We describe the structure of the tensor product of the basic Fock representation of sl(\infty) with its shifted dual. More precisely we prove that this tensor product has a unique decreasing filtration with simple quotients. We use the categorification of this representation via translation functors in the abelain envelope of the Deligne category GL(t) for integral t. We also compute dimensions of standard and tilting objects in this abelian envelope.
In this paper we study the category of Harish-Chandra bimodules $HC_{χ,ψ}$ in the Deligne category $\text{Rep}(GL_t)$. In particular, we answer Question 3.25 posed in Pavel Etingof's paper arXiv:1407.0373 and determine for which central characters $χ$ and $ψ$ this category is not zero.
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.
We construct a faithful categorical representation of an infinite Temperley-Lieb algebra on the periplectic analogue of Deligne's category. We use the corresponding combinatorics to classify thick tensor ideals in this periplectic Deligne category. This allows to determine the objects in the kernel of the monoidal functor going to the module category of the periplectic Lie supergroup. We use this to classify indecomposable direct summands in the tensor powers of the natural representation, determine which are projective and determine their simple top.
We study the characters of simple modules in the parabolic BGG category of the affine Lie algebra in Deligne's category. More specifically, we take the limit of Weyl-Kac formula to compute the character of the irreducible quotient $L(X,k)$ of the parabolic Verma module $M(X,k)$ of level $k$, where $X$ is an indecomposable object of Deligne's category $\underline{\mathrm{Rep}}(GL_t)$, $\underline{\mathrm{Rep}}(O_t)$, or $\underline{\mathrm{Rep}}(Sp_t)$, under conditions that the highest weight of $X$ plus the level gives a fundamental weight, $t$ is transcendental, and the base field $\Bbbk$ ha…
We define a faithful linear monoidal functor from the partition category, and hence from Deligne's category $\mathrm{\underline{Re}p}(S_t)$, to the Heisenberg category. We show that the induced map on Grothendieck rings is injective and corresponds to the Kronecker coproduct on symmetric functions.
Deligne's category $\underline{\rm Rep}(S_t)$ is a tensor category depending on a parameter $t$ "interpolating" the categories of representations of the symmetric groups $S_n$. We construct a family of categories $\mathcal{C}_λ$ (depending on a vector of variables $λ= (λ_1, λ_2, \ldots, λ_l)$, that may be specialised to values in the ground ring) which are module categories over $\underline{\rm Rep}(S_t)$. The categories $\mathcal{C}_λ$ are defined over any ring and are constructed by interpolating permutation representations. Further, they admit specialisation functors to $S_n$-mod which are …
This paper gives an analogue to the classical Schur-Weyl duality in the setting of Deligne categories. Given a finite-dimensional unital vector space V (i.e. a vector space V with a distinguished non-zero vector 1), we give a definition of a complex tensor power of V. This is an Ind-object of the Deligne category Rep(S_t), equipped with a natural action of gl(V). This construction allows us to describe a duality between the abelian envelope of the category Rep(S_t) and a localization of the parabolic category O for gl(V) associated with the pair (V, 1). In particular, we obtain an exact contra…
We study stabilization of finite-dimensional representations of the periplectic Lie superalgebras $\mathfrak{p}(n)$ as $n \to \infty$. The paper gives a construction of the tensor category $Rep(\underline{P})$, possessing nice universal properties among tensor categories over the category $\mathtt{sVect}$ of finite-dimensional complex vector superspaces. First, it is the "abelian envelope" of the Deligne category corresponding to the periplectic Lie superalgebra, in the sense of arXiv:1511.07699. Secondly, given a tensor category $\mathcal{C}$ over $\mathtt{sVect}$, exact tensor functors $Rep(…
Recently P. Deligne introduced the tensor category Rep(S_t) (for t not necessarily an integer) which in a certain precise sense interpolates the categories Rep(S_d) of representations of the symmetric groups S_d. In this paper we describe the blocks of Deligne's category Rep(S_t).
The Deligne category of symmetric groups is the additive Karoubi closure of the partition category. It is semisimple for generic values of the parameter t while producing categories of representations of the symmetric group when modded out by the ideal of negligible morphisms when t is a non-negative integer. The partition category may be interpreted, following Comes, via a particular linearization of the category of two-dimensional oriented cobordisms. The Deligne category and its semisimple quotients admit similar interpretations. This viewpoint coupled to the universal construction of two-d…
We describe indecomposable objects in Deligne's category $\underline{\operatorname{Re}}\!\operatorname{p}(O_δ)$ and explain how to decompose their tensor products. We then classify thick ideals in $\underline{\operatorname{Re}}\!\operatorname{p}(O_δ)$. As an application we classify the indecomposable summands of tensor powers of the standard representation of the orthosymplectic supergroup up to isomorphism.
