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Tilting modules for classical Lie superalgebrasJul 15 2019We study tilting and projective-injective modules in a parabolic BGG category $\mathcal O$ for an arbitrary classical Lie superalgebra. We establish a version of Ringel duality for this type of Lie superalgebras which allows to express the characters ... More

Coulomb branches of quiver gauge theories with symmetrizersJul 15 2019We generalize the mathematical definition of Coulomb branches of $3$-dimensional $\mathcal N=4$ SUSY quiver gauge theories in arXiv:1503.03676, arXiv:1601.03686, arXiv:1604.03625 to the cases with symmetrizers. We obtain generalized affine Grassmannian ... More

More on Periodicity and Duality associated with Jordan partitionsJul 15 2019Let $J_r$ denote a full $r \times r$ Jordan block matrix with eigenvalue $1$ over a field $F$ of characteristic $p$. For positive integers $r$ and $s$ with $r \leq s$, the Jordan canonical form of the $r s \times r s$ matrix $J_{r} \otimes J_{s}$ has ... More

Algebras with finite relative dominant dimension and almost n-precluster tilting modulesJul 15 2019In this paper, we investigate the relative dominant dimension with respect to an injective module and characterize the algebras with finite relative dominant dimension. As an application, we introduce the almost n-precluster tilting module and establish ... More

Distinguished representations of SO(n+1,1) x SO(n,1), periods and branching lawsJul 14 2019Given irreducible representations $\Pi$ and $\pi$ of the rank one special orthogonal groups $G=SO(n+1,1)$ and $G'=SO(n,1)$ with nonsingular integral infinitesimal character, we state in terms of $\theta$-stable parameter necessary and sufficient conditions ... More

Harish-Chandra bimodules for type A rational Cherednik algebrasJul 13 2019We study Harish-Chandra bimodules for the rational Cherednik algebra associated to the symmetric group $S_{n}$. In particular, we show that for any parameter $c \in \mathbb{C}$, the category of Harish-Chandra $H_{c}$-bimodules admits a fully faithful ... More

Cotilting with balanced big Cohen-Macaualay modulesJul 12 2019Over a Cohen-Macaulay local ring admitting a canonical module the definable closure of the class of balanced big Cohen-Macaulay modules is cotilting and is the smallest such class containing the maximal Cohen-Macaulay modules. We describe its cotilting ... More

Deformations of Vector Bundles over Lie GroupoidsJul 12 2019VB-groupoids are vector bundles in the category of Lie groupoids. They encompass several classical objects, including Lie group representations and 2-vector spaces. Moreover, they provide geometric pictures for 2-term representations up to homotopy of ... More

Filiform Lie algebras with low derived lengthJul 12 2019We construct, for any integer n greater than or equal to 5, a family of complex filiform Lie algebras with derived length at most 3 and dimension n. We also give examples of n-dimensional filiform Lie algebras with derived length greater than 3.

Cograde conditions and cotorsion pairsJul 12 2019Let $R$ and $S$ be rings and $_R\omega_S$ a semidualizing bimodule. We study when the double functor $\Tor^S_i(\omega, \Ext^i_{R}(\omega,-))$ preserves epimorphisms and the double functor $\Ext_{R}^i(\omega, \Tor_i^{S}(\omega,-))$ preserves monomorphisms ... More

On L-embeddings and double covers of tori over local fieldsJul 11 2019To every maximal torus T of a connected reductive group G defined over a local field F we associate a canonical double cover of the topological group T(F) of its F-rational points. We further associate an L-group to this double cover and establish a natural ... More

Simple-minded reductions of triangulated categoriesJul 11 2019We will introduce a new reduction process of triangulated category, which is analogue to the silting reduction and Calabi-Yau reduction. For a triangulated category $\cal T$ with a pre-simple-minded collection (=pre-SMC) $\cal R$, we construct a new triangulated ... More

