Modified traces on Deligne’s category \(\underline{\mathrm{Re}}\hspace{0.7pt}\mathrm{p} (S_{t})\)

1.1 Deligne’s category

Let F denote a field of characteristic zero and let t∈F. Recently Deligne gave a definition of a category, \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{t})\), which interpolates among the representations over F of the various symmetric groups [6]. Somewhat more precisely: when t is not a nonnegative integer, the category \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{t})\) is semisimple and when t is a nonnegative integer, then a natural quotient of \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{t})\) is equivalent to the category of representations over F of the symmetric group on t letters.

Axiomatizing Deligne’s construction, Knop gave a number of additional examples of interpolating categories, including representations of finite general linear groups and of wreath products [16, 17]. More recently Etingof defined interpolating categories in other settings which include degenerate affine Hecke algebras and rational Cherednik algebras [8]. Most recently Mathew provided an algebro-geometric setup for studying these categories when the parameter is generic [18]. Comes and Wilson study Deligne’s analogously defined \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(GL_{t})\) and use it to completely describe the indecomposable summands of tensor products of the natural module and its dual for general linear supergroups [5].

We will be interested in Deligne’s \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{t})\). Besides motivating the new direction of research in representation theory discussed above, it is an object of study in its own right. Comes and Ostrik completely describe the indecomposable objects and blocks in \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{t})\) in [4], and classify tensor ideals along the way to proving a conjecture of Deligne in [3]. Recently, Del Padrone used Deligne’s category to answer several questions which arose out of the work of Kahn in studying the rationality of certain zeta functions [7].

1.2 Modified traces in ribbon categories

A reoccurring difficulty in this approach are the objects with categorical dimension zero. These objects necessarily give trivial topological invariants. Tackling this problem Geer and Patureau-Mirand defined modified trace and dimension functions for typical representations of quantum groups associated to Lie superalgebras [11]. With Turaev they generalized this construction to include, for example, the quantum group for \(\mathfrak{sl}(2)\) at a root of unity [13]. Along with various coauthors, they have gone on to vastly generalize their construction and use it to obtain new topological invariants. In particular, they have shown how to use modified traces to give generalized Kashaev and Turaev-Viro-type 3-manifold invariants, to show that these invariants coincide, and that they extend to a relative Homotopy Quantum Field Theory. Especially intriguing, they also show how to use this theory to generalize the quantum dilogarithmic invariant of links appearing in the well-known Volume Conjecture. See [10, 12] and references therein.

On the algebra side of the picture the second author worked jointly with Geer and Patureau-Mirand to provide a ribbon categorical framework for modified trace and dimension functions and considered a number of examples coming from representation theory [9]. They showed that these functions generalize well-known results from representation theory as well as giving entirely new insights. For example, this point of view leads to a natural generalization of a conjecture by Kac and Wakimoto for complex Lie superalgebras. Recently Serganova proved the original Kac–Wakimoto conjecture for the basic classical Lie superalgebras and the generalized Kac–Wakimoto conjecture for \(\mathfrak{gl}(m|n)\) [19]. The generalized Kac–Wakimoto conjecture is in turn used to compute the complexity of the finite dimensional simple supermodules for \(\mathfrak{gl}(m|n)\) by Boe, Nakano, and the second author [2].

Despite the success of this program, it remains mysterious when these modified dimension functions exist. In [9] the authors provide examples which show that rather elementary categories in representation theory (e.g. certain representations of the Lie algebra \(\mathfrak{sl}_{2}(k)\) over field of characteristic p) can fail to have modified dimensions. Motivated by this gap in our understanding and by the aforementioned applications within low-dimensional topology and representation theory, in this paper we investigate modified trace and dimension functions within Deligne’s category \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{t})\).

1.3 The existence of modified traces

Our main result (Theorem 5.9) proves that when t is a nonnegative integer the only nontrivial ideal in \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{t})\) always admits a modified trace. It is worth noting that Deligne’s category, which is only abelian when t is not a nonnegative integer, provides the first example of a nonabelian ribbon category which admits modified traces.

