Quantum Frobenius Heisenberg categorification
aa r X i v : . [ m a t h . R T ] S e p QUANTUM FROBENIUS HEISENBERG CATEGORIFICATION
JONATHAN BRUNDAN, ALISTAIR SAVAGE, AND BEN WEBSTERA bstract . We associate a diagrammatic monoidal category
Heis k ( A ; z , t ), which we call the quantum Frobe-nius Heisenberg category , to a symmetric Frobenius superalgebra A , a central charge k ∈ Z , and invertibleparameters z , t in some ground ring. When A is trivial, i.e. it equals the ground ring, these categories recoverthe quantum Heisenberg categories introduced in our previous work, and when the central charge k is zero theyyield generalizations of the a ffi ne HOMFLY-PT skein category. By exploiting some natural categorical actionsof Heis k ( A ; z , t ) on generalized cyclotomic quotients, we prove a basis theorem for morphism spaces.
1. I ntroduction
Fix a commutative ground ring k and parameters z , t ∈ k × . In this paper, to any central charge k ∈ Z andsymmetric Frobenius superalgebra A , we associate a monoidal supercategory Heis k ( A ; z , t ), which we callthe quantum Frobenius Heisenberg category . The case A = k recovers the quantum Heisenberg categories Heis k ( z , t ) of [BSW20b], which extend the q -deformed Heisenberg category introduced in [LS13]. In allother cases, the categories are new. In this way, the current paper can be viewed as a generalization of allof these earlier treatments of quantum Heisenberg categories. Indeed, in [BSW20b] the proofs of manydiagrammatic lemmas were omitted, deferring them to the current paper, which gives these proofs in themore general setting. Quantum Frobenius Heisenberg categories can also be viewed as quantum analoguesof the Frobenius Heisenberg categories introduced in [RS17, Sav19] and further developed in [BSW20a].In turn, these were generalizations of the
Heisenberg categories of [MS18, Bru18], which were based onthe original construction of [Kho14].Following the approach of [BSW20b], we give three equivalent definitions of
Heis k ( A ; z , t ). All threetake as their starting point the quantum a ffi ne wreath product category AW ( A ; z ), whose morphism spacesare the quantum a ffi ne wreath product algebras introduced in [RS20b]; see Section 2. The next step isto adjoin a right dual ↓ to the generating object ↑ of AW ( A ; z ). This involves adding right cups and capssatisfying zigzag relations. In the first two approaches, described in Sections 3 and 4, the final step is torequire that a certain morphism in the additive envelope is invertible, and then impose one more relationon certain bubbles. The only di ff erence between these two definitions is that the morphism to be invertedinvolves di ff erent crossings (positive or negative). The third definition of Heis k ( A ; z , t ), given in Section 5, isthe most explicit. Here we adjoin additional generating morphisms, the left cups and caps, subject to furtherrelations which are often useful in practice. The proof that all three approaches yield isomorphic categoriesinvolves a systematic “unpacking” of the inversion relations.The categories Heis k ( A ; z , t ) have many desirable properties, analogous to those of the other Heisenbergcategories mentioned above. For example, they are strictly pivotal, and we have curl relations, bubble slide Mathematics Subject Classification.
Primary 18M05; Secondary 20C08.
Key words and phrases.
Categorification, Frobenius algebra, Heisenberg algebra, monoidal category, diagrammatic calculus.J.B. supported in part by NSF grant DMS-1700905.A.S. supported by Discovery Grant RGPIN-2017-03854 from the Natural Sciences and Engineering Research Council ofCanada.B.W. supported by Discovery Grant RGPIN-2018-03974 from the Natural Sciences and Engineering Research Council ofCanada. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute issupported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada andby the Province of Ontario through the Ministry of Research, Innovation and Science.
JONATHAN BRUNDAN, ALISTAIR SAVAGE, AND BEN WEBSTER relations, and infinite Grassmannian relations. When k ,
0, they act naturally on categories of modules for quantum cyclotomic wreath product algebras , a fact which is easiest to see in the inversion relation approachof the first two definitions of
Heis k ( A ; z , t ); see Section 6. We also prove a basis theorem (Theorem 9.2) forthe morphisms spaces in Heis k ( A ; z , t ). Following the methods of [BSW18, BSW20b, BSW20a], our proofof this basis theorem uses a categorical comultiplication (see Section 7) to construct a su ffi ciently largemodule category starting from the actions on modules over the cyclotomic wreath product algebras forgeneric parameters. We also introduce certain generalized cyclotomic quotients ; see Section 8.The central charge zero quantum Heisenberg category Heis ( z , t ) from [BSW20b] is the a ffi ne HOMFLY-PT skein category from [Bru17, §4]. Omitting the dot generator, we obtain the HOMFLY-PT skein cate-gory , which is the ribbon category underpinning the definition of the HOMFLY-PT invariant of an orientedlink. The central charge zero quantum Frobenius Heisenberg category
Heis ( A ; z , t ) yields a Frobenius al-gebra analog of the a ffi ne HOMFLY-PT skein category. By analogy with the A = k case, we expect that Heis ( A ; z , t ) should act on categories of modules for some natural Frobenius algebra analogue of the quan-tized enveloping algebra U q ( gl n ). On omitting the dot generator from Heis ( A ; z , t ), we obtain the FrobeniusHOMFLY-PT skein category , from which one can define a Frobenius algebra analogue of the HOMFLY-PTlink invariant.The name
Heisenberg category comes from Khovanov’s original conjecture that his Heisenberg categorycategorifies a central reduction of the infinite rank Heisenberg algebra. This conjecture was proved in[BSW18, Th. 1.1] and then extended in [BSW20a, Th. 10.5] to show that the Grothendieck ring of theFrobenius Heisenberg category is isomorphic to a lattice Heisenberg algebra associated to the Frobeniusalgebra A . We expect that the Grothendieck ring of the quantum Frobenius Heisenberg category is alsoisomorphic to this lattice Heisenberg algebra. The obstruction to proving this conjecture, present even inthe A = k case, is that we do not know how to compute the split Grothendieck group of the quantum a ffi newreath product algebra.There is an alternative framework for decategorification using trace (or zeroth Hochschild homology) inplace of the Grothendieck ring. In [CLL + q -deformed Heisenberg category of [LS13]was related to the elliptic Hall algebra. It is thus natural to expect that the trace of Heis k ( A ; z , t ) shouldbe related to a Frobenius algebra generalization of the elliptic Hall algebra. This would be the quantumanalogue of the situation for Frobenius Heisenberg categories, where a computation of the trace in [RS20a]suggested a Frobenius algebra generalization of the W-algebra W + ∞ , extending results from [CLLS18].2. T he quantum affine wreath product category A fundamental ingredient in the definition of the quantum Frobenius Heisenberg category is the quantuma ffi ne wreath product algebra introduced in [RS20b]. In this section we recall the definition of this algebra,together with a few key results needed in the current paper. We then switch our point of view to monoidalcategories, introducing categories whose endomorphism spaces are quantum a ffi ne wreath product algebras. Monoidal supercategories.
We fix a commutative ground ring k . All vector spaces, algebras, categories,and functors will be assumed to be linear over k unless otherwise specified. Unadorned tensor productsdenote tensor products over k . Almost everything in the paper will be enriched over the category SVec ofvector superspaces with parity-preserving morphisms. We write ¯ v for the parity of a homogeneous vector v in a vector superspace. When we write formulae involving parities, we assume the elements in question arehomogeneous; we then extend by linearity.For superalgebras A = A ¯0 ⊕ A ¯1 and B = B ¯0 ⊕ B ¯1 , multiplication in the superalgebra A ⊗ B is defined by(2.1) ( a ′ ⊗ b )( a ⊗ b ′ ) = ( − ¯ a ¯ b a ′ a ⊗ bb ′ UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 3 for homogeneous a , a ′ ∈ A , b , b ′ ∈ B . The opposite superalgebra A op is a copy { a op : a ∈ A } of the vectorsuperspace A with multiplication defined from(2.2) a op b op : = ( − ¯ a ¯ b ( ba ) op . The center of A is the subalgebra(2.3) Z ( A ) : = Span k { a ∈ A homogeneous : ab = ( − ¯ a ¯ b ba for all b ∈ A } The cocenter C ( A ), which, in general, is merely a vector superspace not an superalgebra, is(2.4) C ( A ) : = A / Span k { ab − ( − ¯ a ¯ b ba : a , b ∈ A homogeneous } Throughout this paper we will work with strict monoidal supercategories , in the sense of [BE17]. Werefer the reader to [BSW20a, §2] for a summary of this topic well adapted to the current work, or to [BE17]for a thorough treatment. We summarize here a few crucial properties that play an important role in thepresent paper.A supercategory means a category enriched in
SVec . Thus, its morphism spaces are vector superspacesand composition is parity-preserving. A superfunctor between supercategories induces a parity-preservinglinear map between morphism superspaces. For superfunctors F , G : A → B , a supernatural transformation α : F ⇒ G of parity r ∈ Z / α X ∈ Hom B ( FX , GX ) r for each X ∈ A such that G f ◦ α X = ( − r ¯ f α Y ◦ F f for each homogeneous f ∈ Hom A ( X , Y ). Note when r is odd that α is not a naturaltransformation in the usual sense due to the sign. A supernatural transformation α : F ⇒ G is of the form α = α ¯0 + α ¯1 , with each α r being a supernatural transformation of parity r .In a strict monoidal supercategory , morphisms satisfy the super interchange law :(2.5) ( f ′ ⊗ g ) ◦ ( f ⊗ g ′ ) = ( − ¯ f ¯ g ( f ′ ◦ f ) ⊗ ( g ◦ g ′ ) . We denote the unit object by and the identity morphism of an object X by 1 X . We will use the usualcalculus of string diagrams, representing the horizontal composition f ⊗ g (resp. vertical composition f ◦ g )of morphisms f and g diagrammatically by drawing f to the left of g (resp. drawing f above g ). Care isneeded with horizontal levels in such diagrams due to the signs arising from the super interchange law:(2.6) f g = f g = ( − ¯ f ¯ g f g . If A is a supercategory, the category SEnd k ( A ) of superfunctors A → A and supernatural transformationsis a strict monoidal supercategory. A module supercategory over a strict monoidal supercategory C is asupercategory A together with a monoidal superfunctor C → SEnd k ( A ).If R is a superalgebra, we let smod- R denote the supercategory of right R -supermodules and let psmod- R denote the supercategory of finitely-generated projective right R -supermodules. Quantum a ffi ne wreath product algebras. Fix z ∈ k × . (In fact, we do not need the assumption that z is invertible until we introduce quantum Frobenius Heisenberg categories in Section 3, but we make thisassumption from the beginning to be uniform.) Let A be a symmetric Frobenius superalgebra with eventrace map tr : A → k . Thus tr( ab ) = ( − ¯ a ¯ b tr( ba ) , a , b ∈ A . The definition of Frobenius superalgebra gives that A possesses a homogeneous basis B A and a left dualbasis { b ∨ : b ∈ B A } such that(2.7) tr( b ∨ c ) = δ b , c , b , c ∈ B A . It follows that, for all a ∈ A , we have a = X b ∈ B A tr( b ∨ a ) b = X b ∈ B A tr( ab ) b ∨ , (2.8) JONATHAN BRUNDAN, ALISTAIR SAVAGE, AND BEN WEBSTER X b ∈ B A ab ⊗ b ∨ = b ⊗ b ∨ a , X b ∈ B A ( − ¯ a ¯ b ba ⊗ b ∨ = X b ∈ B A ( − ¯ a ¯ b b ⊗ ab ∨ . (2.9)Note that ¯ b = b ∨ , and that the dual basis to { b ∨ : b ∈ B A } is given by(2.10) ( b ∨ ) ∨ = ( − ¯ b b . For the remainder of the paper we adopt the following summation convention: any expression involvingboth the symbols b and b ∨ includes an implicit sum over b ∈ B A . We adopt an analogous convention when b is replaced by a , c , etc. Thus, for instance, ab ⊗ b ∨ = X b ∈ B A ab ⊗ b ∨ = ac ⊗ c ∨ . For any homogeneous a ∈ A , we define(2.11) a † : = ( − ¯ a ¯ b zbab ∨ = z X b ∈ B A ( − ¯ a ¯ b bab ∨ , which is well-defined independent of the choice of the basis B A . Definition 2.1 ([RS20b, Def. 2.1]) . For n ∈ N , n ≥
2, the quantum wreath product algebra (or
FrobeniusHecke algebra ) W n ( A ; z ) is the free product A ⊗ n ∗ h σ i : 1 ≤ i ≤ n − i of the superalgebra A ⊗ n and the freeassociative superalgebra with even generators σ , . . . , σ n − , modulo the relations σ i σ j = σ j σ i , ≤ i , j ≤ n − , | i − j | > , (2.12) σ i σ i + σ i = σ i + σ i σ i + , ≤ i ≤ n − , (2.13) σ i = z τ i σ i + , ≤ i ≤ n − , (2.14) σ i a = s i ( a ) σ i , a ∈ A ⊗ n , ≤ i ≤ n − , (2.15)where(2.16) τ i : = ⊗ ( n − i − b ⊗ b ∨ ⊗ ⊗ ( i − , ≤ i , j ≤ n − , and s i ( a ) denotes the action of the simple transposition s i on a by superpermutation of the factors: s i ( a n ⊗ · · · ⊗ a ) = ( − a i a i + a n ⊗ · · · ⊗ a i + ⊗ a i ⊗ a i + ⊗ a i − ⊗ · · · ⊗ a . We define(2.17) a ( i ) : = ⊗ ( n − i ) ⊗ a ⊗ ⊗ ( i − , a ∈ A , ≤ i ≤ n . Note that here, and throughout the paper, we number factors in the tensor product from right to left . It isstraightforward to verify that τ i does not depend on the choice of basis B A . We adopt the conventions thatW ( A ; z ) : = A and W ( A ; z ) : = k . Definition 2.2 ([RS20b, Def. 2.5]) . For n ∈ N , n ≥
1, we define the quantum a ffi ne wreath product algebra (or a ffi ne Frobenius Hecke algebra ) AW n ( A ; z ) to be the free product k [ x ± , . . . , x ± n ] ∗ W n ( A ; z ) of the Laurentpolynomial algebra viewed as a superalgebra with all x i being even and the superalgebra W n ( a ; z ) fromDefinition 2.1, modulo the relations σ i x j = x j σ i , ≤ i ≤ n − , ≤ j ≤ n , j , i , i + , (2.18) σ i x i σ i = x i + , ≤ i ≤ n − , (2.19) x i a = a x i , ≤ i ≤ n , a ∈ A ⊗ n . (2.20)We adopt the convention that AW ( A ; z ) : = k . UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 5
Remark 2.3.
When A = k = C ( q ) and z = q − q − , then W n ( A ; z ) (resp. AW n ( A ; z )) is the Iwahori–Heckealgebra (resp. a ffi ne Hecke algebra) of type A n − . More generally, let C d be a cyclic group of order d . If k = C ( q ), z = ( q − q − ) / d , and A = k C d , with trace map given by projection onto the identity element of thegroup, then W n ( k C d ; z ) (resp. AW n ( k C d ; z )) is the Yokonuma–Hecke algebra (resp. a ffi ne Yokonuma–Heckealgebra) from [CPd14]. Remark 2.4.
The opposite superalgebra A op is again a symmetric Frobenius superalgebra, with the tracemap being the same underlying linear map as for A . It follows that the elements τ i in W n ( A op ) or AW n ( A op )are given by the exactly the same formula (2.16) as for W n ( A ) or AW n ( A ). The quantum wreath product category.
