Categorical geometric skew Howe duality

We categorify the R-matrix isomorphism between tensor products of minuscule representations of U_q(\mathfraksl_n)U_{q}({\mathfrak{sl}}_{n}) by constructing an equivalence between the derived categories of coherent sheaves on the corresponding convolution products in the affine Grassmannian. The main step in the construction is a categorification of representations of U_q(\mathfraksl₂)U_{q}({\mathfrak{sl}}_{2}) which are related to representations of U_q(\mathfraksl_n)U_{q}({\mathfrak{sl}}_{n}) by quantum skew Howe duality. The resulting equivalence is part of the program of algebro-geometric categorification of Reshitikhin-Turaev tangle invariants developed by the first two authors.     

The categorified Heisenberg algebra I: A combinatorial representation

We show how Khovanov's categorification of the Heisenberg algebra arises as a linearization of a discrete combinatorial structure in the bicategory of spans of groupoids. We also treat a categorification of $U({\mathrm{sl}}_n)$ in a similar way.     

Sheaves and duality

It has long been known in universal algebra that any distributive sublattice of congruences of an algebra which consists entirely of commuting congruences yields a sheaf representation of the algebra. In this paper we provide a generalization of this fact and prove a converse of the generalization. To be precise, we exhibit a one-to-one correspondence (up to isomorphism) between soft sheaf representations of universal algebras over stably compact spaces and frame homomorphisms from the dual frames of such spaces into subframes of pairwise commuting congruences of the congruence lattices of the universal algebras. For distributive-lattice-ordered algebras this allows us to dualize such sheaf representations.     

Discretisation and duality of optimal Skorokhod embedding problems

We prove a strong duality result for a linear programming problem which has the interpretation of being a discretised optimal Skorokhod embedding problem, and we recover this continuous time problem as a limit of the discrete problems. With the discrete setup we show that for a suitably chosen objective function, the optimiser takes the form of a hitting time for a random walk. In the limiting problem we then reprove the existence of the Root, Rost, and cave embedding solutions of the Skorokhod embedding problem.The main strength of this approach is that we can derive properties of the discrete problem more easily than in continuous time, and then prove that these properties hold in the limit. For example, a consequence of the strong duality result is that dual optimisers exist, and our limiting arguments can be used to derive properties of the continuous time dual functions. These arguments are applied in Cox and Kinsley (2017), where the existence of dual solutions is required to prove characterisation results for optimal barriers in a financial application.     

Local duality for structured ring spectra

We use the abstract framework constructed in our earlier paper [8] to study local duality for Noetherian ${\mathbb{E}}_{\infty }$-ring spectra. In particular, we compute the local cohomology of relative dualizing modules for finite morphisms of ring spectra, thereby generalizing the local duality theorem of Benson and Greenlees. We then explain how our results apply to the modular representation theory of compact Lie groups and finite group schemes, which recovers the theory previously developed by Benson, Iyengar, Krause, and Pevtsova.     

Exact completion of path categories and algebraic set theory: Part I: Exact completion of path categories

We introduce the notion of a “category with path objects”, as a slight strengthening of Kenneth Brown's classical notion of a “category of fibrant objects”. We develop the basic properties of such a category and its associated homotopy category. Subsequently, we show how the exact completion of this homotopy category can be obtained as the homotopy category associated to a larger category with path objects, obtained by freely adjoining certain homotopy quotients. In a second part of this paper, we will present an application to models of constructive set theory. Although our work is partly motivated by recent developments in homotopy type theory, this paper is written purely in the language of homotopy theory and category theory, and we do not presuppose any familiarity with type theory on the side of the reader.     

Asymptotic behavior of representations of graded categories with inductive functors

In this paper we describe inductive machinery to investigate asymptotic behavior of homology groups and related invariants of representations of certain graded combinatorial categories over a commutative Noetherian ring k, via introducing inductive functors which generalize important properties of shift functors of FI-modules. In particular, a sufficient criterion for finiteness of Castelnuovo–Mumford regularity of finitely generated representations of these categories is obtained. As applications, we show that a few important infinite combinatorial categories appearing in representation stability theory (for example ${\mathrm{FI}}^d$, ${\mathrm{OI}}^d$, ${\mathrm{FI}}_d$, ${\mathrm{OI}}_d$) are equipped with inductive functors, and hence the finiteness of Castelnuovo–Mumford regularity of their finitely generated representations is guaranteed. We also prove that truncated representations of these categories have linear minimal resolutions by relative projective modules, which are precisely linear minimal projective resolutions when k is a field of characteristic 0.     

Mixed quantum skew Howe duality and link invariants of type A

We define a ribbon category $\mathsf{Sp}(\beta )$, depending on a parameter β, which encompasses Cautis, Kamnitzer and Morrison's spider category, and describes for $\beta =m?n$ the monoidal category of representations of $U_q({\mathfrak{gl}}_{m|n})$ generated by exterior powers of the vector representation and their duals. We identify this category $\mathsf{Sp}(\beta )$ with a direct limit of quotients of a dual idempotented quantum group ${\dot{\mathbf{U}}}_q({\mathfrak{gl}}_{r+s})$, proving a mixed version of skew Howe duality in which exterior powers and their duals appear at the same time. We show that the category $\mathsf{Sp}(\beta )$ gives a unified natural setting for defining the colored ${\mathfrak{gl}}_{m|n}$ link invariant (for $\beta =m?n$) and the colored HOMFLY-PT polynomial (for β generic).     