1.4 A graded variant

In Sect. 6.1 we define a graded variant of Deligne’s category, \(\mathrm{g}\underline{\operatorname{rRe}}\hspace{0.7pt}\mathrm{p}(S_{t})_{q}\), and prove that there is a “degrading” functor \(\mathcal {F}: \mathrm{g}\underline{\operatorname{rRe}}\hspace{0.7pt}\mathrm{p}(S_{t})_{q} \to \underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{t})\). Using this functor we can lift the modified trace functions on \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{t})\) to \(\mathrm{g}\underline{\operatorname{rRe}}\hspace{0.7pt}\mathrm{p}(S_{t})_{q}\). In particular, the graded category has a nonsymmetric braiding and the modified trace function defines a nontrivial knot invariant. In this way we can use Deligne’s category to recover the well-known invariant of framed knots known as the writhe.

1.5 Further questions

The results of this paper raise a number of intriguing questions. As mentioned above, Deligne’s construction naturally generalizes to a wide variety of settings within representation theory. We expect that modified traces should exist for many of these other categories and it would be interesting to investigate this question. Deligne’s category \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{t})\) has a relatively elementary structure (for example, it has a single nontrivial tensor ideal) and we expect that studying modified traces in these other settings will be significantly more involved.

2.1 Ribbon categories

For notation and the general setup of ribbon categories our references are [20] and [14]. A tensor category \(\mathcal {C}\) is a category equipped with a covariant bifunctor

$$\otimes :\mathcal {C}\times \mathcal {C}\rightarrow \mathcal {C}$$

called the tensor product, a unit object Open image in new window , an associativity constraint, and left and right unit constraints such that the Triangle and Pentagon Axioms hold (see [14, XI.2]). In particular, for any V in \(\mathcal {C}\), Open image in new window and Open image in new window are canonically isomorphic to V.

A tensor category \(\mathcal {C}\) has duality if for each object V in \(\mathcal {C}\) there exits an object V ^∗ and coevaluation and evaluation morphisms¹

Open image in new window

satisfying relations [14, XIV.2 (2.1)].

A ribbon category is a braided tensor category with duality and twists. A fundamental feature of ribbon categories is the fact that morphisms in the category can be represented diagrammatically and that isotopic diagrams correspond to equal morphisms. For the sake of brevity, we do not give the graphical calculus here but encourage the interested reader to refer to [14].

In a ribbon category it is convenient to also define the morphisms

Open image in new window

which are given by

$$\operatorname {coev}'_V =({\operatorname {Id}_{V^*}}\otimes {\theta_V}) \circ c_{V,V^*} \circ \operatorname {coev}_{V}\quad \hbox{and}\quad \operatorname {ev}'_V = \operatorname {ev}_{V} \circ c_{V,V^*} \circ (\theta_V\otimes \operatorname {Id}_{V^*}).$$

Finally, the ground ring of a ribbon category \(\mathcal {C}\) is

Open image in new window

We assume K is a field, that the category is K-linear, and that the tensor product is bilinear. Later references to linearity will always be with respect to K. Ultimately the ground ring will be a fixed field F of characteristic zero and the categories in question will be F-linear.

2.2 Ideals in \(\mathcal {C}\)

There are two closely related notions of an ideal within a ribbon category. The first we discuss is used in [9] and defined via objects. We discuss the second notion in Sect. 2.4. Note that here and elsewhere if f and g are morphisms, then we write fg for the composition f∘g.

Trivially, if \(\mathcal {I}\) consists of just the zero object or \(\mathcal {I}= \mathcal {C}\), then \(\mathcal {I}\) is an ideal of the category. We say an ideal \(\mathcal {I}\) is a proper ideal if it contains a nonzero object and is not all of \(\mathcal {C}\).

2.3 Traces in ribbon categories

For an object J, let \(\mathcal {I}_{J}\) denote the ideal whose objects are all objects which are retracts of J⊗X for some X in \(\mathcal {C}\).

2.4 Tensor ideals in a ribbon category

A somewhat different notion of ideal is used in [3, 4]. As we need both, we define it here and discuss the relationship with the earlier definition. To distinguish the two we call these tensor ideals. They are defined via morphisms as follows.