Matlis category equivalences for a ring epimorphismJul 11 2019Under mild assumptions, we construct the two Matlis additive category equivalences for an associative ring epimorphism $u\colon R\to U$. Assuming that the ring epimorphism is homological of flat/projective dimension $1$, we discuss the abelian categories ... More

Demazure and local Weyl modules for twisted hyper current algebrasJul 10 2019In this paper, we study local graded Weyl modules and Demazure modules for twisted hypercurrent algebras. We prove that local graded Weyl modules for a twisted hypercurrent algebra are isomorphic to the corresponding level 1 Demazure modules, and moreover, ... More

Characterizations of nested GVZ-groups by central seriesJul 10 2019Many properties of groups can be defined by the existence of a particular normal series. The classic examples being solvability, supersolvability and nilpotence. Among the nilpotent groups are the so-called nested GVZ-groups --- groups where the centers ... More

Degeneration of spectral sequences and complex Lagrangian submanifoldsJul 10 2019There is a local-to-global $\mathrm{Ext}$ spectral sequence $\mathrm{E}_2^{p,q} = \mathrm{H}^p(\mathrm{L}, \Omega^q_\mathrm{L}) \Rightarrow \mathrm{Ext}^{p+q}(i_*\mathscr{O}_\mathrm{L}, i_*\mathscr{O}_{\mathrm{L}})$ for a smooth Lagrangian subvariety ... More

A functorial approach to monomorphism categories for species IJul 10 2019For any generalised species over a locally bounded quiver we investigate abstract versions of the monomorphism category as studied by Ringel and Schmidmeier. We prove that analogues of the kernel and cokernel functor send almost split sequences over the ... More

Connections between vector-valued and highest weight Jack and Macdonald polynomialsJul 10 2019We analyze conditions under which a projection from the vector-valued Jack or Macdonald polynomials to scalar polynomials has useful properties, especially commuting with the actions of the symmetric group or Hecke algebra, respectively, and with the ... More

Alvis-Curtis Duality for Representations of Reductive Groups with Frobenius MapsJul 10 2019We generalize Alvis-Curtis duality to the abstract representations of reductive groups with Frobenius maps. Analogous the case of representation of finite reductive groups, we show that the Alvis-Curtis duality of infinite type which we defined in this ... More

On the fibres of Mishchenko-Fomenko systemsJul 09 2019This work is concerned with Mishchenko and Fomenko's celebrated theory of completely integrable systems on a complex semisimple Lie algebra $\mathfrak{g}$. Their theory associates a maximal Poisson-commutative subalgebra of $\mathbb{C}[\mathfrak{g}]$ ... More

BGG complexes in singular blocks of category OJul 09 2019Using translation from the regular block, we construct and analyze properties of BGG complexes in singular blocks of BGG category ${\mathcal{O}}$. We provide criteria, in terms of the Kazhdan-Lusztig-Vogan polynomials, for such complexes to be exact. ... More

On the first Hochschild cohomology of cocommutative Hopf algebras of finite representation typeJul 09 2019Let $\mathscr{B}_0(\mathcal{G})\subseteq k\mathcal{G}$ be the principal block algebra of the group algebra $k\mathcal{G}$ of an infinitesimal group scheme $\mathcal{G}$ over an algebraically closed field $k$ of characteristic ${\rm char}(k)=:p\geq 3$. ... More

A product formula for homogeneous characteristic functionsJul 09 2019A bounded linear operator $T$ on a Hilbert space is said to be homogeneous if $\varphi(T)$ is unitarily equivalent to $T$ for all $\varphi$ in the group M\"{o}b of bi-holomorphic automorphisms of the unit disc. A projective unitary representation $\sigma$ ... More

On the $\mathbb{A}^1$-Degree of a Weyl CoverJul 08 2019The notion of $\mathbb{A}^1$-degree provides an arithmetic refinement of the usual notion of degree in algebraic geometry. In this note, we compute $\mathbb{A}^1$-degrees of certain finite covers $f\colon \mathbb{A}^n\to \mathbb{A}^n$ induced by quotients ... More