Drinfeld suggested the definition of a certain endo-functor, called the pseudo-identity functor, on the category of D-modules on an algebraic stack. We extend this definition to an arbitrary DG category, and show that if certain finiteness conditions are satisfied, this functor is the inverse of the Serre functor. We apply this to the category of (g,K)-modules, and we stipulate that the pseudo-identity functor is the analog of the Deligne-Lusztig functor. In order to support this, we show that the pseudo-identity functor for (g,K)-modules is isomorphic to the composition of cohomological and c…
We study the categorical type A action on the Deligne category $\mathcal{D}_t=\underline{Rep}(GL_t)$ (here $t \in \mathbb{C}$) and its "abelian envelope" $\mathcal{V}_t$ constructed in arXiv:1511.07699. For $t \in \mathbb{Z}$, this action categorifies an action of the Lie algebra $\mathfrak{sl}_{\mathbb{Z}}$ on the tensor product of the Fock space $\mathfrak{F}$ with $\mathfrak{F}_t^{\vee}$, its restricted dual "shifted" by $t$, as was suggested by I. Losev. In fact, this action makes the category $\mathcal{V}_t$ the tensor product (in the sense of Losev and Webster, arXiv:1303.1336) of catego…
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}…
We classify indecomposable summands of mixed tensor powers of the natural representation for the general linear supergroup up to isomorphism. We also give a formula for the characters of these summands in terms of composite supersymmetric Schur polynomials, and give a method for decomposing their tensor products. Along the way, we describe indecomposable objects in Rep(GL_δ) and explain how to decompose their tensor products.
For each integer $t$ a tensor category $V_t$ is constructed, such that exact tensor functors $V_t \longrightarrow C$ classify dualizable $t$-dimensional objects in $C$ not annihilated by any Schur functor. This means that $V_t$ is the "abelian envelope" of the Deligne category $Rep(GL_t)$. Any tensor functor $Rep(GL_t)\longrightarrow C$ is proved to factor either through $V_t$ or through one of the classical categories $Rep(GL(m|n))$ with $m-n=t$. The universal property of $V_t$ implies that it is equivalent to the categories $Rep_{Rep(GL_{t_1})\otimes Rep(GL_{t_2})}(GL(X),ε)$, ($t=t_1+t_2$, $…
We classify the "finite-dimensional" irreducible representations of the Yangians $Y(\mathfrak{gl}_t)$ and $Y(\mathfrak{sl}_t)$. These are associative ind-algebras in the Deligne category $\textbf{Rep}(GL_t)$, which generalize the regular Yangians $Y(\mathfrak{gl}_n)$ and $Y(\mathfrak{sl}_n)$ to complex rank. They were first defined by Etingof in his paper "Representation theory in complex rank". Here we solve Problem 7.2 from this paper. We work with the Deligne category $\textbf{Rep}(GL_t)$ using the ultraproduct approach introduced by Deligne and discussed by Nate Harman in arXiv:1601.03426.
We give negative answers to certain questions on abelian semisimple tensor categories raised by Bruno Kahn and Charles A. Weibel in connection with the preprint of Kahn "On the multiplicities of a motive" (arXiv:math/0610446), now published as Parts I and IV of "Zeta functions and motives", Pure Appl. Math. Q. 5 (2009), no. 1, part 2, 507-570. For the most interesting examples we used Deligne's category Rep(S_t,F) of representations of the "symmetric group S_t with t not an integer" with F any algebraically closed field of characteristic zero. This is an interesting family of tensor categories…
We categorify various Fock space representations on the algebra of symmetric functions via the category of polynomial functors. In a prequel, we used polynomial functors to categorify the Fock space representations of type A affine Lie algebras. In the current work we continue the study of polynomial functors from the point of view of higher representation theory. Firstly, we categorify the Fock space representation of the Heisenberg algebra on the category of polynomial functors. Secondly, we construct commuting actions of the affine Lie algebra and the level p action of the Heisenberg algebr…
We give a formula for the relative Deligne tensor product of two indecomposable finite semisimple module categories over a pointed braided fusion category over an algebraically closed field.
This note is mostly an exposition of an unpublished result of Deligne, which introduces an analogue of perverse $t$-structure on the derived category of coherent sheaves on a Noetherian scheme with a dualizing complex. Construction extends to the category of coherent sheaves equivariant under an action of an algebraic group; though proof of the general statement in this case does not require new ideas, it provides examples (such as sheaves on the nilpotent cone of a semi-simple group equivariant under the adjoint action) where construction of coherent "intersection cohomology" sheaves works.
For a reductive group G, we study the Drinfeld-Gaitsgory functor of the category of conjugation-equivariant D-modules on G. We show that this functor is an equivalence of categories, and that it has a filtration with layers expressed via parabolic induction of parabolic restriction. We use this to provide a conceptual definition of the Deligne-Lusztig involution on the set of isomorphism classes of irreducible character D-modules, which was defined previously in [Lu1, section 15].
