0 r<0k≥0 Y Y Y Y Proof. The defining relations of W given in 3.2 imply that the associated graded algebras W> 1+∞ 1+∞ and W< are freely generated by the images of w . The proposition now follows easily since the 1+∞ l,k image of w has bidegree (l,k). f (cid:3) l,k f Lemma 8. [SV, Lemma F.5] (1) W is generated by w ,w ,w . 1+∞ −1,0 1,0 0,2 (2) W> is generated by w for l ≥0. 1+∞ 1,l (3) W< is generated by w for l ≥0. 1+∞ −1,l For c,d ∈ C let C be a one dimensional representation of W≥ where w acts by zero for c,d 1+∞ k,l (k,l)6=(0,0), C acts by c and w acts by d. 0,0 Let M :=IndW1+∞(C ). c,d W≥ c,d 1+∞ Proposition 9. [AFMO,FKRW] The induced module M has a uniqueirreducible quotient V . As c,d c,d a vector space V is isomorphic to C[w ,w ,...]. 1,0 −1,0 −2,0 Proposition 10. [SV, PropositionF.6] The action of W /(C−1,w ) is faithful on V . 1+∞ 0,0 1,0 Proof. This is Proposition F.6 of [SV] since their proof does not depend on how the central elements C and w act. The proof uses the faithfulness of the action of the Heisenberg algebra on Fock space 0,0 together with the filtration on W . (cid:3) 1+∞ Lemma 11. [SV, Lemma F.8] The action of w on V is given by the operator 0,2 1,0 (w w w +w w w )−w . −l,0 −k,0 k+l,0 −l−k,0 l,0 k,0 0,1 k,l>0 X Proof. This is a straightforward computation using the relations of W and the definition of V . 1+∞ 1,0 For more details see the proof of [SV, Lemma F.8]. There is a correction term of −w in the lemma 0,1 because we take w =0 while in [SV] it is taken to be −1. (cid:3) 0,0 2 8 SABINCAUTIS,AAROND.LAUDA,ANTHONYM.LICATA,ANDJOSHUASUSSAN 4. Trace of H 4.1. Definitions and conventions. The trace, or zeroth Hochschild homology, Tr(C) of a k-linear category C is the k-vector space given by Tr(C)= C(X,X) /Span {fg−gf},   k X∈Ob(C) M   where C(X,X)= End (X), f and g run through all pairs of morphisms f: X →Y, g: Y →X with C X,Y ∈Ob(C). The trace is invariant under passage to the Karoubi envelope Kar(C). Proposition 12. [BHLZ,Proposition3.2] The natural map Tr(Kar(C))→Tr(C) induced by inclusion of categories is an isomorphism. Proposition 13. [BHLW, Lemma 2.1] Let C be a -linear additive category. Let S ⊂ Ob(C) be a k subset such that every object in C is isomorphic to the direct sum of finitely many copies of objects in S. Let C| denote the full subcategory of C with Ob(C| ) = S. Then, the inclusion functor C| → C S S S induces an isomorphism (4.1) Tr(C|S)∼=Tr(C) Recall that a categorical representation of the category C is a k-linear category V and a k-linear functorC →End(V)sendingeachobjectX toanendofunctorF(X): V →V. Eachmorphismf: X → Y is sent to a natural transformation of functors F(f): F(X) → F(Y). A categorical representation gives rise to a representation (4.2) ρF: Tr(C)→Vectk sending the class [f] ∈ Tr(C) corresponding to a map f: X → X in C to the endomorphism of the k-vector space Tr(V) defined by sending ψ: U →U in Tr(V) to ρ ([f])([ψ])=[F(X)(ψ)◦F(f) ]=[F(f) ◦F(X)(ψ)]. F U U Here,notethatF(f) denotesthecomponentofthenaturaltransformationF(f): F(X)⇒F(X): V → U V corresponding to the object