1 INTRODUCTION
In the seminal paper [14], P. Deligne constructed symmetric tensor categories , where can be any complex number, which interpolate the categories of representations over the symmetric groups , . These categories, and their relatives for other series of groups, have proven interesting to the study of symmetric tensor categories, as well as to the study of stability phenomena in representation theory, for example, see [3, 8, 9, 19-21]. The category can be constructed using the combinatorics of partitions (see Section 2.2) and has a universal property with respect to Frobenius algebra objects of dimension in symmetric monoidal categories.
In addition to giving this combinatorial definition and universal property Deligne observed that for transcendental can be realized as an ultraproduct, a sort of model-theoretic limit, of the categories , as grows to infinity. This understanding was extended to all values of by the second listed author in [26, Theorem 1.1] by viewing as a limit of the categories , varying the rank as well as the characteristic . We recall the main ideas of this approach in Sections 2.3, 2.4.
In this paper, we apply these ultrafilter techniques to prove several results on the monoidal centers of Deligne's categories. The monoidal or Drinfeld center [29, 33] of a monoidal category is a universal construction of a braided monoidal category with a forgetful functor and instrumental in the construction of quantum groups and solutions to the quantum Yang–Baxter equation, see, for example, [2, 30, 32].
In [24], the first and third listed authors started the investigation of , showed that this is a ribbon category, and obtained invariants of framed links as an application. It was shown that the braided categories interpolate the braided categories in the sense that is the semisimplification of for . In the present paper, we answer several open structural questions about the categories including a classification of their indecomposable objects and computation of the (associated graded) Grothendieck rings.
We start the paper by providing some required background results on in Section 2. Many of the results stated that there are known to experts but are sometimes not available in the literature. Thus, we have included proofs when appropriate.
A key statement we prove in Section 3.1 displays as a model-theoretic limit in rank and characteristic, similar to . For this, we define a bi-filtration on the center, with the filtration layer being the full subcategory on objects which are in the preimage of under the forgetful functor . Here, consists of objects that are direct summands of objects of the form , with , where is the generating object of . With respect to this filtration, we show that is equivalent to the ultraproduct of the categories with partially defined monoidal and additive structures, see Proposition 3.1. In particular, this enables us to solve the question of semisimplicity of raised in [24, Question 3.31]. Recall that is semisimple if and only if .
Theorem 3.3.The category is semisimple if and only if .
Next, we construct a
-linear functor
(1.1) for every
and
. This functor
is a
separable Frobenius monoidal functor compatible with the braidings, see Proposition
3.7. It enables us to classify the indecomposable objects in
. Let
be an integer,
a singleton free partition of
,
the centralizer of an element
of cycle type
,
an
-module, and
an object in
. We denote the image of the object
under
by
. Up to isomorphism the object
does not depend on the choice of
, and only depends on the isomorphism classes of
and
.
Theorem 3.9. is indecomposable if and only if and are, and the objects for as above with and indecomposable form a complete list of all indecomposable objects in up to isomorphism.
The question of classifying the indecomposable objects in
naturally emerged from the paper [
24] but was independently raised by P. Etingof in his research statement. Moreover, in Corollary
3.14 we describe the blocks of the category
as
where
is a block of
as classified in [
8], and the pair
is as above (parametrizing blocks of
not induced from
with
).
We also classify the indecomposable and indecomposable projective objects in , for the abelian envelope of , , as constructed in [9], see Section 3.7. Abelian envelopes and their general theory have been receiving an increasing amount of attention recently [4, 10, 19]. We show that indeed satisfies the universal property of the abelian envelope of in the sense of [4], see Corollary 3.24 and Appendix A.
We note that the first paper [24] on already contained a general construction of objects via explicit idempotents and half-braidings using the combinatorial description of by partitions. We identify the objects constructed in [24] in the image of in Section 3.6 and prove that these objects generate as a Karoubian tensor category but are, in general, not indecomposable.