From weight structures to (orthogonal) $t$-structures and backJul 08 2019A $t$-structure $t=(C_{t\le 0},C_{t\ge 0})$ on a triangulated category $C$ is right adjacent to a weight structure $w=(C_{w\le 0}, C_{w\ge 0})$ if $C_{t\ge 0}=C_{w\ge 0}$; then $t$ can be uniquely recovered from $w$ and vice versa. We prove that if $C$ ... More

Quiver Schur algebras and cohomological Hall algebrasJul 08 2019We establish a connection between a generalization of KLR algebras, called quiver Schur algebras, and the cohomological Hall algebras of Kontsevich and Soibelman. More specifically, we realize quiver Schur algebras as algebras of multiplication and comultiplication ... More

The meromorphic R-matrix of the YangianJul 08 2019Let g be a complex semisimple Lie algebra and Yg the Yangian of g. The main goal of this paper is to clarify the analytic nature of Drinfeld's universal R-matrix of Yg. It is known that the radius of convergence of R(s) on the tensor product of two finite-dimensional ... More

The Principal Representations of Reductive Groups with Frobenius MapsJul 07 2019We introduce the principal representation category of reductive groups with Frobenius maps and show that this category is a highest weight category when the ground field is complex field $\mathbb{C}$. We also study certain kind of bound quiver algebras ... More

Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional typesJul 07 2019Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\mathfrak{b}$ be a Borel subalgebra of $\mathfrak{g}$, $\mathfrak{n}$ be the nilradical of $\mathfrak{b}$, and $U(\mathfrak{n})$ be the universal enveloping algebra of $\mathfrak{n}$. We study primitive ... More

A constructive proof of Pokrzywa's theorem about perturbations of matrix pencilsJul 07 2019Our purpose is to give new proofs of several known results about perturbations of matrix pencils. Andrzej Pokrzywa (1986) described the closure of orbit of a Kronecker canonical pencil $A-\lambda B$ in terms of inequalities with pencil invariants. In ... More

The tamely ramified Fundamental Local Equivalence at integral levelJul 06 2019Let $G$ be an almost simple algebraic group with Langlands dual $\check{G}$, and fix a noncritical integral level $\kappa$ for $G$, with dual level $\check{\kappa}$ for $\check{G}$. We prove an equivalence between $\kappa$-twisted Whittaker $D$-modules ... More

On a method to construct exponential families by representation theoryJul 06 2019Exponential family plays an important role in information geometry. In arXiv:1811.01394, we introduced a method to construct an exponential family $\mathcal{P}=\{p_\theta\}_{\theta\in\Theta}$ on a homogeneous space $G/H$ from a pair $(V,v_0)$. Here $V$ ... More

Rigidity and derived isomorphism problem for enveloping algebrasJul 05 2019Jul 11 2019We prove that there are no injective homomorphisms between enveloping algebras of non-isomorphic semi-simple Lie algebras of the same dimension. We also describe the center of reduction modulo large prime $p$ of the enveloping algebra of an algebraic ... More

Rigidity and derived isomorphism problem for enveloping algebrasJul 05 2019We prove that there are no injective homomorphisms between enveloping algebras of non-isomorphic semi-simple Lie algebras of the same dimension. We also describe the center of reduction modulo large prime $p$ of the enveloping algebra of an algebraic ... More

Cuspidal irreducible representations of quaternionic forms of p-adic classical groups for odd pJul 05 2019Given a quaternionic form G of a p-adic classical group ($p$ odd) we classify all cuspidal irreducible complex representations of G. It is a straight forward generalization of the classification in the p-adic classical group case. We prove two theorems: ... More

Kazama-Suzuki coset construction and its inverseJul 04 2019We study the representation theory of the Kazama-Suzuki coset vertex operator superalgebra associated with the pair of a complex simple Lie algebra and its Cartan subalgebra. In the case of type $A_{1}$, B.L. Feigin, A.M. Semikhatov, and I.Yu. Tipunin ... More