Trivially, for every pair of objects X and Y one can take J(X,Y)=0 and obtain a tensor ideal; similarly, for every pair of objects one can take \(J(X,Y) = \operatorname {Hom}_{\mathcal {C}}(X,Y)\). A tensor ideal J is called proper if J(X,Y) is a proper nonzero subspace of \(\operatorname {Hom}_{\mathcal {C}}(X,Y)\) for at least one pair of objects X and Y in \(\mathcal {C}\).

2.5 Relating the two notions of ideals

If \(\mathcal {I}\) is an ideal of \(\mathcal {C}\) in the sense of Definition 2.1, then one can define subspaces

Open image in new window

Then J forms a tensor ideal and we write \(J(\mathcal {I})\) for this tensor ideal.

Conversely, if J is a tensor ideal, then one can define \(\mathcal {I}\) to be the full subcategory consisting of all objects V in \(\mathcal {C}\) such that \(\operatorname {Id}_{V} \in J(V,V)\). This is an ideal of \(\mathcal {C}\) and we write \(\mathcal {I}(J)\) for this ideal.

In the following lemma we record the basic properties relating these two notions of an ideal. The proofs are elementary arguments using the definitions and previous parts of the lemma.

2.6 Negligibles

When t∈ℤ_≥0, \(\mathcal {N}\) is a proper tensor ideal of Deligne’s category \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{t})\). The quotient of \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{t})\) by this tensor ideal is equivalent to the category of finite dimensional representations over F of the symmetric group S _t (see [4, Theorem 3.24]). It is in this sense that Deligne’s category interpolates among the representations of the various symmetric groups.

4.1 Defining the trace function

Consider the indecomposable object L(□) in the category \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{0})\). We will define a trace on the ideal \(\mathcal {I}_{L(\Box)}\) by verifying by explicit computation that L(□) admits an ambidextrous trace.

In order to define such a trace, we study the endomorphisms of the object L(□)⊗L(□). By [4, Proposition 6.1] we can identify \(\operatorname {End}(L(\Box))\) with the partition algebra FP ₁(0). Hence \(\operatorname {End}(L(\Box)\otimes L(\Box))=FP_{2}(0)\). Consider the following table:

Open image in new window

From the table above we have the following: A linear map \(\operatorname {\mathsf {t}}:\operatorname {End}(L(\Box))\to F\) satisfies

Open image in new window

for all partition diagrams π∈FP ₂(0) if and only if \(\operatorname {\mathsf {t}}\) is constant on the two partition diagrams in FP ₁(0). Therefore there is a unique ambidextrous trace function for L(□) up to a constant multiple. Hence by Theorem 2.4 there is a unique trace on \(\mathcal {I}_{L(\Box)}\) up to constant multiple. We normalize by setting \(\operatorname {\mathsf {t}}\) to be the trace function with \(\operatorname {\mathsf {t}}(\operatorname {Id}_{L(\Box)})=1\). By Theorem 2.4, the map \(\operatorname {\mathsf {t}}\) uniquely extends to a trace on \(\mathcal {I}_{L(\Box)}\). In summary, we have the following result.

We note that by the classification of tensor ideals in \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{0})\) in [3] there is a unique proper tensor ideal in the category and it contains all indecomposable objects except L(∅). This is the tensor ideal \(\mathcal {N}\) of negligible morphisms. Using Lemma 2.6 it follows that there is a unique proper ideal. That is, \(\mathcal {I}_{L(\Box )}\) is the unique proper ideal and it equals \(\mathcal {I}(\mathcal {N})\).

4.2 Dimensions in the non-semisimple block

By [4, Theorem 6.4] the category \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{0})\) has a unique nontrivial block. The indecomposables in this block can be described explicitly and are denoted by L _n =L((1 ⁿ )) for n∈ℤ_≥0. In this section we compute the modified dimensions \(\operatorname {\mathsf {d}}_{\operatorname {\mathsf {t}}}( L_{n})\) for all n>0.

