Latest in math.rt

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Crystal of affine $\widehat{\mathfrak{sl}}_{\ell}$ and Hecke algebras at a primitive $2\ell$th root of unityMay 17 2019Let $\ell\in\mathbb{N}$ with $\ell>2$ and $I:=\mathbb{Z}/2\ell\mathbb{Z}$. In this paper we give a new realization of the crystal of affine $\widehat{\mathfrak{sl}}_{\ell}$ using the modular representation theory of the affine Hecke algebras $H_n$ of ... More
Group-twisted Alexander-Whitney and Eilenberg-Zilber mapsMay 16 2019We define group-twisted Alexander-Whitney and Eilenberg-Zilber maps for converting between bimodule resolutions of skew group algebras. These algebras are the natural semidirect products recording actions of finite groups by automorphisms. The group-twisted ... More
Finite quotients of powers of an elliptic curveMay 16 2019Let $E$ be an elliptic curve. When the symmetric group $\Sigma_{g+1}$ of order $(g+1)!$ acts on $E^{g+1}$ in the natural way, the subgroup $E_0^{g+1}$, consisting of those $(g+1)$-tuples whose coordinates sum to zero, is stable under the action of $\Sigma_{g+1}$. ... More
From the potential to the first Hochschild cohomology group of a cluster tilted algebraMay 16 2019The objective of this paper is to give a concrete interpretation of the dimension of the first Hochschild cohomology space of a cyclically oriented or tame cluster tilted algebra in terms of a numerical invariant arising from the potential.
Local Finiteness of the Twisted Bruhat Orders on Affine Weyl GroupsMay 16 2019In this note we prove that any twisted strong Bruhat order on an affine Weyl group is locally finite, strengthening a result of Dyer in J. Algebra, 163, 861--879 (1994). We also show that for a non-finite and non-cofinite biclosed set $B$ in the positive ... More
Dualizable link homologyMay 16 2019We modify our previous construction of link homology in order to include a natural duality functor $\mathfrak{F}$. To a link $L$ we associate a triply-graded module $HXY(L)$ over the graded polynomial ring $R(L)=\mathbb{C}[x_1,y_1,\dots,x_\ell,y_\ell]$. ... More
Semi-infinite cohomology and the linkage principle for $W$-algebrasMay 16 2019Let $\mathfrak{g}$ be a simple Lie algebra, and let $W_\kappa$ be the affine ${W}$-algebra associated to a principal nilpotent element of $\mathfrak{g}$ and level $\kappa$. We explain a duality between the categories of smooth ${W}$ modules at levels ... More
Tame cuspidal representations in non-defining characteristicsMay 15 2019Let k be a non-archimedean local field of odd residual characteristic p. Let G be a (connected) reductive group that splits over a tamely ramified field extension of k. We revisit Yu's construction of smooth complex representations of G(k) from a slightly ... More
Temperley-Lieb, Brauer and Racah algebras and other centralizers of su(2)May 15 2019In the spirit of the Schur-Weyl duality, we study the connections between the Racah algebra and the centralizers of tensor products of three (possibly different) irreducible representations of su(2). As a first step we show that the Racah algebra always ... More
The PBW basis of $U_{q,\bar{q}}(\ddot{\mathfrak{gl}}_n)$May 15 2019We consider the PBW basis of the type A quantum toroidal algebra developed by the author, and prove commutation relations between its generators akin to the ones studied by Burban-Schiffmann for n=1. This gives rise to a new presentation of the quantum ... More
The Super Orbit ChallengeMay 15 2019When using the generally adopted definition of a super unitary representation, there are lots of super Lie groups for which the regular representation is not super unitary. I propose a new definition of a super unitary representation for which all regular ... More
On tame strongly simply connected algebrasMay 15 2019In this survey we present the criterion for tameness of strongly simply connected algebras due to Br\"ustle, de la Pe\~na and Skowro\'nski. We recall relevant concepts of representation theory and discuss some applications and connections to other problems. ... More
Doubly transitive lines II: Almost simple symmetriesMay 15 2019We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. This paper, the second in a series, classifies those lines that exhibit almost simple symmetries. To perform this classification, ... More