Extensions of finite irreducible modules of Lie conformal algebras $\mathcal{W}(a,b)$ and some Schrödinger-Virasoro type Lie conformal algebrasJul 04 2019Lie conformal algebras $\mathcal{W}(a,b)$ are the semi-direct sums of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one. In this paper, we give a complete classification of extensions of finite irreducible conformal modules ... More

Weak integral forms and the sixth Kaplansky conjectureJul 04 2019It is a short unpublished note from 1998. I make it public because Cuadra and Meir refer to it in their paper. We precisely state and prove a folklore result that if a finite dimensional semisimple Hopf algebra admits a weak integral form then it is of ... More

Conformal modules and their extensions of a Lie conformal algebra related to a 2-dimensional Novikov algebraJul 03 2019Let $\mathcal{R}$ be a free Lie conformal algebra of rank $2$ with $\mathbb{C}[\partial]$-basis $\{L,I\}$ and relations \begin{eqnarray*} \left[L_{\lambda} L\right]=(\partial+2 \lambda) (L+I),\ \left[L_{\lambda} I\right]=(\partial+\lambda) I, \ \left[I_{\lambda} ... More

Quaternionic Analysis, Representation Theory and Physics IIJul 02 2019We develop further quaternionic analysis introducing left and right doubly regular functions. We derive Cauchy-Fueter type formulas for these doubly regular functions that can be regarded as another counterpart of Cauchy's integral formula for the second ... More

On the structure of modules over walled Brauer algebra via normal form and random walksJul 02 2019We analyze cyclic cell modules over walled Brauer algebra in terms of a certain normal form. The latter allows us to decompose the algebra into the generating set and annihilator ideal of a certain cyclic vector. In addition, we show that the numbers ... More

Computing the real Weyl groupJul 02 2019Let g be a semisimple Lie algebra over the real numbers. We describe an explicit combinatorial construction of the real Weyl group of g with respect to a given Cartan subalgebra. An efficient computation of this Weyl group is important for the classification ... More

Stable equivalences of Morita type for $Φ$-Beilinson-Green algebrasJul 02 2019In this paper, we present a method to construct new stable equivalences of Morita type. Suppose that a stable equivalence of Morita type between finite dimensional algebras $A$ and $B$ is defined by a $B$-$A$-bimodule $N$. Then, for any finite admissible ... More

Stable equivalences of Morita type for $Φ$-Beilinson-Green algebrasJul 02 2019Jul 08 2019In this paper, we present a method to construct new stable equivalences of Morita type. Suppose that a stable equivalence of Morita type between finite dimensional algebras $A$ and $B$ is defined by a $B$-$A$-bimodule $N$. Then, for any finite admissible ... More

Classification of the finite-dimensional unitary representations of type B rational Cherednik algebrasJul 01 2019We compare crystal combinatorics of the level $2$ Fock space with the classification of unitary representations of type $B$ rational Cherednik algebras to show that any finite-dimensional unitary irreducible representation of such an algebra is labeled ... More

Classification of the finite-dimensional unitary representations of type B rational Cherednik algebrasJul 01 2019Jul 05 2019We compare crystal combinatorics of the level $2$ Fock space with the classification of unitary representations of type $B$ rational Cherednik algebras to show that any finite-dimensional unitary irreducible representation of such an algebra is labeled ... More

Kazhdan-Lusztig R-polynomials for pirconsJul 01 2019The purpose of this work is to provide a common combinatorial framework for some of the analogues and generalizations of Kazhdan-Lusztig R-polynomials that have appeared since the introduction of these remarkable polynomials (e.g., parabolic Kazhdan-Lusztig ... More

Abstract induced modules for reductive groups with Frobenius mapsJul 01 2019Let ${\bf G}$ be a connected reductive group defined over the finite field $\mathbb{F}_q$ of $q$ elements, and ${\bf B}$ be a Borel subgroup of ${\bf G}$ defined over $\mathbb{F}_q$. We show that the abstract induced module $\mathbb{M}(\theta)=\Bbbk{\bf ... More

Mellin-Barnes presentations for Whittaker wave functionsJul 01 2019We obtain certain Mellin-Barnes integrals which present Whittaker wave functions related to classical real split forms of simple complex Lie groups.