In Section
4, we address the question of describing the associated graded Grothendieck ring
which was suggested by V. Ostrik. To this end, we introduce an induction tensor product structure on the direct sum of categories
. Namely, we define in Section
4.1 the abelian monoidal category
with the tensor product of an object
in
and
in
given by
Note that
is a tower of centers,
not the center of a tower of representation categories. Here,
is the usual induction of group representations with additional half-braiding defined in Proposition
B.1. This induction product on the sum (or
tower) of centers can be applied to other series of groups and may be of independent interest. More generally, in Appendix
B we show that induction produces separable Frobenius monoidal functors
if
is a subgroup.
From (
1.1) we obtain an oplax monoidal functor
(1.2) see Section
4.2, and a description of the associated graded of the
additive Grothendieck ring
.
Theorem 4.4.The functor from (1.2) induces an isomorphism of graded rings
where the associated graded of
is taken with respect to the filtration induced by the filtration
of
.
An analogous statement holds for the abelian envelope if , see Theorem 4.5. Computations in — and hence in — can be carried out by computing induction of modules over centralizer groups of symmetric groups. Some sample computations are included in Section 4.4.
A particularly important class of tensor categories are modular (fusion) categories which find applications in topological field theory, see [41] and references therein. In particular, the center , for a finite group and its cocycle twists appear in Dijkgraaf–Witten theory [16]. Modular categories and some of their applications have been generalized to non-semisimple finite tensor categories [5, 25]. In this generality, a modular category is a non-degenerate finite ribbon tensor category.
The categories , for generic, and , for , are infinite analogs of modular categories. This interpretation follows from [24, Theorem 3.27] where was shown to be a ribbon category and Section 3.8 where we prove that and are non-degenerate braided tensor categories. We note that these categories are also factorizable braided tensor categories by [22, Proposition 8.6.3]. In the finite case, non-degeneracy and factorizability of braided tensor categories are equivalent [39].
As and are infinite tensor categories, the concept of modular category and applications to topological field theory have not been developed for these categories. However, we note that these categories satisfy all conditions (beside finiteness) imposed on modular categories. For generic, the category is, moreover, semisimple. Hence, the results on of this paper and [24] give interesting infinite yet locally finite analogs of modular categories which are not equivalent to (co)modules over (quasi)-Hopf algebras.
2 BACKGROUND
2.1 Notation and conventions
In the following, denotes a field and the category of finite-dimensional -representations over , with . Given a prime , we abbreviate .
The categories considered in this paper are, at the very least, -linear Karoubian rigid monoidal categories. The Karoubian envelope of a -linear category is the idempotent completion of the closure under finite direct sums, and a category is Karoubian if the inclusion into its Karoubian envelope gives an equivalence of categories. In general, the symbol of two such categories denotes the external product as in [31, 35, Section 2.2] which is the Karoubian envelope of the naive -linear tensor product which has objects , for , . Note that in most cases studied, is a finite semisimple (abelian) category, which implies that is a finite direct sum of copies of . If, in addition, is abelian, then is abelian.
For terminology on monoidal, braided and symmetric monoidal categories we follow [22]. In particular, an (abelian) tensor category is a -linear rigid monoidal category which is locally finite, abelian, with , for the tensor unit . A Karoubian tensor category shall satisfy the same conditions with one exception: instead of being abelian, we require it only to be additive and idempotent-complete.
An important technical tool used in this paper is that of ultraproducts of categories (see, for example, [11]). For simplicity, we assume that the categories considered here are small so that objects and morphisms form sets. We replace representation-theoretic categories by equivalent small ones, noting that up to equivalence the ultraproduct will not depend on the choice of equivalent small categories. We assume throughout that is a fixed (non-principal) ultrafilter on and refer the reader to, for example, [37] for generalities on this concept. We may think of as a collection of subsets of , each of which contains ‘almost all’ numbers.
Given an ultrafilter and a collection of categories we can defined their ultraproduct . Its objects are sequences of objects of defined for all in a set belonging to the ultrafilter. Two such sequences are equal if they agree on all indices of some set belonging to . Similarly, morphisms are sequences of morphisms which are defined on a set belonging to the ultrafilter and identified when they agree on some set of the ultrafilter . We refer the reader to [11] or [26] for further explanations. If all categories are -linear, then is linear over the ultraproduct of fields . If all are monoidal categories, then is naturally a monoidal category.
