# The indecomposable objects in the center of Deligne's category $$\protect\underline{{\rm Re}}\!\operatorname{p}S_t$$

## Abstract

We classify the indecomposable objects in the monoidal center of Deligne's interpolation category $$\underline{{\rm Re}}\!\operatorname{p}S_t$$ by viewing $$\underline{{\rm Re}}\!\operatorname{p}S_t$$ as a model-theoretic limit in rank and characteristic. We further prove that the center of $$\underline{{\rm Re}}\!\operatorname{p}S_t$$ is semisimple if and only if $$t$$ is not a non-negative integer. In addition, we identify the associated graded Grothendieck ring of this monoidal center with that of the graded sum of the centers of representation categories of finite symmetric groups with an induction product. We prove analogous statements for the abelian envelope.

## 1 INTRODUCTION

In the seminal paper [14], P. Deligne constructed symmetric tensor categories $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$, where $$t$$ can be any complex number, which interpolate the categories of representations over the symmetric groups $$S_d$$, $$d\in \mathbb {Z}_{\geqslant 0}$$. 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 $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ 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 $$t$$ in symmetric monoidal categories.

In addition to giving this combinatorial definition and universal property Deligne observed that for $$t$$ transcendental $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ can be realized as an ultraproduct, a sort of model-theoretic limit, of the categories $$\operatorname{Rep}S_d$$, as $$d\in \mathbb {Z}_{\geqslant 0}$$ grows to infinity. This understanding was extended to all values of $$t$$ by the second listed author in [26, Theorem 1.1] by viewing $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ as a limit of the categories $$\underline{\mathrm{Re}}\!\operatorname{p}_{p} S_{d}$$, varying the rank $$d$$ as well as the characteristic $$p$$. 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 $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$ of Deligne's categories. The *monoidal* or *Drinfeld center* $$\mathcal {Z}(\mathcal {C})$$ [29, 33] of a monoidal category $$\mathcal {C}$$ is a universal construction of a braided monoidal category with a forgetful functor $$\mathcal {Z}(\mathcal {C})\rightarrow \mathcal {C}$$ 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 $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$, showed that this is a ribbon category, and obtained invariants of framed links as an application. It was shown that the braided categories $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$ interpolate the braided categories $$\mathcal {Z}(\operatorname{Rep}S_d)$$ in the sense that $$\mathcal {Z}(\operatorname{Rep}S_d)$$ is the semisimplification of $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_d)$$ for $$d\in \mathbb {Z}_{\geqslant 0}$$. In the present paper, we answer several open structural questions about the categories $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$ 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 $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ 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 $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$ as a model-theoretic limit in rank and characteristic, similar to $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$. For this, we define a bi-filtration on the center, with the filtration layer $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)^{\leqslant m,k}$$ being the full subcategory on objects which are in the preimage of $$(\underline{\mathrm{Re}}\!\operatorname{p}S_t)^{\leqslant m,k}$$ under the forgetful functor $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)\rightarrow \underline{\mathrm{Re}}\!\operatorname{p}S_t$$. Here, $$(\underline{\mathrm{Re}}\!\operatorname{p}S_t)^{\leqslant m,k}$$ consists of objects that are direct summands of objects of the form $$X^{\otimes m_1}\oplus \ldots \oplus X^{\otimes m_k}$$, with $$m_i\leqslant m$$, where $$X$$ is the generating object of $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$. With respect to this filtration, we show that $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)^{\leqslant k, m}$$ is equivalent to the ultraproduct of the categories $$\mathcal {Z}(\operatorname{Rep}_{\overline{\mathbb {F}}_{p_i}}S_{n_i})^{\leqslant k,m}$$ with partially defined monoidal and additive structures, see Proposition 3.1. In particular, this enables us to solve the question of semisimplicity of $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$ raised in [24, Question 3.31]. Recall that $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ is semisimple if and only if $$t\not\in \mathbb {Z}_{\geqslant 0}$$.

Theorem 3.3.The category $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$ is semisimple if and only if $$t\notin \mathbb {Z}_{\geqslant 0}$$.

*separable Frobenius monoidal functor*compatible with the braidings, see Proposition 3.7. It enables us to classify the indecomposable objects in $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$. Let $$n\geqslant 0$$ be an integer, $$\mu$$ a singleton free partition of $$n$$, $$Z(\mu )$$ the centralizer of an element $$\sigma \in S_n$$ of cycle type $$\mu$$, $$V$$ an $$Z(\mu )$$-module, and $$U$$ an object in $$\underline{\mathrm{Re}}\!\operatorname{p}S_{t-n}$$. We denote the image of the object $$\operatorname{Ind}_{Z(\mu )}^{S_n}(V)\boxtimes U$$ under $$\underline{\operatorname{Ind}}$$ by $$\hspace{0.55542pt}\underline{\hspace{-0.55542pt}W\hspace{-3.88885pt}}\hspace{3.88885pt}_{\hspace{-3.88885pt}\mu ,V,U}$$. Up to isomorphism the object $$\hspace{0.55542pt}\underline{\hspace{-0.55542pt}W\hspace{-3.88885pt}}\hspace{3.88885pt}_{\hspace{-3.88885pt}\mu ,V,U}$$ does not depend on the choice of $$\sigma$$, and only depends on the isomorphism classes of $$U$$ and $$V$$.

Theorem 3.9.$$\hspace{0.55542pt}\underline{\hspace{-0.55542pt}W\hspace{-3.88885pt}}\hspace{3.88885pt}_{\hspace{-3.88885pt}\mu ,V,U}$$ is indecomposable if and only if $$V$$ and $$U$$ are, and the objects $$\hspace{0.55542pt}\underline{\hspace{-0.55542pt}W\hspace{-3.88885pt}}\hspace{3.88885pt}_{\hspace{-3.88885pt}\mu ,V,U}$$ for $$n,\mu ,V,U$$ as above with $$V$$ and $$U$$ indecomposable form a complete list of all indecomposable objects in $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t )$$ up to isomorphism.

We also classify the indecomposable and indecomposable projective objects in $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}^{\mathrm{ab}} S_d)$$, for the *abelian envelope* $$\underline{\mathrm{Re}}\!\operatorname{p}^{\mathrm{ab}} S_d$$ of $$\underline{\mathrm{Re}}\!\operatorname{p}S_d$$, $$d\in \mathbb {Z}_{\geqslant 0}$$, 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 $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}^{\mathrm{ab}} S_d)$$ indeed satisfies the universal property of the abelian envelope of $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_d)$$ in the sense of [4], see Corollary 3.24 and Appendix A.

We note that the first paper [24] on $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$ already contained a general construction of objects via explicit idempotents and half-braidings using the combinatorial description of $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ by partitions. We identify the objects constructed in [24] in the image of $$\underline{\operatorname{Ind}}$$ in Section 3.6 and prove that these objects generate $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$ as a Karoubian tensor category but are, in general, not indecomposable.

*not*the center of a tower of representation categories. Here, $$\mathcal {Z}\operatorname{Ind}_{S_n\times S_m}^{S_{n+m}}(V\boxtimes W)$$ 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

*additive*Grothendieck ring $$K_0^{\oplus }$$.