Representability of permutation representations on coalgebras and the isomorphism problemJul 01 2019Let $G$ be a group and let $\rho\colon G\to\operatorname{Sym}(V)$ be a permutation representation of $G$ on a set $V$. We prove that there is a faithful $G$-coalgebra $C$ such that $G$ arises as the image of the restriction of $\operatorname{Aut}(C)$ ... More

The Grothendieck group of unipotent representations: a new basisJun 30 2019Let G(F_q) be the group of rational points of a simple algebraic group defined and split over a finite field F_q. In this paper we define a new basis for the Grothendieck group of unipotent representations of G(F_q).

Isomorphism problems for tensors, groups, and cubic forms: completeness and reductionsJun 30 2019In this paper we consider the problems of testing isomorphism of tensors, $p$-groups, cubic forms, algebras, and more, which arise from a variety of areas, including machine learning, group theory, and cryptography. These problems can all be cast as orbit ... More

Lectures on Factorization Homology, Infinity-Categories, and Topological Field TheoriesJun 28 2019These are notes from an informal mini-course on factorization homology, infinity-categories, and topological field theories. The target audience was imagined to be graduate students who are not homotopy theorists.

On Shimura varieties for unitary groupsJun 28 2019This is a largely expository article based on our previous work on arithmetic diagonal cycles on unitary Shimura varieties. We define a class of Shimura varieties closely related to unitary groups which represent a moduli problem of abelian varieties ... More

Theta functions and quiver GrassmanniansJun 28 2019In this article, we use the relationship between cluster scattering diagrams and stability scattering diagrams to relate quiver representations with these diagrams. With a notion of positive crossing of a path $\gamma$, we show that if $\gamma$ has positive ... More

Quantum periodicity and Kirillov-Reshetikhin modulesJun 28 2019We give a proof of the periodicity of quantum $T$-systems of type $A_n\times A_\ell$ with certain spiral boundary conditions. Our proof is based on categorification of the $T$-system in terms of the representation theory of quantum affine algebras, more ... More

Quantum generalized Kac--Moody algebras via Hall algebras of complexesJun 28 2019We establish an embedding of the quantum enveloping algebra of a symmetric generalized Kac--Moody algebra into a localized Hall algebra of $\mathbb Z_2$-graded complexes of representations of a quiver with (possible) loops. To overcome difficulties resulting ... More

On spherical unitary representations of groups of spheromorphisms of Bruhat--Tits treesJun 28 2019Consider an infinite homogeneous tree $T_n$ of valence $n+1$, its group $Aut(T_n)$ of automorphisms, and the group $Hie(T_n)$ of its spheromorphisms (hierarchomorphisms), i.~e., the group of homeomorphisms of the boundary of $T_n$ that locally coincide ... More

On a class of infinite simple Lie conformal algebrasJun 28 2019In this paper, we study a class of infinite simple Lie conformal algebras associated to a class of generalized Block type Lie algebras. The central extensions, conformal derivations and free intermediate series modules of this class of Lie conformal algebras ... More

Tilting subcategories in extriangulated categoriesJun 28 2019Extriangulated category was introduced by Nakaoka and Palu to give a unification of properties in exact categories and triangulated categories. A notion of tilting (or cotilting) subcategories in an extriangulated category is defined in this paper. We ... More

Irregular vertex algebrasJun 28 2019We introduce the notion of irregular vertex (operator) algebras. The irregular versions of fundamental properties, such as Goddard uniqueness theorem, associativity and operator product expansions are formulated and proved. We also give some elementary ... More

Factorization problems in complex reflection groupsJun 27 2019We enumerate factorizations of a Coxeter element in a well generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our approach is fully ... More