We will make essential use of Łoś' theorem [37, Theorem 1.3.2] which allows us to transfer any first-order logical statement from the categories to their ultraproduct . See also [28, Section 1] for examples on how this theorem is used. We further need Steinitz' theorem that states that an uncountable algebraically closed field is determined, up to isomorphism, by its characteristic and uncountable cardinality [40]. This theorem implies the existence of isomorphisms of the ultraproducts of fields , or , for a sequence of primes with , and the complex numbers, see, for example, [37, Chapter 1]. In the following, we consider equivalences of monoidal categories linear over an ultraproduct of algebraically closed fields, which we regard as equivalences of -linear monoidal categories under an isomorphism of fields obtained from Steinitz' theorem.
2.2 Deligne's categories as diagrammatic categories
In [14], Deligne constructs a class of Karoubian -linear symmetric monoidal categories depending on a parameter in the field of characteristic zero. It is a well-known observation that every simple complex -module appears as a direct summand of a tensor power of the -dimensional permutation representation of for some . In other words, the category of finite-dimensional -modules is the Karoubian envelope of the monoidal category generated by the single object , cf. [14, §1.7–1.8]. The morphism spaces are given by the -invariants of and can be described combinatorially using the partition algebras , see, for example, [8, Section 2] for details. To define , is now replaced by a general parameter .
Deligne's category
can be constructed using a graphical calculus: it has a distinguished object
represented by a point and its tensor powers
for
represented by
points. Morphisms between two such tensor powers
and
are represented using diagrams consisting of
upper points labeled
and
lower points labeled
, and an arbitrary number of strings connecting points. Two such diagrams are considered equivalent, if the partitions of the points
given by the connected components of each string diagram coincide. The morphism spaces between
and
are defined as the free
-vector space spanned by the equivalence classes of such diagrams, so tensor products and compositions can be defined on diagrams and extended linearly. The following is a typical string diagram representing a morphism in
:
The tensor product of two diagrams is given by stacking the diagrams horizontally. The composition of two diagrams is achieved by first stacking them vertically, and identifying the lower points of with the upper points of . Then these identified points are removed from the string diagram, leaving only the upper points of , the lower points of , and a number of strings. Each connected component of this string diagram which does not contain upper points of or lower points of (such components may arise when removing the identified points) is removed and the resulting string diagram is multiplied by a factor , where is the number of connected components removed in the process. The category is now defined as the Karoubian envelope of the category defined on the objects , for . We refer the reader to [14] and [8, Section 2] for details on the combinatorial construction of .
In the generic case, that is, if
,
is a semisimple symmetric tensor category. If
is a non-negative integer, there is an essentially surjective full monoidal functor
(2.1) which maps
to the
-dimensional standard
-module
and, consequently,
to
for any
. Choosing a basis
in
provides a basis
indexed by tuples
for
, for any
, and the image
of any diagram
with
upper and
lower points is the
-module morphism sending
(2.2) where the coefficient
is 1 if the indices
induce a partition on the upper and lower points of the diagram
which refines the one given by the connected components of
, and 0 otherwise. The functors
allow us to view
as an interpolation category for the classical symmetric tensor categories
.
Further, we recall the (recursive) definition of the morphism
from [
8, Equation (2.1)],
(2.3) where the sum is taken over all partitions
strictly coarser than
. Note that the set
, for all partitions of
, gives a basis for
. If, again,
, then
is the
-module morphism
(2.4) where now the coefficient
is 1 if the indices
induce the same partition on the upper and lower points of the diagram
as the one given by the connected components of
, and 0 otherwise.
Indecomposable objects in are classified by partitions, see [8, Section 3.1].
Theorem 2.1.There is a bijection between partitions , for , and indecomposable objects of . The object is a direct summand of , but not of for .
For the objects behave uniformly in . Their dimensions and character values are given by polynomials in , and they can be cut out of by primitive idempotents which are -linear combinations of partition diagrams.
At non-negative integral values , some of the rational functions defining these primitive idempotents develop poles and the idempotents no longer exist. As such, some surviving idempotents which are generically not primitive become primitive at these special values, and the corresponding generically simple objects get ‘glued together’ in a sense.