Theorem 4.4.The functor $${\underline{\operatorname{Ind}}}$$ from (1.2) induces an isomorphism of graded rings

An analogous statement holds for the abelian envelope $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d)$$ if $$t=d\in \mathbb {Z}_{\geqslant 0}$$, see Theorem 4.5. Computations in $$\mathcal {Z}\underline{\mathrm{Re}}\!\operatorname{p}S_{\geqslant 0}$$ — and hence in $$K^{\oplus }_0(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$ — 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 $$\mathcal {Z}(\operatorname{Rep}G)$$, for $$G$$ 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 $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$, for $$t$$ generic, and $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d)$$, for $$d\in \mathbb {Z}_{\geqslant 0}$$, are *infinite* analogs of modular categories. This interpretation follows from [24, Theorem 3.27] where $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$ was shown to be a ribbon category and Section 3.8 where we prove that $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$ and $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d)$$ 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 $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$ and $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d)$$ 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 $$t$$ generic, the category $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$ is, moreover, semisimple. Hence, the results on $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$ 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, $$\mathbb {k}$$ denotes a field and $$\operatorname{Rep}_\mathbb {k}G$$ the category of finite-dimensional $$G$$-representations over $$\mathbb {k}$$, with $$\operatorname{Rep}G=\operatorname{Rep}_\mathbb {C}G$$. Given a prime $$p$$, we abbreviate $$\operatorname{Rep}_p G=\operatorname{Rep}_{\overline{\mathbb {F}}_p} G$$.

The categories considered in this paper are, at the very least, $$\mathbb {k}$$-linear Karoubian rigid monoidal categories. The *Karoubian envelope* of a $$\mathbb {k}$$-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 $$\mathcal {C}\boxtimes \mathcal {D}$$ of two such categories $$\mathcal {C},\mathcal {D}$$ denotes the external product as in [31, 35, Section 2.2] which is the Karoubian envelope of the naive $$\mathbb {k}$$-linear tensor product which has objects $$X\boxtimes Y$$, for $$X\in \mathcal {C}$$, $$Y\in \mathcal {D}$$. Note that in most cases studied, $$\mathcal {C}$$ is a finite semisimple (abelian) category, which implies that $$\mathcal {C}\boxtimes \mathcal {D}$$ is a finite direct sum of copies of $$\mathcal {D}$$. If, in addition, $$\mathcal {D}$$ is abelian, then $$\mathcal {C}\boxtimes \mathcal {D}$$ is abelian.

For terminology on monoidal, braided and symmetric monoidal categories we follow [22]. In particular, an (abelian) tensor category is a $$\mathbb {k}$$-linear rigid monoidal category which is locally finite, abelian, with $$\operatorname{End}(\mathbf {1})=\mathbb {k}$$, for the tensor unit $$\mathbf {1}$$. 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 $$\mathcal {U}$$ is a fixed (non-principal) *ultrafilter* on $$\mathbb {N}$$ and refer the reader to, for example, [37] for generalities on this concept. We may think of $$\mathcal {U}$$ as a collection of subsets of $$\mathbb {N}$$, each of which contains ‘almost all’ numbers.

Given an ultrafilter $$\mathcal {U}$$ and a collection of categories $$(\mathcal {C}_i)_{i\in \mathbb {N}}$$ we can defined their *ultraproduct* $$\prod _{\mathcal {U}}\mathcal {C}_i$$. Its objects are sequences $$\prod _\mathcal {U}V_i$$ of objects $$V_i$$ of $$\mathcal {C}_i$$ defined for all $$i$$ in a set belonging to the ultrafilter. Two such sequences are equal if they agree on all indices of some set belonging to $$\mathcal {U}$$. Similarly, morphisms are sequences $$\prod _{\mathcal {U}}f_i\colon \prod _\mathcal {U}V_i\rightarrow \prod _\mathcal {U}W_i$$ of morphisms $$f_i\colon V_i\rightarrow W_i$$ which are defined on a set belonging to the ultrafilter and identified when they agree on some set of the ultrafilter $$\mathcal {U}$$. We refer the reader to [11] or [26] for further explanations. If all categories $$\mathcal {C}_i$$ are $$\mathbb {k}_i$$-linear, then $$\prod _\mathcal {U}\mathcal {C}_i$$ is linear over the ultraproduct of fields $$\prod _\mathcal {U}\mathbb {k}_i$$. If all $$\mathcal {C}_i$$ are monoidal categories, then $$\prod _\mathcal {U}\mathcal {C}_i$$ 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 $$\mathcal {C}_i$$ to their ultraproduct $$\prod _\mathcal {U}\mathcal {C}_i$$. 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 $$\prod _{\mathcal {U}}\mathbb {C}$$, or $$\prod _{\mathcal {U}}\overline{\mathbb {F}}_{p_i}$$, for a sequence of primes with $$\liminf _i p_i=\infty$$, and the complex numbers, see, for example, [37, Chapter 1]. In the following, we consider equivalences of monoidal categories linear over an ultraproduct $$\prod _{\mathcal {U}}\mathbb {k}_i$$ of algebraically closed fields, which we regard as equivalences of $$\mathbb {C}$$-linear monoidal categories under an isomorphism of fields $$\prod _{\mathcal {U}}\mathbb {k}_i\cong \mathbb {C}$$ obtained from Steinitz' theorem.

### 2.2 Deligne's categories as diagrammatic categories

In [14], Deligne constructs a class of Karoubian $$\mathbb {k}$$-linear symmetric monoidal categories $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ depending on a parameter $$t$$ in the field $$\mathbb {k}$$ of characteristic zero. It is a well-known observation that every simple complex $$S_n$$-module appears as a direct summand of a tensor power $$X_n^{\otimes k}$$ of the $$n$$-dimensional permutation representation $$X_n$$ of $$S_n$$ for some $$k\geqslant 1$$. In other words, the category $$\operatorname{Rep}S_n$$ of finite-dimensional $${\mathbb {C}} S_n$$-modules is the *Karoubian envelope* of the monoidal category generated by the single object $$X_n$$, cf. [14, §1.7–1.8]. The morphism spaces $$\operatorname{Hom}(X_n^{\otimes k},X_n^{\otimes l})$$ are given by the $$S_n$$-invariants of $$X_n^{\otimes (k+l)}$$ and can be described combinatorially using the partition algebras $$P_k(n)$$, see, for example, [8, Section 2] for details. To define $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$, $$n$$ is now replaced by a general parameter $$t\in \mathbb {k}$$.

The tensor product of two diagrams is given by stacking the diagrams horizontally. The composition of two diagrams $$\pi ,\mu$$ is achieved by first stacking them vertically, and identifying the lower points of $$\pi$$ with the upper points of $$\mu$$. Then these identified points are removed from the string diagram, leaving only the upper points of $$\pi$$, the lower points of $$\mu$$, and a number of strings. Each connected component of this string diagram which does not contain upper points of $$\pi$$ or lower points of $$\mu$$ (such components may arise when removing the identified points) is removed and the resulting string diagram is multiplied by a factor $$t^\ell$$, where $$\ell \geqslant 0$$ is the number of connected components removed in the process. The category $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ is now defined as the Karoubian envelope of the category defined on the objects $$X^{\otimes k}$$, for $$k\geqslant 0$$. We refer the reader to [14] and [8, Section 2] for details on the combinatorial construction of $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$.

Indecomposable objects in $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ are classified by partitions, see [8, Section 3.1].