On spectral measures for certain unitary representations of R. Thompson's group FMay 14 2019The Hilbert space $\mathcal H$ of backward renormalisation of an anyonic quantum spin chain affords a unitary representation of Thompson's group $F$ via local scale transformations. Given a vector in the canonical dense subspace of $\mathcal H$ we show ... More
Finite direct sums of cyclic embeddingsMay 14 2019In this paper we generalize Kaplansky's combinatorial characterization of the isomorphism types of embeddings of a cyclic subgroup in a finite abelian group given in his 1951 book ``Infinite Abelian Groups''. For this we introduce partial maps on Littlewood-Richardson ... More
Embedding Deligne's category $\mathrm{Rep}(S_t)$ in the Heisenberg categoryMay 14 2019We define a faithful linear monoidal functor from the partition category, and hence from Deligne's category $\mathrm{Rep}(S_t)$, to the Heisenberg category. We show that the induced map on Grothendieck rings is injective and corresponds to the Kronecker ... More
Stability conditions on morphisms in a categoryMay 14 2019Let $\mathbf D$ be the homotopy category of a stable infinity category. Then the category $\mathbf D^{\Delta^1}$ is also triangulated. Hence the space $\mathsf{Stab}\,{ \mathbf D^{\Delta^1}}$ of stability conditions on $\mathbf D^{\Delta^1}$ is well-defined ... More
A study of Kostant-Kumar modules via Littelmann pathsMay 13 2019We obtain, in Littelmann's language of paths, a character formula and a decomposition rule for Kostant-Kumar modules, which by definition are certain submodules of the tensor product of two irreducible finite dimensional representations of a complex semisimple ... More
A geometric $q$-character formula for snake modulesMay 13 2019Let $\mathscr{C}$ be the category of finite dimensional modules over the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ of a simple complex Lie algebra ${\mathfrak{g}}$. Let $\mathscr{C}^-$ be the subcategory introduced by Hernandez and Leclerc. ... More
Bijective proofs of skew Schur polynomial factorizationsMay 13 2019In a recent paper, Ayyer and Behrend present for a wide class of partitions factorizations of Schur polynomials with an even number of variables where half of the variables are the reciprocals of the others into symplectic and/or orthogonal group characters, ... More
On calibrated representations of the degenerate affine periplectic Brauer algebraMay 13 2019We initiate the representation theory of the degenerate affine periplectic Brauer algebra on $n$ strands by constructing its finite-dimensional calibrated representations when $n=2$. We show that any such representation that is indecomposable and does ... More
Projective generation for equivariant $D$-modulesMay 13 2019We investigate compact projective generators in the category of equivariant $D$-modules on a smooth affine variety. For a reductive group $G$ acting on a smooth affine variety $X$, there is a natural set of compact projective objects indexed by finite ... More
Cycle-finite modules over artin algebrasMay 13 2019We describe the representation theory of finitely generated indecomposable modules over artin algebras which do not lie on cycles of indecomposable modules involving homomorphisms from the infinite Jacobson radical of the module category.
A Batalin-Vilkovisky structure on the complete cohomology ring of a Frobenius algebraMay 13 2019We study the exsistence of a Batalin-Vilkovisky differential on the complete cohomology ring of a Frobenius algebra. We construct a Batalin-Vilkovisky differential on the complete cohomology ring in the case of Frobenius algebras with diagonalizable Nakayama ... More
An intrinsic characterization of cofree representations of reductive groupsMay 13 2019We formulate and partially verify a conjecture characterizing cofree representations of connected reductive groups. As we explain, this conjecture may be viewed as a natural generalization of the Chevalley-Shepard-Todd theorem from the case of finite ... More
The capacity of quiver representations and Brascamp-Lieb constantsMay 12 2019Let $Q$ be a bipartite quiver, $V$ a real representation of $Q$, and $\sigma$ an integral weight of $Q$ orthogonal to the dimension vector of $V$. In this paper, we introduce the Brascamp-Lieb operator $T_{V,\sigma}$ associated to $(V,\sigma)$ and study ... More
Kac-Wakimoto conjecture for the periplectic Lie superalgebraMay 12 2019We prove the Kac-Wakimoto conjecture for the periplectic Lie superalgebra $\mathfrak{p}(n)$, stating that any simple module lying in a block of non-maximal atypicality has superdimension zero.