Define the quantum wreath product category W ( A ; z ) to be thestrict k -linear monoidal supercategory generated by one object ↑ and morphisms(2.21) , : ↑ ⊗ ↑→↑ ⊗ ↑ , a : ↑→↑ , a ∈ A , where the crossings are even and the parity of the morphism a is the same as the parity of a . We refer tothe generators above as a positive crossing , a negative crossing , and a token , respectively. The relations areas follows: = , λ a + µ b = λ a + µ b , ba = ab , a , b ∈ A , λ, µ ∈ k , (2.22) = = , = , a = a , a = a , (2.23) − = z b b ∨ . (2.24)Of course (2.24) should be interpreted using the usual summation convention; we call it the Frobenius skeinrelation . The relations (2.22) imply that the map A → End W ( A ; z ) ( ↑ ) , a a , is a superalgebra homomorphism. Using (2.6) and (2.23), it also follows that(2.25) a = a , a = a , a b = ( − ¯ a ¯ b a b . We introduce the teleporters (2.26) = = : = z b b ∨ (2.10) = z b ∨ b . We do not insist that the tokens in a teleporter are drawn at the same horizontal level, the convention whenthis is not the case being that b is on the higher of the tokens and b ∨ is on the lower one. We will also drawteleporters in larger diagrams. When doing so, we add a sign of ( − y ¯ b in front of the b summand in (2.26),where y is the sum of the parities of all morphisms in the diagram vertically between the tokens labeled b and b ∨ . For example, a c = ( − (¯ a + ¯ c )¯ b z a cb b ∨ . This convention ensures that one can slide the endpoints of teleporters along strands: a c = a c = a c = a c . Using teleporters, the Frobenius skein relation (2.24) can be written as(2.27) − = JONATHAN BRUNDAN, ALISTAIR SAVAGE, AND BEN WEBSTER
It follows from (2.9) that tokens can “teleport” across teleporters (justifying the terminology) in the sensethat, for a ∈ A , we have(2.28) a = a , a = a , where the strings can occur anywhere in a diagram (i.e. they do not need to be adjacent). The endpoints ofteleporters slide through crossings and they can teleport too. For example we have(2.29) = , = = . For more discussion of teleporters, see [BSW20a, § W ( A ; z ) are {↑ ⊗ n : n ∈ N } . There are no nonzero morphisms ↑ ⊗ m →↑ ⊗ n for m , n .Furthermore, we have an isomorphism of superalgebrasW n ( A ; z ) (cid:27) −→ End W ( A ; z ) ( ↑ ⊗ n )sending a ( i ) , a ∈ A , to a token labelled a on the i -th strand and σ i to a positive crossing of the i -th and( i + from right to left . The quantum a ffi ne wreath product category. We define the quantum a ffi ne wreath product category AW ( A ; z ) to be the strict k -linear monoidal supercategory obtained by adjoining to W ( A ; z ) an additionalinvertible even morphism : ↑→↑ and imposing the additional relations(2.30) = , = , a = a , a ∈ A . Note that, in fact, one only needs to impose one of the first two relations in (2.30); the other then follows bycomposing on the top and bottom by the appropriate crossings. It also follows that endpoints of teleporterspass through dots: = . The objects of AW ( A ; z ) are {↑ ⊗ n : n ∈ N } . There are no nonzero morphisms ↑ ⊗ m →↑ ⊗ n for m , n .Furthermore, we have an isomorphism of superalgebrasAW n ( A ; z ) (cid:27) −→ End AW ( A ; z ) ( ↑ ⊗ n ) , sending a ( i ) , a ∈ A , to a token labeled a on the i -th strand, x i to a dot on the i -th strand, and σ i to a positivecrossing of the i -th and ( i + Remark 2.5.
Throughout this paper, we assume that the Frobenius superalgebra A is symmetric for simplic-ity of exposition. We expect that all the results can be extended to the case where A is a general Frobeniussuperalgebra. In this setup, the Nakayama automorphism ψ will appear in some of the relations. For exam-ple, the last relation in (2.30) becomes(2.31) a = ψ ( a ) and one must also apply a power of ψ to tokens as they travel over the left caps and cups to be introducedbelow. We refer the reader to [Sav19] for a treatment of Frobenius Heisenberg categories in this level of gen-erality. The main case of interest that our assumption on A excludes is the Cli ff ord superalgebra, where the UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 7 quantum a ffi ne wreath product algebra would correspond to the a ffi ne Hecke–Cli ff ord superalgebra. How-ever, this case is more naturally treated by considering an odd a ffi nization of the quantum wreath productalgebra and the resulting Heisenberg category. We hope to explore this in future work.3. F irst approach In this section we give our first definition of the quantum Frobenius Heisenberg category
Heis k ( A ; z , t ).We will give two more, equivalent definitions in Sections 4 and 5.We begin by adjoining a right dual ↓ to the object ↑ . Thus we have additional generating morphisms: →↓ ⊗ ↑ and : ↑ ⊗ ↓→ subject to the right adjunction relations(3.1) = , = . We define the downward dots and tokens(3.2) : = , a : = a , a ∈ A . Using (3.1), we immediately have that(3.3) A op → End( ↓ ) , a op a , is a homomorphism of superalgebras, i.e.(3.4) ba = ( − ¯ a ¯ b ba , a , b ∈ A , We introduce teleporters on downward strings as in (2.26) (with the orientation of strings reversed); seeRemark 2.4. Tokens teleport across these new downward teleporters in just the same way as in (2.28).We may also draw teleporters with endpoints on oppositely oriented strings, interpreting them as usual byputting the label b on the higher of the tokens and b ∨ on the lower one, summing over all b ∈ B A . The waythat dots teleport across these is slightly di ff erent; for example, we have that a = a , a = a . We define the positive and negative right crossings by(3.5) : = , : = . We then define positive and negative downwards crossings by(3.6) : = , : = . It follows that the rightwards and downwards Frobenius skein relations hold:(3.7) − = , (3.8) − = . In addition, for all a ∈ A , we have = , = , a = a , a = a , (3.9) = , = , = , = , (3.10) = , = , = , = . (3.11) JONATHAN BRUNDAN, ALISTAIR SAVAGE, AND BEN WEBSTER
Furthermore, attaching right caps to the top and right cups to the bottom of the relations (2.23), (2.25),and (2.30) gives the following relations for all a ∈ A : = = , = , = , = , (3.12) a = a , a = a , a = a , a = a , (3.13) a = a , a = a , a = a , a = a , (3.14) = , = , = , = , (3.15) a = a . (3.16)For n ∈ Z , we let n denote the composition of n dots if n ≥ | n | inverse dots if n < Lemma 3.1.
The following relations hold for n ∈ Z : n = n − X r + s = nr , s > r s if n > , n + X r + s = nr , s ≤ r s if n ≤ n = n + X r + s = nr , s ≥ r s if n ≥ , n − X r + s = nr , s < r s if n < n = n + X r + s = nr , s > r s if n > , n − X r + s = nr , s ≤ r s if n ≤ n = n − X r + s = nr , s ≥ r s if n ≥ , n + X r + s = nr , s < r s if n < . (3.18) Proof.
For n ≥
0, this follows from repeated application of (2.27) and (2.30). The cases for n < n > (cid:3) Using (3.15), we have rightwards and downwards analogues of Lemma 3.1. Similarly, after proving theleftwards dot slide relation (4.5) below, we also have a leftwards analogue. In what follows, we will simplyrefer to these rotated relations using the equation numbers of Lemma 3.1.Now we fix the index set J : = { ⋆ } ∪ { ( r , b ) : 0 ≤ r ≤ | − k | − , b ∈ B A } and we fix a total order on this set such that ⋆ < ( r , b ) for all r , b . We can then naturally speak of 1 × J , J × J × J matrices. Definition 3.2.
The quantum Frobenius Heisenberg category
Heis k ( A ; z , t ) is the strict k -linear monoidalsupercategory obtained from AW ( A ; z , t ) by adjoining a right dual ↓ to ↑ , together with the relation that the UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 9 following matrix of morphisms is invertible in the additive envelope of
Heis k ( A ; z , t ):(3.19) (cid:20) rb ∨ , ≤ r ≤ k − , b ∈ B A (cid:21) T : ↑↓→↓↑ ⊕ ⊕ k dim A if k ≥ , (cid:20) rb ∨ , ≤ r ≤ − k − , b ∈ B A (cid:21) : ↑↓ ⊕ ⊕ ( − k dim A ) →↓↑ if k < . (The above matrices are of size J × × J in the cases k ≥ k <
0, respectively.) We denote thematrix entries of the two-sided inverse of (3.19) as follows:(3.20) _ ♦ ( r , b ) , ≤ r ≤ k − , b ∈ B A = (cid:20) rb ∨ , ≤ r ≤ k − , b ∈ B A (cid:21) T ! − if k ≥ , r ♥ ( r , b ) , ≤ r ≤ − k − , b ∈ B A T = (cid:20) rb ∨ , ≤ r ≤ − k − , b ∈ B A (cid:21) − if k < . We extend the definition of the decorated left cups and caps by linearity in the second argument of the label.In other words, for a ∈ A , we define _ ♦ ( r , a ) = tr( b ∨ a ) _ ♦ ( r , b ) , if k > , r ♥ ( r , a ) = tr( b ∨ a ) r ♥ ( r , b ) , if k < . We define the left cup and cap by(3.21) : = − t − z _ ♦ ( k − , − if k > , t if k = , t − − k if k < , : = t k if k ≥ , − t − z r ♥ (0 , if k < . We then impose one additional relation:(3.22) a = tz tr( a )1 if k > , a = t − t − z tr( a )1 if k = , a − k = tz tr( a )1 if k < . This concludes the definition of
Heis k ( A ; z , t ).One left crossing has been defined as the first entry in the matrix appearing in (3.20). We define the otherleft crossing so that the Frobenius skein relation holds:(3.23) − = . For a ∈ A , we also set r ♥ (0 , a ) : = _ ♦ (0 , a ) + z a if k > , r ♥ ( n , a ) : = _ ♦ ( n , a ) if 0 < n < k , (3.24) _ ♦ (0 , a ) : = r ♥ (0 , a ) + ( − ¯ a z a if k < , _ ♦ ( n , a ) : = r ♥ ( n , a ) if 0 < n < − k . (3.25)For n ≤ a ∈ A , we define the ( + ) -bubbles :(3.26) + a n : = − tz _ ♦ ( − n , a ) k if n > − k , tz tr( a )1 if n = − k , n < − k , + an : = t − z _ ♦ ( − n , a ) − k if n > k , − t − z tr( a )1 if n = k , n < k . For a ∈ A , we define + an : = an if n > , + a n : = a n if n > . (3.27)Then we define the ( − ) -bubbles so that, for all a ∈ A , n ∈ Z , an = + an + − an , a n = + a n + − a n for all n ∈ Z . (3.28)It follows from the definitions that the ( ± )-bubbles are linear in the label a ∈ A . Note that although ournotation for the ( ± )-bubbles involves tokens and dots, these bubbles are not built from cups, caps, tokens,and dots in general. Furthermore, we allow ourselves the freedom to draw the tokens and dots on ( ± )-bubblesin any position, and we use the relations (2.22), (3.4), and (3.9) to interpret ( ± )-bubbles with multiple tokens.For example,(3.29) + nab = + abn , − n = − b b ∨ n . When k =
0, the assertion that (3.19) and (3.20) are two-sided inverses means that(3.30) = if k = , = if k = . When k >
0, the inversion relation implies that, for all a ∈ A , = if k > , = − k − X r = rb ∨ _ ♦ ( r , b ) if k > , (3.31) a = a = ar = , ≤ r < k , ar = − δ r , k t − z tr( a )1 , < r ≤ k . (3.32)Recall, in (3.31) and in analogous expressions below, our convention that there is an implicit sum over b ∈ B A .Using (2.8), (3.7), (3.17), (3.22), and (3.32) we also have(3.33) n = δ n , t and n = δ n , k t − for 0 ≤ n ≤ k . When k <
0, we have = − − k − X r = r b ∨ r ♥ ( r , b ) if k < , = if k < , (3.34) = k < , ra = ≤ r < − k , a r = − δ r , t − z tr( a )1 if 0 ≤ r < − k , (3.35)for all a ∈ A . Lemma 3.3.
The following relations hold for all a ∈ A: (3.36) a = a , a = a , a = a , a = a . Proof.
First suppose k ≥
0. The first relation in (3.36) follows from composing the second relation in (3.13)on the top and bottom with the negative leftwards crossing, then using (3.30) to (3.32). The fourth relation in(3.36) follows similarly from composing the third relation in (3.13) on the top and bottom with the negativeleftwards crossing, then using (3.30) to (3.32). Then the second and third relations in (3.36) follow from thefirst and second using (2.28) and (3.23). The case k < (cid:3) UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 11
Now consider the analogue of the morphism (3.19) defined using the negative instead of the positiverightward crossing:(3.37) (cid:20) rb ∨ , ≤ r ≤ k − , b ∈ B A (cid:21) T : ↑↓→↓↑ ⊕ ⊕ k dim A if k ≥ , (cid:20) r b ∨ , ≤ r ≤ − k − , b ∈ B A (cid:21) : ↑↓ ⊕ ⊕ ( − k dim A ) →↓↑ if k < . Lemma 3.4.
The morphism (3.37) is invertible with two-sided inverse (3.38) r ♥ ( r , b ) , ≤ r ≤ k − , b ∈ B A : ↓↑ ⊕ ⊕ k dim A →↑↓ if k > , _ ♦ ( r , b ) , ≤ r ≤ − k − , b ∈ B A T : ↓↑→↑↓ ⊕ ⊕ ( − k dim A ) if k ≤ . Moreover, we have that ak = − t − z tr( a )1 if k > , a = t − t − z tr( a )1 if k = , a = − t − z tr( a )1 if k < , (3.39) = tz r ♥ (0 , if k > , t − − k if k ≤ , = t k if k > , t − if k = , tz − _ ♦ ( − k − , if k < . (3.40) Proof.
First consider the case k =
0. Note that a (3.21) = t a (3.36) = (3.9) t a (3.21) = a (3.22) = t − t − z tr( f )1 , proving (3.39). Also, (3.21) = t (3.23) = (3.39)(2.8) t − ( t − , from which the left-hand relation in (3.40) follows. The proof of the right-hand relation in (3.40) is analo-gous.Now suppose k >
0. By (3.7), we have (cid:20) rb ∨ , ≤ r ≤ k − , b ∈ B A (cid:21) T = M (cid:20) rb ∨ , ≤ r ≤ k − , b ∈ B A (cid:21) T , where M is the J × J matrix whose ( ⋆, ⋆ ) and ( ⋆, (0 , b )) entries areand − z b , whose (( r , b ) , ( s , a )) entry is δ r , s δ a , b for all 0 ≤ r , s ≤ k − a , b ∈ B A , and whose other entries are zero. Thematrix M is upper unitriangular, and hence invertible. Its inverse is the matrix whose ( ⋆, ⋆ ) and ( ⋆, (0 , b ))entries are and z b , whose (( r , b ) , ( s , a )) entry is δ r , s δ a , b for all 0 ≤ r , s ≤ k − a , b ∈ B A , and whose other entries are zero. Thus, (cid:20) rb ∨ , ≤ r ≤ k − , b ∈ B A (cid:21) T ! − = _ ♦ ( r , b ) , ≤ r ≤ k − , b ∈ B A M −
12 JONATHAN BRUNDAN, ALISTAIR SAVAGE, AND BEN WEBSTER(3.24) = r ♥ ( r , b ) , ≤ r ≤ k − , b ∈ B A . The relation (3.39) follows from (3.32), and the right-hand relation in (3.40) is the right-hand relation in(3.21). To prove the left-hand relation in (3.40), we compose both sides on the top with the isomorphism(3.37), to see that it su ffi ces to show that = , b ∨ n = δ n , tz tr( b ∨ )1 , ≤ n < k , b ∈ B A . These relations follow immediately from (3.22) and (3.32).The case k < k > (cid:3)
4. S econd approach
We now give a second definition of
Heis k ( A ; z , t ). Intuitively, this di ff ers from Definition 3.2 by replacingthe positive crossing in the inversion relation by the negative crossing. Definition 4.1.