Theorem 2.1.There is a bijection between partitions $$\lambda \vdash n$$, for $$n\geqslant 0$$, and indecomposable objects $$X_\lambda$$ of $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$. The object $$X_\lambda$$ is a direct summand of $$X^{\otimes n}$$, but not of $$X^{\otimes i}$$ for $$i<n$$.

For $$t \notin \mathbb {Z}_{\geqslant 0}$$ the objects $$X_\lambda$$ behave uniformly in $$t$$. Their dimensions and character values are given by polynomials in $$t$$, and they can be cut out of $$X^{\otimes n}$$ by primitive idempotents which are $$\mathbb {Q}(t)$$-linear combinations of partition diagrams.

At non-negative integral values $$t = d \in \mathbb {Z}_{\geqslant 0}$$, 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 $$X_\lambda$$ at $$t=d$$, and describes how it splits apart when we deform it to nearby semisimple values of $$t$$. 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 $$X\in \underline{\mathrm{Re}}\!\operatorname{p}S_d$$ there is an object $$\operatorname{Lift}_d(X)$$ in $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$, uniquely defined up to isomorphism, for $$t$$ in a formal neighborhood of $$d$$ such that for all $$X,Y\in \underline{\mathrm{Re}}\!\operatorname{p}S_d$$,

- (1) $$\operatorname{Lift}_d(X\oplus Y)\cong \operatorname{Lift}_d(X)\oplus \operatorname{Lift}_d(Y)$$;
- (2) $$\operatorname{Lift}_d(X\otimes Y)\cong \operatorname{Lift}_d(X)\otimes \operatorname{Lift}_d(Y)$$;
- (3) for any indecomposable object $$X_\lambda \in \underline{\mathrm{Re}}\!\operatorname{p}S_d$$, either
- (a)
$$\operatorname{Lift}_d(X_\lambda )$$ remains indecomposable and is still labeled by $$X_\lambda$$

or

- (b) $$\operatorname{Lift}_d(X_\lambda )$$ decomposes as $$X_\lambda \oplus X_{\lambda ^{\prime }}$$ for a partition $$\lambda ^{\prime }$$ depending on $$d$$ and $$\lambda$$ satisfying $$|\lambda | > |\lambda ^{\prime }|$$.

- (a)

Remark 2.3.We believe that over the complex numbers one can check that this lifting operation can be defined for $$t$$ in an analytic neighborhood of $$d$$, 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 $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$, for *any* $$t\in \mathbb {C}$$, as an ultraproduct of the representation categories $$\operatorname{Rep}_p S_d$$, cf. Section 2.1. The case for $$t$$ 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 $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ up to equivalence of symmetric monoidal categories (under isomorphisms of fields $$\mathbb {C}\rightarrow \mathbb {C}$$ that send one transcendental parameter to the other).

Definition 2.4.For any object $$X$$ in a Karoubian or abelian tensor category, we denote by $$\langle X\rangle$$ the full Karoubian tensor subcategory generated by $$X$$.

Note that $$\langle X\rangle$$ is generated by duals, tensor products, direct sums, and direct summands involving $$X$$. In the situations we discuss, $$X$$ 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 $$\mathbf {t}=(t_i)_{i\in \mathbb {N}}$$ and a sequence of fields $$(\mathbb {k}_i)_{i\in \mathbb {N}}$$. We denote by $$X$$ the object $$\prod _{\mathcal {U}} X_{t_i}$$ in $$\prod _{\mathcal {U}}\operatorname{Rep}_{\mathbb {k}_i}S_{t_i}$$, and by $$t$$ the complex number corresponding to $$\prod _{\mathcal {U}} t_i$$ under the respective isomorphism of fields $$\prod _{\mathcal {U}}\mathbb {k}_i\cong \mathbb {C}$$.

- (1) $$t$$ transcendental: Consider a sequence $$\mathbf {t}=(t_i)_{i\in \mathbb {N}}$$ with $$\liminf _{i} t_i=\infty$$ and the ultraproduct $$\prod _{\mathcal {U}}\operatorname{Rep}S_{t_i}$$. Then, under an isomorphism of fields $$\prod _{\mathcal {U}}\mathbb {C}\cong \mathbb {C}$$, $$\langle X\rangle \subset \prod _\mathcal {U}\operatorname{Rep}S_{t_i}$$ is equivalent to $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ as a $$\mathbb {C}$$-linear symmetric monoidal category.
- (2) $$t\in \overline{\mathbb {Q}}\setminus \mathbb {Z}_{\geqslant 0}$$, with minimal polynomial $$m_t$$: We can choose increasing sequences of positive integers $$\mathbf {t}=(t_i)_{i\in \mathbb {N}}$$ and primes $$\mathbf {p}=(p_i)_{i\in \mathbb {N}}$$ such that $$t_i<p_i$$ for any $$i\in \mathbb {N}$$, satisfying
$$$\begin{align*} m_t(t_i)\equiv 0 \mod {p_i}, \qquad\forall i\in \mathbb {N}. \end{align*}$$$Then, under an isomorphism of fields $$\prod _{\mathcal {U}}\overline{\mathbb {F}}_{p_i}\cong \mathbb {C}$$, $$\langle X \rangle \subset \prod _{\mathcal {U}}\operatorname{Rep}_{p_i}S_{t_i}$$ is equivalent to $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ as a $$\mathbb {C}$$-linear symmetric monoidal category.
- (3) $$t=d\in \mathbb {Z}_{\geqslant 0}$$: Set $$t_i=p_i+d$$ for $$p_i$$ the $$i$$th prime number. Then again, under an isomorphism of fields $$\prod _{\mathcal {U}}\overline{\mathbb {F}}_{p_i}\cong \mathbb {C}$$, $$\langle X \rangle \subset \prod _{\mathcal {U}}\operatorname{Rep}_{p_i}S_{p_i+d}$$ is equivalent to $$\underline{\mathrm{Re}}\!\operatorname{p}S_d$$ as a $$\mathbb {C}$$-linear symmetric monoidal category.

We remark that under the chosen isomorphisms of fields, the respective functors $$ \underline{\mathrm{Re}}\!\operatorname{p}S_t\rightarrow \langle X \rangle$$ are obtained by the universal property of $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ by sending the tensor generator of $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ to the object $$X=\prod _{\mathcal {U}}X_{t_i}$$ in the ultraproduct.

Since $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ is defined in terms of its generating object $$X$$, there is a natural exhaustive filtration $$(\underline{\mathrm{Re}}\!\operatorname{p}S_t)^{\leqslant k , m}$$ encoding the complexity of objects in terms of this generator. Explicitly, $$(\underline{\mathrm{Re}}\!\operatorname{p}S_t)^{\leqslant k , m}$$ is the full subcategory of objects which are direct summands of objects $$X^{\otimes j_1} \oplus \dots \oplus X^{\otimes j_\ell }$$, where each $$j_i$$ is at most $$k$$ and the number of terms $$\ell$$ is at most $$m$$. We may similarly filter the categories $$\operatorname{Rep}_{p_i} S_{n_i}$$ in terms of the defining $$n_i$$-dimensional representation of $$S_{n_i}$$, analogously calling the filtered pieces $$(\operatorname{Rep}_{p_i} S_{n_i})^{\leqslant k , m}$$.

