Latest in math.rt

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Representations of Simple Hom-Lie algebrasMar 21 2019The purpose of this paper is to study representations of simple multiplicative Hom-Lie algebras. First, we provide a new proof using Killing form for characterization theorem of simple Hom-Lie algebras given by Chen and Han, then discuss the representations ... More
Research topics in finite groups and vertex algebrasMar 21 2019We suggest a few projects for studying vertex algebras with emphasis on finite group viewpoints.
Cohomologie des fibrés en droite sur SL3 /B en caractéristique positive : deux filtrations et conséquencesMar 20 2019In the paper, I will prove the existence of two filtrations of the cohomology of line bundles on SL_3/B. The first one is a two-step filtration that exists for $H^1(\mu)$ and $H^2(\mu)$ if $\mu$ is in the Griffith region. The second one exists for all ... More
Trace decategorification of tensor product algebrasMar 20 2019We show that in ADE type the trace of Webster's categorification of a tensor product of irreducibles for the quantum group is isomorphic to a tensor product of Weyl modules for the current algebra $\dot{U}(\mathfrak{g}[t])$. This extends a result of Beliakova, ... More
On the Lie algebra structure of $HH^1(A)$ of a finite-dimensional algebra $A$Mar 20 2019Let $A$ be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the Ext-quiver of $A$ is a simple directed graph, then $HH^1(A)$ is a solvable Lie algebra. The second main result shows that ... More
Monomial $G$-posets and their Lefschetz invariantsMar 20 2019Let $G$ be a finite group, and $C$ be an abelian group. We introduce the notions of $C$-monomial $G$-sets and $C$-monomial $G$-posets, and state some of their categorical properties. This gives in particular a new description of the $C$-monomial Burnside ... More
Castelnuovo-Mumford regularity of $\mathrm{FI}^m$-modules presented in finite degreesMar 20 2019Let $V$ be a representation of the category $\mathrm{FI}^m$, a product of $m$ copies of the category of finite sets and injections, over an arbitrary commutative coefficient ring. We show in this paper that $V$ has finite Castelnuovo-Mumford regularity ... More
Generalized Slices for Minuscule CocharactersMar 19 2019Let $G$ be a connected reductive complex algebraic group with a maximal torus $T$. We denote by $\Lambda$ the cocharacter lattice of $(T,G)$. Let $\Lambda^+ \subset \Lambda$ be the submonoid of dominant coweights. For $\lambda \in \Lambda^+,\,\mu \in ... More
Holomorphic functions of exponential type on connected complex Lie groupsMar 19 2019Holomorphic functions of exponential type on a complex Lie group $G$ (introduced by Akbarov) form a locally convex algebra, which is denoted by $\cO_{exp}(G)$. Our aim is to describe the structure of $\cO_{exp}(G)$ in the case when $G$ is connected. The ... More
Frobenius bimodules and flat-dominant dimensionsMar 19 2019We establish relations between Frobenius parts and between flat-dominant dimensions of algebras linked by Frobenius bimodules. This is motivated by the Nakayama conjecture and an approach of Martinez-Villa to the Auslander-Reiten conjecture on stable ... More
Generators for Coulomb branches of quiver gauge theoriesMar 18 2019We study the Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories, focusing on the generators for their quantized coordinate rings. We show that there is a surjective map from a shifted Yangian onto the quantized Coulomb branch, once the deformation ... More
Kronecker positivity and 2-modular representation theoryMar 18 2019This paper consists of two prongs. Firstly, we prove that any Specht module labelled by a 2-separated partition is semisimple and we completely determine its decomposition as a direct sum of graded simple modules. Secondly, we apply these results and ... More
Indecomposable Jordan types of Loewy length $2$Mar 18 2019Let $k$ be an algebraically closed field, $\mathop{char}(k) = p \geq 2$ and $E_r$ be a $p$-elementary abelian group of rank $r \geq 2$. Let $(c,d) \in \mathbb{N}^2$. We show that there exists an indecomposable module of constant Jordan type $[1]^c [2]^d$ ... More
Type classification of extremal quantized charactersMar 18 2019The notion of quantized characters is introduced in our previous paper as a natural quantization of characters in the context of asymptotic representation theory for compact quantum groups. As in the case of ordinary groups, the representation associated ... More
On solvability of the first Hochschild cohomology of a finite-dimensional algebraMar 18 2019For an arbitrary finite-dimensional algebra $A$, we first provide a sufficient condition for the solvability of its first Hochschild cohomology, considered as a Lie algebra, in terms of a corresponding separated quiver. If $A$ is moreover of non-wild ... More