The quantum Frobenius Heisenberg category
Heis k ( A ; z , t ) is the strict k -linear monoidalsupercategory obtained from AW ( A ; z , t ) by adjoining a right dual ↓ to ↑ , together with matrix entries of(3.38), which we declare to be a two-sided inverse to (3.37). In addition, we impose the relation (3.39) forthe leftwards cups and caps, which are defined in this approach by (3.40).Introduce the other leftward crossing not appearing in (3.38) so that (3.23) holds, and set _ ♦ (0 , a ) : = r ♥ (0 , a ) − z a if k > , _ ♦ ( n , a ) : = r ♥ ( n , a ) if 0 < n < k , (4.1) r ♥ (0 , a ) : = _ ♦ (0 , a ) − ( − ¯ a z a if k < , r ♥ ( n , a ) : = _ ♦ ( n , a ) if 0 < n < − k . (4.2)Finally, we define the fake bubbles from (3.26) to (3.28) as before. Theorem 4.2.
Definitions 3.2 and 4.1 give two di ff erent presentations for the same monoidal supercategory,and all of the named morphisms introduced in the two definitions are the same. Moreover, there is a uniqueisomorphism of k -linear monoidal supercategories (4.3) Ω k : Heis k ( A ; z , t ) → Heis − k ( A ; z , t − ) op , determined by a a , a ∈ A , ,
7→ − , , . The isomorphism Ω k acts on the other morphisms as follows: a a , a ∈ A , ,
7→ − ,
7→ − ,
7→ − ,
7→ − ,
7→ − ,
7→ − ,
7→ − , _ ♦ ( n , a ) ( − ¯ a _ ♦ ( n , a ) , _ ♦ ( n , a ) ( − ¯ a _ ♦ ( n , a ) , r ♥ ( n , a ) ( − ¯ a r ♥ ( n , a ) , r ♥ ( n , a ) ( − ¯ a r ♥ ( n , a ) ,
7→ − ,
7→ − , ± a n
7→ − ± an . In particular, Ω k is the identity. Note that the isomorphism Ω k is given by reflecting diagrams in a horizontal plane and multiplying by( − c + d + ( y ), where c is the number of crossings, d is the number of left cups and caps (including left cupsand caps in fake bubbles, but not ones labelled by ♦ or ♥ ), and y is the number of odd tokens. UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 13
Proof.
To avoid confusion, we denote the category
Heis k ( A ; z , t ) from Definition 3.2 by Heis old k ( A ; z , t )and the one from Definition 4.1 by Heis new k ( A ; z , t ). The relations and other definitions for the category Heis new k ( A ; z , t ) in Definition 4.1 and the ones for Heis old k ( A ; z , t − ) from Definition 3.2 are related by reflect-ing all diagrams in a horizontal plane and multiplying by ( − c + d + ( y ), where c is the number of crossingsand d is the number of left cups and caps (including left cups and caps in fake bubbles, but not ones labelledby ♦ or ♥ ), and y is the number of odd tokens. It follows that there are mutually inverse isomorphisms Heis old − k ( A ; z , t − ) Ω − ⇄ Ω + Heis new k ( A ; z , t ) op both defined in the same way as the functor Ω k in the statement of the theorem. Now, by Lemma 3.4and Definition 4.1, there exists a strict k -linear monoidal functor Θ k : Heis new k ( A ; z , t ) → Heis old k ( A ; z , t )that is the identity on diagrams. This functor is an isomorphism because it has a two-sided inverse, namely Ω − ◦ Θ − k ◦ Ω − . Thus, using Θ k , we may identify Heis new k ( A ; z , t ) and Heis old k ( A ; z , t ). Finally, Ω k : = Ω + givesthe required symmetry. (cid:3) We now prove some important relations. In particular, we will show how we can remove the decoratedleft cups and caps from our string diagrams. Then we prove the important infinite Grassmannian relations.
Lemma 4.3.
For all a ∈ A, we have (4.4) _ ♦ ( k − , a ) = − zt a if k > , _ ♦ ( − k − , a ) = ( − ¯ a zt − a if k < . Proof.
To prove the first relation in (4.4), compose the second relation in (3.31) on the bottom with a and use (3.15) and (3.32). The second relation in (4.4) then follows by applying Ω k . (cid:3) Lemma 4.4.
The following relations hold for all f ∈ F: = , = , (4.5) a = a , a = a , = , = . (4.6) Proof.
Using Ω k , it su ffi ces to consider the case k ≥
0. First suppose k =
0. Composing the first relation in(3.15) on the top with the leftward negative crossing and on the bottom with the leftward positive crossing,we have = (3.30) = ⇒ = . The proof of the second relation in (4.5) is obtained similarly from composing the second relation in (3.15)on the top with the leftward negative crossing and on the bottom with the leftward positive crossing. Therelations in (4.6) then follow from the definition (3.21), together with (3.9), (3.36), and (4.5).Now suppose k >
0. Composing the first relation in (3.15) on the top and bottom with the leftwardnegative crossing, we have0 = − (3.31) = (3.7) − k − X r = r + b ∨ _ ♦ ( r , b ) − − z bb ∨ = (3.31) − kb ∨ _ ♦ ( k − , b ) − (3.23) = − z b b ∨ − kb ∨ _ ♦ ( k − , b ) − (3.36)(3.21) = (4.4) − . The proof of the second relation in (4.5) is similar, starting with the second relation in (3.15).The first relation in (4.6) follows immediately from the definition (3.21), together with (2.30) and (3.36).To prove the second relation in (4.6), compose the second relation in (3.31) on the bottom with a anduse (3.15) and (3.32) to obtain a = − z − t − _ ♦ ( k − , a ) (4.4) = a . Then we add an inverse dot to the top of the left strand.To prove the third relation in (4.6), we note that (3.21) = t k (4.5) = (3.9) t k + = t k (3.21) = . Finally, to prove the fourth relation in (4.6), we compose the second relation in (3.31) on the bottom withand use (3.15) and (3.32) to get = − z − t − _ ♦ ( k − ,
1) (4.4) = . (cid:3) From now on, we will slide tokens and dots over cups and caps, and tokens through crossings, withoutciting the relevant relations.
Theorem 4.5 (Infinite Grassmannian relations) . For any n ∈ Z , we have (4.7) X r + s = n + + a br s = X r + s = n − − a br s = − δ n , z tr( ab )1 . Furthermore, + a n = δ n , − k tz tr( a )1 if n ≤ − k , + an = − δ n , k t − z tr( a )1 if n ≤ k , (4.8) − an = δ n , tz tr( a )1 if n ≥ , − a n = − δ n , t − z tr( a )1 if n ≥ . (4.9) Proof.
Using Ω k , it su ffi ces to consider the cases k ≥
0. In addition, using (2.28) and the fact that the( ± )-bubbles are linear in a , it su ffi ces to consider (4.7) in the case where a = k =
0. Then (4.8) follows immediately from (3.26), while (4.9) follows from (3.22), (3.27),(3.28), and (3.39). If n ≤
0, it follows from (2.8) and (3.26) that the left-hand side of (4.7) is equal to theright-hand side. If n >
0, we have t − n b (3.21) = n b (3.17) = nb + z X r + s = nr , s > rcs c ∨ b = t bn + z X r + s = nr , s > nb , where the final equality follows from the fact that nb (3.23) = nb + z n cc ∨ b (3.21) = (3.39) t − bn + ( t − t − ) bn = t bn . UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 15
Thus we have 0 = − t − z n b + tz n b + X r + s = nr , s > rcs c ∨ b (3.26) = (3.27) X r , s ≥ r + s = n + c r + c ∨ bs . If n ≥
0, then the ( − )-bubble sum in (4.7) is equal to − δ n , z − tr( ab ) by (4.9). Now suppose n <
0. Then,as above, we have t − n b (3.21) = n b (3.17) = nb − z X r + s = nr , s ≤ rcs c ∨ b = t bn − z X r + s = nr , s ≤ c rc ∨ bs . Thus, by (3.22), (3.28), (3.39), (4.8), and (4.9), we have X r + s = nr , s < + c r + c ∨ bs = . Now suppose k >
0. Note that(4.10) z − n b (3.21) = z − t nk b (3.31) = z − t bk + n − z − t k − X r = k _ ♦ ( r , c ) c ∨ br + n (3.26) = X r + s = nr ≤ + c rc ∨ bs . Now consider the second relation in (4.8). If n ≤
0, this relation holds immediately by the definition (3.26).On the other hand, if 0 < n ≤ k , then it holds by (3.27) and (3.32). The first relation in (4.8) followsimmediately from (3.26).Now we prove that the left-hand side of (4.7) is equal to the right-hand side. If n <
0, then, by (3.26), wehave + c ∨ bs = s < k and + ac r = r < − k , and so the sum on the left-hand side of (4.7) zero. Similarly, if n =
0, then only the s = k , r = − k termssurvives, and so the sum is equal to − z tr( ac )tr( c ∨ b )1 (2.8) = − z tr( ab )1 . Now assume n >
0. Using (3.17), the left-hand side of (4.10) becomes z − nb − X r + s = nr , s > c rc ∨ bs (3.32) = (3.27) − X r + s = nr , s > + c rc ∨ bs . Then it follows from (3.27) and (4.8) that X r + s = n + c r + c ∨ bs = , as desired.Next we prove that the ( − )-bubble sum in (4.7) is equal to the right-hand side. By (3.27) and (3.28), wehave − ar = + a r = r > . Thus the identity holds when n >
0. Now suppose n ≤
0. Then, using (3.17) on the left-hand side, therelation (4.10) becomes X r + s = nr ≤ + c rc ∨ bs = z − n + X r + s = nr , s ≤ c rc ∨ bs (3.32) = X r + s = nr , s ≤ c rc ∨ bs . Using (3.28), we then have X r + s = nr ≤ + c rc ∨ bs = X r + s = nr , s ≤ + c rc ∨ bs + − c r + c ∨ bs + − c r − c ∨ bs . Thus, by (4.8), we have X r + s = n − c r − c ∨ bs = X r + s = nr ≤ , s > + c rc ∨ bs (4.8) = (3.32) − δ n , z tr( c )tr( c ∨ b )1 (2.8) = − δ n , z tr( b )1 . It remains to prove (4.9). When n >
0, these relations follows immediately from (3.27) and (3.28). Now, tz tr( a )1 (3.22) = a (3.28) = + a + − a = − a . Then, by (4.7) with n =
0, we have − t − z tr( a )1 = zt − − ac − c ∨ = tr( c ∨ ) − ac = − a . (cid:3) It will be useful to expression some of our relations in terms of generating functions. For a ∈ A and anindeterminate w , let(4.11) wa : = t − z X r ∈ Z + a r w − r ∈ tr( a )1 + w k − End
Heis k ( A ; z , t ) ( ) ~ w − (cid:127) , (4.12) w a : = − tz X r ∈ Z + ar w − r ∈ tr( a ) u − k + w − k − End
Heis k ( A ; z , t ) ( ) ~ w − (cid:127) , (4.13) wa : = − tz X r ∈ Z − a r w − r ∈ tr( a ) w k + w End
Heis k ( A ; z , t ) ( ) ~ w (cid:127) , (4.14) w a : = t − z X r ∈ Z − ar w − r ∈ tr( a )1 + w End
Heis k ( A ; z , t ) ( ) ~ w (cid:127) . (4.15) Corollary 4.6.
We have (4.16) w wa b = w wa b = z tr( ab )1 . We adopt the convention that determinants of matrices whose entries lie in a superalgebra are to becomputed using the usual Laplace expansions, keeping the non-commuting variables in each monomialordered in the same way as the columns from which they are taken (see [Sav19, (17)]).
Lemma 4.7.
For all a ∈ A, we have + a r − k = z r − t r + X b ,..., b r − ∈ B A det (cid:18) b ∨ j − b j i − j + k + (cid:19) ri , j = , r ≤ k , (4.17) + ar + k = − z r − t − r − X b ,..., b r − ∈ B A det (cid:18) − b ∨ j − b j i − j − k + (cid:19) ri , j = , r ≤ − k , (4.18) adopting the convention that b ∨ : = a and b r : = . We interpret these determinants as tr( a ) if r = or as ifr < .Proof. It su ffi ces to prove (4.17), since (4.18) then follows by applying Ω k . If k <
0, then the right-handside in (4.17) is zero by our convention, and so the result follows by (4.8). Now suppose k ≥
0. The cases
UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 17 r ≤ r > r −
1. For b , . . . , b r − ∈ B A , define the matrix A = (cid:18) b ∨ j − b j i − j + k + (cid:19) ri , j = . (We leave the dependence on b , . . . , b r − ∈ B A implicit to simplify the notation.) We havedet A = r X m = ( − m + ab m + k det A m , , where A m , is the ( r − × ( r −
1) matrix obtained from A by removing the m -th row and first column. If weconsider A m , as a block matrix with upper-left block of size ( m − × ( m −
1) and lower-right block of size( r − m ) × ( r − m ), we see that it is block lower triangular. By (3.32), the upper-left block is lower triangularwith diagonal entries − t − z − tr( b ∨ b ) , − t − z − tr( b ∨ b ) , . . . , − t − z − tr( b ∨ m − b m ) . On the other hand, the lower-right block is the matrix (cid:18) b ∨ m + j − b m + j i − j + k + (cid:19) r − mi , j = . Thus, using the induction hypothesis and (2.7), we have z r − t r + X b ,..., b r − ∈ B A det A = zt r X m = ab m + k + b ∨ r − k − m (3.27) = zt r X m = + ab m + k + b ∨ r − k − m (4.7)(4.8) = (2.8) + a r − k . (cid:3) Lemma 4.8.
For all r ∈ Z and a , b ∈ A, we have + ab r = ( − ¯ a ¯ b + ba r , + abr = ( − ¯ a ¯ b + bar , (4.19) − ab r = ( − ¯ a ¯ b − ba r , − abr = ( − ¯ a ¯ b − bar . (4.20) Proof.
Writing out the columns of the matrix in (4.17), we have z − r + t − r − + ab r − k (4.17) = X b ,..., b n + k − ∈ B A det (cid:16) abb i + k b ∨ b i − + k · · · b ∨ r − i − r + k + (cid:17) ri = = X b ,..., b r − ∈ B A det (cid:16) ab i + k b ∨ b i − + k · · · b ∨ r − bi − r + k + (cid:17) ri = = X b ,..., b n + k − ∈ B A ( − ¯ b ¯ b r − det (cid:16) ab i + k b ∨ b i − + k · · · bb ∨ r − i − r + k + (cid:17) ri = = X b ,..., b r − ∈ B A ( − ¯ b ¯ b det (cid:16) ab bi + k b ∨ b i − + k · · · b ∨ r − i − r + k + (cid:17) ri = = X b ,..., b n + k − ∈ B A ( − ¯ a ¯ b det (cid:16) bab i + k b ∨ b i − + k · · · b ∨ r − i − r + k + (cid:17) ri = = ( − ¯ a ¯ b z − r + t − r − + ba r − k . This completes the proof of the first relation in (4.19). The second follows by applying Ω k . Finally, (4.20)follows from (3.28) and (4.19). (cid:3) It follows from Lemma 4.8 that we can label the tokens in ( ± )-bubbles by elements of the cocenter C ( A ) of A . The following lemma, together with (3.24) and (4.2), allows us to eliminate the leftwards capsdecorated by ♦ and ♥ from any diagram. Lemma 4.9.