The Nakayama automorphism of a self-injective preprojective algebraJun 27 2019We give a simple proof, using Auslander-Reiten theory, that the preprojective algebra of a basic hereditary algebra of finite representation type is Frobenius. We then describe its Nakayama automorphism, which is induced by the Nakayama functor on the ... More

Some implications of a conjecture of Zabrocki to the action of $S_{n}$ on polynomial differential formsJun 27 2019The symmetric group acts on polynomial differential forms on $\mathbb{R}^{n}$ through its action by permuting the coordinates. In this paper the $S_{n}% $-invariants are shown to be freely generated by the elementary symmetric polynomials and their exterior ... More

The Racah algebra as a subalgebra of the Bannai--Ito algebraJun 27 2019Assume that $\mathbb F$ is a field with ${\rm char\,}\mathbb F\not=2$. The Racah algebra $\Re$ is a unital associative $\mathbb F$-algebra defined by generators and relations. The generators are $A,B,C,D$ and the relations assert that $$ [A,B]=[B,C]=[C,A]=2D ... More

The Racah algebra as a subalgebra of the Bannai--Ito algebraJun 27 2019Jun 28 2019Assume that $\mathbb F$ is a field with ${\rm char\,}\mathbb F\not=2$. The Racah algebra $\Re$ is a unital associative $\mathbb F$-algebra defined by generators and relations. The generators are $A,B,C,D$ and the relations assert that $$ [A,B]=[B,C]=[C,A]=2D ... More

A reduction theorem for the Galois-McKay conjectureJun 27 2019We generalize the theory of ordering character triples, developed by Navarro and Sp\"ath, by taking into account the action of Galois automorphisms on characters. This new technique, together with previous results of Ladisch and Turull, allows us to reduce ... More

A note on vertices of indecomposable tensor productsJun 27 2019G. Navarro raised the question under what circumstancs two vertices of two indecomposable modules over a finite group algebra generate a Sylow $p$-subgroup. The present note provides a sufficient criterion for when this is the case. This generalises a ... More

Some questions on global distinction for $\mathrm{SL}(n)$Jun 27 2019Let $E/F$ be a quadratic extension of number fields and let $\pi$ be an $\mathrm{SL}_n(\mathbb{A}_F)$-distinguished cuspidal automorphic representation of $\mathrm{SL}_n(\mathbb{A}_E)$. Using an unfolding argument, we prove that an element of the $\mathrm{L}$-packet ... More

The normal shapes of the symplectic and contact forms over algebras of divided powersJun 27 2019This text is the English translation of a 1986 manuscript which gives the classification of the differential forms parametrizing the finite-dimensional Lie algebras of hamiltonian and contact Cartan types over fields of positive characteristic. The results ... More

2-representations of Soergel bimodulesJun 27 2019In this paper we study the graded 2-representation theory of Soergel bimodules for a finite Coxeter group. We establish a precise connection between the graded 2-representation theory of this non-semisimple 2-category and the 2-representation theory of ... More

Expository paper on Clifford algebras ,representations , and the octonion algebraJun 27 2019This paper is meant to be an informative introduction to spinor representations of Clifford algebras. In this paper we will have a look at Clifford algebras and the octonion algebra. We begin the paper looking at the quaternion algebra $\mathbb{H}$ and ... More

Some extension algebras for standard modules over KLR algebras of type $A$Jun 26 2019Khovanov-Lauda-Rouquier algebras $R_\theta$ of finite Lie type are affine quasihereditary with standard modules $\Delta(\pi)$ labeled by Kostant partitions of $\theta$. Let $\Delta$ be the direct sum of all standard modules. It is known that the Yoneda ... More

Resolutions of standard modules over KLR algebras of type $A$Jun 26 2019Khovanov-Lauda-Rouquier algebras $R_\theta$ of finite Lie type are affine quasihereditary with standard modules $\Delta(\pi)$ labeled by Kostant partitions $\pi$ of $\theta$. In type $A$, we construct explicit projective resolutions of standard modules ... More