Finite-dimensional representations of hyper multicurrent and multiloop algebrasMay 12 2019We investigate the categories of finite-dimensional representations of multicurrent and multiloop hyperalgebras in positive characteristic, i.e., the hyperalgebras associated to the multicurrent algebras $\mathfrak g\otimes\mathbb{C}[t_1,\ldots,t_n]$ ... More
Coherent sheaves and quantum Coulomb branches I: tilting bundles from integrable systemsMay 12 2019In this paper, we consider how the approach of Bezrukavnikov and Kaledin to understanding the categories of coherent sheaves on symplectic resolutions can be applied to the Coulomb branches introduced by Braverman, Finkelberg and Nakajima. In particular, ... More
Symmetrization of representations of $GL_N$May 11 2019In this article, we develop a process to symmetrize the irreducible admissible representation of $GL_N(\mathbb{Q}_p)$, as a consequence we obtain a more geometric understanding of the coefficient $m(\mathbf{b}, \mathbf{a})$ appearing in the decomposition ... More
The Nullstellensatz for supersymmetric polynomialsMay 10 2019In this paper we prove a Nullstellensatz for supersymmetric polynomials. This gives a bijection between radical ideals and superalgebraic sets. These are algebraic sets which are invariant under the Weyl groupoid of Sergeev and Veselov, \cite{SV2}. Note ... More
Defect 2 spin blocks of symmetric groups and canonical basis coefficientsMay 10 2019This paper addresses the decomposition number problem for spin representations of symmetric groups in odd characteristic. Our main aim is to find a combinatorial formula for decomposition numbers in blocks of defect $2$, analogous to Richards's formula ... More
Gorenstein-projective and semi-Gorenstein-projective modules. IIMay 10 2019Let k be a field and q a non-zero element of k. In Part I, we have exhibited a 6-dimensional k-algebra A = A(q) and we have shown that if q has infinite multiplicative order, then A has a 3-dimensional local module which is semi-Gorenstein-projective, ... More
Functorial transfer of Cohomological Representations from $SP(4,\mathbb{R})$ to $GL(5,\mathbb{R})$May 10 2019Let $G=Sp(4,\mathbb{R})$ and let $\pi$ be an irreducible, unitary representation of $G$ which is cohomological with respect to trivial coefficients. Using the inclusion from ${}^LSp(4,\mathbb{R})^\circ = SO(5,\mathbb{C})$ to ${}^LGL(4,\mathbb{R})^\circ ... More
Socle deformations of selfinjective orbit algebras of tilted typeMay 09 2019We survey recent development of the study of finite-dimensional selfinjective algebras over a field which are socle equivalent to selfinjective orbit algebras of tilted type.