The advantage of these filtrations is that Theorem 2.5 tells us that $$(\underline{\mathrm{Re}}\!\operatorname{p}S_t)^{\leqslant k , m}$$ is equivalent to the ultraproduct of $$(\operatorname{Rep}_{p_i} S_{n_i})^{\leqslant k , m}$$, with no need to further cut down the ultraproduct. These subcategories $$(\underline{\mathrm{Re}}\!\operatorname{p}S_t)^{\leqslant k , m}$$ 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 $$m \in \mathbb {N}$$ we obtain a coarser filtration $$\mathcal {C}^{\leqslant k}$$, for either $$\mathcal {C}=\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ or $$\mathcal {C}= \operatorname{Rep}_{p_i} S_{n_i}$$. These filtered pieces are additive and in fact abelian in the case where $$t \notin \mathbb {Z}_{\geqslant 0}$$. These are not monoidal subcategories but they satisfy the condition that if $$V \in \mathcal {C}^{\leqslant k}$$ and $$W \in \mathcal {C}^{\leqslant k^{\prime }}$$, then $$V\otimes W\in \mathcal {C}^{\leqslant k+k^{\prime }}$$, which makes the Grothendieck ring a filtered ring.

The descriptions of $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ can be used to transfer any first-order statement in the signature of symmetric monoidal categories with a distinguished object $$X$$ from the classical (modular) representation theory to the limit, that is, the interpolation category $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ using Łoś' theorem. For more details and applications of this philosophy, see [26, 28].

The combinatorial description of $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ recalled in Section 2.2 can be matched with the above characterization through ultrafilters in Theorem 2.5, giving an evaluation of the equivalence $$\langle X \rangle \simeq \underline{\mathrm{Re}}\!\operatorname{p}S_t$$ on morphisms.

We may extend the assignment $$\pi \mapsto \prod _{\mathcal {U}}\pi _i$$ using the fixed isomorphism of fields $$\mathbb {C}\stackrel{\sim }{\longrightarrow }\prod _{\mathcal {U}}\overline{\mathbb {F}}_{p_i}, \alpha _i\mapsto \prod _{\mathcal {U}}\alpha _i$$ by $$\alpha \pi \mapsto \prod _{\mathcal {U}}\alpha _i\pi _i$$, for $$\alpha \in \mathbb {C}$$. We note that $$m\in \mathbb {Z}$$ corresponds to $$\prod _\mathcal {U}m$$ under the isomorphism of fields, but for a general complex number $$\alpha$$, $$\alpha _i$$ is only defined for almost all $$i$$.

### 2.4 Representations of a fixed group in large characteristic

In addition to symmetric groups, we will often have an auxiliary finite group $$G$$ for which we want to compare representations across different large characteristics.

Let $$G$$ be a fixed finite group, and let $$X$$ be a faithful representation of $$G$$ defined over the integers (for example, the permutation representation). In a slight abuse of notation we will use $$X$$ to denote the corresponding base changes to $$\operatorname{Rep}G$$ as well as to $$\operatorname{Rep}_p G$$. As $$X$$, in particular, labels an object in each of the categories $$\operatorname{Rep}_{p_i} G$$ for any sequence of primes $$(p_i)_{i\in \mathbb {N}}$$, we also have an object $$\prod _\mathcal {U}X$$ in the ultraproduct $$\prod _\mathcal {U}\operatorname{Rep}_{p_i} G$$, which we will also denote by $$X$$. The following result of Crumley realizes the characteristic zero representation theory of $$G$$ as an ultraproduct of the representations of $$G$$ in large characteristic.

Theorem 2.6. ([[11], Section 9.5.1])Let $$\mathbf {p} = (p_i)_{i\in \mathbb {N}}$$ be an arbitrary increasing sequence of primes. Under an isomorphism of fields $$\prod _{\mathcal {U}}\overline{\mathbb {F}}_{p_i}\cong \mathbb {C}$$, there exists an equivalence of $$\mathbb {C}$$-linear symmetric monoidal categories between $$\langle X \rangle \subset \prod _{\mathcal {U}} \operatorname{Rep}_{p_i}G$$ and $$\operatorname{Rep}G$$.

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 $$p$$ does not divide $$|G|$$, then the representation theory of $$G$$ over an algebraically closed field of characteristic $$p$$ 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 $$\mathcal {O}$$ is the ring of integers in a number field $$\mathbb {k}$$ such that every irreducible complex representation of $$G$$ is defined over $$\mathcal {O}$$, $$p$$ is a prime number not dividing $$|G|$$ or the discriminant of $$\mathcal {O}$$, and let $$\mathfrak {p}$$ be a prime ideal of $$\mathcal {O}$$ lying above $$p$$. If $$V$$ is an irreducible complex representation of $$G$$, choose an integral form $$V_\mathcal {O}$$ defined over $$\mathcal {O}$$ and define its reduction modulo $$p$$ as $$V_\mathfrak {p} := V_\mathcal {O} \otimes _{\mathcal {O}} \mathcal {O}/\mathfrak {p}$$.

- (1) $$V_\mathfrak {p}$$ is an
*(*absolutely*)*irreducible representation of $$G$$ over $$\mathcal {O}/\mathfrak {p}$$. - (2) The isomorphism class of $$V_\mathfrak {p}$$ is independent of the choice of the integral form $$V_\mathcal {O}$$.
- (3) If $$\mathfrak {p}^{\prime }$$ is another prime lying above $$p$$, then there exists an automorphism $$\sigma$$ of $$\mathbb {k}$$ sending $$\mathfrak {p}$$ to $$\mathfrak {p}^{\prime }$$. The induced isomorphism $$\tilde{\sigma }: \mathcal {O}/\mathfrak {p} \rightarrow \mathcal {O}/\mathfrak {p}^{\prime }$$ identifies $$V_\mathfrak {p}$$ with $$V_{\mathfrak {p}^{\prime }}$$.
- (4) Every irreducible representation of $$G$$ over $$\mathcal {O}/\mathfrak {p}$$ arises this way.
- (5) If $$V$$ and $$W$$ are two irreducible complex representations, then $$V_\mathfrak {p} \cong W_\mathfrak {p}$$ if and only if $$V \cong W$$.

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 $$C \subset G$$ is a conjugacy class, then the element $$\sum _{g \in C} g \in Z(\mathbb {Z}G)$$ defines an endomorphism of the identity functor in $$\operatorname{Rep}G$$ and $$\operatorname{Rep}_{p}G$$ for all primes $$p$$. Evaluating the trace of these endomorphisms on a representation $$V$$ gives the central character of $$V$$. Over the complex numbers the central character of $$V$$ is just a rescaling of the ordinary character, and in particular determines $$V$$ up to isomorphism.

Fix $$\mathcal {O}$$ as in Theorem 2.7, and for each prime $$p_i$$ as in Theorem 2.6 fix a prime ideal $$\mathfrak {p}_i$$ lying above $$p_i$$ as well as an algebraic closure $$\overline{\mathcal {O}/\mathfrak {p}_i}$$ of $$\mathcal {O}/\mathfrak {p}_i$$. We may choose our isomorphism $$\mathbb {C} \cong \prod _\mathcal {U} \overline{\mathcal {O}/\mathfrak {p}_i}$$ such that on $$\mathcal {O}$$ it is the ultraproduct of the natural quotient maps $$\mathcal {O} \rightarrow \mathcal {O}/\mathfrak {p}_i$$.