A sharp upper bound for the size of Lusztig seriesJun 26 2019The paper is concerned with the character theory of finite groups of Lie type. The set of irreducible characters of a group $G$ of Lie type is partitioned into the Lusztig series. We determine the maximum of the sizes of such series for classical groups ... More

Primitive characters of odd order groupsJun 26 2019Let $G$ be a finite group of odd order. We show that if $\chi$ is an irreducible primitive character of $G$ then for all primes $p$ dividing the order of $G$ there is a conjugacy class such that the $p-$part of $\chi(1)$ divides the size of that conjugacy ... More

Burnside rings for Real $2$-representation theory: The linear theoryJun 26 2019Jul 02 2019This paper is a fundamental study of the Real $2$-representation theory of $2$-groups. It also contains many new results in the ordinary (non-Real) case. Our framework relies on a $2$-equivariant Morita bicategory, where a novel construction of induction ... More

Burnside rings for Real $2$-representation theory: The linear theoryJun 26 2019This paper is a fundamental study of the Real $2$-representation theory of $2$-groups. It also contains many new results in the usual (non-Real) case. Our framework relies on a $2$-equivariant Morita bicategory, where a novel construction of induction ... More

Proper classes and Gorensteinness in extriangulated categoriesJun 26 2019Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. A notion of proper class in an extriangulated category is defined in this paper. Let $\mathcal{C}$ be an extriangulated ... More

The generating rank of a polar GrassmannianJun 25 2019In this paper we compute the generating rank of $k$-polar Grassmannians defined over commutative division rings. Among the new results, we compute the generating rank of $k$-Grassmannians arising from Hermitian forms of Witt index $n$ defined over vector ... More

The Quantum DELL SystemJun 25 2019We propose quantum Hamiltonians of the double elliptic many-body integrable system (DELL) and study its spectrum. These Hamiltonians are certain elliptic functions of coordinates and momenta. Our results provide quantization of the classical DELL system ... More

Projective normality of torus quotients of homogeneous spacesJun 24 2019Let $G=SL_n(\mathbb C)$ and $T$ be a maximal torus in $G$. We show that the quotient $T \backslash \backslash G/{P_{\alpha_1}\cap P_{\alpha_2}}$ is projectively normal with respect to the descent of a suitable line bundle, where $P_{\alpha_i}$ is the ... More

Supergroup $OSP(2,2n)$ and super Jacobi polynomialsJun 24 2019Coefficients of super Jacobi polynomials of type $B(1,n)$ are rational functions in three parameters $k,p,q$. At the point $(-1,0,0)$ these coefficient may have poles. Let us set $q=0$ and consider pair $(k,p)$ as a point of $\Bbb A^2$. If we apply blow ... More

Logarithmic concavity of Schur and related polynomialsJun 23 2019We show that normalized Schur polynomials are strongly log-concave. As a consequence, we obtain Okounkov's log-concavity conjecture for Littlewood-Richardson coefficients in the special case of Kostka numbers.

Irreducibility of the Wysiwyg representations of Thompson's groupsJun 23 2019We prove irreducibility and mutual inequivalence for certain unitary representations of R. Thompson's groups F and T.

Differential graded bocses and $A_{\infty}$-modulesJun 22 2019We introduce and study the category of twisted modules over a triangular differential graded bocs. We show that in this category idempotents split, that it admits a natural structure of a Frobenius category, that a twisted module is homotopically trivial ... More

Conjectures P1-P15 for hyperbolic Coxeter groups of rank 3Jun 22 2019We prove Lusztig's conjectures P1-P15 for hyperbolic Coxeter groups of rank 3 with any positive weight function. Combined with Guilhot and Parkinson's work on affine Weyl groups of type $ \tilde{B}_2 $, $ \tilde{G}_2 $, this completes the proof of P1-P15 ... More