Integral presentations of the shifted convolution problem and subconvexity estimates for $\operatorname{GL}_n$-automorphic $L$-functionsMar 18 2019Fix $n \geq 2$ an integer, and let $F$ be a totally real number field. We reduce the shifted convolution problem for $L$-function coefficients of $\operatorname{GL}_n({\bf{A}}_F)$-automorphic forms to the better-understood setting of $\operatorname{GL}_2({\bf{A}}_F)$ ... More
A note on Muller's irreducibility criterion for generalized principal seriesMar 18 2019In this paper, via Casselman--Tadic's Jacquet module machine, we reprove I. Muller's irreducibility criterion for principal series, and extend it to generalized principal series. An analogous criterion for covering groups is readily obtained. At last, ... More
Categorification of two-dimensional cohomological Hall algebrasMar 18 2019In the present paper, we provide a full categorification, at the level of stable $\infty$-categories, of two-dimensional cohomological Hall algebras of curves and surfaces. This is achieved by producing a suitable derived enhancement of the relevant moduli ... More
Relative $B$-groupsMar 17 2019This paper extends the notion of $B$-group to a relative context. For a finite group $K$ and a field $\mathbb{F}$ of characteristic 0, the lattice of ideals of the Green biset functor $\mathbb{F}B_K$ obtained by shifting the Burnside functor $\mathbb{F}B$ ... More
On the Spectrum of Finite Rooted Homogeneous TreesMar 17 2019In this paper we study the spectrum of families of finite rooted trees with regular branching properties. In particular, we show that in the case of constant branching the eigenvalues are realized as the roots of a family of generalized Fibonacci polynomials ... More
Higher gentle algebrasMar 17 2019We introduce higher gentle algebras. Our definition allows us to determine the singularity categories and subsequently show that higher gentle algebras are Iwanaga-Gorenstein. Under extra assumptions, we show that cluster-tilted algebras (in the sense ... More
Description of infinite orbits on multiple projective spacesMar 16 2019Let $G$ be the general linear group of the degree $n\geq 2$ over the field $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. In this article, we give a description of orbit decomposition of the multiple projective space $G^m/P^m$ under the diagonal action of $G$ ... More
Classification of irreducible modules over gap-$p$ Virasoro algebrasMar 16 2019We prove that any irreducible Harish-Chandra modules for a class of Lie algebras, which we call gap-$p$ Virasoro algebras, must be a highest weight module, a lowest weight module, or a module of intermediate series.These algebras are closely related to ... More
Unitary representations with non-zero Dirac cohomology for $E_{6(-14)}$Mar 16 2019Up to equivalence, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology for the Hermitian symmetric Lie group $E_{6(-14)}$. Along the way, we have also obtained all the fully supported irreducible unitary representations ... More
The Betti Numbers of a Determinantal VarietyMar 16 2019We determine the Poincar\'e polynomial of the determinantal variety $\{\det = 0\}$ in the projective space associated with the monoid of $n\times n$ matrices.
Orthogonal multiple flag varieties of finite type II : even degree caseMar 15 2019Let $G$ be the split orthogonal group of degree $2n$ over an arbitrary infinite field $\mathbb{F}$ of chararcteristic not $2$. In this paper, we classify multiple flag varieties $G/P_1\times\cdots\times G/P_k$ of finite type. Here a multiple flag variety ... More
Cohomological representations of parahoric subgroupsMar 14 2019We generalize a cohomological construction of representations due to Lusztig from the hyperspecial case to arbitrary parahoric subgroups of a reductive group over a local field, which splits over an unramified extension. We compute the character of these ... More
Kazhdan-Lusztig representations and Whittaker space of some genuine representationsMar 14 2019We propose a conjectural formula for the dimension of Whittaker functionals of irreducible constituents of a regular unramified genuine principal series for covering groups. The formula explicitly relates such dimension to the Kazhdan-Lusztig representations ... More
Gabriel-Roiter measure, representation dimension and rejective chainsMar 13 2019The Gabriel-Roiter measure is used to give an alternative proof of the finiteness of the representation dimension for Artin algebras, a result established by Iyama in 2002. The concept of Gabriel-Roiter measure can be extended to abelian length categories ... More
Representations of principal $W$-algebra for the superalgebra $Q(n)$Mar 13 2019We classify irreducible representations of finite $W$-algebra of the queer Lie superalgebra $Q(n)$ associated with the principal nilpotent coadjoint orbits.