Proposition 2.8.Under the equivalence of categories defined in Theorem 2.6, the image of an irreducible representation $$V \in \operatorname{Rep}G$$ is isomorphic to $$\prod _{\mathcal {U}} V_{\mathfrak {p}_i} \in \prod _{\mathcal {U}} \operatorname{Rep}_{\mathfrak {p}_i} G$$.

Proof.A priori we know that under this equivalence of categories $$\prod _{\mathcal {U}} V_{p_i}$$ gets identified with some irreducible complex representation $$V^{\prime }$$; hence, it suffices to check that $$V$$ and $$V^{\prime }$$ have the same central character. $$V$$ is defined over $$\mathcal {O}$$, so therefore its central character is as well. Moreover, by construction the central character of $$V_{\mathfrak {p_i}}$$ is the reduction modulo $$\mathfrak {p_i}$$ of the central character of $$V$$. Since we chose our identification $$\mathbb {C} \cong \prod _\mathcal {U} \overline{\mathcal {O}/\mathfrak {p}_i}$$ to identify $$\mathcal {O}$$ with the product of its reductions modulo $$\mathfrak {p}_i$$, we see that indeed these characters agree.$$\Box$$

### 2.5 The Grothendieck ring of $$\protect\underline{\mathrm{Re}}\!\operatorname{p}S_t$$

*additive Grothendieck ring*, that is, the quotient the free ring generated by all isomorphism classes of objects in $$\mathcal {C}$$ by the ideal of relations given through direct sums and tensor products. If we denote by $$[Y]$$ the symbol of an object $$Y$$ of $$\mathcal {C}$$ inside of the ring $$K_0^\oplus (\mathcal {C})$$, these relations are

The Grothendieck ring of $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ is a filtered ring: Recall the filtration on the category $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ from Section 2.3. Here, an object $$Y\in \underline{\mathrm{Re}}\!\operatorname{p}S_t$$ is in $$(\underline{\mathrm{Re}}\!\operatorname{p}S_t)^{\leqslant k}$$, for $$k\geqslant 0$$, if $$Y$$ is isomorphic to a direct summand of a sum of objects $$X^{\otimes l}$$ with $$l\leqslant k$$, where $$X$$ is the tensor generator of $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$. This defines a filtration of the ring $$K_0^\oplus (\underline{\mathrm{Re}}\!\operatorname{p}S_t )$$.

Let us denote the irreducible complex $$S_n$$-module corresponding uniquely up to isomorphism to a partition $$\lambda \vdash n$$ by $$S^\lambda$$.

Lemma 2.9. ([[14], Proposition 5.11], [[8], Proposition 3.12], [[27], Theorem 3.3])Sending $$[X_\lambda ]\mapsto [S^\lambda ]$$ induces an isomorphism $$\operatorname{gr}K_0^\oplus (\underline{\mathrm{Re}}\!\operatorname{p}S_t)\cong \bigoplus _{n\geqslant 0} K_0(\operatorname{Rep}S_n)$$, 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 $$t$$, this result exhibits the Grothendieck ring of $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ as a filtered deformation of that ring.

### 2.6 Induction and restriction between Deligne categories

*restriction functor*

*induction*and

*co-induction*functors

Given a partition $$\lambda$$ of size $$n$$ we recall that $$S^\lambda$$ denotes the corresponding irreducible representation of $$S_n$$. We will use $$X_\lambda$$ to denote the corresponding indecomposable object of $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$, see Section 2.1. Note that by Theorem 2.1, $$X_\lambda \in (\underline{\mathrm{Re}}\!\operatorname{p}S_t)^{\leqslant |\lambda |}$$ but not in $$(\underline{\mathrm{Re}}\!\operatorname{p}S_t)^{\leqslant |\lambda |-1}$$.

If $$\lambda = (\lambda _1, \lambda _2, \dots \lambda _\ell )$$ is a partition, then for $$n$$ sufficiently large define the *padded partition* $$\lambda [n] = (n- |\lambda |, \lambda _1, \lambda _2, \dots \lambda _\ell )$$. In terms of Young diagrams, this padding operation just adds a long first row to $$\lambda$$ to make it have $$n$$ total boxes. The relevance for our purposes is that Comes and Ostrik showed that if $$n-|\lambda |\geqslant \lambda _1$$ (that is, if $$\lambda [n]$$ defines a Young diagram), then $$X_\lambda \in \underline{\mathrm{Re}}\!\operatorname{p}S_n$$ gets mapped to $$S^{\lambda [n]} \in \operatorname{Rep}S_n$$ under the specialization functor $$\mathcal {F}_n\colon \underline{\mathrm{Re}}\!\operatorname{p}S_n\rightarrow \operatorname{Rep}S_n$$ [8, Proposition 3.25].

For a partition $$\lambda$$ let $$\lambda - h.s$$. denote the set of partitions $$\mu$$ which can be obtained from $$\lambda$$ 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 $$X_\lambda$$ and certain easy to work with induced objects.

Lemma 2.10.

- (1) For $$t \not\in \mathbb {Z}_{\geqslant 0}$$ or for $$t \in \mathbb {Z}$$ with $$t \gg |\lambda |$$
$$$\begin{equation*} \underline{\operatorname{Ind}}_{S_k \times S_t}^{S_{t+k}}(S^{\lambda } \boxtimes \mathbf {1}) = \bigoplus _{\mu \in \lambda -h.s.} X_\mu . \end{equation*}$$$In particular, the right-hand side is $$X_\lambda$$ plus terms $$X_\mu$$ with $$|\mu | < |\lambda |$$.
- (2) For $$t \in \mathbb {Z}_{\geqslant 0}$$ the object $$\underline{\operatorname{Ind}}_{S_k \times S_t}^{S_{t+k}}(S^{\lambda } \boxtimes \mathbf {1})$$ decomposes as a multiplicity-free direct sum of indecomposable objects $$X_\mu$$, such that:
- (a) $$X_\lambda$$ occurs with multiplicity one,
- (b) Each $$X_\mu$$ that appears has $$\mu \in \lambda -\text{h.s.}$$
*(*but not all such $$\mu$$ need appear*)*.

Proof.The ordinary Pieri rule for symmetric groups tells us that in characteristic 0 or $$p > n+k$$

Now suppose $$t=d \in \mathbb {Z}_{\geqslant 0}$$. A priori one knows that $$\operatorname{Ind}_{S_k \times S_d}^{S_{d+k}}(S^{\lambda } \boxtimes \mathbf {1})$$ decomposes as a direct sum of indecomposable objects $$X_\nu$$ with some multiplicities $$c_\nu$$. If we apply the Comes–Ostrik lifting operator $$\operatorname{Lift}_d$$ 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 $$X_\lambda$$ appears as a direct summand of $$\operatorname{Lift}_{d}(X_\nu )$$ for $$\nu = \lambda$$ and in that case, at most one other partition $$\lambda ^{\prime }$$, with $$|\lambda ^{\prime }| > |\lambda |$$ appears as a summand giving $$\operatorname{Lift}_{d}(X_{\lambda ^{\prime }}) = X_{\lambda ^{\prime }} \oplus X_\lambda$$. Since $$\lambda$$ is the largest partition appearing in $$\lambda -h.s$$. and it appears with multiplicity one, we see that $$X_\lambda$$ must occur at $$t=d$$ with multiplicity one, proving part (1).