Recall that any indecomposable object in \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{0})\) is of the form ([n],e) for some primitive idempotent e∈FP _n (0) (see [4, Proposition 2.20(2)]). Moreover, A:=([n],e) is the direct summand of \(B:=([n],\operatorname {Id}_{n})=L(\Box)\otimes ([n-1], \operatorname {Id}_{n-1})\) where the inclusion map A→B and the projection map B→A are both given by e. Hence

Open image in new window

Let S _n be the symmetric group on n letters whose elements are viewed as endomorphisms in Deligne’s category [4, Remark 2.14], and let \(\operatorname {sgn}: S_{n} \to \{\pm 1 \}\) be the usual sign function. Recall from [4, Proposition 6.1] that L _n ≅([n],s _n ) where

$$ s_n=\frac{1}{n!}\sum_{\sigma\in S_n}\operatorname {sgn}(\sigma)\sigma.$$

(4.5)

In particular,

Open image in new window

For example,

Open image in new window

so that

Open image in new window

More generally, given σ∈S _n ,

Open image in new window

Hence

$$\operatorname {\mathsf {d}}_{\operatorname {\mathsf {t}}}(L_n)=(-1)^{n+1}\frac{(\mathrm{number~of} \ n\hbox{-}\mathrm{cycles~in} \ S_n)}{n!}=\frac{(-1)^{n+1}}{n}.$$

5.1 Notation

Given σ∈S _n and I⊂{1,…,n}, let σ _I denote the partition diagram obtained from the partition diagram for σ by removing all edges adjacent to top vertices labeled by elements of I. Also, for i∈{1,…,n} we write σ _i =σ _{i}. For example, if

Open image in new window

Given n∈ℕ, write \(x_{n}=(1_{S_{n}})_{n}\) where \(1_{S_{n}}\) is the identity permutation in S _n . Finally, let \(S_{n}^{-}:=\{\sigma_{i}~|~\sigma\in S_{n}, 1\leq i\leq n\}\).

5.2 The object M _n

Let \(M_{n}:=([n], s_{n})\in \underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{t})\) where s _n is as in (4.5).

In the remainder of this section we examine the endomorphisms of M _n .

5.3 An ambidextrous trace on M _n

For the remainder of this section assume n>0. Given a partition diagram π∈P _2n , write π _L (respectively, π _R ) for the partition diagram in P _n obtained by restricting the partition π to the vertices {1,1′,…,n,n′} (respectively, {n+1,(n+1)′,…,2n,(2n)′}). For example,

Open image in new window

Finally, let \(\Theta_{1}, \Theta_{2}: P_{2n}\to \operatorname {End}_{\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{t})}(M_{n})\) be the maps given by

Open image in new window

The following lemma is the first of three lemmas concerning Θ₁ and Θ₂ which will be used to show that \(\operatorname {\mathsf {t}}_{n}\) is an ambidextrous trace.

Proof

If \(\pi_{L}\notin S_{n}\sqcup S_{n}^{-}\) then (arguing as in the proof of Proposition 5.3) there exists a transposition τ∈S _n with \(\pi(\tau\otimes \operatorname {Id}_{n})=\pi\) or \((\tau\otimes \operatorname {Id}_{n})\pi=\pi\). Thus

Open image in new window

In either case, by Proposition 5.3(1), Θ₁(π)=−Θ₁(π) and hence Θ₁(π)=0 as the characteristic of F is not 2. Moreover,

Open image in new window

Using Proposition 5.3(1) again, we have Θ₂(π)=−Θ₂(π) so that Θ₂(π)=0. The proof when \(\pi_{R}\notin S_{n}\sqcup S_{n}^{-}\) is similar. □

Now for the second lemma concerning Θ₁ and Θ₂. In this lemma, the symbol ≥ refers to the partial order on partition diagrams found in [4, Sect. 2.1].