Representations and cohomology of a family of finite supergroup schemesMay 08 2019We examine the cohomology and representation theory of a family of finite supergroup schemes of the form $(\mathbb G_a^-\times \mathbb G_a^-)\rtimes (\mathbb G_{a(r)}\times (\mathbb Z/p)^s)$. In particular, we show that a certain relation holds in the ... More
Higher Spherical AlgebrasMay 08 2019We introduce and study higher spherical algebras, an exotic family of finite-dimensional algebras over an algebraically closed field. We prove that every such an algebra is derived equivalent to a higher tetrahedral algebra studied in [7], an hence that ... More
Gluing two affine Yangians of $\mathfrak{gl}_1$May 08 2019We construct a four-parameter family of affine Yangian algebras by gluing two copies of the affine Yangian of $\mathfrak{gl}_1$. Our construction allows for gluing operators with arbitrary (integer or half integer) conformal dimension and arbitrary (bosonic ... More
Koszul Algebras and Flow LatticesMay 08 2019We provide a homological algebraic realization of the lattices of integer cuts and integer flows of graphs. To a finite 2-edge-connected graph $\Gamma$ with a spanning tree $T$, we associate a finite dimensional Koszul algebra $A_{\Gamma,T}$. Under the ... More
On the Adjoint Representation of a Hopf AlgebraMay 08 2019We consider the adjoint representation of a Hopf algebra $H$ focusing on the locally finite part, $H_{\text{adfin}}$, defined as the sum of all finite-dimensional subrepresentations. For cocommutative $H$, we show that $H_{\text{adfin}}$ is a Hopf subalgebra ... More
The ascent-descent property for $2$-term silting complexesMay 08 2019We will prove that over commutative rings the silting property of $2$-term complexes induced by morphisms between projective modules is preserved and reflected by faithfully flat extensions.
Twisted superpotentials of 3-dimensional quadratic AS-regular algebrasMay 07 2019Let $k$ be an algebraically closed field of characteristic $0$ and $A$ a graded $k$-algebra finitely generated in degree $1$. In this paper, for $3$-dimensional quadratic AS-regular algebras except for Type EC, we give a complete list of twisted superpotentials ... More
A construction of projective bases for irreducible representations of multiplicative groups of division algebras over local fieldsMay 07 2019Let $F$ be a local non-archimedian field of positive characteristic, $D$ be a skew-field with center $F$ and $ G=D^{\star}$ be the multiplicative group of $D$. The goal of this paper is to provide a canonical decomposition of any complex irreducible representation ... More
Burch ideals and Burch ringsMay 07 2019May 09 2019We introduce the notion of Burch ideals and Burch rings. They are easy to define, and can be viewed as generalization of many well-known concepts, for example integrally closed ideals of finite colength and Cohen--Macaulay rings of minimal multiplicity. ... More
Burch ideals and Burch ringsMay 07 2019We introduce the notion of Burch ideals and Burch rings. They are easy to define, and can be viewed as generalization of many well-known concepts, for example integrally closed ideals of finite colength and Cohen--Macaulay rings of minimal multiplicity. ... More
Bessel Descents and Branching ProblemsMay 07 2019We discuss the theory of automorphic descents of Bessel type and its relation to automorphic version of branching problem and its relevant reciprocal branching problem.
The wall-chamber structures of the real Grothendieck groupsMay 06 2019For a finite-dimensional algebra $A$ over a field $K$, the real Grothendieck group $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}:=K_0(\operatorname{\mathsf{proj}} A) \otimes_\mathbb{Z} \mathbb{R}$ gives stability conditions of King. We study the associated ... More
On Representations of Reductive $p$--adic Groups over $\mathbb Q$--algebrasMay 06 2019May 10 2019In this paper we study certain category of smooth modules for reductive $p$--adic groups analogous to the usual smooth complex representations but with the field of complex numbers replaced by a $\mathbb Q$--algebra. We prove some fundamental results ... More