The following relations hold: _ ♦ ( n , a ) = − z X r ≥ r b + b ∨ a − r − n if ≤ n < k , (4.21) _ ♦ ( n , a ) = − ( − ¯ a z X r ≥ rb ∨ + ab − r − n if ≤ n < − k . (4.22) Proof.
We first prove (4.21), and so we assume 0 ≤ n < k . Composing the second relation in (3.31) with theright side of (4.21) gives − z X r ≥ r b + b ∨ a − r − n = − z X r ≥ br + b ∨ a − r − n − z X r ≥ k − X s = _ ♦ ( s , c ) c ∨ br + s + b ∨ a − r − n . By (4.8), the ( + )-bubble in the first sum on the right side above is zero unless r ≤ k . But for these valuesof r this term is zero by (3.33). Thus the first sum on the right side above vanishes. Now, in the second(double) sum, (3.27) allows us to replace the clockwise bubble by + c ∨ br + s . We can also replace the sumover r ≥ r ∈ Z , since the additional terms are zero by (4.8). Then (4.21) follows by (2.8)and (4.7).Now, applying to Ω k to (4.21), we have, for 0 ≤ n < − k , _ ♦ ( n , a ) = − ( − ¯ b + ¯ b ¯ a + ¯ a z X r ≥ rb + b ∨ a − r − n (4.19) = (2.10) − ( − ¯ a z X r ≥ rb ∨ + ab − r − n . (cid:3) Corollary 4.10.
The following relations hold for all a ∈ A: _ ♦ ( n , a ) = _ ♦ ( n , a , r ♥ ( n , a ) = r ♥ ( n , a , ≤ n < k , (4.23) _ ♦ ( n , a ) = ( − ¯ a _ ♦ ( n , a , r ♥ ( n , a ) = ( − ¯ a r ♥ ( n , a , ≤ n < − k . (4.24) Proof.
The relations involving the ♦ -decorated left cup and cap follow immediately from (2.9), (4.21),and (4.22). Then the relations involving the ♥ -decorated left cup and cap follow from (3.24) and (3.25). (cid:3) The final goal of this section is to endow
Heis k ( A ; z , t ) with the structure of a strict pivotal category. Lemma 4.11.
We have (4.25) = and = if k ≥ . Proof.
Suppose k ≥
0. We first claim that(4.26) = , (4.27) = . To prove (4.26), we note that, composing on the top of the two leftmost strands with the invertible map(3.19), it su ffi ces to prove that(4.28) = , (4.29) na = na , ≤ n < k , a ∈ A . To see (4.28), we compute
UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 19(2.27) = − (3.12) = − (3.33) = t − t (2.27) = t (3.10) = (2.23) t (3.33) = . Now consider (4.29). If n =
0, we have a (3.11) = a (2.23) = a (3.22) = a . (We note that the above step is the reason we need to impose the relation (3.22).) On the other hand, if0 < n < k , we have (recall the notation a † from (2.11)) na (3.11) = na (2.27) = na + a † n (3.18) = a n + X r + s = nr , s > a † r s (2.23) = na (3.32) = (3.32) = na , where, in the third and fourth equalities, we used that(4.30) s (3.11) = s (3.33) = ≤ s < k . To prove (4.27), we note that, composing on the top of the two rightmost strands with the invertible map(3.19), it su ffi ces to prove that(4.31) = , (4.32) na = na , ≤ n < k , a ∈ A . The proofs of (4.31) and (4.32) are similar to those of (4.28) and (4.29) and are left to the reader.Now, to prove the left-hand relation in (4.25), we compose (4.26) on the top with to get = Applying (3.30) and (3.31) gives − k − X n = nb ∨ _ ♦ ( n , b ) = . Then the left relation in (4.25) follows from (4.30). The right relation in (4.25) follows similarly fromcomposing (4.27) on the top with . (cid:3)
Lemma 4.12.
The following relations hold: (4.33) = , = . Proof.
It su ffi ces to consider the case k ≥
0, since the case k < Ω k . For thefirst relation, we have (3.21) = t k (4.25) = t k (3.11) = t k (3.33) = (3.1) = . For the second relation, we have (3.21) = t k (4.25) = t k (3.11) = t k (3.33) = (3.1) = . (cid:3) Lemma 4.13.
The following relations hold: = , = , = , = , (4.34) = , = , = , = . (4.35) Proof.
It su ffi ces to consider the case k ≥
0, since the case k < Ω k . The secondand third relations in (4.34) are (4.25). To see the first relation in (4.34), we compute (2.27) = + (4.25) = + (3.23) = (4.33) . The proof of the fourth relation in (4.34) is analogous.The relations (4.35) now follow by attaching left caps to the relations in (4.34) and using (4.33). Forexample, attaching left caps to the top left and top right strands of the first relation in (4.34) gives = . Then the first relation in (4.35) follows from (4.33). (cid:3)
It follows from (4.6), (4.34), and (4.35) that the category
Heis k ( A ; z , t ) is strictly pivotal, with dualityfunction(4.36) ∗ : Heis k ( A ; z , t ) (cid:27) −→ (cid:16)(cid:0) Heis k ( A ; z , t ) (cid:1) op (cid:17) rev defined by rotating diagrams through 180 ◦ and multiplying by ( − y ), where y is the number of odd tokensin the diagram. Intuitively, this means that morphisms are invariant under isotopy fixing the endpoints(multiplying by the appropriate sign when odd elements change height). For this reason, we will allowourselves to draw tokens and dots at the critical points of cups and caps, and this does not give rise to anyambiguity. 5. T hird approach In this section we give our third definition of
Heis k ( A ; z , t ). In this approach, the inversion relation isreplaced by explicit relations amongst generators. To do this, we must add a left cup and cap as generatingmorphisms. Definition 5.1.
The quantum Frobenius Heisenberg category
Heis k ( A ; z , t ) is the strict k -linear monoidal su-percategory obtained from AW ( A ; z , t ) by adjoining a right dual ↓ to ↑ , plus two more generating morphismsand , subject to the following additional relations: = − t − + X r , s > + − r − srs , (5.1) = + t + X r , s > + − r − s rs , (5.2) = δ k , t − if k ≥ , an = δ n , t − δ n , k t − z tr( a )1 if 0 ≤ n ≤ k , (5.3) UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 21 = δ k , t if k ≤ , an = δ n , − k t − δ n , t − z tr( a )1 if 0 ≤ n ≤ − k . (5.4)Here we have used the leftward crossings, which are defined in this approach by(5.5) : = , : = , and the ( + )-bubbles, which are defined in this approach by (4.17) and (4.18) when they carry a token labelled a ∈ A and a dot labeled n ≤
0. Finally, we define the ( + )-bubbles with n > n dots as in (3.27), and then define the ( − )-bubbles, with an arbitrary number of dots, so that (3.28)holds.We will prove that Definition 5.1 is equivalent to Definitions 3.2 and 4.1. But before doing so, we makesome remarks about the relations (4.17), (4.18), and (5.1) to (5.5). If k ≤
1, it follows immediately from(4.17) that relation (5.1) is equivalent to(5.6) = − t − if k ≤ . Similarly, when k ≥ −
1, relation (5.2) is equivalent to(5.7) = + t if k ≥ − . In addition, using (3.5) and (3.7), we obtain the following relations from (5.3) and (5.4): = δ k , t − , = t , = δ k , t − , = t if k ≥ , (5.8) = δ k , t , = t − , = δ k , t , = t − , if k ≤ . (5.9)Then, using (5.9) and (3.23) to convert the negative crossings in (5.6) to positive ones, we see that, when k <
0, (5.6) is equivalent to(5.10) = if k < . Similarly, when k >
0, (5.7) is equivalent to(5.11) = if k > . Finally, when k =
0, (5.6) and (5.7) together are equivalent to the single assertion(5.12) = (cid:16) (cid:17) − , i.e. both of the relations (3.30). Theorem 5.2.
The category
Heis k ( A ; z , t ) of Definition 5.1 is the same as the one of Definitions 3.2 and 4.1,with all morphisms introduced in Definition 5.1 being the same as the ones of Definitions 3.2 and 4.1.Proof. To avoid confusion in the proof, we denote the category from the equivalent Definitions 3.2 and 4.1by
Heis old k ( A ; z , t ), and the one from Definition 5.1 by Heis new k ( A ; z , t ). It is clear from the symmetry in therelations (4.17), (4.18), and (5.1) to (5.5) that there is an isomorphism Heis new k ( A ; z , t ) → Heis new − k ( A ; z , t − ) op2 JONATHAN BRUNDAN, ALISTAIR SAVAGE, AND BEN WEBSTER that reflects diagrams in a horizontal plane and multiplies by ( − c + l + ( y ), where c is the number of crossings, d is the number of left cups and caps, and y is the number of odd tokens. Thus, by (4.3), it su ffi ces to provethe theorem in the case k ≤ Heis old k ( A ; z , t ), so that we have a naturalstrict k -linear monoidal functor Θ : Heis new k ( A ; z , t ) → Heis old k ( A ; z , t ) , which is the identity on diagrams. • (5.1) and (5.2): When k =
0, (5.1) and (5.2) are equivalent to (5.12), which holds by (3.30). Nowsuppose k <
0. Then (5.1) is equivalent to (5.10), which holds by the second relation in (3.34). To check(5.2), we start with the first relation in (3.34) and expand the ♥ -decorated left caps using (3.21) and (4.24)when r =
0, or (3.25) and (4.22) when r > • (5.3) and (5.4): These relations follow easily from (3.21), (3.22), (3.35), (3.39), and (3.40). • (5.5): This holds in Heis old k ( A ; z , t ) since we have shown this category is strictly pivotal.Now we want to show that Θ is an isomorphism. We do this by using the presentation from Definition 3.2to construct a two-sided inverse Φ : Heis old k ( A ; z , t ) → Heis new k ( A ; z , t ) , still assuming k ≤
0. We define Φ on morphisms be declaring that it takes the rightwards cup, the right-wards cap, and all the tokens, dots, and crossings (with any orientation) to the corresponding morphisms in Heis new k ( A ; z , t ), and also (see (3.21), (4.2), (4.22), and (4.24)) Φ r ♥ (0 , a ) ! = − ( − ¯ a tz a if k < , Φ r ♥ ( n , a ) ! = − ( − ¯ a z X r ≥ b ∨ r + ab − r − n if 0 < n < − k . To see that Φ is well defined, we must verify the relations from Definition 3.2. For (3.22), this amounts tochecking that, in Heis new k ( A ; z , t ), we have t a = t − t − z tr( a )1 if k = , a − k = tz tr( a )1 if k < . The first relation follows from (5.3) and (5.9), while the second follows from (5.4). Thus, it remains to showthat the images under Φ of the morphisms (3.19) and (3.20) are two-sided inverses in Heis new k ( A ; z , t ). When k =
0, this is immediate from (5.12), so suppose k <
0. The images under Φ of the two equations in (3.34)are precisely (5.2) and (5.10). We are left with checking that the images under Φ of the relations a m = , r ♥ ( n , a ) = , a m r ♥ ( n , b ) = δ m , n tr( ab )1 , (5.13)hold in Heis new k ( A ; z , t ) for all a , b ∈ A and 0 ≤ m , n < − k . For the first relation, we can slide the tokenthrough the crossing, and so it su ffi ces to prove it without the token. When m =
0, it then follows by (5.9).When 0 < m < − k , we have m (3.23) = (5.4) m (3.18)(5.5) = (5.4) m (5.9) = . When n =
0, the second and third relations in (5.13) follow from (5.4) and (5.9). To prove them when0 < n < − k , we must show that X r ≥ b ∨ r + ab − r − n = , X r ≥ + bc − r − nc ∨ a r + m = − ( − ¯ b δ m , n z δ m , n tr( ab )1 (5.14) UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 23 in Heis new k ( A ; z , t ). For the first identity, note that the terms with r ≥ − k vanish since the ( + )-bubble is zeroby (4.17). The terms with 0 < r < − k also vanish, as can be seen by using the skein relation to flip thecrossing, sliding the dots past the crossing, and then using (5.4) and (5.9). To prove the second identity in(5.13), we first note that, by (2.9), it su ffi ces to consider the case where b =
1. When m ≤ n , the identityfollows from (4.18) and (5.4). When m > n , we compute: + am − n + k (4.18) = − z m − n − t n − m − X b ,..., b m − n − ∈ B A det (cid:18) − b ∨ j − b j i − j − k + (cid:19) m − ni , j = = − zt − m − n X s = ab ∨ s − k + bm − n + k , where, in the final equality, we expanded the determinant along the first column as in proof of Lemma 4.7.By (4.18), it follows that0 = m − n X s = ab s − k + b ∨ m − n + k = m − n X s = ( − ¯ b ba s − k + b ∨ m − n + k = m − n X s = + b ∨ m − n + kb ∨ a s − k . Then the second identity in (5.14) follows by letting r = s − m − k .To complete the proof, we must show that Θ and Φ are indeed two-sided inverses. To verify that Θ ◦ Φ = id, the only nontrivial thing to check is that Θ Φ r ♥ ( n , a ) !! = r ♥ ( n , a ) for a ∈ A , ≤ n < − k . When n =
0, this follows immediately from (3.21) and (4.24). On the other hand, if 0 < n < − k , it followsfrom (3.25) and (4.22). To verify that Φ ◦ Θ = id, the only nontrivial thing to check is that Φ (cid:16) (cid:17) = and Φ (cid:16) (cid:17) = . The first equality follows from (3.21) and (5.8). The second follows from (3.21) when k =
0. When k < Φ (cid:16) (cid:17) (3.21) = t − − k (3.23) = (5.4)(2.8) − k + (3.18)(5.5) = (5.4)(5.9) . (cid:3) Remark 5.3.
The Frobeneius Heisenberg categories studied in [Sav19, BSW20a] depend only, up to iso-morphism, on the underlying algebra A , and not on the trace map; see [BSW20a, Lem. 5.3]. However, inthe quantum setting of the current paper, there do not seem to be obvious isomorphisms between quantumFrobenius Heisenberg categories corresponding to the same algebra but with di ff erent trace maps. This istrue even for the quantum a ffi ne wreath product algebras; see [RS20b, Rem. 2.2]. Lemma 5.4.
Suppose that C is a strict k -linear monoidal category containing objects ↑ and ↓ and mor-phisms a , , , , , and satisfying (2.22) , (2.23) , (2.27) , and (2.30) . Then C contains at mostone pair of morphisms and satisfying (5.1) to (5.4) (for sideways crossings and the ( + ) -bubblesdefined via (3.5) , (4.17) , (4.18) , and (5.5) ).Proof. The proof is analogous to that of [BSW20b, Lem. 4.3]. (cid:3)
We now prove some further useful relations that hold in
Heis k ( A ; z , t ). We will state some of theserelations with and without the language of generating functions. To state the relations in terms of generatingfunctions, we need to unify our notation for dots and tokens, adopting the notation ax n : = an and ax n : = an for a ∈ A , n ∈ Z . Then we can also label tokens by polynomials a n x n + · · · + a x + a ∈ A [ x ], meaning thesum of morphisms defined by the tokens labelled a n x n , a x , . . . , x , or even by Laurent series in A [ x ](( w − )) or A [ x ](( w )). For example, expanding in A [ x ](( w − )), we have wx ( w − x ) − = w − + w − + w − + w − + · · · . The caveat here is that, whereas tokens labelled by elements of A (( w − )) slide through crossings and canbe teleported, the more general tokens labelled by elements of A [ x ](( w − )) no longer have these propertiesin general. For a Laurent series p ( w ), we let [ p ( w )] w r denote its w r -coe ffi cient, and we write [ p ( w )] w < for P n < [ p ( w )] w n u n . We adopt the convention that, in any equation involving the generating functions (4.14)and (4.15), we expand all rational functions as Laurent series in A [ x ](( w )). In all other equations, we expandrational functions as Laurent series in A [ x ](( w − )). Lemma 5.5 (Curl relations) . The following relations hold for all n ∈ Z : n = X r ≥ r + n − r − X r > − r − n + r , ( w − x ) − = t ( w − x ) − w w < , (5.15) n = X r > r + n − r − X r ≥ − r − n + r , (5.16) n = X r ≥ − r − n + r − X r > r + n − r , (5.17) n = X r > − r − n + r − X r ≥ r + n − r , ( w − x ) − = t − ( w − x ) − w w < . (5.18) Proof.