Proof

Suppose \(\pi_{L}\in S_{n}^{-}\) so that π _L =σ _i for some σ∈S _n and i∈{1,…,n}. Let π′∈P _2n be the partition diagram obtained from π by adding an edge between the vertices labeled i and σ(i)′. Then π′≥π which implies \(\pi'_{R}\geq \pi_{R}\). Also, \(\pi'_{L}=\sigma\in S_{n}\). Moreover, given τ∈S _n , the connected components of the graph

Open image in new window

(5.6)

are all cycles. Hence, as the graph of

Open image in new window

(5.7)

is obtained from (5.6) by deleting one edge, the partitions of {1,1′,…,n,n′} corresponding to the connected components of (5.6) and (5.7) are equal. Therefore,

Open image in new window

for every τ∈S _n . Hence

Open image in new window

which implies Θ₂(π′)=Θ₂(π).

Now, fix ρ∈S _n and let \(\mu_{\rho}, \mu_{\rho}'\in P_{n}\), ℓ(ρ),ℓ(ρ)′∈ℤ_≥0 be such that

Open image in new window

Notice that ℓ(ρ) (respectively, ℓ(ρ)′) is the number of connected components of the partition diagram \(\pi(\operatorname {Id}_{n}\otimes \rho)\) (respectively, \(\pi'(\operatorname {Id}_{n}\otimes \rho)\)) which only contain vertices labeled by integers greater than n. The connected components of π and π′ (and hence of \(\pi(\operatorname {Id}_{n}\otimes \rho)\) and \(\pi'(\operatorname {Id}_{n}\otimes \rho)\)) which only contain vertices labeled by integers greater than n are identical. Hence ℓ(ρ)=ℓ(ρ)′. Also, it is easy to see that μ _ρ ≥π _L . Thus, as \(\pi_{L}\in S_{n}^{-}\), there are three cases: (i) μ _ρ =π _L , (ii) \(\mu_{\rho}=\pi'_{L}\), (iii) \(\mu_{\rho}\notin S_{n}\sqcup S_{n}^{-}\). Next, we show that \(\operatorname {\mathsf {t}}_{n}(s_{n}\mu_{\rho}s_{n})=\operatorname {\mathsf {t}}_{n}(s_{n}\mu'_{\rho}s_{n})\) in each of the three cases above:

(i) If μ _ρ =π _L =σ _i then it is easy to see that \(\mu'_{\rho}=\pi'_{L}=\sigma\). Hence, by Proposition 5.3(1)&(3) and the definition of \(\operatorname {\mathsf {t}}_{n}\), \(\operatorname {\mathsf {t}}_{n}(s_{n}\mu_{\rho}s_{n})=\operatorname {sgn}(\sigma)=\operatorname {\mathsf {t}}_{n}(s_{n}\mu'_{\rho}s_{n})\).

(ii) If \(\mu_{\rho}=\pi'_{L}\) then \(\mu'_{\rho}=\pi_{L}'\) too. Hence \(\operatorname {\mathsf {t}}_{n}(s_{n}\mu_{\rho}s_{n})=\operatorname {\mathsf {t}}_{n}(s_{n}\mu'_{\rho}s_{n})\).

(iii) If \(\mu_{\rho}\notin S_{n}\sqcup S_{n}^{-}\) then \(\mu'_{\rho}\notin S_{n}\sqcup S_{n}^{-}\) too. Therefore, by Proposition 5.3(2), \(\operatorname {\mathsf {t}}_{n}(s_{n}\mu_{\rho}s_{n})=0=\operatorname {\mathsf {t}}_{n}(s_{n}\mu'_{\rho}s_{n})\).

As ρ was an arbitrary element of S _n , we have

Open image in new window

The statement of the lemma with \(\pi_{L}\in S_{n}^{-}\) follows. The proof when \(\pi_{R}\in S_{n}^{-}\) is similar. □

Before proving the third and final lemma concerning Θ₁ and Θ₂ we need to introduce a bit more notation. Suppose π∈P _2n is such that π _L ,π _R ∈S _n . Let I=I _π ⊂{1,…,n} denote the set of all i∈{1,…,n} which correspond to vertices in π whose parts are of size two. Now let π _L−R =σ _I where σ∈S _n is any permutation with σ(i)=j whenever the top vertices labeled by i and n+j are in the same part of π. For example,

Open image in new window

The following proposition concerning π _L−R will be used in the proof of the final lemma concerning Θ₁ and Θ₂.