New expression of unramified local L-functions by certain Hecke operatorsMar 12 2019In this paper, we express the local L-functions of unramified representations of a split connected reductive group over a non-archimedean local field as the inverse of the product of characteristic polynomials of certain test functions in the Hecke algebra. ... More
Modules for twisted affine Lie superalgebrasMar 12 2019Mar 13 2019This work provides the first step toward the classification of irreducible finite weight modules over twisted affine Lie superalgebras. We study all such modules whether the canonical central element acts as a nonzero multiple of the identity map or not. ... More
Modules for twistes affine Lie superalgebrasMar 12 2019This work provides the first step toward the classification of irreducible finite weight modules over twisted affine Lie superalgebras. We study all such modules whether the canonical central element acts as a nonzero multiple of the identity map or not. ... More
A characterization of representation infinite quiver settingsMar 12 2019We characterize pairs (Q,d) consisting of a quiver Q and a dimension vector d, such that over a given algebraically closed field k there are infinitely many representations of Q of dimension vector d. We also present an application of this result to the ... More
Odd singular vector formula for general linear Lie superalgebrasMar 12 2019We establish a closed formula for a singular vector of weight $\lambda-\beta$ in the Verma module of highest weight $\lambda$ for Lie superalgebra $\mathfrak{gl}(m|n)$ when $\lambda$ is atypical with respect to an odd positive root $\beta$. It is further ... More
Syzygy Filtrations of Cyclic Nakayama AlgebrasMar 11 2019We introduce a method "syzygy filtration" to give building blocks of syzygies appearing in projective resolutions of indecomposable $\Lambda$-modules where $\Lambda$ is a cyclic Nakayama algebra. We interpret homological invariants of $\Lambda$ including ... More
Rationality of Rigid Quiver GrassmanniansMar 11 2019We show that any quiver Grassmannian associated with a rigid representation of a quiver is a rational variety using torus localization techniques.
The Exterior Cubic L-function of GU(6) and Unitary Automorphic InductionMar 11 2019In this paper, we extend Ginzburg-Rallis' integral representation for the exterior cube automorphic $L$-function of ${\rm GL}_6\times {\rm GL}_1$ to that of the quasi-split unitary similitude group ${\rm GU}_6$ and establish its analytic properties to ... More
Cyclic posets and triangulation clustersMar 11 2019Triangulated categories coming from cyclic posets were originally introduced by the authors in [IT15b] as a generalization of the constructions of various triangulated categories with cluster structures. We give an overview, then analyze triangulation ... More
From conjugacy classes in the Weyl group to semisimple conjugacy classesMar 11 2019Suppose $G$ is a connected complex semisimple group and $W$ is its Weyl group. The lifting of an element of $W$ to $G$ is semisimple. This induces a well-defined map from the set of elliptic conjugacy classes of $W$ to the set of semisimple conjugacy ... More
LLT polynomials, elementary symmetric functions and melting lollipopsMar 10 2019We conjecture an explicit positive combinatorial formula for the expansion of unicellular LLT polynomials in the elementary symmetric basis. This is an analogue of the Stanley--Stembridge conjecture and previously studied by Panova and the author in 2018. ... More
Compact quantum groups generated by their toriMar 09 2019Associated to any closed subgroup $G\subset U_N^+$ is a family of toral subgroups $T_Q\subset G$, indexed by the unitary matrices $Q\in U_N$. The family $\{T_Q|Q\in U_N\}$ is expected to encode the main properties of $G$, and there are several conjectures ... More
Conformal embeddings in affine vertex superalgebrasMar 09 2019This paper is a natural continuation of our previous work on conformal embeddings of vertex algebras [6], [7], [8]. Here we consider conformal embeddings in simple affine vertex superalgebra $V_k(\mathfrak g)$ where $\mathfrak g=\mathfrak g_{\bar 0}\oplus ... More
On the moduli spaces of commuting elements in the projective unitary groupsMar 09 2019We provide descriptions for the moduli spaces $\Rep(\Gamma, PU(m))$, where $\Gamma$ is any finitely generated abelian group and $PU(m)$ is the group of $m\times m$ projective unitary matrices. As an application we show that for any connected CW--complex ... More
Schur-Weyl type duality for quantized gl(1|1),the Burau representation of braid groups and invariants of tangled graphsMar 08 2019We show that the Schur-Weyl type duality between $gl(1|1)$ and $GL_n$ gives a natural representation-theoretic setting for the relation between reduced and non-reduced Burau representations.