As part of their analysis of blocks, Comes and Ostrik described a process of ‘lifting’ which takes an indecomposable object at , and describes how it splits apart when we deform it to nearby semisimple values of . The combinatorics of this process is completely described by Comes and Ostrik [8], but we will just use the following simplified version.
Theorem 2.2. ([[8], Proposition 3.10, Proposition 3.12(a), Lemma 5.20])For every object there is an object in , uniquely defined up to isomorphism, for in a formal neighborhood of such that for all ,
- (1) ;
- (2) ;
- (3) for any indecomposable object , either
Moreover each arises as a summand of for at most one partition with .
Remark 2.3.We believe that over the complex numbers one can check that this lifting operation can be defined for in an analytic neighborhood of , rather than just a formal one. For our purposes though this will not be necessary, so we will not pursue this direction further.
2.3 Deligne's categories as limits in characteristic and rank
We now recall [26, Theorem 1.1] to display , for any , as an ultraproduct of the representation categories , cf. Section 2.1. The case for transcendental was already contained in Deligne's seminal paper [14]. We are interested in three cases of this theorem to describe all cases of Deligne's category up to equivalence of symmetric monoidal categories (under isomorphisms of fields that send one transcendental parameter to the other).
Definition 2.4.For any object in a Karoubian or abelian tensor category, we denote by the full Karoubian tensor subcategory generated by .
Note that is generated by duals, tensor products, direct sums, and direct summands involving . In the situations we discuss, will often be a self-dual object.
Theorem 2.5. ([[14, 26], Theorem 1.1])In each case below, we specify an increasing sequence of positive integers and a sequence of fields . We denote by the object in , and by the complex number corresponding to under the respective isomorphism of fields .
- (1) transcendental: Consider a sequence with and the ultraproduct . Then, under an isomorphism of fields , is equivalent to as a -linear symmetric monoidal category.
- (2) , with minimal polynomial : We can choose increasing sequences of positive integers and primes such that for any , satisfying
Then, under an isomorphism of fields , is equivalent to as a -linear symmetric monoidal category.
- (3) : Set for the th prime number. Then again, under an isomorphism of fields , is equivalent to as a -linear symmetric monoidal category.
We remark that under the chosen isomorphisms of fields, the respective functors are obtained by the universal property of by sending the tensor generator of to the object in the ultraproduct.
Since is defined in terms of its generating object , there is a natural exhaustive filtration encoding the complexity of objects in terms of this generator. Explicitly, is the full subcategory of objects which are direct summands of objects , where each is at most and the number of terms is at most . We may similarly filter the categories in terms of the defining -dimensional representation of , analogously calling the filtered pieces .
The advantage of these filtrations is that Theorem 2.5 tells us that is equivalent to the ultraproduct of , with no need to further cut down the ultraproduct. These subcategories are not monoidal or even additive, as taking a tensor product or direct sum will possibly land you in a higher term of the filtration, but the ultraproduct does respect those products and sums that are defined.
Taking the union over all we obtain a coarser filtration , for either or . These filtered pieces are additive and in fact abelian in the case where . These are not monoidal subcategories but they satisfy the condition that if and , then , which makes the Grothendieck ring a filtered ring.
The descriptions of can be used to transfer any first-order statement in the signature of symmetric monoidal categories with a distinguished object from the classical (modular) representation theory to the limit, that is, the interpolation category using Łoś' theorem. For more details and applications of this philosophy, see [26, 28].
The combinatorial description of recalled in Section 2.2 can be matched with the above characterization through ultrafilters in Theorem 2.5, giving an evaluation of the equivalence on morphisms.
Given a partition
of
viewed as a morphism
,
corresponds to the ultraproduct
described in the following. Using the notation
as in (
2.2),
is defined by
(2.5) for all
-tuples
and all
-tuples
of integers in
, where the coefficient
is the same as in (
2.2), the complex case.
We may extend the assignment using the fixed isomorphism of fields by , for . We note that corresponds to under the isomorphism of fields, but for a general complex number , is only defined for almost all .
2.4 Representations of a fixed group in large characteristic
In addition to symmetric groups, we will often have an auxiliary finite group for which we want to compare representations across different large characteristics.