U ∈Ob(V). By naturality this map is well defined. For f ∈ End(H) denote its class in Tr by [f]. We view the element [f] ∈ Tr by drawing f in an annulus and then closing up the diagram for f. f 7→ f Let w ∈S Define n h ⊗(xj1···xjn):=[f ] w 1 n w;j1,...,jn where f ∈End(Pn) is given by: w;j1,...,jn W-ALGEBRAS FROM HEISENBERG CATEGORIES 9 j ··· j 1 n f := w w;j1,...,jn ··· Define h ⊗(xj1···xjn):=[f ] −w 1 n −w;j1,...,jn where f ∈End(Qn) is given by: −w;j1,...,jn j ··· j 1 n f := w −w;j1,...,jn ··· Of particular importance is the element σ =s ···s ∈S . We then abbreviate n 1 n−1 n j1 j2 j3 ... jn (4.3) h ⊗(xj1···xjn):=h ⊗(xj1···xjn)= n 1 n σn 1 n   ...     j1 j2 j3 ... jn h ⊗(xj1···xjn):=h ⊗(xj1···xjn)= −n 1 n −σn 1 n   ...     The next lemma allows us to express generators hn⊗(xj11···xjnn) in terms of generators hr⊗xk1. Lemma 14. For n≥1 and 1≤i≤n−1 we have h ⊗x =h ⊗x ±(h ⊗1)(h ⊗1). ±n i ±n i+1 ±i ±(n−i) Proof. This is immediate from (2.6) and properties of the trace. (cid:3) The next lemma allows us to express generators [f ] in terms of the more elementary gener- w;j1,...,jn ators h ⊗p where p is a polynomial in variables x ,...,x . r 1 r Lemma 15. Let w ∈S and (n ,...,n ) a sequence of natural numbers which sum to n. Then n 1 r [f ]= d (h ⊗p )···(h ⊗p ) ±w;j1,...,jn n1,...,nr n1 n1 nr nr for some constants d ∈C and poXlynomials p in i variables. n1,...,nr ni Proof. We prove this by induction on j +···+j . The base case where all of the j equal zero is 1 n i trivial because then f is just an element in the symmetric group and thus its class [f ] w;j1,...,jn w;j1,...,jn in the trace is determined by its conjucacy class in S which of course could be written as a product n of disjoint cycles. Choose an element g ∈S such that n gwg−1 =(s ···s )···(s ···s ). 1 n1−1 n1+···nr−1 n1+···nr−1 Let p = xj11···xjnn. Thus fw;j1,...,jn = pw. Now we conjugate this element by g to get gpwg−1 = (g(cid:5)p)gwg−1+p wg−1 where p is a polynomial of degree less than j +···j and g(cid:5)p is some other L L 1 n polynomial of degree j +···j . The lemma follows by applying induction to the second term and 1 n noting that in the first term gwg−1 is a product of cycles. (cid:3) 10 SABINCAUTIS,AAROND.LAUDA,ANTHONYM.LICATA,ANDJOSHUASUSSAN Corollary 16. Let w∈S and (n ,...,n ) a sequence of natural numbers which sum to n. Then n 1 r [f ]= d (h ⊗xl1)···(h ⊗xlr) ±w;j1,...,jn n1,...,nr ±n1 1 ±nr 1 for some constants d ∈C and soXme non-negative integers l ,...,l . n1,...,nr 1 r Proof. This follows from Lemma 14 and Lemma 15. (cid:3) 4.2. Elements p(n) and q(n). In this section we will see how the Heisenberg algebra h in the presen- tation given in terms of p(n),q(n) from Section 2 appears in Tr(H). Define p(n)⊗1 and q(n)⊗1 in Tr(H) by 1 (4.4) p(n)⊗1= [f ] w;0,...,0 n! wX∈Sn 1 (4.5) q(n)⊗1= [f ] −w;0,...,0 n! wX∈Sn We depict the elements p(n)⊗1 and q(n)⊗1 by (n) (n) We have for example the relation (q(1) ⊗1)(p(1) ⊗1) = (p(1) ⊗1)(q(1) ⊗1)+1. The composition (q(1)⊗1)(p(1)⊗1) is Now, the graphical relations in H allow us to pull the inner circle outside as follows = +