On Lenagan's Theorem for finite length bimodulesMar 08 2019We offer a self-contained proof of Lenagan's Theorem which does not rely on Goldie's Theorem
Geometrisation of Quantum Dynamics by a Coherent States TransformMar 08 2019We propose a novel concept of coherent states geometrising a time evolution of quantum systems. The respective coherent state transforms reduce certain Hamiltonians to first-order differential operators, thus the dynamics can be explicitly expressed through ... More
Test vectors for Rankin-Selberg $L$-functionsMar 08 2019We study the local zeta integrals attached to a pair of generic representations $(\pi,\tau)$ of $GL_n\times GL_m$, $n>m$, over a $p$-adic field. Through a process of unipotent averaging we produce a pair of corresponding Whittaker functions whose zeta ... More
Candidate for the crystal $B(-\infty)$ for the queer Lie superalgebraMar 08 2019It is shown that the direct limit of the semistandard decomposition tableau model for polynomial representations of the queer Lie superalgebra exists, which is believed to be the crystal for the upper half of the corresponding quantum group. An extension ... More
Fixed rings in quotients of completed group ringsMar 07 2019Let $k$ be $\mathbb{F}_p$ or $\mathbb{Z}_p$, let $G$ be a compact $p$-adic analytic group, and form its completed group algebra $kG$. Take a closed subgroup $\Gamma$ of $G$. We analyse the structure of the fixed ring of $kG/I$ under the conjugation action ... More
Restricted shifted Yangians and restricted finite $W$-algebrasMar 07 2019We study the truncated shifted Yangian $Y_{n,l}(\sigma)$ over an algebraically closed field $\mathbb{k}$ of characteristic $p > 0$, which is known to be isomorphic to the finite $W$-algebra $U(\mathfrak{g}, e)$ associated to a corresponding nilpotent ... More
Reparameterizing Distributions on Lie GroupsMar 07 2019Reparameterizable densities are an important way to learn probability distributions in a deep learning setting. For many distributions it is possible to create low-variance gradient estimators by utilizing a `reparameterization trick'. Due to the absence ... More
The motive of a smooth proper connective DG-algebra is of unit typeMar 07 2019We prove that the motive of a smooth, proper, and connective DG-algebra over a field $k$ is isomorphic to the motive of its degree $0$ cohomology. Along the way we show that the degree $0$ cohomology of such a DG-algebra is necessarily also smooth. If ... More
Fock space representation of the circle quantum groupMar 07 2019In [ arXiv:1711.07391 ] we have defined quantum groups $\mathbf{U}_\upsilon(\mathfrak{sl}(\mathbb{R}))$ and $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1))$, which can be interpreted as continuous generalizations of the quantum groups of the Kac-Moody Lie algebras ... More
Self-dual cuspidal representationsMar 07 2019Let $G$ be a connected reductive group over a finite field $\mathfrak{f}$ of order $q$. When $q$ is small, we make further assumptions on $G$. Then we determine precisely when $G(\mathfrak{f})$ admits irreducible, cuspidal representations that are self-dual, ... More
A relationship between Gelfand-Tsetlin bases and Chari-Loktev bases for irreducible finite dimensional representations of special linear Lie algebrasMar 07 2019We consider two bases for an arbitrary finite dimensional irreducible representation of a complex special linear Lie algebra: the classical Gelfand-Tsetlin basis and the relatively new Chari-Loktev basis. Both are parametrized by the set of (integral ... More
FI-hyperhomology and ordered configuration spacesMar 07 2019Using a result of Gan and Li on FI-hyperhomology and a semi-simplicial resolution of configuration spaces due to Randal-Williams, we establish an improved representation stability stable range for configuration spaces of distinct ordered points in a manifold. ... More