Let be a fixed finite group, and let be a faithful representation of defined over the integers (for example, the permutation representation). In a slight abuse of notation we will use to denote the corresponding base changes to as well as to . As , in particular, labels an object in each of the categories for any sequence of primes , we also have an object in the ultraproduct , which we will also denote by . The following result of Crumley realizes the characteristic zero representation theory of as an ultraproduct of the representations of in large characteristic.
Theorem 2.6. ([[11], Section 9.5.1])Let be an arbitrary increasing sequence of primes. Under an isomorphism of fields , there exists an equivalence of -linear symmetric monoidal categories between and .
This result is nice in that it involves an equivalence of categories, and closely parallels the ultraproduct construction of Deligne categories above. However, in this case we can actually be very explicit about what happens at the level of objects, but first let us recall a bit about the representation theory of finite groups in large characteristic.
It was first observed by Dickson in 1902 [15] that if does not divide , then the representation theory of over an algebraically closed field of characteristic is ‘the same’ as over the complex numbers. Translated into a more modern set-up the following theorem explicitly describes this relationship. We refer to [38, Part III] for basic facts about the modular representation theory of finite groups.
Theorem 2.7.Suppose that is the ring of integers in a number field such that every irreducible complex representation of is defined over , is a prime number not dividing or the discriminant of , and let be a prime ideal of lying above . If is an irreducible complex representation of , choose an integral form defined over and define its reduction modulo as .
- (1) is an (absolutely) irreducible representation of over .
- (2) The isomorphism class of is independent of the choice of the integral form .
- (3) If is another prime lying above , then there exists an automorphism of sending to . The induced isomorphism identifies with .
- (4) Every irreducible representation of over arises this way.
- (5) If and are two irreducible complex representations, then if and only if .
We are about ready to give an explicit description of what Crumley's equivalence does, but first let us recall basics about central characters. If is a conjugacy class, then the element defines an endomorphism of the identity functor in and for all primes . Evaluating the trace of these endomorphisms on a representation gives the central character of . Over the complex numbers the central character of is just a rescaling of the ordinary character, and in particular determines up to isomorphism.
Fix as in Theorem 2.7, and for each prime as in Theorem 2.6 fix a prime ideal lying above as well as an algebraic closure of . We may choose our isomorphism such that on it is the ultraproduct of the natural quotient maps .
Proposition 2.8.Under the equivalence of categories defined in Theorem 2.6, the image of an irreducible representation is isomorphic to .
Proof.A priori we know that under this equivalence of categories gets identified with some irreducible complex representation ; hence, it suffices to check that and have the same central character. is defined over , so therefore its central character is as well. Moreover, by construction the central character of is the reduction modulo of the central character of . Since we chose our identification to identify with the product of its reductions modulo , we see that indeed these characters agree.
2.5 The Grothendieck ring of
Given a
-linear additive monoidal category
, let
denote its
additive Grothendieck ring, that is, the quotient the free ring generated by all isomorphism classes of objects in
by the ideal of relations given through direct sums and tensor products. If we denote by
the symbol of an object
of
inside of the ring
, these relations are
for all
.
The Grothendieck ring of is a filtered ring: Recall the filtration on the category from Section 2.3. Here, an object is in , for , if is isomorphic to a direct summand of a sum of objects with , where is the tensor generator of . This defines a filtration of the ring .
Let us denote the irreducible complex -module corresponding uniquely up to isomorphism to a partition by .
Lemma 2.9. ([[14], Proposition 5.11], [[8], Proposition 3.12], [[27], Theorem 3.3])Sending induces an isomorphism , where the product on the right-hand side is given by induction, or equivalently, by Littlewood–Richardson coefficients.
As the ring on the right-hand side is a graded ring which does not depend on , this result exhibits the Grothendieck ring of as a filtered deformation of that ring.
2.6 Induction and restriction between Deligne categories
The universal property of
gives a natural exact symmetric monoidal
restriction functor
which sends the defining object
to
. If
is a non-negative integer, this descends to the ordinary restriction functor from
to
, under the fiber functors
,
to the semisimple categories from (
2.1).