The right-hand relations in (5.15) and (5.18) are simply reformulations of the n ≥ ffi ces to prove (5.15)and (5.18). Now, given one of (5.15) or (5.18) for k ≥
0, we can rotate through 180 ◦ (using the strictlypivotal structure), and then apply Ω k to obtain the other relation for k ≤
0. Thus, we are reduced to proving(5.18) when k ≥ k > k ≥
0. When n ≥
0, we have n (3.18) = n − X r + s = nr , s ≥ r s (3.33) = (3.28) t n − X r + s = nr , s ≥ r + s − n − = − X r + s = nr , s ≥ r + s (4.8) = − X r ≥ r + n − r . This gives (5.18), since the ( − )-bubbles there are zero when n ≥
0, by (3.27) and (3.28). When n <
0, wehave n (3.18) = n + X r + s = − nr , s > − r − s (3.33) = (4.8)(3.28) t n + − n − X r = − r − n + r (4.9) = X r > − r − n + r . This gives (5.18), since the ( + )-bubbles there are zero when n <
0, by (3.28) and (4.8).Now suppose k >
0. When n =
0, we have (3.21) = t k (3.31) = t k − t k − X r = rb ∨ k _ ♦ ( r , b ) (3.26) = z k X r = rb ∨ + b − r . UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 25
Using the strictly pivotal structure, this gives (5.15) since the ( − )-bubbles appearing there are zero by (3.27)and (3.28).When n >
0, we have n (2.27) = n + n (3.18) = (3.27) n + X r + s = nr , s > s r + + n (3.27) = X r ≥ + − r r + n + n − X r = + n − r r = X r ≥ + n − r r , where, in the second-to-last equality we used the n = n > − )-bubbles appearing there are zero by (3.27) and (3.28).Finally, suppose n <
0. Then n (3.17) = n − X r + s = nr , s < s r = X r > + − r r + n − − − − X r = n + n − r r (3.28) = X r ≥ + − r r + n − − X r = n n − r r (3.28) = (3.27) X r ≥ + n − r r − X r > − n + r − r , where, in the second equality, we used the n = (cid:3) Recall the definition (2.11) of the notation a † . Lemma 5.6 (Bubble slides) . The following relations hold for all n ∈ Z and a ∈ A: + an = + an − X r > r + n − ra † r , w a = w a − wa † xw ( w − x ) − (5.19) + a n = + a n − X r > r + n − r a † r , wa = wa − w a † xw ( w − x ) − , (5.20) − an = − an − X r > r − n + ra † − r , w a = w a − wa † xw ( w − x ) − , (5.21) − a n = − a n − X r > r − n + r a † − r , wa = wa − w a † xw ( w − x ) − . (5.22) Proof.
Note that, in each of (5.19) to (5.22), the right-hand relation is simply a restatement of the left-handrelation in terms of generating functions. It su ffi ces to prove (5.19) to (5.22) for k ≥
0, since the identitiesfor k < ◦ and applying Ω k . So we assume k ≥ n ≤ k , by (4.8). Thus, we suppose n > k . Then we have + an (3.27) = an (3.30)(3.31) = (3.13) an (4.34)(3.11) = (3.17) na + X r + s = nr , s ≥ a † r s (2.27) = an + n X r = a † r n − r (2.23) = (5.18) an − n X r = X s ≥ + n − r − sa † r + s (3.27) = + an − X t > t + n − ta † t , where, in the final equality, we have used the fact that the ( + )-bubble there is zero when t > n by (4.8),together with our assumptions that k ≥ Next we prove (5.20). Take the second relation in (5.19) with a replaced by b ∨ c , tensor on the left with wb and on the right with wc ∨ a , multiply by z , and then sum over b , c ∈ B A to give ww w a = w w w a − ( − ¯ a ¯ d w w w add ∨ xw ( w − x ) − . Then the second relation in (5.20) follows after applying (4.16).The identity (5.21) is trivial when n ≥ n <
0, the proof is almost identical to the proofof (5.19) given above. We then obtain the second relation in (5.22) from the second relation in (5.21) justas we did for the ( + )-bubbles. (cid:3) Lemma 5.7 (Alternating braid relation) . The following relation holds: − = X r , s ≥ t > + − r − s − t rs t if k ≥ , (5.23) − = X r , s ≥ t > + − r − s − trst if k ≤ . (5.24) Proof.
We prove (5.23), since (5.24) then follows by rotating through 180 ◦ and applying Ω k . So we suppose k ≥
0. Consider the relation = from (3.12) and compose on the top and bottom with and respectively. Then we have = (3.30) = ⇒ (3.31) = − k − X n = _ ♦ ( n , b ) nb ∨ (2.27)(3.17) = ⇒ (3.30)(3.31) + = + − k − X n = _ ♦ ( n , b ) nb ∨ | {z } = + X s + t = ns , t ≥ s _ ♦ ( n , b ) tb ∨ (3.23)(3.30) = ⇒ (3.31)(3.33) − − k − X n = _ ♦ ( n , b ) b ∨ n = − − k − X n = _ ♦ ( n , b ) nb ∨ − k − X n = X s + t = ns > , t ≥ s _ ♦ ( n , b ) tb ∨ (3.30)(3.31) = ⇒ (3.17)(3.33) + k − X n = _ ♦ ( n , b ) b ∨ = + k − X n = _ ♦ ( n , b ) b ∨ n − k − X n = _ ♦ ( n , b ) nb ∨ − k − X n = X s + t = ns > , t ≥ _ ♦ ( n , b ) tb ∨ s . Since _ ♦ ( n , b ) b ∨ n (4.23) = c _ ♦ ( n , c ∨ b ) b ∨ n (2.9) = c _ ♦ ( n , b ) b ∨ c ∨ n (2.28) = _ ♦ ( n , b ) n b ∨ , UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 27 we have = − k − X n = X s + t = ns ≥ , t > _ ♦ ( n , b ) tb ∨ s − k − X n = _ ♦ ( n , b ) b ∨ (4.21) = + k − X n = X s + t = ns ≥ , t > X r ≥ + − r − n rs t + k − X n = X t ≥ + − t − n n t , and (5.23) follows. (cid:3)
6. A ction on modules over quantum cyclotomic wreath product algebras
In this section we describe a natural categorical action of
Heis k ( A ; z , t ) on categories of modules overcyclotomic quotients of quantum a ffi ne wreath product algebras.Recall from (2.3) the center Z ( A ) of A , with even part Z ( A ) ¯0 . We fix a polynomial(6.1) f ( w ) = f w l + f w l − + · · · + f l ∈ Z ( A ) ¯0 [ w ] , of degree l ≥ f = f l = t . When l ≥
1, the quantum cyclotomic wreath product algebra W fn = W fn ( A ; z ) of level l asssociated to the polynomial f ( w ) is the quotient of AW n ( A ; z ) by the two-sidedideal generated by f ( x ). We also allow n =
0, with the convention that W f = k . When f ( w ) =
1, we havethat W f = k and W fn = n > n x r · · · x r n n a σ g : 0 ≤ r , . . . , r n < l , a ∈ B A ⊗ n , g ∈ S n o is a basis of W fn , where σ g = σ i · · · σ i m for some reduced expression g = s i · · · s i m for g in the symmetricgroup S n . (The element σ g is independent of the choice of reduced expression.) It follows that we have aninjective homomorphism W fn → W fn + , sending the generators x i , σ j to the elements of W fn + with the samenames and sending a ∈ A ⊗ n to 1 ⊗ a ∈ A ⊗ ( n + . In this way, we identify W fn with a subalgebra of W fn + . Wethen have induction and restriction superfunctorsind n + n : = − ⊗ W fn W fn + : smod-W fn → smod-W fn + , (6.3) res n + n : smod-W fn + → smod-W fn . (6.4)Our goal is to endow the abelian category L n ≥ smod-W fn with the structure of a left Heis − l ( A ; z , t )-module category, with ↑ and ↓ acting as induction and restriction, respectively. The key algebraic result thatallows us to do this is the cyclotomic Mackey theorem [RS20b, Prop. 4.13], which states that we have anisomorphism of (W fn , W fn )-bimodules(6.5) W fn ⊗ W fn − W fn ⊕ M ≤ r < lb ∈ B A W fn → W fn + , (cid:0) u ⊗ v , ( w r , b ) (cid:1) u σ n v + X ≤ r < lb ∈ B A x rn + b ( n + w r , where we recall that b ( n + = b ⊗ ⊗ n . As explained in [RS20b, §4.5], there is a unique homomorphism of(W fn , W fn )-bimodules tr fn + : W fn + → W fn such that tr fn + ( σ n ) =
0, tr fn + ( a ( n + x rn + ) = δ r , tr( a ) for a ∈ A , 0 ≤ r < l . (Recall that tr is the trace map ofthe Frobenius algebra A .) Lemma 6.1.
For any n ≥ , we have tr fn ( f ( x n )) = .Proof. The proof is almost identical to that of [BSW20b, Lem. 6.1]. (cid:3)
Theorem 6.2.
There is a unique strict k -linear monoidal superfunctor Ψ f : Heis − l ( A ; z , t ) → SEnd k M n ≥ smod-W fn sending the generating object ↑ (resp. ↓ ) to the additive endosuperfunctor that takes a W fn -module M to ind n + n M (resp. res nn − M), and the generating morphisms to the supernatural transformations defined on the W fn -module M as follows: • Ψ f ( a ) M : M ⊗ W fn W fn + → M ⊗ W fn W fn + , u ⊗ v ( − ¯ a ¯ u u ⊗ a ( n + v; • Ψ f ( ) M : M ⊗ W fn W fn + → M ⊗ W fn W fn + , u ⊗ v u ⊗ x n + v; • Ψ f (cid:16) (cid:17) M : M ⊗ W fn W fn + → M ⊗ W fn W fn + , u ⊗ v u ⊗ σ n + v (where we have identified ind n + n + ◦ ind n + n with ind n + n in the obvious way); • Ψ f (cid:16) (cid:17) M : M → res n + n (cid:16) M ⊗ W fn W fn + (cid:17) , v v ⊗ , i.e. it is the unit of the canonical adjunctionmaking (ind n + n , res n + n ) into an adjoint pair of superfunctors; • Ψ f (cid:16) (cid:17) M : (res nn − M ) ⊗ W fn − W fn → M, u ⊗ v uv, i.e. it is the counit of the canonical adjunctionmaking (ind nn − , res nn − ) into an adjoint pair of superfunctors.Proof. The proof is similar to the proof of [BSW20b, Th. 6.2], using the presentation of
Heis − l ( A ; z , t ) fromDefinition 3.2. The inversion relation follows from (6.5), while the relation (3.22) follows from Lemma 6.1. (cid:3) We can reformulate Theorem 6.2 in terms of Heisenberg categories of positive central charge by switch-ing the roles of induction and restriction. In fact, it is somewhat more natural to replace the inductionsuperfunctor ind n + n , which is the canonical left adjoint to restriction, with the coinduction superfunctor (6.6) coind n + n : = Hom W fn (cid:16) W fn + , − (cid:17) : smod-W fn → smod-W fn + , which is its canonical right adjoint. Theorem 6.3.
There is a unique strict k -linear monoidal superfunctor Ψ ∨ f : Heis l ( A op ; z , t − ) → SEnd k M n ≥ smod-W fn sending the generating object ↑ (resp. ↓ ) to the additive endosuperfunctor that takes a W fn -module M to res n + n M (resp. coind nn − M), and the generating morphisms to the natural transformations defined on the W fn -module M as follows: • Ψ ∨ f ( a ) M : res nn − M → res nn − M, v ( − ¯ a ¯ v va ( n ) ; • Ψ ∨ f ( ) M : res nn − M → res nn − M, v vx n ; • Ψ ∨ f (cid:16) (cid:17) M : res nn − M → res nn − M, v
7→ − v σ − n − ; • Ψ ∨ f (cid:16) (cid:17) M : M → Hom W fn − (W fn , res nn − M ) , v ( u vu ) , i.e. it is the unit of the canonicaladjunction making (res n + n , coind n + n ) into an adjoint pair of superfunctors; • Ψ ∨ f (cid:16) (cid:17) M : res n + n (cid:16) Hom W fn (W fn + , M ) (cid:17) → M, θ θ (1) , i.e. it is the counit of the canonical adjunc-tion making (res nn − , coind nn − ) into an adjoint pair of superfunctors.Proof. The proof is similar to the proof of Theorem 6.2, using instead the presentation for
Heis l ( A op ; z , t − )from Definition 4.1; see also [BSW20b, Th. 6.3]. (cid:3) UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 29
In fact, we have that ind n + n (cid:27) coind n + n . This follows from the fact that W fn + is a Frobenius extensionof W fn ; see [RS20b, Prop. 4.18]. It also follows from the results of the current paper and the uniqueness ofadjoints, since (4.33) and Theorem 6.2 (resp. Theorem 6.3) imply that ind n + n is right adjoint to restrictionas well as being left adjoint (resp. coind n + n is left adjoint to restriction as well as being right adjoint). Asa consequence, all three functors (induction, coinduction, and restriction) send finitely-generated projectivemodules to finitely-generated projective modules. This gives the following result. Lemma 6.4.
The restrictions of the functors Ψ f and Ψ ∨ f from Theorems 6.2 and 6.3 give strict k -linearmonoidal superfunctors Ψ f : Heis − l ( A ; z , t ) → SEnd k M n ≥ psmod-W fn , Ψ ∨ f : Heis l ( A op ; z , t − ) → SEnd k M n ≥ psmod-W fn , recalling that psmod-W fn denotes the category of finite-generated projective right W fn -supermodules.
7. C ategorical comultiplication
In this section, we construct a quantum analog of the categorical comultiplication of [BSW20a, Th. 5.12];see also [BSW20b, Th. 8.9] and [BSW18, Th. 5.4].Given a diagram D representing a morphism in Heis k ( A ; z , t ) and two generic points P and Q on thisdiagram, we will denote the morphism represented by D − ( D with a negative dot at P and a positive dot at Q )by labelling the points with dots joined by a dotted line oriented from P to Q . We call this oriented dottedline a spear . For example,(7.1) : = − − . It follows that dots and tokens pass through the endpoints of spears. For example,(7.2) a = a , a = a , = , = . We also have(7.3) − = − , = + . Now define the neutral crossing (7.4) : = − − = − − = − = + . Direct computation shows that(7.5) a = a , a = a , = , = , = , = + . (The proof of the fifth relation is lengthy but straightforward.) The relations (7.5) show that the neutralcrossing is an analogue of the intertwining operators that play an important role in the study of (degenerate)a ffi ne Hecke algebras. In addition, the neutral crossing is a quantum analogue of the intertwining operatorsfor a ffi ne wreath product algebras introduced in [Sav20, §4.7].We would like use the neutral crossing as a generator instead of the positive and negative crossings.However, in order to write the positive and negative crossings in terms of the neutral crossings, we needto invert the spear. Let AW ( A ; z ) denote the linear monoidal supercategory obtained from AW ( A ; z ) byadjoining a two-sided inverse to the spear:(7.6) : = (cid:18) (cid:19) − . We call this inverse a dart . Using (7.3), we then automatically have the other dart : = (cid:16) (cid:17) − . It alsofollows from (7.3) that − = − , = + , n = n + X r + s = nr > , s ≥ r s − X r + s = nr ≤ , s < r s , (7.7) n + n = X r + s = nr , s ≥ r s − X r + s = nr , s < r s , n + n = − X r + s = nr , s > r s + X r + s = nr , s ≤ r s , (7.8)for all n ∈ Z . Theorem 7.1.