The classification of blocks in BGG category OMar 07 2019We classify all equivalences between the indecomposable abelian categories which appear as blocks in BGG category O for reductive Lie algebras. Our classification implies that a block in category O only depends on the Bruhat order of the relevant parabolic ... More
Gauge modules for the Lie algebras of vector fields on affine varietiesMar 06 2019For a smooth irreducible affine algebraic variety we study a class of gauge modules admitting compatible actions of both the algebra $A$ of functions and the Lie algebra $\mathcal{V}$ of vector fields on the variety. We prove that a gauge module corresponding ... More
On the residue method for period integralsMar 06 2019Mar 08 2019By applying the residue method for period integrals and Langlands-Shahidi's theory for residues of Eisenstein series, we study the period integrals for six spherical varieties. For each spherical variety, we prove a relation between the period integrals ... More
On the residue method for period integralsMar 06 2019By applying the residue method for period integrals and Langlands-Shahidi's theory for residues of Eisenstein series, we study the period integrals for six spherical varieties. For each spherical variety, we prove a relation between the period integrals ... More
Characterisation of the poles of the $\ell$-modular Asai $L$-factorMar 06 2019Mar 10 2019Let $E/F$ be a quadratic extension of non-archimedean local fields, and let $\ell$ be a prime number different from the residual characteristic of $F$. For a complex cuspidal representation $\pi$ of $GL(n,E)$, the Asai $L$-factor $L^+(X,\pi)$ has a pole ... More
Characterisation of the poles of the $\ell$-modular Asai $L$-factorMar 06 2019Let $E/F$ be a quadratic extension of non-archimedean local fields, and let $\ell$ be a prime number different from the residual characteristic of $F$. For a complex cuspidal representation $\pi$ of $GL(n,E)$, the Asai $L$-factor $L^+(X,\pi)$ has a pole ... More
Characterisation of the poles of the $\ell$-modular Asai $L$-factorMar 06 2019Mar 16 2019Let $E/F$ be a quadratic extension of non-archimedean local fields, and let $\ell$ be a prime number different from the residual characteristic of $F$. For a complex cuspidal representation $\pi$ of $GL(n,E)$, the Asai $L$-factor $L^+(X,\pi)$ has a pole ... More
Unified products of Leibniz conformal algebrasMar 06 2019The aim of this paper is to provide an answer to the $\mathbb{C}[\partial]$-split extending structures problem for Leibniz conformal algebras, which asks that how to describe all Leibniz conformal algebra structures on $E=R\oplus Q$ up to an isomorphism ... More
Cohomologies of a Lie algebra with a derivation and applicationsMar 06 2019The main object of study of this paper is the notion of a LieDer pair, i.e. a Lie algebra with a derivation. We introduce the concept of a representation of a LieDer pair and study the corresponding cohomologies. We show that a LieDer pair is rigid if ... More
Test Vectors for Nonarchimedean Godement-Jacquet Zeta IntegralsMar 05 2019Given an induced representation of Langlands type $(\pi,V)$ of $\mathrm{GL}_n(F)$ with $F$ nonarchimedean, we show that there exist explicit choices of matrix coefficient $\beta$ and Schwartz-Bruhat function $\Phi$ for which the Godement-Jacquet zeta ... More
Bispectral dual difference equations for the quantum Toda chain with boundary perturbationsMar 05 2019We show that hyperoctahedral Whittaker functions---diagonalizing an open quantum Toda chain with one-sided boundary potentials of Morse type---satisfy a dual system of difference equations in the spectral variable. This extends a well-known bispectral ... More
Morita theory of systemsMar 05 2019We present the rudiments of the Morita theory of module systems (over semirings), paralleling the classical Morita theory over associative rings.