Proof

Write I=I _π as above and let σ∈S _n be any permutation with π _L−R =σ _I . Let μ,μ′∈P _2n be the following partition diagrams

Open image in new window

It is easy to check that \(\mu_{L}=\pi_{R}=\mu'_{L}\), \(\mu_{R}=\pi_{L}=\mu'_{R}\), and \(\mu_{L-R}=(\sigma^{-1})_{J}=\mu'_{L-R}\) where J={σ(i) | i∈I}. Hence, by Proposition 5.7, μ=μ′. Also, by Proposition 5.3(1), Θ₁(π)=Θ₁(μ). Therefore Θ₁(π)=Θ₁(μ)=Θ₁(μ′)=Θ₂(π). □

Now we prove the main result of this section.

Proof

Given \(h\in \operatorname {End}_{\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{t})}(M_{n}\otimes M_{n})=(s_{n}\otimes s_{n})FP_{2n}(t)(s_{n}\otimes s_{n})\) we can write

$$h=\sum_{\pi\in P_{2n}}h_\pi(s_n\otimes s_n)\pi(s_n\otimes s_n)$$

for some h _π ∈F. Hence

Open image in new window

(where the right equality uses the fact that s _n is an idempotent). A similar calculation shows \(\operatorname {tr}_{L}(h)=\sum_{\pi\in P_{2n}}h_{\pi}\Theta_{2}(\pi)\). Thus, it suffices to show \(\operatorname {\mathsf {t}}_{n}(\Theta_{1}(\pi))=\operatorname {\mathsf {t}}_{n}(\Theta_{2}(\pi))\) for all π∈P _2n .

If either π _L or π _R is not in \(S_{n}\sqcup S_{n}^{-}\) then the result follows from Lemma 5.5. Hence we can assume \(\pi_{L}, \pi_{R}\in S_{n}\sqcup S_{n}^{-}\). If \(\pi_{L}\in S_{n}^{-}\) then by Lemma 5.6 there exists π′∈P _2n with \(\pi'_{L}\in S_{n}\) and \(\operatorname {\mathsf {t}}_{n}(\Theta_{i}(\pi'))=\operatorname {\mathsf {t}}_{n}(\Theta_{i}(\pi))\) for i=1,2. If \(\pi'_{R}\notin S_{n}\sqcup S_{n}^{-}\) then by Lemma 5.5 we have \(\operatorname {\mathsf {t}}_{n}(\Theta_{i}(\pi))=\operatorname {\mathsf {t}}_{n}(\Theta_{i}(\pi'))=0\) for i=1,2; hence we can assume \(\pi'_{R}\in S_{n}\sqcup S_{n}^{-}\). If \(\pi'_{R}\in S_{n}^{-}\) then by Lemma 5.6 there exists π″∈P _2n with \(\pi''_{R}\in S_{n}\), \(\pi''_{L}\geq \pi'_{L}\), and \(\operatorname {\mathsf {t}}_{n}(\Theta_{i}(\pi''))=\operatorname {\mathsf {t}}_{n}(\Theta_{i}(\pi'))\) for i=1,2. If \(\pi''_{L}\notin S_{n}\sqcup S_{n}^{-}\) then by Lemma 5.5 we have \(\operatorname {\mathsf {t}}_{n}(\Theta_{i}(\pi))=\operatorname {\mathsf {t}}_{n}(\Theta_{i}(\pi''))=0\) for i=1,2; hence we can assume \(\pi''_{L}\in S_{n}\sqcup S_{n}^{-}\). Also, since \(\pi''_{L}\geq \pi'_{L}\) and \(\pi'_{L}\in S_{n}\) it follows that \(\pi''_{L}\notin S_{n}^{-}\). Thus, we can assume \(\pi''_{L}\in S_{n}\). In this case, by Lemma 5.8, \(\operatorname {\mathsf {t}}_{n}(\Theta_{1}(\pi))=\operatorname {\mathsf {t}}_{n}(\Theta_{1}(\pi''))=\operatorname {\mathsf {t}}_{n}(\Theta_{2}(\pi''))=\operatorname {\mathsf {t}}_{n}(\Theta_{2}(\pi))\). □

6.1 A graded variant

Fix t,q∈F with q≠0. We then let \(\mathrm{g}\underline{\operatorname{rRe}}\hspace{0.7pt}\mathrm{p}_{0}(S_{t})=\mathrm{g}\underline{\operatorname{rRe}}\hspace{0.7pt}\mathrm{p}_{0}(S_{t})_{q}\) be the category defined as follows. The objects are all pairs [a,b] for all a,b∈ℤ_≥0. We put a ℤ-grading on the objects of the category by setting the degree of [a,b] to be a−b.

