Deligne Category Research Articles

The Structure of the Grothendieck Rings of Wreath Product Deligne Categories and their Generalisations

Abstract Given a tensor category $\mathcal{C}$ over an algebraically closed field of characteristic zero, we may form the wreath product category $\mathcal{W}_n(\mathcal{C})$. It was shown in [10] that the Grothendieck rings of these wreath product categories stabilise in some sense as $n \to \infty $. The resulting “limit” ring, $\mathcal{G}_\infty ^{\mathbb{Z}}(\mathcal{C})$, is isomorphic to the Grothendieck ring of the wreath product Deligne category $S_t(\mathcal{C})$ as defined by [9] (although it is also related to $FI_G$-modules). This ring only depends on the Grothendieck ring $\mathcal{G}(\mathcal{C})$. Given a ring $R$ that is free as a $\mathbb{Z}$-module, we construct a ring $\mathcal{G}_\infty ^{\mathbb{Z}}(R)$ that specialises to $\mathcal{G}_\infty ^{\mathbb{Z}}(\mathcal{C})$ when $R = \mathcal{G}(\mathcal{C})$. We give a description of $\mathcal{G}_\infty ^{\mathbb{Z}}(R)$ using generators very similar to the basic hooks of [5]. We also show that $\mathcal{G}_\infty ^{\mathbb{Z}}(R)$ is a $\lambda $-ring wherever $R$ is and that $\mathcal{G}_\infty ^{\mathbb{Z}}(R)$ is (unconditionally) a Hopf algebra. Finally, we show that $\mathcal{G}_\infty ^{\mathbb{Z}}(R)$ is isomorphic to the Hopf algebra of distributions on the formal neighbourhood of the identity in $(W\otimes _{\mathbb{Z}} R)^\times $, where $W$ is the ring of Big Witt Vectors.

Finite-Dimensional Representations of Yangians in Complex Rank

Abstract We classify the “finite-dimensional” irreducible representations of the Yangians $Y(\mathfrak{g}\mathfrak{l}_t)$ and $Y(\mathfrak{s}\mathfrak{l}_t)$. These are associative ind-algebras in the Deligne category $\textrm{Rep}(GL_t)$, which generalize the regular Yangians $Y(\mathfrak{g}\mathfrak{l}_n)$ and $Y(\mathfrak{s}\mathfrak{l}_n)$ to complex rank. They were first defined in the paper [14]. Here we solve [14, Problem 7.2]. We work with the Deligne category $\textrm{Rep}(GL_t)$ using the ultraproduct approach introduced in [7] and [16].

Deligne categories and representations of the infinite symmetric group

We establish a connection between two settings of representation stability for the symmetric groups Sn over C. One is the symmetric monoidal category Rep(S∞) of algebraic representations of the infinite symmetric group S∞=⋃nSn, related to the theory of FI-modules. The other is the family of rigid symmetric monoidal Deligne categories Rep_(St), t∈C, together with their abelian versions Rep_ab(St), constructed by Comes and Ostrik.We show that for any t∈C the natural functor Rep(S∞)→Rep_ab(St) is an exact symmetric faithful monoidal functor, and compute its action on the simple representations of S∞. Considering the highest weight structure on Rep_ab(St), we show that the image of any object of Rep(S∞) has a filtration with standard objects in Rep_ab(St).As a by-product of the proof, we give answers to the questions posed by P. Deligne concerning the cohomology of some complexes in the Deligne category Rep_(St), and their specializations at non-negative integers n.

Stable Grothendieck rings of wreath product categories

Let k be an algebraically closed field of characteristic zero, and let $${\mathcal {C}} = {\mathcal {R}} -\hbox {mod}$$ be the category of finite-dimensional modules over a fixed Hopf algebra over k. One may form the wreath product categories $$ {\mathcal {W}}_{n}({\mathcal {C}}) = ( {\mathcal {R}} \wr S_n)-\hbox {mod}$$ whose Grothendieck groups inherit the structure of a ring. Fixing distinguished generating sets (called basic hooks) of the Grothendieck rings, the classification of the simple objects in $$ {\mathcal {W}}_{n}({\mathcal {C}}) $$ allows one to demonstrate stability of structure constants in the Grothendieck rings (appropriately understood), and hence define a limiting Grothendieck ring. This ring is the Grothendieck ring of the wreath product Deligne category $$S_t({\mathcal {C}})$$ . We give a presentation of the ring and an expression for the distinguished basis arising from simple objects in the wreath product categories as polynomials in basic hooks. We discuss some applications when $$ {\mathcal {R}} $$ is the group algebra of a finite group, and some results about stable Kronecker coefficients. Finally, we explain how to generalise to the setting where $${\mathcal {C}}$$ is a tensor category.