Tensor product of correspondence functorsMar 05 2019As part of the study of correspondence functors, the present paper investigates their tensor product and proves some of its main properties. In particular, the correspondence functor associated to a finite lattice has the structure of a commutative algebra ... More
Periods of automorphic forms over reductive subgroupsMar 05 2019We present a regularization procedure of period integrals of automorphic forms on a group $G$ over an arbitrary reductive subgroup $G' \subset G$. As a consequence we obtain an explicit $G'(\mathbb{A})$-invariant functional on the space of automorphic ... More
Periods of automorphic forms over reductive subgroupsMar 05 2019Mar 08 2019We present a regularization procedure of period integrals of automorphic forms on a group $G$ over an arbitrary reductive subgroup $G' \subset G$. As a consequence we obtain an explicit $G'(\mathbb{A})$-invariant functional on the space of automorphic ... More
Morita equivalences for cyclotomic Hecke algebras of type B and DMar 04 2019We give a Morita equivalence theorem for cyclotomic Hecke algebras of type B and D, in the spirit of a classical result of Dipper-Mathas in type A for Ariki-Koike algebras. The main step in the proof consists in a decomposition theorem for generalisations ... More
Fibers of maps to totally nonnegative spacesMar 04 2019This paper undertakes a study of the structure of the fibers of a family of maps $f_{(i_1,\dots ,i_d)}$ arising from representation theory, motivated both by connections to Lusztig's theory of canonical bases and also by the fact that these fibers encode ... More
Quantization of continuum Kac-Moody algebrasMar 04 2019Continuum Kac-Moody algebras have been recently introduced by the authors and O. Schiffmann. These are Lie algebras governed by a continuum root system, which can be realized as uncountable colimits of Borcherds-Kac-Moody algebras. In this paper, we prove ... More
Linear periods of automorphic sheaves for GL_{2n}Mar 04 2019Mar 11 2019We study, in the framework of the geometric Langlands program, the periods of cuspidal automorphic sheaves for GL_{2n} along the Levi subgroup GL_n\times\GL_n. We solve the corresponding local problem, and discuss the global applications.
Linear periods of automorphic sheaves for GL_{2n}Mar 04 2019We study, in the framework of the geometric Langlands program, the periods of cuspidal automorphic sheaves for GL_{2n} along the Levi subgroup GL_n\times\GL_n. We solve the corresponding local problem, and discuss the global applications.
Lie structure on the Hochschild cohomology of a family of subalgebras of the Weyl algebraMar 04 2019For each nonzero $h\in \mathbb{F}[x]$, where $\mathbb{F}$ is a field, let $\mathsf{A}_h$ be the unital associative algebra generated by elements $x,y$, satisfying the relation $yx-xy = h$. This gives a parametric family of subalgebras of the Weyl algebra ... More
Non-crossing partitionsMar 04 2019Non-crossing partitions have been a staple in combinatorics for quite some time. More recently, they have surfaced (sometimes unexpectedly) in various other contexts from free probability to classifying spaces of braid groups. Also, analogues of the non-crossing ... More
Uglov bipartitions and extended Young diagramsMar 04 2019We study the class of Uglov bipartitions and prove a generalization of a conjecture by Dipper, James and Murphy. We give two consequences concerning the computation of canonical bases in affine type A and the description of decomposition matrices for ... More
A graphical categorification of the two-variable Chebyshev polynomials of the second kindMar 04 2019We show that the $A_2$ clasps in the Karoubi envelope of $A_2$ spider satisfy the recursive formula of the two-variable Chebyshev polynomials of the second kind associated with a root system of type $A_2$. The $A_2$ spider is a diagrammatic description ... More
The generalized Auslander-Reiten duality on a module categoryMar 04 2019We characterize the generalized Auslander--Reiten duality on the category of finitely presented modules over some certain Hom-finite category. Examples of such category include FI and VI.
Periodicities in cluster algebras and cluster automorphism groupsMar 03 2019In this article, we study the relations between groups related to cluster automorphism groups which are defined by Assem, Schiffler and Shamchenko in \cite{ASS}. We establish the relationship among (strict) direct cluster automorphism groups and those ... More
Quantum cluster algebras via Hall algebras of morphismsMar 03 2019We realize the quantum cluster algebra with principal coefficients as a subquotient of certain Hall algebra involving the category of morphisms between projectives.