On Representations of Reductive $p$--adic Groups over $\mathbb Q$--algebrasMay 06 2019In this paper we define and study certain category of smooth modules for reductive $p$--adic groups analogous to the usual smooth complex representations but with the field of complex numbers replaced by a $\mathbb Q$--algebra. We prove many fundamental ... More
Non-associative magnetic translations: A QFT constructionMay 06 2019The non-associativity of translations in a quantum system with magnetic field background has received renewed interest in association with topologically trivial gerbes over $\mathbb{R}^n.$ The non-associativity is described by a 3-cocycle of the group ... More
Convexity properties of gradient maps associated to real reductive representationsMay 06 2019Let G be a connected real reductive Lie group acting linearly on a finite dimensional vector space V over R. This action admits a Kempf-Ness function and so we have an associated gradient map. If G is Abelian we explicitly compute the image of G orbits ... More
New symmetries for the $U_q(sl_N)$ 6-j symbols from the Eigenvalue conjectureMay 06 2019In the present paper we discuss the eigenvalue conjecture, suggested in 2012, in the particular case of $U_q(sl_2)$. The eigenvalue conjecture provides a certain symmetry for Racah coefficients and we prove that \textbf{the eigenvalue conjecture is provided ... More
Three results for tau-rigid modulesMay 06 2019$\tau$-rigid modules are essential in the $\tau$-tilting theory introduced by Adachi, Iyama and Reiten. In this paper, we give equivalent conditions for Iwanaga-Gorenstein algebras with self-injective dimension at most one in terms of $\tau$-rigid modules. ... More
On integrability of transverse Lie-Poisson structure to nilpotent elementsMay 06 2019We apply the argument shift method to transverse Poisson structures to nilpotent elements of Lie-Poisson structures in simple Lie algebras. Examples show that this method always leads to a simple construction of completely integrable system. We provide ... More
The regular representation of $U_v(\mathfrak{gl}_{m|n})$May 06 2019Using quantum differential operators, we construct a super representation of $U_v(\mathfrak{gl}_{m|n})$ on a certain polynomial superalgebra. We then extend the representation to its formal power series algebra which contains a $U_v(\mathfrak{gl}_{m|n})$-submodule ... More
Comparison of quiver varieties, loop Grassmannians and nilpotent cones in type AMay 06 2019In type A we find equivalences of geometries arising in three settings: Nakajima's (``framed'') quiver varieties, conjugacy classes of matrices and loop Grassmannians. These are now all given by explicit formulas. As an application we provide a geometric ... More
Homological branching law for $(\mathrm{GL}_{n+1}(F), \mathrm{GL}_n(F))$: projectivity and indecomposabilityMay 05 2019This paper studies homological properties of irreducible representations restricted from $\mathrm{GL}_{n+1}(F)$ to $\mathrm{GL}_n(F)$. We establish the following: (1) classify irreducible smooth representations of $\mathrm{GL}_{n+1}(F)$ which are projective ... More
Enhanced zeta distributions and its functional equationsMay 05 2019We consider an ``enhanced symmetric space'', which is a prehomogeneous vector space. This vector space is intimately related to a double flag variety studied in \cite{NO.2018}. On a distinguished open orbit called ``enhanced positive cone'', we consider ... More
Representations and Modules of Rota-Baxter AlgebrasMay 04 2019We give a broad study of representation and module theory of Rota-Baxter algebras, motivated by Rota-Baxter matrix representations in the renormalization of quantum field theory and by geometric connections. Regular-singular decompositions of Rota-Baxter ... More
Binomial arrays and generalized Vandermonde identitiesMay 04 2019In previous work on Clebsch-Gordan coefficients, certain remarkable hexagonal arrays of integers are constructed that display behaviors found in Pascal's Triangle. We explain these behaviors further using the binomial transform and discrete convolution. ... More