To show part (2) we then inductively apply the same logic as above to the next largest partitions $$\mu \in \lambda -h.s$$. Note that as we repeat this argument for some $$\mu \in \lambda -{h.s.}$$, the term $$X_\mu$$ might come from $$\operatorname{Lift}_d(X_{\mu ^{\prime }})$$ for some larger $$\mu ^{\prime } \in \lambda -{h.s.}$$ already accounted for, in which case $$X_\mu$$ does not appear in the induction at $$t=d$$.$$\Box$$

Note that this lemma gives us an alternative way of characterizing the indecomposable object $$X_\lambda$$ as the unique direct summand of $$\operatorname{Ind}_{S_k \times S_t}^{S_{t+k}}(S^{\lambda } \boxtimes \mathbf {1})$$ not occurring as a summand of $$\operatorname{Ind}_{S_j \times S_t}^{S_{t+j}}(S^{\mu } \boxtimes \mathbf {1})$$ for any partition $$\mu$$ with $$|\mu | < |\lambda |$$.

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])

### 2.7 The abelian envelope of Deligne's categories

An (abelian) multitensor category $$\mathcal {D}$$ is called an *abelian envelope* of a Karoubian rigid monoidal tensor category $$\mathcal {C}$$ if it contains $$\mathcal {C}$$ and for any multitensor category $$\mathcal {D}^{\prime }$$, the category of tensor functors $$\mathcal {D}\rightarrow \mathcal {D}^{\prime }$$ is equivalent to the category of faithful monoidal functors $$\mathcal {C}\rightarrow \mathcal {D}^{\prime }$$ 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 $$d\in \mathbb {Z}_{\geqslant 0}$$, the abelian envelope $$\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d$$ of $$\underline{\mathrm{Re}}\!\operatorname{p}S_d$$ ([4, Example 2.45(1)]) has several explicit constructions. The first construction uses a symmetric monoidal functor $$\underline{\mathrm{Re}}\!\operatorname{p}S_d\rightarrow \underline{\mathrm{Re}}\!\operatorname{p}S_{-1}$$ and displays the abelian envelope as a category of representations over an algebraic group object internal to the semisimple category $$\underline{\mathrm{Re}}\!\operatorname{p}S_{-1}$$. The second construction introduces a $$t$$-structure on complexes of objects in $$\underline{\mathrm{Re}}\!\operatorname{p}S_d$$ and displays the abelian envelope as the heart of this $$t$$-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 $$S_{p_i+d}$$ in finite characteristic $$p_i$$ (rather than just those in $$\langle X_{p_i+d}\rangle$$) 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 $$\langle X\rangle$$ under subquotients.

Theorem 2.12. ([[26], Theorem 1.1(b)])Set $$t_i=p_i+d$$ for $$p_i$$ the $$i$$th prime number. Then there are equivalences of additive categories

Comes and Ostrik showed ([9, Proposition 2.9, Corollary 4.6]) that each block of $$\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d$$ is equivalent to a block in the category of representations of quantum $$SL(2)$$, with the objects in $$\underline{\mathrm{Re}}\!\operatorname{p}S_d$$ corresponding to tilting objects. As a consequence, $$\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d$$ itself forms a highest weight category with the indecomposable objects $$X_\lambda \in \underline{\mathrm{Re}}\!\operatorname{p}S_d$$ as the indecomposable tilting objects. The simple objects $$D^\lambda$$ of $$\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d$$ are again indexed by partitions, with $$D^\lambda$$ appearing as a composition factor of $$X_{\lambda }$$ with multiplicity one, and all other composition factors $$X_{\lambda }$$ are of the form $$D^\mu$$ with $$\mu$$ satisfying $$|\mu | < |\lambda |$$.

Under the ultrafilter identification the indecomposable module $$X_\lambda$$ corresponds to the so-called Young modules $$Y(\lambda [p_i+d])$$, which arise as direct summands of the permutation representation. The simple objects $$D^\lambda$$ of $$\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d$$ correspond to ultraproducts of simple objects $$D^{\lambda [p_i+d]}$$. We note that while in general $$\operatorname{Rep}S_n$$ is not a highest weight category, the truncated categories $$(\operatorname{Rep}S_n)^{\leqslant k}$$ with $$k$$ less than the characteristic $$p$$ 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 $$\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d$$ 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 $$\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d$$. We can, however, relax it by passing to the Grothendieck ring and instead keeping track of composition multiplicities.

For an abelian category $$\mathcal {A}$$, the *(abelian) Grothendieck ring* $$K_0(\mathcal {A})$$ is the quotient of the additive Grothendieck ring $$K_0^{\oplus }(\mathcal {A})$$ from Section 2.5 by relations obtained from short exact sequences. If $$\mathcal {A}$$ is semisimple, then $$K_0(\mathcal {A})=K_0^{\oplus }(\mathcal {A})$$.

Corollary 2.13.In $$K_0(\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d)$$ one has

Proof.This follows immediately from Corollary 2.11 by passing to the Grothendieck ring and then substituting $$[X_\lambda ] = [D^\lambda ] + \lbrace l.o.t\rbrace$$ everywhere and collecting all of the lower order terms to one side.$$\Box$$

We have the following analog of Lemma 2.9 for the abelian Grothendieck ring of $$\underline{\mathrm{Re}}\!\operatorname{p}S_d$$.

Lemma 2.14.Sending $$[D^\lambda ]\mapsto [S^\lambda ]$$ induces an isomorphism of rings from $$\operatorname{gr}K_0(\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d)$$ to $$\bigoplus _{n\geqslant 0} K_0(\operatorname{Rep}S_n)$$, 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 $$[X_\lambda ] = [ D^\lambda ] + \lbrace \text{l.o.t.}\rbrace$$ implies that these $$[X_\lambda ]$$ form a basis for $$K_0(\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d)$$, and therefore the inclusion $$K_0^\oplus (\underline{\mathrm{Re}}\!\operatorname{p}S_d) \hookrightarrow K_0(\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d)$$ is in fact an isomorphism. Moreover, this upper triangularity property implies that the assignment $$[D^\lambda ] \rightarrow [X_\lambda ]$$ defines an isomorphism of associated graded rings $$ \operatorname{gr}K_0(\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d) \cong \operatorname{gr}K_0^\oplus (\underline{\mathrm{Re}}\!\operatorname{p}S_d)$$.

Next, the first two properties of lifting in Theorem 2.2 say that the map $$[X] \mapsto [\operatorname{Lift}_d(X)]$$ defines a ring homomorphism $$K_0^\oplus (\underline{\mathrm{Re}}\!\operatorname{p}S_d) \rightarrow K_0(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$ at a nearby transcendental value of $$t$$. The third property in Theorem 2.2 is again an upper triangularity property, which implies that this homomorphism is an isomorphism, but moreover that $$[X_\lambda ] \mapsto [X_\lambda ]$$ defines an isomorphism of associated graded rings $$\operatorname{gr}K_0^\oplus (\underline{\mathrm{Re}}\!\operatorname{p}S_d) \rightarrow \operatorname{gr}K_0(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$ at generic $$t$$.

Finally we use the isomorphism $$\operatorname{gr}K_0(\underline{\mathrm{Re}}\!\operatorname{p}S_t) \cong \bigoplus _{n\geqslant 0} K_0(\operatorname{Rep}S_n)$$ of Lemma 2.9, which sends $$[X_\lambda ]$$ to $$[S^\lambda ]$$. Composing these three isomorphisms gives the desired result.$$\Box$$

### 2.8 The projective objects in the abelian envelope

We now turn to describing the projective objects in $$\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d$$ for later use. Key to understanding of the projectives will be the projective cover of the tensor unit.