The strict k -linear monoidal supercategory AW ( A ; z ) is generated by one object ↑ and mor-phisms (7.9) , : ↑ ⊗ ↑→↑ ⊗ ↑ , a : ↑→↑ , a ∈ A , where the crossings are even and the parity of the morphism a is the same as the parity of A. A completeset of relations is given by (2.22) , (7.5) , and (7.6) (where in (7.6) we use the definition (7.1) of the spear).Proof. Let C be the category whose presentation is given in the statement of the theorem. As noted above,we have a strict monoidal superfunctor C → AW ( A ; z ) sending each generating morphism to the morphismof AW ( A ; z ) represented by the same string diagram. Its inverse is given by mapping tokens and dots to thesame tokens and dots, and + ,
7→ − , (7.10)where the left-pointing spear is defined using the first relation in (7.7). Direct computation shows that thisrespects the relations (2.23) and (2.27), hence is well defined. (cid:3) Define the box dumbbell (7.11) : = + , so that = . Straightforward computation shows that the box dumbbell is central in End AW ( A ; z ) ( ↑ ⊗ ):(7.12) a = a , a = a , = , = , = , = , = . We also introduce the box darts (7.13) : = = + , and similarly for the other orientation of the box darts or, more generally, a box dart joining any two genericpoints in a string diagram.The categorical comultiplication to be defined shortly will go from Heis k ( A ; z , t ) to a certain monoidalsupercategory built from Heis l ( A ; z , u ) and Heis m ( A ; z , v ) for l , m ∈ Z and u , v ∈ k × chosen so that(7.14) k = l + m , t = uv . To avoid confusion between these di ff erent categories, the reader will want to view the subsequent materialin this section in color.Let Heis l ( A ; z , u ) ⊙ Heis m ( A ; z , v ) be the symmetric product of Heis l ( A ; z , u ) and Heis m ( A ; z , v ) as definedin [BSW18, § k -linear monoidal category defined by first taking the free product of Heis l ( A ; z , u ) and Heis m ( A ; z , v ), i.e. the strict k -linear monoidal category defined by the disjoint union of UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 31 the given generators and relations of
Heis l ( A ; z , u ) and of Heis m ( A ; z , v ), then adjoining even isomorphisms σ X , Y : X ⊗ Y (cid:27) −→ Y ⊗ X for each pair of objects X ∈ Heis l ( A ; z , u ) and Y ∈ Heis m ( A ; z , v ) subject to the relations σ X ⊗ X , Y = ( σ X , Y ⊗ X ) ◦ (1 X ⊗ σ X , Y ) , σ X , Y ◦ ( f ⊗ Y ) = (1 Y ⊗ f ) ◦ σ X , Y ,σ X , Y ⊗ Y = (1 Y ⊗ σ X , Y ) ◦ ( σ X , Y ⊗ Y ) , σ X , Y ◦ (1 X ⊗ g ) = ( g ⊗ X ) ◦ σ X , Y , for all X , X , X ∈ Heis l ( A ; z , u ), Y , Y , Y ∈ Heis m ( A ; z , v ) and f : X → X , g : Y → Y . Morphismsin Heis l ( A ; z , u ) ⊙ Heis m ( A ; z , v ) are linear combinations of diagrams colored both blue and red. In thesediagrams, as well as the generating morphisms of Heis l ( A ; z , u ) and Heis m ( A ; z , v ), we have the additionaltwo-color crossings , , , which represent the isomorphisms σ X , Y for X ∈ {↑ , ↓} and Y ∈ {↑ , ↓} , and their inverses , , , . We also have the symmetric product AW ( A ; z ) ⊙ AW ( A ; z ), defined in an analogous manner, where we nowonly have the upward two-colored crossings.Let Heis l ( A ; z , u ) ⊙ Heis m ( A ; z , v ) be the strict k -linear monoidal category obtained from Heis l ( A ; z , u ) ⊙ Heis m ( A ; z , v ) by localizing at the spear . This means that we adjoin a two-sided inverse to this morphism,which we denote again by a dart:(7.15) : = (cid:18) (cid:19) − . As with the dumbbell defined in [BSW18, §4,§5], all morphisms whose string diagram is that of an identitymorphism with a spear joining two points of di ff erent colors are also automatically invertible in the localizedcategory. We also denote the inverses of such morphisms by using a dart in place of the spear. For instance: = = − = (cid:18) (cid:19) − . We will use two-colored box dumbbells and box darts, defined in the obvious way.We have the following relations to move darts past one-color crossings: = − , = + , (7.16) = − , = + , (7.17) = − , = + , (7.18) = − , = + . (7.19)The same relations hold if we interchange the colors of the strands. Theorem 7.2.
There is a strict k -linear monoidal superfunctor ∆ : AW ( A ; z ) → Add (cid:16) AW ( A ; z ) ⊙ AW ( A ; z ) (cid:17) such that ↑7→ ↑ ⊕ ↑ , ↓7→ ↓ ⊕ ↓ , and on morphisms (7.20) + , a a + a , + + + . Proof.
It is straightforward to verify that the defining relations from Theorem 7.1 are preserved. For example ∆ (cid:16) (cid:17) ◦ ∆ (cid:16) (cid:17) = + + + = + + + = ∆ ! . (cid:3) Corollary 7.3.
There is a strict k -linear monoidal superfunctor ∆ : AW ( A ; z ) → Add (cid:16) AW ( A ; z ) ⊙ AW ( A ; z ) (cid:17) such that ↑7→ ↑ ⊕ ↑ , ↓7→ ↓ ⊕ ↓ , and on morphisms + , a a + a , a ∈ A , (7.21) + + + + + , + + + − − . (7.22) Proof.
We compose the canonical superfunctor AW ( A ; z ) → AW ( A ; z ) with the superfunctor of Theo-rem 7.2 and compute ∆ ! = ∆ + ! = + + + + + + += + + + + + . The computation of the image of the negative crossing is similar. (cid:3)
Remark 7.4.
The categorical comultiplications of Theorem 7.2 and Corollary 7.3 are not unique. Forexample, we could place the box dumbbell on the other two-colored crossing in (7.20). In addition, when A = k and z = q − q − for some q ∈ k × , we can factor the box dumbbell as = (cid:16) q − q − − (cid:17) (cid:16) q − q − − (cid:17) . Placing the first factor at the bottom of the first two-colored crossing in (7.20) and the second factor at thetop of the second two-colored crossing in (7.20) (and removing the box dumbbell) yields the categoricalcomultiplication of [BSW20b, Th. 8.9].Our goal is to extend the categorical comultiplication of Corollary 7.3 to Frobenius Heisenberg cate-gories. We will need the following morphisms, which we refer to as internal bubbles :: = X r ≥ r + − r − , : = X r ≥ r + − r − (7.7) = (3.28)(4.9) X r > r + − r + − u , (7.23) : = X r ≥ r + − r − , : = X r ≥ r + − r − (7.7) = (3.28)(4.9) X r > r + − r − − v . (7.24)The category Heis l ( A ; z , u ) ⊙ Heis m ( A ; z , v ) has several useful symmetries. Coming from (4.3), we havethe strict k -linear monoidal isomorphism(7.25) Ω l | m : Heis l ( A ; z , u ) ⊙ Heis m ( A ; z , v ) (cid:27) −→ (cid:16) Heis − l ( A ; z , u − ) ⊙ Heis − m ( A ; z , v − ) (cid:17) op , which reflects a diagram in a horizontal plane and multiplies by ( − c + d + ( y ), where c is the number of one-colored crossings, d is the number of left cups and caps (including ones in ( + )-, ( − )-, and internal bubbles),and y is the number of odd tokens. We also have(7.26) flip : Heis l ( A ; z , u ) ⊙ Heis m ( A ; z , v ) (cid:27) −→ Heis l ( A ; z , u ) ⊙ Heis m ( A ; z , v ) UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 33 defined on diagrams by switching the colors blue and red. Finally, the category
Heis l ( A ; z , u ) ⊙ Heis m ( A ; z , v )is strictly pivotal, with duality superfunctor ∗ : Heis l ( A ; z , u ) ⊙ Heis m ( A ; z , v ) (cid:27) −→ (cid:16)(cid:16) Heis l ( A ; z , u ) ⊙ Heis m ( A ; z , v ) (cid:17) op (cid:17) rev defined, as in (4.36), by rotating diagrams through 180 ◦ and multiplying by ( − y ), where y is the numberof odd tokens in the diagram.We denote the duals of the internal bubbles (7.23) and (7.24) by , , , . It follows that internal bubbles slide past all cups and caps.We now state the main result of the section, which is an extension of the categorical comultiplication ofCorollary 7.3 to the quantum Frobenius Heisenberg category. The proof of Theorem 7.5 will be based on aseries of lemmas.
Theorem 7.5.
For k = l + m and t = uv, there is a unique strict k -linear monoidal superfunctor ∆ l | m : Heis k ( A ; z , t ) → Add (cid:16)
Heis l ( A ; z , u ) ⊙ Heis m ( A ; z , v ) (cid:17) such that ↑7→ ↑ ⊕ ↑ , ↓7→ ↓ ⊕ ↓ , and on morphisms by (7.21) , (7.22) and + , + . (7.27) Moreover, we have that ∆ l | m (cid:16) (cid:17) = + , ∆ l | m (cid:16) (cid:17) = − − . (7.28) Also, the following hold for all n ∈ Z and a ∈ A: ∆ l | m (cid:16) + a n (cid:17) = X r ∈ Z + + a r n − r , ∆ l | m (cid:16) + an (cid:17) = − X r ∈ Z + + a r n − r , (7.29) ∆ l | m (cid:16) − a n (cid:17) = − X r ∈ Z − − a r n − r , ∆ l | m (cid:16) − an (cid:17) = X r ∈ Z − − a r n − r . (7.30) Equivalently, in terms of the generating functions (4.12) to (4.15) and their analogues in Heis l ( A ; z , u ) and Heis m ( A ; z , v ) : ∆ l | m (cid:16) wa (cid:17) = z − w wa , ∆ l | m (cid:16) wa (cid:17) = z − w wa , (7.31) ∆ l | m (cid:16) wa (cid:17) = z − w wa , ∆ l | m (cid:16) wa (cid:17) = z − w wa . (7.32) Lemma 7.6.
We have = − ! − . Proof.
We first compute (7.7) = − − = − − − = (7.7) − (5.15) = (5.17) X r ≥ + r − r + X r ≥ + − rr . Thus (7.24) = X r , s ≥ ++ − r − sr + s + X r ≥ + r − r − X r ≥ + r − r + X r ≥ + − rr − X r ≥ + − rr (7.7) = X r , s ≥ ++ − r − sr + s + X r > s ≥ + r − r − ss + X r ≥ s > + − r − ssr = X r , s ∈ Z ++ rsr + s (4.7) = − . (cid:3) Lemma 7.7.
For any n ∈ Z and a ∈ A, we have n a + n a = X r ∈ Z + + a n − r r − X r ∈ Z − − a n − r r . Proof.
We have n a + n a = X r ≥ an + r + − r + X r ≥ + a − rn + r − an − an (7.8) = X r ≥ an + r + − r + X r ≥ + a − rn + r + X r + s = nr , s > asr − X r + s = nr , s ≤ asr (3.28) = (4.8)(4.9) X r ∈ Z + an − r + r − X r ∈ Z − an − r − r . (cid:3) Lemma 7.8.
The following relations hold: = − − X r > s ≥ + − r − sr s , = − − X r ≥ s > + − r − s − r − sr s . Proof.
We have (7.24) = X n ≥ + − nn − (3.17) = (7.16) X n ≥ + − n n − X r > s ≥ + − r − s − r − sr s − − , which is equal to the right-hand side of the first relation. The second relation is proved similarly. (cid:3) Lemma 7.9.
We have = − X r ≥ s ∈ Z ++ s − r − sr . Proof.
Using Lemma 7.8, we have = − − X r ≥ s > + − r − ssr (5.18) = (7.24) − X r , s ≥ ++ − s − rr + s + X r ≥ ++ − rr − − X r ≥ s > + − r − ssr (7.23) = + X r ≥ ++ − rr − X r ≥ ++ − rr − X r ≥ s > + − r − ssr − X r , s ≥ ++ − s − rr + s (7.7) = − X r ≥ s ∈ Z ++ s − r − sr . (cid:3) There is an error in [BSW20b, Lem. 8.3]. Both sums there should be over all b ∈ Z , so that the statement matches the A = k case of the current lemma. UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 35
Lemma 7.10.
We have = − t − + X r , s > p ∈ Z ++ rs − r − p − s + p . Proof.
Using Lemma 7.8, we have = − − X r > s ≥ + s − r − sr (7.16) = − − − X r > s ≥ + s − r − sr (7.23) = (5.17) − + X r > + r − r − u − u + X r > + r − r − u + − rr + X r , s > p ≤ ++ rs − r − p − s + p (7.8) = (7.7) − − u − X r , p > + r − r − pp − u X r > + − rr + X r , s > p ≤ ++ rs − r − p − s + p (5.2) = (7.24) − uv − + X r , p > + − r − ppr − X r , p > + r − r − pp + X r , s > p ≤ ++ rs − r − p − s + p (7.24) = − t − + X r , s > p ≤ s ++ rs − r − p − s + p − X r , p > + r − r − pp − X r , p > + r − r − pp (7.8) = − t − + X r , s > p ∈ Z ++ rs − r − p − s + p . (cid:3) Lemma 7.11.
We have + = X r , s > p ∈ Z ++ r s − r − p − s + p − t . Proof.
First note that, using Lemma 7.9, we have (2.27) = + = + − X r ≥ s ∈ Z ++ s − r − sr . Thus + (7.17) = + + += + − X r ≥ s ∈ Z ++ rs − r − s + + + − X r ≥ s ∈ Z ++ r s − r − s + (7.7) = (7.8) + + X r , s > p ∈ Z ++ r s − r − p − s + p − X r ∈ Z ++ r − r = X r , s > p ∈ Z ++ r s − r − p − s + p − t , where, in the last equality, we used Lemma 7.7, (4.9), and the fact that t = uv . (cid:3) Lemma 7.12.
We have = − − . Proof.
First, a lengthy but straightforward computation using (5.1), (5.2), (7.7), (7.16), and (7.17) showsthat(7.33) a = a + a † − X r ≥ r + − ra † − X r ≥ r + s + − r − sa † . Then, (7.24) = + − rr − (5.19) = (7.33) X r ≥ − X r ≥ s > s + − r − s rs − + X r ≥ + − rr + X r ≥ s > + − r − sr + s (7.24) = + − X r ≥ + − r r − X r ≥ s > s + − r − s rs + X r ≥ + − rr + X r ≥ s > + − r − sr + s (7.7) = (7.8) + . Then, adding a counter-clockwise interior bubble to bottom of the blue strand completes the proof of thelemma. (cid:3)
Proof of Theorem 7.5.