Nori Motives of Curves With Modulus and Laumon 1-motives

AbstractLet k be a number field. We describe the category of Laumon 1-isomotives over k as the universal category in the sense of M. Nori associated with a quiver representation built out of smooth proper k-curves with two disjoint effective divisors and a notion of for such “curves with modulus”. This result extends and relies on a theorem of J. Ayoub and L. Barbieri-Viale that describes Deligne's category of 1-isomotives in terms of Nori's Abelian category of motives.

Deligne Categories and the Limit of Categories Rep(GL(m|n))

Abstract For each integer $t$ a tensor category $\mathcal{V}_t$ is constructed, such that exact tensor functors $\mathcal{V}_t\rightarrow \mathcal{C}$ classify dualizable $t$-dimensional objects in $\mathcal{C}$ not annihilated by any Schur functor. This means that $\mathcal{V}_t$ is the “abelian envelope” of the Deligne category $\mathcal{D}_t=\operatorname{Rep}(GL_t)$. Any tensor functor $\operatorname{Rep}(GL_t)\longrightarrow \mathcal{C}$ is proved to factor either through $\mathcal{V}_t$ or through one of the classical categories $\operatorname{Rep}(GL(m|n))$ with $m-n=t$. The universal property of $\mathcal{V}_t$ implies that it is equivalent to the categories $\operatorname{Rep}_{\mathcal{D}_{t_1}\otimes \mathcal{D}_{t_2}}(GL(X),\epsilon )$, ($t=t_1+t_2$, $t_1$ not an integer) suggested by Deligne as candidates for the role of abelian envelope.

Categorical actions and multiplicities in the Deligne category [formula omitted

We study the categorical type A action on the Deligne category Dt=Rep_(GLt) (t∈C) and its “abelian envelope” Vt constructed in [13].For t∈Z, this action categorifies an action of the Lie algebra slZ on the tensor product of the Fock space F with Ft∨, its restricted dual “shifted” by t, as was suggested by I. Losev. In fact, this action makes the category Vt the tensor product (in the sense of Losev and Webster, [20]) of categorical slZ-modules Pol and Polt∨. The latter categorify F and Ft∨ respectively, the underlying category in both cases being the category of stable polynomial representations (also known as the category of Schur functors), as described in [16,18].When t∉Z, the Deligne category Dt is abelian semisimple, and the type A action induces a categorical action of slZ×slZ. This action categorifies the slZ×slZ-module F⊠F∨, making Dt the exterior tensor product of the categorical slZ-modules Pol, Pol∨.Along the way we establish a new relation between the Kazhdan–Lusztig coefficients and the multiplicities in the standard filtrations of tilting objects in Vt.

Thick ideals in Deligne's category [formula omitted

We describe indecomposable objects in Deligne's category Re_p(Oδ) and explain how to decompose their tensor products. We then classify thick ideals in Re_p(Oδ). As an application we classify the indecomposable summands of tensor powers of the standard representation of the orthosymplectic supergroup up to isomorphism.