Lemma 2.15.Let $$P\cong \prod _{\mathcal {U}} P_i$$ be an object in $$\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d$$. Then $$P$$ is projective in $$\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d$$ if and only if there exists an integer $$k_0\geqslant 0$$ such that for each integer $$k\geqslant k_0$$, almost all $$P_i$$ are projective objects in $$(\operatorname{Rep}_{p_i} S_{t_{i}})^{\leqslant k}$$.

Proof.By virtue of $$P$$ being an object of $$\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d$$, there exists $$k_0\geqslant 0$$ such that $$P\in (\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d)^{\leqslant k_0}$$. In particular, $$P$$ is contained in $$P\in (\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d)^{\leqslant k}$$ for any $$k\geqslant k_0$$. Hence, for fixed $$k$$, almost all $$P_i$$ are contained in $$(\operatorname{Rep}_{p_i} S_{t_{i}})^{\leqslant k}$$. Assume that $$P$$ is projective. The property that $$P$$ is projective is defined by the functor $$\operatorname{Hom}(P,-)$$ being right exact, as a functor on $$(\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d)^{\leqslant k}$$, which can be expressed as a first-order property. Thus, $$P_i$$ is a projective object in $$(\operatorname{Rep}_{p_i} S_{t_{i}})^{\leqslant k}$$, for almost all $$i$$.

Conversely, again using Łoś' theorem, we see that if $$P_i$$ is a projective object in $$(\operatorname{Rep}_{p_i} S_{t_{i}})^{\leqslant k}$$, for almost all $$i$$, then $$P$$ is projective in $$(\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d)^{\leqslant k}$$. Since $$\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d$$ is the filtered colimit of all of these categories, projectivity of $$P$$ holds in the entire category $$\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d$$.$$\Box$$

Recall the indecomposable objects $$X_{\lambda }$$ of $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$, see Section 2.2 and consider the case $$\lambda =(d+1)$$.

Lemma 2.16.The indecomposable object $$X_{(d+1)}\in (\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d)^{\leqslant d+1}$$ is the projective cover of the tensor unit $$\mathbf {1}$$.

Proof.Recall from Theorem 2.12 that $$(\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d)^{\leqslant k}$$ is equivalent to the ultraproduct of the categories $$(\operatorname{Rep}_{p_i}S_{p_i+d})^{\leqslant k}$$. Observe that $$\underline{\operatorname{Ind}}$$ is left adjoint of a functor-preserving epimorphisms and thus preserves projective objects; therefore, any object of the form $$\operatorname{Ind}_{S_{d+1}\times S_{p_i-1}}^{S_{p_i+d}}(V\boxtimes W)$$ is projective in $$\operatorname{Rep}_{p_i}S_{p_i+d}$$ as it is induced from a semisimple category where all objects are projective. In particular, observe that

By Lemma 2.10, $$P\cong X_{(d+1)}\oplus \text{(l.o.t)}$$. The indecomposables appearing as lower order terms are of the form $$X_{(k)}$$, with $$k\leqslant d+1$$. The combinatorial description of objects in the block of $$X_{(d+1)}$$ from [9, Theorem 2.6] implies that of the objects $$X_{(k)}$$, only $$X_{(0)}=\mathbf {1}$$ belongs to this block and $$\operatorname{Hom}(P,Y)=0$$ for objects $$Y$$ from other blocks. However, $$\mathbf {1}$$ is not projective as $$\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d$$ is not semisimple. Thus, $$X_{(d+1)}$$ is the projective cover of $$\mathbf {1}$$.$$\Box$$

Proposition 2.17.Let $$X\in (\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d)^{\leqslant k}$$. Then $$X$$ is projective in $$(\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d)^{\leqslant (k+d+1)}$$ if and only if $$X$$ is projective in $$\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d$$.

Proof.Given an indecomposable object $$X$$ in $$(\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d)^{\leqslant k}$$, we can use Lemma 2.16 to see that the projective cover of $$X$$ is contained in $$(\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d)^{\leqslant (k+d+1)}$$. Indeed, $$X\otimes X_{(d+1)}$$ is a projective object in $$(\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d)^{\leqslant (k+d+1)}$$ with an epimorphism to $$X\otimes \mathbf {1}\cong X$$. Now, $$X$$ is projective in an additive subcategory $$\mathcal {C}$$ of $$\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d$$ if and only if its projective cover is contained in $$\mathcal {C}$$ and $$X$$ is isomorphic to its projective cover. By the above observation and Lemma 2.15, this condition holds in $$\mathcal {C}=\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d$$ if and only if it holds in $$\mathcal {C}=(\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d)^{\leqslant (k+d+1)}$$.$$\Box$$

The advantage of the above proposition is that projectivity of an object in $$\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d$$ can be checked using the components of an ultrafilter presentation, as well as the following observation.

Corollary 2.18.All projective objects of $$\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d$$ are contained in the subcategory $$\underline{\mathrm{Re}}\!\operatorname{p}S_d$$.

Proof.This follows directly from [9, Remark 4.8] since for any object $$P$$ of $$\underline{\mathrm{Re}}\!\operatorname{p}^\mathrm{ab}S_d$$ there exists a surjective map $$\bigoplus _{i=1}^m X^{\otimes n_i}\twoheadrightarrow P$$. Hence, if $$P$$ is projective, it is a direct summand of an object in $$\underline{\mathrm{Re}}\!\operatorname{p}S_d$$.$$\Box$$

### 2.9 Indecomposable Yetter–Drinfeld modules over $$S_n$$ in arbitrary characteristic

In this section, let $$\mathbb {k}$$ be an algebraically closed field and $$G$$ a finite group. We now turn to background results on the monoidal center $$\mathcal {Z}(\operatorname{Rep}_\mathbb {k}G)$$ of the category $$\operatorname{Rep}_\mathbb {k}G$$. We employ the equivalent description of this braided tensor category as *Yetter–Drinfeld modules* [43, Definition 3.6], that is, $$G$$-graded $$G$$-representations $$V=\bigoplus _{g\in G}V_g$$ such that $$g\cdot V_h=V_{ghg^{-1}}$$, see, for example, [32, Proposition 7.1.6]. If $$v\in V_g$$, we write $$|v|=g$$ for the *degree* of $$v$$. Equivalently, $$\mathcal {Z}(\operatorname{Rep}_\mathbb {k}G)$$ can be described as modules over the *Drinfeld double* $$\operatorname{Drin}(G)$$ of $$G$$ [18]. We recall the following result found in [16] for $$\mathbb {k}=\mathbb {C}$$ and in [42, Corollary 2.3] for general characteristic.

Theorem 2.19. (Dijkgraaf–Pasquier–Roche, Witherspoon)Let $$G$$ be a finite group and $$\mathbb {k}$$ be an algebraically closed field. A complete list of indecomposable *(*respectively, irreducible*)* objects in $$\mathcal {Z}(\operatorname{Rep}_\mathbb {k}G)$$ is given by modules of the form $$W_{\sigma ,V}=\operatorname{Ind}_Z^G(V)$$ where $$\sigma$$ is a representative of a conjugacy class of elements in $$G$$ and $$V$$ is an indecomposable *(*respectively, irreducible*)* module over the centralizer $$Z=Z(\sigma )$$ of $$\sigma$$ in $$G$$.