The morphisms are given by

Open image in new window

The composition of morphisms is given by the same vertical concatenation of diagrams rule as in the definition of Deligne’s category \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}_{0}(S_{t})\). By definition the morphisms preserve the ℤ-grading.

The unit object is then Open image in new window and the unit constraints [0,0]⊗[a,b]→[a,b] and [a,b]⊗[0,0]→[a,b] are given by the identity.

The dual of the object [a,b] is given by

$$[a,b]^{*}=[b,a].$$

The evaluation morphism Open image in new window is given by a diagram in FP _2a+2b,0 which gives the evaluation morphism Open image in new window in \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}_{0}(S_{t})\). Similarly, the coevaluation morphism Open image in new window is given by the coevaluation in \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}_{0}(S_{t})\).

A direct verification of the axioms shows that the above tensor product, unit, duality, braiding, and twists make \(\mathrm{g}\underline{\operatorname{rRe}}\hspace{0.7pt}\mathrm{p}_{0}(S_{t})\) into a ribbon category.

Write \(\mathrm{g}\underline{\operatorname{rRe}}\hspace{0.7pt}\mathrm{p}(S_{t})\) for the Karoubian envelope of the additive envelope of \(\mathrm{g}\underline{\operatorname{rRe}}\hspace{0.7pt}\mathrm{p}_{0}(S_{t})\). The ribbon category structure on \(\mathrm{g}\underline{\operatorname{rRe}}\hspace{0.7pt}\mathrm{p}_{0}(S_{t})\) defines a ribbon category structure on \(\mathrm{g}\underline{\operatorname{rRe}}\hspace{0.7pt}\mathrm{p}(S_{t})\) just as it does going from \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}_{0}(S_{t})\) to \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{t})\). We also note that using the definition of the additive and Karoubian envelopes we see that the category \(\mathrm{g}\underline{\operatorname{rRe}}\hspace{0.7pt}\mathrm{p}(S_{t})\) inherits a ℤ-grading and all morphisms are grading preserving.

We have the following “degrading” functor between the graded and ungraded versions of Deligne’s category.

Remark 6.2

It is straightforward to verify that \(\mathcal {F}\) is a tensor functor (i.e. \(\mathcal {F}(X \otimes Y) =\mathcal {F}(X) \otimes \mathcal {F}(Y)\) for all objects X and Y, Open image in new window , and preserves the associativity and unit constraints) and that it preserves duals (i.e. \(\mathcal {F}(X^{*})=\mathcal {F}(X)^{*}\) for all objects X and preserves the evaluation and coevaluation morphisms). If V,W are homogeneous objects in \(\mathrm{g}\underline{\operatorname{rRe}}\hspace{0.7pt}\mathrm{p}(S_{t})_{q}\) with V of degree r∈ℤ and W of degree s∈ℤ, then

$$\mathcal {F}(c_{V,W}) = q^{r s} c_{\mathcal {F}(V), \mathcal {F}(W)},$$

where \(c_{\mathcal {F}(V), \mathcal {F}(W)}\) is the braiding in the category \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{t})\). Similarly,

$$\mathcal {F}(\theta_{V}) = q^{r^{2}}\theta_{\mathcal {F}(V)},$$

where \(\theta_{\mathcal {F}(V)}\) is the twist in the category \(\underline{\operatorname{Re}}\hspace{0.7pt}\mathrm{p}(S_{t})\).

Thus the functor \(\mathcal {F}\) preserves braidings and twists if and only if q=1 but, as we will soon see, it is close enough for our purposes.