Schur–Weyl duality for Deligne categories, II : The limit case

This paper is a continuation of a previous paper of the author, which gave an analogue to the classical Schur-Weyl duality in the setting of Deligne categories. Given a finite-dimensional unital vector space $V$ (a vector space $V$ with a chosen non-zero vector $\mathbf{1}$), we constructed a complex tensor power of $V$: an $Ind$-object of the Deligne category $\underline{Rep}(S_t)$ which is a Harish-Chandra module for the pair $(\mathfrak{gl}(V), \bar{\mathfrak{P}}_{\mathbf{1}})$, where $\bar{\mathfrak{P}}_{\mathbf{1}} \subset GL(V)$ is the mirabolic subgroup preserving the vector $\mathbf{1}$. This construction allowed us to obtain an exact contravariant functor $\widehat{SW}_{t, V}$ from the category $\underline{Rep}^{ab}(S_t)$ (the abelian envelope of the category $\underline{Rep}(S_t)$) to a certain localization of the parabolic category $\mathcal{O}$ associated with the pair $(\mathfrak{gl}(V), \bar{\mathfrak{P}}_{\mathbf{1}})$. In this paper, we consider the case when $V = \mathbb{C}^{\infty}$. We define the appropriate version of the parabolic category $\mathcal{O}$ and its localization, and show that the latter is equivalent to a "restricted" inverse limit of categories $\widehat{\mathcal{O}}^{\mathfrak{p}}_{t,\mathbb{C}^N}$ with $N$ tending to infinity. The Schur-Weyl functors $\widehat{SW}_{t, \mathbb{C}^N}$ then give an anti-equivalence between this category and the category $\underline{Rep}^{ab}(S_t)$. This duality provides an unexpected tensor structure on the category $\widehat{\mathcal{O}}^{\mathfrak{p}_{\infty}}_{t, \mathbb{C}^{\infty}}$.

Representation theory in complex rank, II

We define and study representation categories based on Deligne categories Rep(GLt), Rep(Ot), Rep(Sp2t), where t is any (non-integer) complex number. Namely, we define complex rank analogs of the parabolic category O and the representation categories of real reductive Lie groups and supergroups, affine Lie algebras, and Yangians. We develop a framework and language for studying these categories, prove basic results about them, and outline a number of directions of further research.

Semisimple and G-Equivariant Simple Algebras Over Operads

Let G be a finite group. There is a standard theorem on the classification of G-equivariant finite dimensional simple commutative, associative, and Lie algebras (i.e., simple algebras of these types in the category of representations of G). Namely, such an algebra is of the form A=FunH(G,B), where H is a subgroup of G, and B is a simple algebra of the corresponding type with an H-action. We explain that such a result holds in the generality of algebras over a linear operad. This allows one to extend Theorem 5.5 of Sciarappa (arXiv:1506.07565) on the classification of simple commutative algebras in the Deligne category Rep(St) to algebras over any finitely generated linear operad.

Deligne categories and reduced Kronecker coefficients

The Kronecker coefficients are the structural constants for the tensor categories of representations of the symmetric groups, namely, given three partitions $${\lambda }, \mu , \tau $$ź,μ,ź of n, the multiplicity of $$\lambda $$ź in $$\mu \otimes \tau $$μźź is called the Kronecker coefficient $$g^{{\lambda }}_{\mu , \tau }$$gμ,źź. When the first part of each of the partitions is taken to be very large (the remaining parts being fixed), the values of the appropriate Kronecker coefficients stabilize; the stable value is called the reduced (or stable) Kronecker coefficient. These coefficients also generalize the Littlewood---Richardson coefficients and have been studied quite extensively. In this paper, we show that reduced Kronecker coefficients appear naturally as structure constants of Deligne categories $$\underline{Rep}(S_t)$$Repź(St). This allows us to interpret various properties of the reduced Kronecker coefficients as categorical properties of Deligne categories $$\underline{Rep}(S_t)$$Repź(St) and derive new combinatorial identities.

Generators for the representation rings of certain wreath products

Working in the setting of Deligne categories, we generalize a result of Marin that hooks generate the representation ring of symmetric groups to wreath products of symmetric groups with a fixed finite group or Hopf algebra. In particular, when we take the finite group to be cyclic order 2 we recover a conjecture of Marin about Coxeter groups in type B.

Schur–Weyl Duality for Deligne Categories

Schur–Weyl Duality for Deligne Categories

On Deligne’s categoryRepab(Sd)

We prove a universal property of Deligne's category $\uRep^{ab}(S_d)$. Along the way, we classify tensor ideals in the category $\uRep(S_d)$.

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

Deligne has defined a category which interpolates among the representations of the various symmetric groups. In this paper we show Deligne's category admits a unique nontrivial family of modified trace functions. Such modified trace functions have already proven to be interesting in both low-dimensional topology and representation theory. We also introduce a graded variant of Deligne's category, lift the modified trace functions to the graded setting, and use them to recover the well-known invariant of framed knots known as the writhe.

On blocks of Deligne's category [formula omitted

We describe blocks in Deligne's category Re ̲ p ( S t ) .