The following lemma is easily seen from the description of $$\mathcal {Z}(\operatorname{Rep}_\mathbb {k}G)$$ in terms of Yetter–Drinfeld modules over $$G$$.

Lemma 2.20.The functors $$\operatorname{Ind}_Z^G$$ are full and faithful, that is, there are isomorphisms

If $$\sigma$$ and $$\tau$$ are not conjugate, $$V$$ is a $$\mathbb {k}Z(\sigma )$$-module, and $$W$$ is a $$\mathbb {k}Z(\tau )$$-module, then

A direct consequence of this observation is the following: let $$W$$ be a Yetter–Drinfeld $$G$$-module over any field $$\mathbb {k}$$, and let $$W_{g,V}$$ be the simple Yetter–Drinfeld $$G$$-module obtained from $$g$$ in $$G$$ and a simple module $$V$$ over the centralizer $$Z(g)$$ of $$g$$ in $$G$$. Then the multiplicity $$[W:W_{g,V}]$$ in $$\mathcal {Z}(\operatorname{Rep}_\mathbb {k}G)$$ equals the multiplicity $$[W_g:V]$$ in $$\operatorname{Rep}_\mathbb {k}Z(g)$$, where $$W_g$$ is the homogeneous subspace of $$W$$ of degree $$g$$.

Lemma 2.21.$$W_{\sigma ,P}$$ is projective in $$\mathcal {Z}(\operatorname{Rep}_\mathbb {k}G)$$ if and only if $$P$$ is projective in $$\operatorname{Rep}_\mathbb {k}Z(\sigma )$$.

Proof.For $$\sigma \in G$$, consider the regular $$\mathbb {k}Z$$-module and note that $$\operatorname{Ind}_Z^G(\mathbb {k}Z)=\mathbb {k}G$$. As an object in $$\mathcal {Z}(\operatorname{Rep}_\mathbb {k}G)$$, this is the projective $$\operatorname{Drin}(G)$$-submodule $$\operatorname{Drin}(G)\delta _\sigma =\mathbb {k}G\delta _\sigma$$ of $$\operatorname{Drin}(G)$$, where $$\lbrace \delta _g\rbrace _{g\in G}$$ is the basis of $$(\mathbb {k}G)^*$$ dual to the basis $$\lbrace g\rbrace _{g\in G}$$. By functoriality of $$\operatorname{Ind}_Z^G$$, $$P$$ being a direct summand of $$(\mathbb {k}Z)^{\oplus n}$$, it readily follows that $$W_{\sigma ,P}$$ is a direct summand of $$\operatorname{Drin}(G)\delta _\sigma$$ and thus projective.

Assume that $$W:=W_{\sigma ,P}$$ is a direct summand of a direct sum $$R:=((\mathbb {k}G)^*\otimes \mathbb {k}G)^{\oplus m}$$ of the regular module in $$\mathcal {Z}(\operatorname{Rep}_\mathbb {k}G)$$. Note that $$R_\sigma =(\mathbb {k}G\delta _\sigma )^{\oplus m}$$ and $$W_\sigma =P$$. Thus, we obtain that $$P$$ is a direct summand of $$(\mathbb {k}G\delta _\sigma )^{\oplus m}$$ as a $$\mathbb {k}Z$$-module. Choosing a decomposition of $$G$$ into right $$Z$$-cosets, we observe that as a $$\mathbb {k}Z$$-module, $$\mathbb {k}G\delta _\sigma$$ is simply a direct sum of copies of the regular module. Thus, $$P$$ is projective.$$\Box$$

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 $$G$$ a finite group. There is an isomorphism of rings

This corollary is proved in [42, p. 316] and uses a result of G. Lusztig.

## 3 CLASSIFICATION OF INDECOMPOSABLE OBJECTS IN THE CENTER OF $$\protect\underline{\mathrm{Re}}\!\operatorname{p}S_t$$

### 3.1 The center as a model-theoretic limit

Viewing $$\underline{\mathrm{Re}}\!\operatorname{p}S_t$$ as a model-theoretic limit of categories $$\operatorname{Rep}_{p_i} S_{t_i}$$ as in Theorem 2.5 suggests that it may be possible to interpret the center $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$ as a limit of the centers $$\mathcal {Z}(\operatorname{Rep}_{p_i}S_{t_i})$$. An object of the center of a monoidal category consists of the data of an object of the category along with a half-braiding. Therefore, the centers $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)$$ and $$\mathcal {Z}(\operatorname{Rep}_{p_i}S_{t_i})$$ inherit filtrations $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)^{\leqslant k, m}$$ and $$\mathcal {Z}(\operatorname{Rep}_{p_i}S_{t_i})^{\leqslant k,m}$$ just by looking at the underlying object. Precisely, $$\mathcal {Z}(\operatorname{Rep}_{p_i}S_{t_i})^{\leqslant k,m}$$ consists of those objects $$(V,c)$$ where $$V$$ lies in $$(\operatorname{Rep}_{p_i}S_{t_i})^{\leqslant k,m}$$.

Proposition 3.1.Recall the setup of Theorem 2.5(1)–(3). In all cases, the category $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)^{\leqslant k, m}$$ is equivalent to the ultraproduct of the categories $$\mathcal {Z}(\operatorname{Rep}_{p_i}S_{t_i})^{\leqslant k,m}$$ *(*or, $$\mathcal {Z}(\operatorname{Rep}S_{t_i})^{\leqslant k,m}$$*)* as $$\mathbb {C}$$-linear categories with partially defined monoidal and additive structures.

Proof.One direction is fairly clear: If $$(Y_i, c_i)$$ is an object of $$\mathcal {Z}(\operatorname{Rep}_{p_i}S_{t_i})^{\leqslant k,m}$$ for each $$i$$, then we can take the ultraproduct of the underlying objects $$Y_i$$ to obtain an object $$Y$$ in $$(\underline{\mathrm{Re}}\!\operatorname{p}S_t)^{\leqslant k , m}$$. The fact that $$(Y_i, c_i)$$ is an element of $$\mathcal {Z}(\operatorname{Rep}_{p_i}S_{t_i})$$ means that the half-braiding $$c_i$$ is not just defined on $$\mathcal {Z}(\operatorname{Rep}_{p_i}S_{t_i})^{\leqslant k,m}$$, but on all larger filtered pieces as well. Using Łoś' theorem, the ultraproduct $$(Y,c)$$ becomes an object in $$(\underline{\mathrm{Re}}\!\operatorname{p}S_t)^{\leqslant k, m}$$ with half-braiding $$c_Z$$ globally defined for all $$Z\in \underline{\mathrm{Re}}\!\operatorname{p}S_t$$, hence lies in $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)^{\leqslant k, m}$$.

Going the other way requires a bit more work. If we start with an object $$(Y, c)$$ in $$\mathcal {Z}(\underline{\mathrm{Re}}\!\operatorname{p}S_t)^{\leqslant k, m}$$, we can always represent $$Y$$ as an ultraproduct of objects $$Y_i$$ in $$(\operatorname{Rep}_{p_i} S_{t_i})^{\leqslant k , m}$$. However, a priori the half-braiding $$c$$ need not come from an ultraproduct of half-braidings $$c_i$$ globally on each ${Rep}_{{p}_{i}}$