The 2-leg vertex in K-theoretic DT theoryMay 04 2019K-theoretic Donaldson-Thomas counts of curves in toric and many related threefolds can be computed in terms of a certain canonical 3-valent tensor, the K-theoretic equivariant vertex. In this paper we derive a formula for the vertex in the case when two ... More
Local duality for the singularity category of a finite dimensional Gorenstein algebraMay 04 2019A duality theorem for the singularity category of a finite dimensional Gorenstein algebra is proved. It complements a duality on the category of perfect complexes, discovered by Happel. One of its consequences is an analogue of Serre duality, and the ... More
Highest weight vectors in plethysmsMay 04 2019We realize the $\mathrm{GL}_n(\mathbb{C})$-modules $S^k(S^m(\mathbb{C}^n))$ and $\Lambda^k(S^m(\mathbb{C}^n))$ as spaces of polynomial functions on $n\times k$ matrices. In the case $k=3$, we describe explicitly all the $\mathrm{GL}_n(\mathbb{C})$-highest ... More
Regular irreducible represntations of classical reductive groups over finite quotient ringsMay 04 2019A parametrization of irreducible representations associated with a regular adjoint orbit of a reductive group over finite quotient rings of a non-dyadic non-archimedean local filed is presented. The parametrization is given by means of (a subset of) the ... More
Graded torsion-free ${\mathfrak{sl}_2(\mathbb{C})}$-modules of rank 2May 03 2019In this paper we explore the possibility of endowing simple infinite-dimensional ${\mathfrak{sl}_2(\mathbb{C})}$-modules by the structure of the graded module. The gradings on finite-dimensional simple module over simple Lie algebras has been studied ... More
Multiplicity one theorem for $(\mathrm{GL}_{n+1},\mathrm{GL}_n)$ over a local field of positive characteristicMay 03 2019Let $\mathbb{F}$ be a non-archimedean local field of positive characteristic different from 2. We consider distributions on $\mathrm{GL}(n+1,\mathbb{F})$ which are invariant under the adjoint action of $\mathrm{GL}(n,\mathbb{F})$. We prove that any such ... More
Wide subcategories and lattices of torsion classesMay 03 2019In this paper, we study the relationship between wide subcategories and torsion classes of an abelian length category $\mathcal{A}$ from the point of view of lattice theory. Motivated by $\tau$-tilting reduction of Jasso, we mainly focus on intervals ... More
On the Schwartz Space $ \mathcal S(G(k)\backslash G(\mathbb A)) $May 02 2019For a connected reductive group $ G $ defined over a number field $ k $, we construct the Schwartz space $ \mathcal S(G(k)\backslash G(\mathbb A)) $. This space is an adelic version of Casselman's Schwartz space $ \mathcal S(\Gamma\backslash G_\infty) ... More
The cohomology of the Steenrod algebra and the mod $p$ Lannes-Zarati homomorphismMay 02 2019In this paper, we compute ${\rm Ext}_{A}^{s}(\widetilde{H}^*(B\mathbb{Z}/p),\mathbb{F}_p)$ for $s\leq 1$. Using this result, we investigate the behavior of $\varphi_3^{\mathbb{F}_p}$ and $\varphi_s^{\widetilde{H}^*(B\mathbb{Z}/p)}\ (s\leq1)$ for an odd ... More
Combinatorics of faithfully balanced modulesMay 02 2019We study and classify faithfully balanced modules for the algebra of lower triangular $n$ by $n$ matrices. The theory extends known results about tilting modules, which are classified by binary trees, and counted with the Catalan numbers. The number of ... More
Lie subalgebras of Differential Operators in one VariableMay 01 2019Let $\operatorname{Witt}$ be the Lie algebra generated by the set $\{L_i\,\vert\, i \in {\mathbb Z}\}$ and $\operatorname{Vir}$ its universal central extension. Let $\operatorname{Diff}(V)$ be the Lie algebra of differential operators on $V=\mathbb{C}[[z]]$, ... More
Jordan Algebraic Interpretation of Maximal Parabolic Subalgebras : Exceptional Lie AlgebrasMay 01 2019With this paper we start a project which connects two vast scientific areas: Jordan algebras and representation theory. Within representation theory, we focus on non-compact semisimple Lie algebras and groups and on the modern theory of their induced ... More