A construction of lower-bounded generalized twisted modules for a grading-restricted vertex (super)algebraMar 02 2019We give a general, direct and explicit construction of lower-bounded generalized twisted modules satisfying a universal property for a grading-restricted vertex (super)algebra $V$ associated to an automorphism $g$ of $V$. In particular, when $g$ is the ... More
Refined invariants of finite-dimensional Jacobi algebrasMar 02 2019We define and study refined Gopakumar-Vafa invariants of contractible curves in complex algebraic 3-folds, via the cohomological Donaldson--Thomas theory of finite-dimensional Jacobi algebras. These Gopakumar-Vafa invariants can be constructed one of ... More
Second adjointness for tempered admissible representations of a real groupMar 01 2019Mar 05 2019We study second adjointness in the context of tempered admissible representations of a real reductive group. Compared to a recent result of Crisp and Higson, this generalizes from $SL_2$ to a general group, but specializes to only considering admissible ... More
Ext-enhanced monoidal Koszul duality for $\mathrm{GL}_2$Mar 01 2019The Hecke category participates in an equivalence called monoidal Koszul duality, which exchanges it with the category of (Langlands-dual) "free-monodromic tilting sheaves." Motivated by a recent conjecture of Gorsky and the first-named author on HOMFLYPT ... More
On monoidal Koszul duality for the Hecke categoryMar 01 2019We attempt to give a gentle (though ahistorical) introduction to Koszul duality phenomena for the Hecke category, focusing on the form of this duality studied in joint work of Achar, Riche, Williamson, and the author. We illustrate some key phenomena ... More
On double quantum affinization: 1. Type $\mathfrak a_1$Mar 01 2019We define the double quantum affinization $\ddot{\mathrm{U}}_q(\mathfrak a_1)$ of type $\mathfrak{a}_1$ as a topological Hopf algebra. We prove that it admits a subalgebra $\ddot{\mathrm{U}}_q'(\mathfrak a_1)$ whose completion is (bicontinuously) isomorphic ... More
Invariants of the Weyl Group of Type $A_{2l}^{(2)}$Mar 01 2019In this note, we show the polynomiality of the ring of invariants with respect to the Weyl group of type $A_{2l}^{(2)}$.
Isomorphism between the R-matrix and Drinfeld presentations of quantum affine algebra: type CMar 01 2019Mar 15 2019An explicit isomorphism between the $R$-matrix and Drinfeld presentations of the quantum affine algebra in type $A$ was given by Ding and I. Frenkel (1993). We show that this result can be extended to types $B$, $C$ and $D$ and give a detailed construction ... More
Isomorphism between the R-matrix and Drinfeld presentations of quantum affine algebra: type CMar 01 2019An explicit isomorphism between the $R$-matrix and Drinfeld presentations of the quantum affine algebra in type $A$ was given by Ding and I. Frenkel (1993). We show that this result can be extended to types $B$, $C$ and $D$ and give a detailed construction ... More
Infinite-dimensional Polish groups and Property (T)Mar 01 2019We show that all groups of a distinguished class of \guillemotleft large\guillemotright\ topological groups, that of Roelcke precompact Polish groups, have Kazhdan's Property (T). This answers a question of Tsankov and generalizes previous results by ... More
Conjectures P1-P15 for Coxeter groups with complete graphFeb 28 2019We prove Lusztig's conjectures P1-P15 for Coxeter groups with complete graph, using deceasing induction on $ \mathbf{a} $-values and a kind of decomposition formula of Kazhdan-Lusztig basis elements. As a byproduct, we give a description of the left, ... More
A vanishing conjecture: the GL_n caseFeb 28 2019In this article we propose a vanishing conjecture for a certain class of $\ell$-adic complexes on a reductive group G, which can be regraded as a generalization of the acyclicity of the Artin-Schreier sheaf. We show that the vanishing conjecture contains, ... More
Hecke algebras of simply-laced type with independent parametersFeb 28 2019We study the (complex) Hecke algebra $\mathcal{H}_S(\mathbf{q})$ of a finite simply-laced Coxeter system $(W,S)$ with independent parameters $\mathbf{q} \in \left( \mathbb{C} \setminus\{\text{roots of unity}\} \right)^S$. We construct its irreducible ... More
$n$-Cotorsion pairsFeb 28 2019Motivated by some properties satisfied by Gorenstein projective and Gorenstein injective modules over an Iwanaga-Gorenstein ring, we present the concept of left and right $n$-cotorsion pairs in an abelian category $\mathcal{C}$. Two classes $\mathcal{A}$ ... More
A central idempotent in the endomorphism algebra of a finite latticeFeb 27 2019We give a direct construction of a specific idempotent in the endomorphism algebra of a finite lattice $T$. This idempotent is associated with all possible sublattices of $T$ which are total orders.
An exponential lower bound for the degrees of invariants of cubic forms and tensor actionsFeb 27 2019Using the Grosshans Principle, we develop a method for proving lower bounds for the maximal degree of a system of generators of an invariant ring. This method also gives lower bounds for the maximal degree of a set of invariants that define Hilbert's ... More