Howe duality and the quantum general linear group

A Howe duality is established for a pair of quantized enveloping algebras of general linear algebras. It is also shown that this quantum Howe duality implies Jimbo's duality between and the Hecke algebra.

     

Operator-stable and operator-self-similar random fields

Two classes of multivariate random fields with operator-stable marginals are constructed. The random fields $\mathbb{X}=\{X\left(t\right):t\in {\mathbb{R}}^d\}$ with values in ${\mathbb{R}}^m$ are invariant in law under operator-scaling in both the time-domain and the state-space. The construction is based on operator-stable random measures utilizing certain homogeneous functions.     

Koszul duality and equivalences of categories

Let and be dual Koszul algebras. By Positselski a filtered algebra with gr is Koszul dual to a differential graded algebra . We relate the module categories of this dual pair by a Hom adjunction. This descends to give an equivalence of suitable quotient categories and generalizes work of Beilinson, Ginzburg, and Soergel.

     

Derived recollements and generalised AR formulas

The Defect Recollement, Restriction Recollement, Auslander–Gruson–Jensen Recollement, and others, are shown to be instances of a general construction using zeroth derived functors and methods from stable module theory. The right derived functors ${\mathsf{W}}_k:=R_k{(\underset{\_}{})}^{?}$ are computed and it is shown that the functor ${\mathsf{W}}_2:=R_2{(\underset{\_}{})}^{?}$ is right exact and restricts to a duality $\mathsf{W}$ of the defect zero functors. The duality $\mathsf{W}$ satisfies two identities which we call the Generalised Auslander–Reiten formulas. We show that $\mathsf{W}$ induces the generalised Auslander–Bridger transpose and show that the Generalised Auslander–Reiten formulas reduce to the well-known Auslander–Reiten formulas.     

Integrality and gauge dependence of Hennings TQFTs

We provide a general construction of integral TQFTs over a general commutative ring, $\mathbb{k}$, starting from a finite Hopf algebra over $\mathbb{k}$ which is Frobenius and double balanced. These TQFTs specialize to the Hennings invariants of the respective doubles on closed 3-manifolds.We show the construction applies to index 2 extensions of the Borel parts of Lusztig's small quantum groups for all simple Lie types, yielding integral TQFTs over the cyclotomic integers for surfaces with one boundary component.We further establish and compute isomorphisms of TQFT functors constructed from Hopf algebras that are related by a strict gauge transformation in the sense of Drinfeld. Formulas for the natural isomorphisms are given in terms of the gauge twist element.These results are combined and applied to show that the Hennings invariant associated to quantum-${\mathfrak{sl}}_2$ takes values in the cyclotomic integers. Using prior results of Chen et al. we infer integrality also of the Witten–Reshetikhin–Turaev $SO(3)$ invariant for rational homology spheres.As opposed to most other approaches the methods described in this article do not invoke calculations of skeins, knots polynomials, or representation theory, but follow a combinatorial construction that uses only the elements and operations of the underlying Hopf algebras.     

Josef Meixner: His life and his orthogonal polynomials

This paper starts with a biographical sketch of the life of Josef Meixner. Then his motivations to work on orthogonal polynomials and special functions are reviewed. Meixner’s 1934 paper introducing the Meixner and Meixner–Pollaczek polynomials is discussed in detail. Truksa’s forgotten 1931 paper, which already contains the Meixner polynomials, is mentioned. The paper ends with a survey of the reception of Meixner’s 1934 paper.     

Monochromatic tree covers and Ramsey numbers for set-coloured graphs

We consider a generalisation of the classical Ramsey theory setting to a setting where each of the edges of the underlying host graph is coloured with a set of colours (instead of just one colour). We give bounds for monochromatic tree covers in this setting, both for an underlying complete graph, and an underlying complete bipartite graph. We also discuss a generalisation of Ramsey numbers to our setting and propose some other new directions.Our results for tree covers in complete graphs imply that a stronger version of Ryser’s conjecture holds for $k$-intersecting $r$-partite $r$-uniform hypergraphs: they have a transversal of size at most $r?k$. (Similar results have been obtained by Király et al., see below.) However, we also show that the bound $r?k$ is not best possible in general.     

Universal algebra of a Hom-Lie algebra and group-like elements

We construct the universal enveloping algebra of a Hom-Lie algebra and endow it with a Hom-Hopf algebra structure. We discuss group-like elements that we see as a Hom-group integrating the initial Hom-Lie algebra.     

Flow-contractible configurations and group connectivity of signed graphs

Jaeger, Linial, Payan and Tarsi (JCTB, 1992) introduced the concept of group connectivity as a generalization of nowhere-zero flow for graphs. In this paper, we introduce group connectivity for signed graphs and establish some fundamental properties. For a finite abelian group $A$, it is proved that an $A$-connected signed graph is a contractible configuration for $A$-flow problem of signed graphs. In addition, we give sufficient edge connectivity conditions for signed graphs to be $A$-connected and study the group connectivity of some families of signed graphs.