Geometry of central extensions of nilpotent Lie algebrasMay 01 2019We obtain a recurrent and monotone method for constructing and classifying nilpotent Lie algebras by means of successive central extensions. It consists in calculating the second cohomology of an extendable nilpotent Lie algebra with the subsequent study ... More
On the existence of admissible supersingular representations of $p$-adic reductive groupsApr 30 2019Suppose that $\mathbf{G}$ is a connected reductive group over a finite extension $F/\mathbb{Q}_p$, and that $C$ is a field of characteristic $p$. We prove that the group $\mathbf{G}(F)$ admits an irreducible admissible supercuspidal, or equivalently supersingular, ... More
Lifting 1/4-BPS States on K3 and Mathieu MoonshineApr 30 2019The elliptic genus of K3 is an index for the 1/4-BPS states of its sigma model. At the torus orbifold point there is an accidental degeneracy of such states. We blow up the orbifold fixed points using conformal perturbation theory, and find that this ... More
On canonical bases of Letzter algebra $\mathbf U^{\imath}(\mathfrak{sl}_2)$Apr 30 2019Let $\mathbf U^{\imath}\equiv\mathbf U^{\imath} (\mathfrak{sl}_2)$ be Letzter's coideal subalgebra of quantum $\mathfrak{sl}_2$ corresponding to the symmetric pair $(\mathfrak{sl}_2(\mathbb C),\mathbb C)$. As a subalgebra of quantum $\mathfrak{sl}_2$, ... More
Ring Constructions and Generation of the Unbounded Derived Module CategoryApr 30 2019Given the unbounded derived module category of a ring $A$, we consider the triangulated subcategory closed under arbitrary coproducts generated by injective modules. Similarly we also look at the triangulated subcategory closed under arbitrary products ... More
D-modules on rigid analytic spaces III: Weak holonomicity and operationsApr 30 2019We develop a dimension theory for coadmissible D-cap-modules on rigid analytic spaces and study those which are of minimal dimension, in analogy to the theory of holonomic D-modules in the algebraic setting. We discuss a number of pathologies contained ... More
D-modules on rigid analytic spaces III: Weak holonomicity and operationsApr 30 2019May 15 2019We develop a dimension theory for coadmissible D-cap-modules on rigid analytic spaces and study those which are of minimal dimension, in analogy to the theory of holonomic D-modules in the algebraic setting. We discuss a number of pathologies contained ... More
Tensor products of modular representations of $\operatorname{SL}_2(\mathbb{F}_p)$ and a random walk on their indecomposable summandsApr 30 2019In this paper we give a novel, concise and elementary proof of the decomposition of tensor products of simple modular $\operatorname{SL}_2(\mathbb{F}_p)$-representations. This result is used to decompose tensor products involving their projective covers ... More
A generalization of Steinberg theory and an exotic moment mapApr 30 2019For a reductive group $G$, Steinberg established a map from the Weyl group to the set of nilpotent $G$-orbits by using moment maps on double flag varieties. In particular, in the case of the general linear group, it provides a geometric interpretation ... More
Finite-dimensional Leibniz algebra representations of $\mathfrak{sl}_2$Apr 30 2019All finite-dimensional Leibniz algebra bimodules of a Lie algebra $\mathfrak{sl}_2$ over a field of characteristic zero are described.
Jantzen filtration of Weyl modules, product of Young symmetrizers and denominator of Young's seminormal basisApr 30 2019Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic $p>0$, $\Delta(\lambda)$ denote the Weyl module of $G$ of highest weight $\lambda$ and $\iota_{\lambda,\mu}:\Delta(\lambda+\mu)\to \Delta(\lambda)\otimes\Delta(\mu)$ ... More
Enumeration for strong minuscule elements in the Weyl group of type AApr 29 2019In the case of finite-dimensional simple Lie algebra of type A, we enumerate special classes of pre-dominant integral weights and dominant minuscule elements. In addition, as an application, we give a dimension formula for certain Demazure modules.