Etingof ([
20], Section
2.3) considered
induction and
co-induction functors
defined a priori as the left and right adjoints to the restriction functor defined above. Etingof also considered restriction to a product of two Deligne categories, and in that case one needs to pass to an ind-completion in order to define these adjoints, but for our purposes we will only need the finite versions.
It was observed in [
28] that if we think of
and
as the model-theoretic limits of categories
and
, respectively (in the sense described above), then these induction and co-induction functors are limits of the ordinary induction and co-induction functors
corresponding to the embedding of
into
for each
. Moreover, since induction and co-induction are naturally isomorphic for representations of finite groups (over any field), it follows that these Deligne category induction and co-induction functors are naturally isomorphic as well and we can view them as a single two-sided adjoint to the restriction functor which we will refer to just as induction.
These induction functors are also well behaved with respect to the filtrations
defined earlier. In particular we have:
(2.6) moreover, this
shift in the filtration is optimal in a strong sense: If
is an object of
that does not lie in
, then
lies in
but not in
.
If
is a subgroup, then we may further restrict from
to
. By the same reasoning as above, this also admits a two-sided adjoint induction functor
We note that
is naturally isomorphic to
, where we first perform ordinary induction from
to
and then perform Deligne category induction. In particular, it shifts up the filtration by the same amount.
Given a partition of size we recall that denotes the corresponding irreducible representation of . We will use to denote the corresponding indecomposable object of , see Section 2.1. Note that by Theorem 2.1, but not in .
If is a partition, then for sufficiently large define the padded partition . In terms of Young diagrams, this padding operation just adds a long first row to to make it have total boxes. The relevance for our purposes is that Comes and Ostrik showed that if (that is, if defines a Young diagram), then gets mapped to under the specialization functor [8, Proposition 3.25].
For a partition let . denote the set of partitions which can be obtained from by removing a horizontal strip, that is, by removing at most one box from each column. The following Deligne category Pieri rule gives us an upper triangularity property between the simple objects and certain easy to work with induced objects.
Lemma 2.10.
- (1) For or for with
In particular, the right-hand side is plus terms with .
- (2) For the object decomposes as a multiplicity-free direct sum of indecomposable objects , such that:
- (a) occurs with multiplicity one,
- (b) Each that appears has (but not all such need appear).
Proof.The ordinary Pieri rule for symmetric groups tells us that in characteristic 0 or
Part 1 follows since
corresponds to the object
, under both the ultraproduct identification and under the quotient functor from
to
.
Now suppose . A priori one knows that decomposes as a direct sum of indecomposable objects with some multiplicities . If we apply the Comes–Ostrik lifting operator to this sum, we must obtain the answer from Part (1).
Comes and Ostrik's description of lifting (as summarized in Theorem 2.2) tells us that appears as a direct summand of for and in that case, at most one other partition , with appears as a summand giving . Since is the largest partition appearing in . and it appears with multiplicity one, we see that must occur at with multiplicity one, proving part (1).
To show part (2) we then inductively apply the same logic as above to the next largest partitions . Note that as we repeat this argument for some , the term might come from for some larger already accounted for, in which case does not appear in the induction at .
Note that this lemma gives us an alternative way of characterizing the indecomposable object as the unique direct summand of not occurring as a summand of for any partition with .
Once one has established the Deligne category Pieri rule for induction, one can iteratively compute more general induced representations combinatorially. In particular, if we only keep track of the leading order terms, one obtains the following.
Corollary 2.11. (See [[27], Section 2.1])
where
denotes a Littlewood–Richardson coefficient.
2.7 The abelian envelope of Deligne's categories
An (abelian) multitensor category is called an abelian envelope of a Karoubian rigid monoidal tensor category if it contains and for any multitensor category , the category of tensor functors is equivalent to the category of faithful monoidal functors by restriction. If it exists, the abelian envelope is unique up to equivalence. We refer to [4] and [19, Section 9] for details on the abelian envelope and its universal property.
For , the abelian envelope of ([4, Example 2.45(1)]) has several explicit constructions. The first construction uses a symmetric monoidal functor and displays the abelian envelope as a category of representations over an algebraic group object internal to the semisimple category . The second construction introduces a -structure on complexes of objects in and displays the abelian envelope as the heart of this -structure — see [9] for details on these constructions.