In light of the uniqueness from Lemma 5.4, we can take (7.21), (7.22), (7.27),and (7.28) as the definition of ∆ l | m on generating morphisms, and must check that the images of the relations(2.22), (2.23), (2.27), (2.30), (3.1), and (5.1) to (5.4) are all satisfied in Add (cid:16) Heis l ( A ; z , u ) ⊙ Heis m ( A ; z , v ) (cid:17) .We must also check (7.29) and (7.30). Relations (2.22), (2.23), (2.27), and (2.30) follow from Corollary 7.3and (3.1) is straightforward; so it remains to check the others. UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 37
We first check (7.29) and (7.30). First assume k ≥
0. Consider the bubble + an . When n <
0, bothsides of the second relation in (7.29) are zero by (4.8) and our assumption that k ≥
0. When n =
0, we have ∆ l | m (cid:16) + an (cid:17) (4.8) = − δ , k t − z tr( a )1 = − δ , k u − v − z tr( a )1 (4.8) = (2.8) − + + a n n − r . Now assume n >
0, so that + an = an . Then, using Lemma 7.7, we have ∆ l | m (cid:16) + an (cid:17) = ∆ l | m (cid:16) an (cid:17) = − n a − n a = − X r ∈ Z + + a r n − r . This establishes the right-hand identity in (7.29), hence the right-hand identity in (7.31). Next we compute ∆ l | m (cid:16) wab (cid:17) ◦ (cid:16) w wb ∨ (cid:17) = z ∆ l | m (cid:16) wab (cid:17) ◦ ∆ l | m (cid:16) wb ∨ (cid:17) = ∆ l | m (cid:16) w wa (cid:17) (4.16) = z tr( a )1 , z − w wab w wb ∨ (2.29) = z − w ww wabb ∨ (4.16) = (2.8) z tr( a )1 . Now we observe that there is a unique morphism in f ( a ; w ) ∈ End
Heis l ( A ; z , u ) ⊙ Heis m ( A ; z , v ) ( ) ~ w − (cid:127) such that f ( ab ; w ) (cid:16) w wb ∨ (cid:17) = z tr( a )1 ; see the proof of [BSW20a, Lem. 7.1] for a similar situation. It followsthat f ( a ; w ) = ∆ l | m (cid:16) wa (cid:17) = z − w wa . This establishes the left-hand identity in (7.31), hence the left-identity in (7.29). Next consider the bubble + a n . When n >
0, both sides of the second relation in (7.30) are zero by (4.9), and this relation isstraightforward to verify when n =
0, again using (4.9). Now assume n <
0, so that − an = an .Then, using Lemma 7.7, we have ∆ l | m (cid:16) − an (cid:17) = ∆ l | m (cid:16) an (cid:17) = − n a − n a = X r ∈ Z − − a r n − r . Then we complete the proof of (7.30) as above, using (4.16) and (7.32). It remains to treat the case k < + )- and ( − )-bubblesusing the identities obtained by applying Ω l | m to Lemma 7.7, then gets the clockwise ones using (4.16).Now consider (5.3) and (5.4). The relations involving bubbles follow easily from (7.29) and (7.30). Nextconsider the right curl relation in (5.3), so k ≥
0. Considering the image of this relation under ∆ l | m , we mustshow that(7.34) − − + + = δ k , t − + δ k , t − . This follows from Lemma 7.9, together with its image under flip, noting that the only nonzero term in thesummation on the right-hand side of that identity is the one with r = s =
0, due to (4.8) and the assumptionthat k ≥
0. For the left curl relation in (5.4), we see, after applying ∆ l | m , that we must show, for k ≤
0, that + + + = δ k , t + δ k , t . This follows from (7.34) for
Heis − l ( A ; z , u − ) ⊙ Heis − m ( A ; z , v − ) after applying ∗ ◦ Ω ( − l ) | ( − m ) .Now consider (5.2). By definitions (3.5) and (5.5), we see that ∆ l | m (cid:18) (cid:19) = + + + + + , ∆ l | m (cid:18) (cid:19) = − − − − − − . We can then compute the image under ∆ l | m of both sides of (5.2). Comparing the matrix entries of theresulting morphisms, we are reduced to verifying the identities − − = + t − X r , s > p ∈ Z ++ rs p − r − s − p , − − = t − X r , s > p ∈ Z ++ rs p − r − s − p , − = , − = , plus the images of the first two under the symmetry flip. To prove the first two identities, simplify themby multiplying on the bottom of the left string by a clockwise internal bubble and using Lemma 7.6. Theresulting identities then follow from Lemmas 7.10 and 7.11. For the third identity, composing on the top ofboth sides of the identity by , then multiplying on the top of the blue strands by a counter-clockwise in-ternal bubble, on the bottom of the blue strands by a clockwise internal bubble, and finally using Lemma 7.6and (7.13), we see that we must show that − − = . This follows from Lemma 7.12. The proof of the fourth identity is similar. (cid:3)
8. G eneralized cyclotomic quotients
We now construct some important strict module supercategories over
Heis k ( A ; z , t ) known as generalizedcyclotomic quotients.We fix a supercommutative superalgebra R over our usual ground ring k . Defining the R -superalgebra A R : = R ⊗ k A , we can extend the trace map of A to an R -linear map tr R = id ⊗ tr : A R → R . We can then alsobase change to obtain Heis k ( A R ; z , t ) = R ⊗ k Heis k ( A ; z , t ), which is a strict R -linear monoidal supercategory.Since scalars in R are not necessarily even, we must take extra care with the potential additional signs. Indiagrams for morphisms in Heis k ( A R ; z , t ), we can label tokens by elements of A R . Recall also the generatingfunction formalism introduced in Section 5. Lemma 8.1.
For a polynomial p ( w ) ∈ A R [ w ] , we have p ( x ) = " ( w − x ) − p ( w ) w − , p ( x ) = " ( w − x ) − p ( w ) w − , (8.1) p ( x ) = tz − tr( p (0))1 − t − z − h w p ( w ) i w , p ( x ) = tz − h w p ( w ) i w − t − z − tr( p (0))1 , (8.2) p ( x ) = t − ( w − x ) − wp ( u ) w − , p ( x ) = t ( w − x ) − wp ( w ) w − . (8.3) UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 39
Proof.
By linearity, it su ffi ces to consider the case p ( x ) = ax n , a ∈ A R , n ≥
0. In that case, (8.1) and (8.2)follow immediately from computing the w − coe ffi cient on the right-hand side. For (8.2), we use that a = tz − tr( a )1 + + a and a = + a − t − z − tr( a )1 . by (3.28) and (4.9). Finally, (8.3) follows from (5.15), (5.18), and (8.1) (cid:3) The even part Z ( A R ) ¯0 of the center Z ( A R ) = R ⊗ k Z ( A ) of A R is a commutative algebra. Fix a pair ofmonic polynomials f ( w ) = f w l + f w l − + · · · + f l ∈ Z ( A R ) ¯0 [ w ] , (8.4) g ( w ) = g w m + g w m − + · · · + g m ∈ Z ( A R ) ¯0 [ w ](8.5)such that k = m − l , f l , g m ∈ k × , and t = f l / g m . Define O + ( w ) : = t − z X n ∈ Z O + n w − n : = g ( w ) / f ( w ) = w k + w k − Z ( A R ) ¯0 ~ w − (cid:127) , (8.6) ˜ O + ( w ) : = − tz X n ∈ Z ˜ O + n w − n : = f ( w ) / g ( w ) = w − k + w − k − Z ( A R ) ¯0 ~ w − (cid:127) , (8.7) O − ( w ) : = − tz X n ∈ Z O − n w − n : = t g ( w ) / f ( w ) = + wZ ( A R ) ¯0 ~ w (cid:127) , (8.8) ˜ O − ( w ) : = t − z X n ∈ Z ˜ O − n w − n : = t − f ( w ) / g ( w ) = + wZ ( A R ) ¯0 ~ w (cid:127) , (8.9)cf. (4.12) to (4.15). Lemma 8.2.
The R-linear left tensor ideal I R ( f | g ) of Heis k ( A R ; z , t ) generated by (8.10) f ( x ) and + a n − tr R ( O + n a )1 , − k < n < l , a ∈ A R , is equal to the R-linear left tensor ideal I R ( f | g ) of Heis k ( A R ; z , t ) generated by (8.11) g ( x ) and + an − tr R ( ˜ O + n a )1 , k < n < m , a ∈ A R . Moreover, this ideal contains (8.12) + a n − tr R ( O + n a )1 , + an − tr R ( ˜ O + n a )1 , − a n − tr R ( O − n a )1 , − an − tr R ( ˜ O − n a )1 , for all a ∈ A, n ∈ Z .Proof. The proof is similar to those of [BSW20b, Lem. 9.2] and [BSW20a, Lem. 6.2]. (cid:3)
The generalized cyclotomic quotient of Heis k ( A R ; z , t ) corresponding to f , g is the R -linear category(8.13) H R ( f | g ) : = Heis k ( A R ; z , t ) / I R ( f | g ) . Assume that we are given a factorization t = uv − for u , v ∈ k × such that u = tr( f l ) and t g m = f l , whichimplies that v = tr( g m ). Let(8.14) V ( f ) : = M n ≥ psmod-W fn ( A R ; z ) and V ( g ) ∨ : = M n ≥ psmod-W fn ( A op R ; z ) , viewed as supermodule categories over Heis − l ( A R ; z , u ) and Heis m ( A R ; z , v − ) via the monoidal superfunctors Ψ f and Ψ ∨ g from Lemma 6.4. Let(8.15) V ( f | g ) : = V ( f ) ⊠ R V ( g ) ∨ be their linearized Cartesian product, i.e. the R -linear supercategory with objects that are pairs ( X , Y ) for X ∈ V ( f ), Y ∈ V ( g ) ∨ , and morphismsHom V ( f | g ) (( X , Y ) , ( U , V )) : = Hom V ( f ) ( X , Y ) ⊗ R Hom V ( g ) ∨ ( Y , V ) , with the obvious composition law. There is an equivalence of categories V ( f | g ) → M n , m ≥ psmod- (cid:16) W fn ( A R ; z ) ⊗ R W gm ( A op R ; z ) (cid:17) , hence V ( f | g ) is additive and idempotent complete. Furthermore, V ( f | g ) is a module category over thesymmetric product Heis − l ( A R ; z , u ) ⊙ Heis m ( A R ; z , v − ). We assume for the remainder of this section that the base ring R is a finite-dimensional supercommutativesuperalgebra over an algebraically closed field K ⊇ k ; by “eigenvalue” we mean eigenvalue in this field K .Recall the definition of τ i from (2.16). The element z τ induces a K -linear endomorphism of A R ⊗ R A R by leftmultiplication. Since A R ⊗ R A R is finite dimensional over K , this endomorphism has a minimal polynomial m ( w ) ∈ K [ w ]. Let Γ R be the multiplicative subgroup of K generated by the finite set(8.16) ( µµ − η : µ, η ∈ K , m ( η ) = , µ − µη − = ) . For example, if A = k and z = q − q − for some q ∈ k × then τ = ⊗
1, hence m ( w ) = w − z , and it followsthat Γ R = { q n : n ∈ Z } . Schur’s lemma implies that Z ( A R ) ¯0 acts on any irreducible A R -supermodule L via acentral character χ L : Z ( A R ) ¯0 → K . For p ( w ) ∈ Z ( A R ) ¯0 [ w ], we have χ L ( p ( w )) ∈ K [ w ]. Let Ξ p be the set ofall roots of the polynomials χ L ( p ( w )) for all irreducible A R -supermodules L . Lemma 8.3.
Let V be a finite-dimensional AW ( A R ; z ) -module. All eigenvalues of x on V are of the form λ or γλ for γ ∈ Γ R and an eigenvalue λ of x on V.Proof. Let λ be an eigenvalue of x . Since x , x , and z τ all commute, we can choose a vector y ∈ V in the λ -eigenspace of x that is also an eigenvector of x with some eigenvalue λ and an eigenvector of z τ with some eigenvalue η . First suppose that y is an eigenvector for σ with eigenvalue µ . It followsfrom (2.27) that the element σ ∈ AW n ( A R ; z ) satisfies the equation σ − z τ σ − =
0. Thus we have0 = ( σ − z τ σ − y = ( µ − µη − y . So µ, η satisfy the conditions in the definition of the set (8.16).Furthermore, it follows from (2.27) and (2.30) that σ x = x σ − zx τ . Thus µλ = µλ − ηλ =
0, whichimplies that λ = µµ − η λ .On the other hand, if y is not an eigenvalue for σ , then y and σ y are linearly independent. Thenthe matrix describing the action of x on the subspace with basis { y , σ y } is λ − ηλ λ ! . Hence λ is aneigenvalue for the action of x on V . (cid:3) Corollary 8.4.
The eigenvalues of x , . . . , x n on any W fn ( A R ; z ) -module lie in Γ R Ξ f . Lemma 8.5.
Suppose that f ( w ) , g ( w ) ∈ Z ( A R ) ¯0 [ w ] are generic in the sense that the sets Γ R Ξ f and Γ R Ξ g aredisjoint. Then the categorical action of the symmetric product Heis − l ( A R ; z , u ) ⊙ Heis m ( A R ; z , v − ) on V ( f | g ) defined above extends to an action of the localization Heis − l ( A R ; z , u ) ⊙ Heis m ( A R ; z , v − ) from Section 7.Proof. The proof is analogous to that of [BSW20b, Lem. 9.4], using Corollary 8.4. (cid:3)
When the genericity assumption of Lemma 8.5 is satisfied, the lemma implies that there is a strict k -linear monoidal superfunctor Ψ f ⊙ Ψ ∨ g : Heis − l ( A R ; z , u ) ⊙ Heis m ( A R ; z , v − ) → SEnd R ( V ( f | g )). Composingthis superfunctor with the superfunctor ∆ − l | m from Theorem 7.5 yields a strict R -linear monoidal endofunctor(8.17) Ψ f | g : = (cid:16) Ψ f ⊙ Ψ ∨ g (cid:17) ◦ ∆ − l | m : Heis z ( A R ; z , t ) → SEnd R ( V ( f | g )) . In this way, we make V ( f | g ) into a module supercategory over Heis k ( A R ; z , t ). Lemma 8.6.
Recalling the ( + ) -bubble generating functions from (4.12) and (4.13) , we have Ψ f | g (cid:16) wa (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) (W f ( A R ; z ) , W g ( A op R ; z )) = tr R (cid:16) g ( w ) f ( w ) − a (cid:17) ∈ w m − l R ~ w − (cid:127) , UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 41 Ψ f | g (cid:16) w a (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) (W f ( A R ; z ) , W g ( A op R ; z )) = tr R (cid:16) f ( w ) g ( w ) − a (cid:17) ∈ w l − m R ~ w − (cid:127) . Proof.
The proof is almost identical to that of [BSW20a, Lem. 6.8], using Lemma 8.2 and (7.31). (cid:3)
Theorem 8.7.
Suppose f ( w ) , g ( w ) are generic as in Lemma 8.5. Let Ev :
SEnd R ( V ( f | g )) → V ( f | g ) be theR-linear superfunctor defined by evaluation on (cid:16) W f ( A R ; z ) , W g ( A op R ; z ) (cid:17) ∈ V ( f | g ) . Then Ev ◦ Ψ f | g factorsthrough the generalized cyclotomic quotient H R ( f | g ) to induce an equivalence of Heis k ( A R ; z , t ) -modulesupercategories ψ f | g : Kar( H R ( f | g ) π ) → V ( f | g ) , where Kar( H R ( f | g ) π ) denotes the Karoubi envelope of the Π -envelope H R ( f | g ) π of H R ( f | g ) (see [BE17,Def. 1.10] ).Proof. It is clear that f ( x ) acts as zero on (cid:16) W f ( A R ; z ) , W g ( A op R ; z ) (cid:17) . Together with Lemma 8.6, this impliesthat Ev ◦ Ψ f | g factors through the quotient H R ( f | g ). Since V ( f | g ) is an idempotent complete Π -supercategory(in the sense of [BE17, Def. 3.1]) we obtain the induced R -linear superfunctor ψ f | g from the statement ofthe theorem. To show that ψ f | g is an equivalence, it remains to prove that it is full, faithful and dense. Thisargument is almost identical to the one in the proof of [BSW20b, Th. 9.5]. (cid:3) Remark 8.8.