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 \cite{Ryba} 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 \cite{Mori}. This ring only depends on the Grothendieck ring $\mathcal{G}(\mathcal{C})$. Given a ring $R$ which …
We establish a connection between two settings of representation stability for the symmetric groups $S_n$ over $\mathbb{C}$. One is the symmetric monoidal category ${\rm Rep}(S_{\infty})$ of algebraic representations of the infinite symmetric group $S_{\infty} = \bigcup_n S_n$, related to the theory of ${\bf FI}$-modules. The other is the family of rigid symmetric monoidal Deligne categories $\underline{\rm Rep}(S_t)$, $t \in \mathbb{C}$, together with their abelian versions $\underline{\rm Rep}^{ab}(S_t)$, constructed by Comes and Ostrik. We show that for any $t \in \mathbb{C}$ the natural fu…
We define the i-restriction and i-induction functors on the category O of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space.
The category-valued trace assigns to a bimodule category over a linear monoidal category a linear category. It generalizes Drinfeld centers of monoidal categories and the relative Deligne product of bimodule categories. In this article, we study bimodule categories that are given as categories of bicomodules over a Hopf algebra. Our main result is a representation-theoretic realization of the category-valued trace as a category of generalized Hopf bimodules.
We derive several tools for classifying tensor ideals in monoidal categories. We use these results to classify tensor ideals in Deligne's universal categories RepO, RepGL and RepP. These results are then used to obtain new insight into the second fundamental theorem of invariant theory for the algebraic supergroups of types A,B,C,D,P. We also find short proofs for the classification of tensor ideals in RepS and in the category of tilting modules for SL2(k) with char(k)>0 and for Uq(sl2) with q a root of unity. In general, for a simple Lie algebra g of type ADE, we show that the lattice of such…
The purpose of this short note is to understand the Grothendieck ring of the Deligne category $\underline{\text{Rep}}(GL_t)$. We give a formula for the class of an indecomposable object in terms of exterior powers of the tautological object and its dual.
For an arbitrary commutative ring k and t in k, we construct a 2-functor S_t which sends a tensor category to a new tensor category. By applying it to the representation category of a bialgebra we obtain a family of categories which interpolates the representation categories of the wreath products of the bialgebra. This generalizes the construction of Deligne's category Rep(S_t,k) for representation categories of symmetric groups.
Geometry of buildings is used to prove some homological properties of the category of smooth representations of a reductive p-adic group (Kazhdan's "pairing conjecture", Bernstein's description of homological duality in terms of Deligne-Lusztig duality). A different proof had been obtained a little earlier by Schneider and Stuhler.
Many key invariants in the representation theory of classical groups (symmetric groups $S_n$, matrix groups $GL_n$, $O_n$, $Sp_{2n}$) are polynomials in $n$ (e.g., dimensions of irreducible representations). This allowed Deligne to extend the representation theory of these groups to complex values of the rank $n$. Namely, Deligne defined generically semisimple families of tensor categories parametrized by $n\in \mathbb{C}$, which at positive integer $n$ specialize to the classical representation categories. Using Deligne's work, Etingof proposed a similar extrapolation for many non-semisimple …
A fundamental theorem of P. Deligne (2002) states that a pre-Tannakian category over an algebraically closed field of characteristic zero admits a fiber functor to the category of supervector spaces (i.e., is the representation category of an affine proalgebraic supergroup) if and only if it has moderate growth (i.e., the lengths of tensor powers of an object grow at most exponentially). In this paper we prove a characteristic p version of this theorem. Namely we show that a pre-Tannakian category over an algebraically closed field of characteristic p>0 admits a fiber functor into the Verlinde…
Starting from an abelian category A such that every object has only finitely many subobjects we construct a semisimple tensor category T. We show that T interpolates the categories Rep(Aut(p),K) where p runs through certain projective (pro-)objects of A. The main example is A=finite dimensional F_q-vector spaces. Then T can be considered as the category of representations of GL(n,F_q) where n is not a natural number. This work extends a construction of Deligne for symmetric groups.
We give a purely geometric categorification of tensor products of finite-dimensional simple $U_q(sl_2)$-modules and $R$-matrices on them. The work is developed in the framework of category of perverse sheaves and the categorification theorems are understood as consequences of Deligne's theory of weights.
The paper explores the relation between noncommutative power series and topological theories of one-dimensional cobordisms decorated by labelled zero-dimensional submanifolds. These topological theories give rise to several types of tensor envelopes of noncommutative recognizable power series, including the categories built from the syntactic algebra and syntactic ideals of the series and the analogue of the Deligne category.
We determine internal characterisations for when a tensor category is (super) tannakian, for fields of positive characteristic. This generalises the corresponding characterisations in characteristic zero by P. Deligne. We also explore notions of Frobenius twists in tensor categories in positive characteristic.
We derive a closed formula for the tensor product of a family of mixed tensors using Deligne's interpolating category $Rep(GL_{0})$. We use this formula to compute the tensor product between any two maximal atypical irreducible $GL(2|2)$-representations.
We describe two approaches to classifying the possible monodromy cones C arising from nilpotent orbits in Hodge theory. The first is based upon the observation that C is contained in the open orbit of any interior point N in C under an associated Levi subgroup determined by the limit mixed Hodge structure. The possible relations between the interior of C its faces are described in terms of signed Young diagrams. The second approach is to understand the Tannakian category of nilpotent orbits via a category D introduced by Deligne in a letter to Cattani and Kaplan. In analogy with Hodge theory, …