On morphisms killing weights and Hurewicz-type theoremsApr 29 2019We study "canonical weight decompositions" slightly generalizing that defined by J. Wildeshaus. For an triangulated category $C$, any integer $n$, and a weight structure $w$ on $C$ a triangle $LM\to M\to RM\to LM[1]$, where $LM$ is of weights at most ... More
Polynomial functors and two-parameter quantum symmetric pairsApr 29 2019Apr 30 2019We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups $GL_n$, the two-parameter polynomial functors give a new ... More
Polynomial functors and two-parameter quantum symmetric pairsApr 29 2019We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups $GL_n$, the two-parameter polynomial functors give a new ... More
Polynomial functors and two-parameter quantum symmetric pairsApr 29 2019May 07 2019We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups $GL_n$, the two-parameter polynomial functors give a new ... More
On stratification for spaces with Noetherian mod $p$ cohomologyApr 29 2019Let $X$ be a topological space with Noetherian mod $p$ cohomology and let $C^*(X;\mathbb{F}_p)$ be the commutative ring spectrum of $\mathbb{F}_p$-valued cochains on $X$. The goal of this paper is to exhibit conditions under which the category of module ... More
Virtual representation motivesApr 29 2019Principal $GL_n$-bundles (aka vector bundles) are locally trivial in the Zariski topology, whereas principal $PGL_n$-bundles (aka Azumaya algebras) are not, to the delight of every non-commutative algebraist. Still, this makes the calculation of motives ... More
The Milnor-Moore theorem for $L_\infty$ algebras in rational homotopy theoryApr 29 2019We give a construction of the universal enveloping $A_\infty$ algebra of a given $L_\infty$ algebra, alternative to the already existing versions. As applications, we derive a higher homotopy algebras version of the classical Milnor-Moore theorem, proposing ... More
Center at the critical level for centralizers in type $A$Apr 29 2019We consider the affine vertex algebra at the critical level associated with the centralizer of a nilpotent element in the Lie algebra $\mathfrak{gl}_N$. Due to a recent result of Arakawa and Premet, the center of this vertex algebra is an algebra of polynomials. ... More
Cuspidal modules for the derivation Lie algebra over a rational quantum torusApr 29 2019Let $\mathbb C_Q$ denote a rational quantum torus with $d$ variables, and $\mathcal Z$ be the centre of $\mathbb C_Q$. In this paper we give a explicit description of the structure of the cuspidal modules for the derivation Lie algebra $\mathcal D$ over ... More
Density of $g$-vector cones from triangulated surfacesApr 29 2019We study $g$-vector cones associated with clusters of cluster algebras defined from a marked surface $(S,M)$ of rank $n$. We determine the closure of the union of $g$-vector cones associated with all clusters. It is equal to $\mathbb{R}^n$ except for ... More
Density of $g$-vector cones from triangulated surfacesApr 29 2019May 14 2019We study $g$-vector cones associated with clusters of cluster algebras defined from a marked surface $(S,M)$ of rank $n$. We determine the closure of the union of $g$-vector cones associated with all clusters. It is equal to $\mathbb{R}^n$ except for ... More
BGG category for the quantum Schrödinger algebraApr 29 2019In this paper, we study the BGG category $\mathcal{O}$ for the quantum Schr{\"o}dinger algebra $U_q(\mathfrak{s})$, where $q$ is a nonzero complex number which is not a root of unity. If the central charge $\dot z\neq 0$, using the module $B_{\dot z}$ ... More
Deformations of Loday-type algebras and their morphismsApr 28 2019We study a formal deformation of multiplications in an operad. This closely resemble Gerstenhaber's deformation theory for associative algebras. However, this is applicable to various Loday-type algebras and to their twisted analogues. We explicitly describe ... More
Deformations of Loday-type algebras and their morphismsApr 28 2019May 03 2019We study a formal deformation of multiplications in an operad. This closely resemble Gerstenhaber's deformation theory for associative algebras. However, this is applicable to various Loday-type algebras and to their twisted analogues. We explicitly describe ... More
On the Representation theory of the Infinite Temperley-Lieb algebraApr 28 2019We begin the study of the representation theory of the infinite Temperley-Lieb algebra. We fully classify its finite dimensional representations, then introduce infinite link state representations and classify when they are irreducible or indecomposable. ... More