A third construction, which we will employ, is given using ultrafilters. For this, we consider the ultraproduct of general modules over in finite characteristic (rather than just those in ) in Theorem 2.5(c) and obtain the following modification of the ultraproduct description for the abelian envelope, which is obtained as the closure of under subquotients.
Theorem 2.12. ([[26], Theorem 1.1(b)])Set for the th prime number. Then there are equivalences of additive categories
which provide, under an isomorphism of fields
, an equivalence of
-linear symmetric tensor categories between
and
.
It follows from Theorem
2.12 (or can be deduced directly from the universal property of
) that the induction functor from Section
2.6 extends to the abelian envelope such that
(2.7) is a commutative diagram of
-linear functors. This functor is again left and right adjoint to restriction and hence exact.
Comes and Ostrik showed ([9, Proposition 2.9, Corollary 4.6]) that each block of is equivalent to a block in the category of representations of quantum , with the objects in corresponding to tilting objects. As a consequence, itself forms a highest weight category with the indecomposable objects as the indecomposable tilting objects. The simple objects of are again indexed by partitions, with appearing as a composition factor of with multiplicity one, and all other composition factors are of the form with satisfying .
Under the ultrafilter identification the indecomposable module corresponds to the so-called Young modules , which arise as direct summands of the permutation representation. The simple objects of correspond to ultraproducts of simple objects . We note that while in general is not a highest weight category, the truncated categories with less than the characteristic are highest weight with Specht modules as standard objects and Young modules as tilting objects. One can check that the ultraproduct of these highest weight structures defines a highest weight structure on agreeing with the one defined by Comes and Ostrik.
Induction does not preserve semisimplicity in general, so we cannot expect a version of Corollary 2.11 for induction of simple objects to hold as a direct sum decomposition in . We can, however, relax it by passing to the Grothendieck ring and instead keeping track of composition multiplicities.
For an abelian category , the (abelian) Grothendieck ring is the quotient of the additive Grothendieck ring from Section 2.5 by relations obtained from short exact sequences. If is semisimple, then .
Corollary 2.13.In one has
where
denotes a Littlewood–Richardson coefficient.
Proof.This follows immediately from Corollary 2.11 by passing to the Grothendieck ring and then substituting everywhere and collecting all of the lower order terms to one side.
We have the following analog of Lemma 2.9 for the abelian Grothendieck ring of .
Lemma 2.14.Sending induces an isomorphism of rings from to , where the product on the right-hand side is given by induction, or equivalently, by Littlewood–Richardson coefficients.
Proof.The proof will be via a chain of isomorphisms.
First note that the upper triangularity property implies that these form a basis for , and therefore the inclusion is in fact an isomorphism. Moreover, this upper triangularity property implies that the assignment defines an isomorphism of associated graded rings .
Next, the first two properties of lifting in Theorem 2.2 say that the map defines a ring homomorphism at a nearby transcendental value of . The third property in Theorem 2.2 is again an upper triangularity property, which implies that this homomorphism is an isomorphism, but moreover that defines an isomorphism of associated graded rings at generic .
Finally we use the isomorphism of Lemma 2.9, which sends to . Composing these three isomorphisms gives the desired result.
2.8 The projective objects in the abelian envelope
We now turn to describing the projective objects in for later use. Key to understanding of the projectives will be the projective cover of the tensor unit.
Lemma 2.15.Let be an object in . Then is projective in if and only if there exists an integer such that for each integer , almost all are projective objects in .
Proof.By virtue of being an object of , there exists such that . In particular, is contained in for any . Hence, for fixed , almost all are contained in . Assume that is projective. The property that is projective is defined by the functor being right exact, as a functor on , which can be expressed as a first-order property. Thus, is a projective object in , for almost all .
Conversely, again using Łoś' theorem, we see that if is a projective object in , for almost all , then is projective in . Since is the filtered colimit of all of these categories, projectivity of holds in the entire category .
Recall the indecomposable objects of , see Section 2.2 and consider the case .
Lemma 2.16.The indecomposable object is the projective cover of the tensor unit .