When g ( w ) =
1, the genericity assumption is vacuous. Hence Theorem 8.7 yields an equiva-lence of categories ψ f | : Kar( H R ( f | π ) → V ( f ). So the generalized cyclotomic quotient H R ( f |
1) is Moritaequivalent to the usual cyclotomic quotient, that is, the locally unital superalgebra L n ≥ W fn ( A R ; z ).9. B asis theorem In this final section we prove a basis theorem for the morphism spaces in
Heis k ( A ; z , t ). Unsurprisingly,our method of proof is a hybrid of the proofs for the quantum Heisenberg category [BSW20b, Th. 10.1] andthe Frobenius Heisenberg category [BSW20a, Th. 7.2]; see also [BSW18, Th. 6.4].Recall from (2.4) the definition of the cocenter C ( A ) of A . For a ∈ A , we let ˙ a denote its image in C ( A ). For a general superalgebra A , the cocenter C ( A ) is merely a vector superspace. However, underour assumption that A is a Frobenius superalgebra, the cocenter can be endowed with the structure of anassociative commutative, but not necessarily unital, superalgebra with product ⋄ ; see [RS20a, Prop. 2.1].We will not use this product explicitly in this paper; however we note that a † = za ⋄ Z ( A ) × C ( A ) → k , ( a , ˙ b ) tr( ab ) , a , b ∈ A . (For a proof, see [BSW20a, Lem 4.1].)Let Sym( A ) denote the symmetric superalgebra generated by the vector superspace C ( A )[ x ], where x here is an even indeterminate. For n ∈ Z and a ∈ A , let e n ( a ) ∈ Sym( A ) denote(9.2) e n ( a ) : = n < , tr( a ) if n = , ˙ ax n − if n > . This defines a parity-preserving linear map e n : A → Sym( A ). Lemma 9.1 ([BSW20a, Lem. 7.1]) . For each n ∈ Z , there is a unique parity-preserving linear map h n : A → Sym( A ) such that (9.3) e ( ac ; − u ) h ( c ∨ b ; u ) = tr( ab ) , for all a , b ∈ A , where we are using the generating functions (9.4) e ( a ; u ) : = X n ≥ e n ( a ) u − n , h ( a ; u ) : = X n ≥ h n ( a ) u − n ∈ Sym( A ) ~ u − (cid:127) . By Lemma 9.1 and (4.16), we have a well-defined homomorphism of superalgebras(9.5) β : Sym( A ) ⊗ Sym( A ) → End
Heis k ( A ; z , t ) ( ) , e n ( a ) ⊗ ( − n − t − z + a n − k , ⊗ e n ( a ) ( − n − tz − a − n , (9.6) h n ( a ) ⊗ tz + an + k , ⊗ h n ( a ) t − z − a − n . (9.7)We will show in Corollary 9.3 that β is an isomorphism. We have that, for X , Y ∈ Heis k ( A ; z , t ), the super-space Hom Heis k ( A ; z , t ) ( X , Y ) is a right Sym( A ) ⊗ Sym( A )-module under the action φθ : = φ ⊗ β ( θ ) , φ ∈ Hom
Heis Ak ( X , Y ) , θ ∈ Sym( A ) ⊗ Sym( A ) . Let X = X n ⊗ · · · ⊗ X and Y = Y m ⊗ · · · ⊗ Y be objects of Heis k ( A ; z , t ) for X i , Y j ∈ {↑ , ↓} . An ( X , Y ) -matching is a bijection between the sets { i : X i = ↑} ⊔ { j : Y j = ↓} and { i : X i = ↓} ⊔ { j : Y j = ↑} . A reduced lift of an ( X , Y )-matching is a string diagram representing a morphism X → Y such that • the endpoints of each string are points which correspond under the given matching; • there are no floating bubbles and no dots or tokens on any string; • there are no self-intersections of strings and no two strings cross each other more than once.For each ( X , Y ), fix a set B ( X , Y ) consisting of a choice of reduced lift for each ( X , Y )-matching. Then let B ◦ ( X , Y ) denote the set of all morphisms that can be obtained from the elements of B ( X , Y ) by adding aninteger number of dots near the terminus of each string and one element of B A to each string. Theorem 9.2.
Assume that the ground ring k is an integral domain and that z , t ∈ k × are arbitrary. Forobjects X , Y ∈ Heis k ( A ; z , t ) , the morphism space Hom
Heis k ( A ; z , t ) ( X , Y ) is a free right Sym( A ) ⊗ Sym( A ) -module with basis B ◦ ( X , Y ) .Proof. It su ffi ces to consider the case k ≤
0, since the result for k ≥ Ω k from(4.3). Let X = X r ⊗ · · · ⊗ X and Y = Y s ⊗ · · · ⊗ Y be two objects, with X i , Y i ∈ {↑ , ↓} .The defining relations and the additional relations proved in Sections 3 to 5 give Reidemeister-typerelations modulo terms with fewer crossing, plus a skein relation and bubble, dot, and token sliding relations.These relations allow diagrams for morphisms in Heis k ( A ; z , t ) to be manipulated in a similar way to the wayoriented tangles are simplified in skein categories, modulo diagrams with fewer crossings. Thus, we have astraightening algorithm to rewrite any diagram representing a morphism X → Y as a linear combination ofthe ones in B ◦ ( X , Y ). Hence B ◦ ( X , Y ) spans Hom Heis k ( A ; z , t ) as a right Sym( A ) ⊗ Sym( A )-module.It remains to prove linear independence of B ◦ ( X , Y ). For this, recalling that k is an integral domain byassumption, we can replace k by the algebraic closure of its field of fractions to assume without loss ofgenerality that k is actually an algebraically closed field. We say φ ∈ B ◦ ( X , Y ) is positive if it only involvesnonnegative powers of dots. It su ffi ces to show that the positive morphisms in B ◦ ( X , Y ) are linearly inde-pendent. Indeed, given any linear relation P Ni = φ i ⊗ β ( θ i ) = φ i ∈ B ◦ ( X , Y ) and coe ffi cients θ i ∈ Sym( A ) ⊗ Sym( A ), we can multiply the termini of the strings by su ffi ciently large positive powers ofdots to reduce to the positive case.We begin with the case X = Y = ↑ ⊗ n . Consider a linear relation P Ns = φ s β ( θ s ) for some positive φ s ∈ B ◦ ( X , Y ) and θ s ∈ Sym( A ) ⊗ Sym( A ). Fix a homogeneous basis a , . . . , a r of C ( A ) with a , . . . , a r ′ evenand a r ′ + , . . . , a r odd. Choose m , m ≥ θ s ∈ Sym( A ) are polynomials in e ( a j ) ⊗ , . . . , e m ( a j ) ⊗ ⊗ e ( a j ) , . . . , ⊗ e m ( a j ), 1 ≤ j ≤ r . Then choose l , m ≥ • k = m − l ; • the multiplicities of dots in all φ s arising in the linear relation are < l ; • m + m < m . UANTUM FROBENIUS HEISENBERG CATEGORIFICATION 43
Let u i , j , 1 ≤ i ≤ m , 1 ≤ j ≤ r , and v i , j , 1 ≤ i ≤ m , 1 ≤ j ≤ r , be indeterminates, with u i , j and v i , j evenfor 1 ≤ j ≤ r ′ and odd for r ′ < j ≤ r . Let K be the algebraic closure of k ( u i , j , v p , j : 1 ≤ i ≤ m , ≤ p ≤ m , ≤ j ≤ r ′ ), and define R to be the free supercommutive K -superalgebra generated by u i , j , 1 ≤ i ≤ m , r ′ < j ≤ r , and v i , j , 1 ≤ i ≤ m , r ′ < j ≤ r . Since the u i , j and v i , j are odd for r ′ < j ≤ r , R is finitedimensional over K . We will now work with algebras / categories that are linear over R , as in Section 8. Let a ∨ , . . . , a ∨ r be a basis of Z ( A ) dual to the basis a , . . . , a r of C ( A ) under the pairing of (9.1). Consider thecyclotomic wreath product algebras W fn ( A R ; z ) and W gn ( A op R ; z ) associated to the polynomials f ( w ) : = w l + t , g ( w ) : = w m + ( u , a ∨ + · · · + u , r a ∨ r ) w m − + · · · + ( u m , a ∨ + · · · + w m , r a ∨ r ) w m − m + ( v m , a ∨ + · · · + v m , r a ∨ r ) w m + · · · + ( v , a ∨ + · · · v , r a ∨ r ) w + ∈ Z ( A R ) ¯0 [ w ] . The roots of f ( w ) are contained in k . Also, since A is defined over the algebraically closed field k , the set Γ R from (8.16) is actually contained in k × . For any irreducible A R -module L , the evaluation of χ L ( g ( w )) at anyelement of Γ R involves at least one of the even u i , j or v i , j with a nonzero coe ffi cient, hence is not containedin k . This shows that f ( w ), g ( w ) is generic in the sense of Lemma 8.5. Hence we can use the superfunctor Ψ f | g of (8.17) to make V ( f | g ) into a Heis k ( A R ; z , t )-module supercategory. Using the canonical k -linearmonoidal superfunctor Heis k ( A ; z , t ) → Heis k ( A R ; z , t ), we can view V ( f | g ) as a module supercategory over Heis k ( A ; z , t ). We now evaluate the relation P Ns = φ s ⊗ β ( θ s ) = f ( A R ; z ) , W g ( A op R ; z )) ∈ V ( f | g ) to obtaina relation in W fn ( A R ; z ). It follows from (6.2) and the choice of l that the images of φ , . . . , φ N in W fn ( A R ; z )are linearly independent over R . Thus the image of β ( θ s ) ∈ R is zero for each s . We wish to deduce thateach θ s =
0. By our choice of m , each θ s is a supercommutative polynomial in e ( a j ) ⊗ , . . . , e m ( a j ) ⊗ ⊗ e ( a j ) , . . . , ⊗ e m ( a j ) for 1 ≤ j ≤ r . So it su ffi ces to show that the images of these elements under β generate a free supercommutative superalgebra. In fact, we claim that these images are the elements u i , j and v i , j , respectively, up to a sign. To see this, note that the low degree terms of the series O ± ( w ) from (8.6)and (8.8) are O + ( w ) = w k + ( u , a ∨ + · · · + u , r a ∨ r ) w k − + · · · + ( u m , a ∨ + · · · + w m , r a ∨ r ) w k − m + · · · ∈ w k K ~ w − (cid:127) , O − ( w ) = + ( v , a ∨ + · · · v , r a ∨ r ) w + · · · + ( v m , a ∨ + · · · + v m , r a ∨ r ) w m + · · · ∈ K ~ w (cid:127) . Thus, the claim follows from (8.6), (8.8), and (9.6) and Lemma 8.2.We have now proved the linear independence when X = Y = ↑ ⊗ n . The linear independence for moregeneral X and Y follows from this as in the proof of [BSW20b, Th. 10.1]. (cid:3) Corollary 9.3.
The map β : End Heis k ( A ; z , t ) ( ) (cid:27) Sym( A ) ⊗ Sym( A ) is an isomorphism of superalgebras. R eferences [BE17] J. Brundan and A. P. Ellis. Monoidal supercategories. Comm. Math. Phys. , 351(3):1045–1089, 2017. arXiv:1603.05928 , doi:10.1007/s00220-017-2850-9 .[Bru17] J. Brundan. Representations of oriented skein categories. 2017. arXiv:1712.08953 .[Bru18] J. Brundan. On the definition of Heisenberg category. Algebr. Comb. , 1(4):523–544, 2018. arXiv:1709.06589 , doi:10.5802/alco.26 .[BSW18] J. Brundan, A. Savage, and B. Webster. The degenerate Heisenberg category and its Grothendieck ring. 2018. arXiv:1812.03255 .[BSW20a] J. Brundan, A. Savage, and B. Webster. Foundations of Frobenius Heisenberg categories. 2020. arXiv:2007.01642 .[BSW20b] J. Brundan, A. Savage, and B. Webster. On the definition of quantum Heisenberg category. Algebra Number Theory ,14(2):275–321, 2020. arXiv:1812.04779 , doi:10.2140/ant.2020.14.275 .[CLL +
18] S. Cautis, A. D. Lauda, A. M. Licata, P. Samuelson, and J. Sussan. The elliptic Hall algebra and thedeformed Khovanov Heisenberg category.
Selecta Math. (N.S.) , 24(5):4041–4103, 2018. arXiv:1609.03506 , doi:10.1007/s00029-018-0429-8 .[CLLS18] S. Cautis, A. D. Lauda, A. M. Licata, and J. Sussan. W-algebras from Heisenberg categories. J. Inst. Math. Jussieu ,17(5):981–1017, 2018. arXiv:1501.00589 , doi:10.1017/S1474748016000189 . [CPd14] M. Chlouveraki and L. Poulain d’Andecy. Representation theory of the Yokonuma–Hecke algebra. Adv. Math. ,259:134–172, 2014. arXiv:1302.6225 , doi:10.1016/j.aim.2014.03.017 .[Kho14] M. Khovanov. Heisenberg algebra and a graphical calculus. Fund. Math. , 225(1):169–210, 2014. arXiv:1009.3295 , doi:10.4064/fm225-1-8 .[LS13] A. Licata and A. Savage. Hecke algebras, finite general linear groups, and Heisenberg categorification. Quantum Topol. ,4(2):125–185, 2013. arXiv:1101.0420 , doi:10.4171/QT/37 .[MS18] M. Mackaay and A. Savage. Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification. J.Algebra , 505:150–193, 2018. arXiv:1705.03066 , doi:10.1016/j.jalgebra.2018.03.004 .[RS17] D. Rosso and A. Savage. A general approach to Heisenberg categorification via wreath product algebras. Math. Z. ,286(1-2):603–655, 2017. arXiv:1507.06298 , doi:10.1007/s00209-016-1776-9 .[RS20a] M. Reeks and A. Savage. Frobenius W-algebras and traces of Frobenius Heisenberg categories. 2020. arXiv:2007.02732 .[RS20b] D. Rosso and A. Savage. Quantum a ffi ne wreath algebras. Doc. Math. , 25:425–456, 2020. arXiv:1902.00143 , doi:10.25537/dm.2020v25.425-456 .[Sav19] A. Savage. Frobenius Heisenberg categorification. Algebr. Comb. , 2(5):937–967, 2019. arXiv:1802.01626 , doi:10.5802/alco.73 .[Sav20] A. Savage. A ffi ne wreath product algebras. Int. Math. Res. Not. IMRN , (10):2977–3041, 2020. arXiv:1709.02998 , doi:10.1093/imrn/rny092 .(J.B.) D epartment of M athematics , U niversity of O regon , E ugene , OR, USA E-mail address : [email protected] (A.S.) D epartment of M athematics and S tatistics , U niversity of O ttawa , O ttawa , ON, C anada URL : alistairsavage.ca, ORCiD : orcid.org/0000-0002-2859-0239 E-mail address : [email protected] (B.W.) D epartment of P ure M athematics , U niversity of W aterloo & P erimeter I nstitute for T heoretical P hysics , W aterloo ,ON, C anada E-mail address ::