Proof.Recall from Theorem 2.12 that is equivalent to the ultraproduct of the categories . Observe that is left adjoint of a functor-preserving epimorphisms and thus preserves projective objects; therefore, any object of the form is projective in as it is induced from a semisimple category where all objects are projective. In particular, observe that
and that the surjective morphisms
induce a surjective morphism
, for
.
By Lemma 2.10, . The indecomposables appearing as lower order terms are of the form , with . The combinatorial description of objects in the block of from [9, Theorem 2.6] implies that of the objects , only belongs to this block and for objects from other blocks. However, is not projective as is not semisimple. Thus, is the projective cover of .
Proposition 2.17.Let . Then is projective in if and only if is projective in .
Proof.Given an indecomposable object in , we can use Lemma 2.16 to see that the projective cover of is contained in . Indeed, is a projective object in with an epimorphism to . Now, is projective in an additive subcategory of if and only if its projective cover is contained in and is isomorphic to its projective cover. By the above observation and Lemma 2.15, this condition holds in if and only if it holds in .
The advantage of the above proposition is that projectivity of an object in can be checked using the components of an ultrafilter presentation, as well as the following observation.
Corollary 2.18.All projective objects of are contained in the subcategory .
Proof.This follows directly from [9, Remark 4.8] since for any object of there exists a surjective map . Hence, if is projective, it is a direct summand of an object in .
2.9 Indecomposable Yetter–Drinfeld modules over in arbitrary characteristic
In this section, let be an algebraically closed field and a finite group. We now turn to background results on the monoidal center of the category . We employ the equivalent description of this braided tensor category as Yetter–Drinfeld modules [43, Definition 3.6], that is, -graded -representations such that , see, for example, [32, Proposition 7.1.6]. If , we write for the degree of . Equivalently, can be described as modules over the Drinfeld double of [18]. We recall the following result found in [16] for and in [42, Corollary 2.3] for general characteristic.
Theorem 2.19. (Dijkgraaf–Pasquier–Roche, Witherspoon)Let be a finite group and be an algebraically closed field. A complete list of indecomposable (respectively, irreducible) objects in is given by modules of the form where is a representative of a conjugacy class of elements in and is an indecomposable (respectively, irreducible) module over the centralizer of in .
In fact, for each
, with
, we have functors
The half-braiding on
is given by
, for any
-module
. The
-grading of the associated Yetter–Drinfeld module
here is given by
. This grading does not depend on
, so it is clear that any morphism of
-modules
induces one in
, namely,
. The constructions are independent of the choice of a representative
of a conjugacy class up to isomorphism.
The following lemma is easily seen from the description of in terms of Yetter–Drinfeld modules over .
Lemma 2.20.The functors are full and faithful, that is, there are isomorphisms
for any
-modules
and
.
If and are not conjugate, is a -module, and is a -module, then
Hence, as an abelian category,
decomposes as a direct sum (as defined in [
22, Section 1.3])
where
ranges over a set of representatives of the conjugacy classes of
.
A direct consequence of this observation is the following: let be a Yetter–Drinfeld -module over any field , and let be the simple Yetter–Drinfeld -module obtained from in and a simple module over the centralizer of in . Then the multiplicity in equals the multiplicity in , where is the homogeneous subspace of of degree .
Lemma 2.21. is projective in if and only if is projective in .
Proof.For , consider the regular -module and note that . As an object in , this is the projective -submodule of , where is the basis of dual to the basis . By functoriality of , being a direct summand of , it readily follows that is a direct summand of and thus projective.
Assume that is a direct summand of a direct sum of the regular module in . Note that and . Thus, we obtain that is a direct summand of as a -module. Choosing a decomposition of into right -cosets, we observe that as a -module, is simply a direct sum of copies of the regular module. Thus, is projective.
The Grothendieck ring of the monoidal center of representations over algebraically closed fields of arbitrary characteristic was studied in [42]. See also [17] for some concrete examples.
Corollary 2.22. ([[42], Section 3])For a finite group. There is an isomorphism of rings
where
varies over a set of representatives of the conjugacy classes of
.
This corollary is proved in [42, p. 316] and uses a result of G. Lusztig.