Description of unitary representations of the group of infinite $p$-adic integer matricesJun 22 2019We classify irreducible unitary representations of the group of all infinite matrices over a $p$-adic field ($p\ne 2$) with integer elements equipped with a natural topology. Any irreducible representation passes through a group $GL$ of infinite matrices ... More

On quantum $K$-groups of partial flag manifoldsJun 21 2019We show that the equivariant small quantum $K$-group of a partial flag manifold is a quotient of that of the full flag manifold as a based ring. This yields a variant of the $K$-theoretic analogue of the parabolic version of Peterson's theorem [Lam-Shimozono, ... More

A combinatorial study of affine Schubert varieties in affine GrassmannianJun 21 2019Let $\overline{\mathtt{X}}_\lambda$ be the closure of the $\mathtt{I}$-orbit $\mathtt{X}_\lambda$ in the affine Grassmanian $\mathtt{Gr}$ of a simple algebraic group $G$ of adjoint type, where $\mathtt{I}$ is the Iwahori group and $\lambda$ is a coweight ... More

A combinatorial study of affine Schubert varieties in affine GrassmannianJun 21 2019Jun 28 2019Let $\overline{\mathtt{X}}_\lambda$ be the closure of the $\mathtt{I}$-orbit $\mathtt{X}_\lambda$ in the affine Grassmanian $\mathtt{Gr}$ of a simple algebraic group $G$ of adjoint type, where $\mathtt{I}$ is the Iwahori group and $\lambda$ is a coweight ... More

Characteristic cycles, micro local packets and packets with cohomologyJun 21 2019Relying on work of Kashiwara-Schapira and Schmid-Vilonen, we describe the behaviour of characteristic cycles with respect to the operation of geometric induction, the geometric counterpart of taking parabolic or cohomological induction in representation ... More

Finite-dimensional modules of the Racah algebra and the additive DAHA of type $(C_1^\vee,C_1)$Jun 21 2019Assume that $\mathbb F$ is an algebraically closed field with characteristic zero. The Racah algebra $\Re$ is a unital associative $\mathbb F$-algebra defined by generators and relations. The generators are $A,B, C, D$ and the relations state that $$ ... More

Simplicity of spectra for Bethe subalgebras in $Y(\mathfrak{gl}_2)$Jun 21 2019We consider Bethe subalgebras B(C) in the Yangian $Y(\mathfrak{gl}_2)$ with $C$ regular $2\times 2$ matrix. We study the action of Bethe subalgebras of $Y(\mathfrak{gl}_2)$ on finite-dimensional representations of $Y(\mathfrak{gl}_2)$. We prove that $B(C)$ ... More

Dynamical zeta functions of Reidemeister type and representations spacesJun 21 2019In this paper we continue to study the Reidemeister zeta function. We prove P\'olya -- Carlson dichotomy between rationality and a natural boundary for analytic behavior of the Reidemeister zeta function for a large class of automorphisms of Abelian groups. ... More

Representations of the Infinite-Dimensional $p$-Adic Affine GroupJun 21 2019We introduce an infinite-dimensional $p$-adic affine group and construct its irreducible unitary representation. Our approach follows the one used by Vershik, Gelfand and Graev for the diffeomorphism group, but with modifications made necessary by the ... More

Octonions, Albert vectors and the group $\mathrm{E}_6(F)$Jun 20 2019We present a uniform approach to the construction of the groups of type $\mathrm{E}_6$ over arbitrary fields without using Lie theory. This gives a simple description of the group generators and some of the subgroup structure. In the finite case our approach ... More

Explicit Combinatorial Formulas for Some Irreducible Characters of the $GL_k\times \mathbb{S}_n$-module of multivariate diagonal harmonicsJun 20 2019We give an explicit combinatorial formula for some irreducible components of $GL_k\times \mathbb{S}_n$-modules of multivariate diagonal harmonics. To this end we introduce a new path combinatorial object $T_{n,s}$ allowing us to give the formula directly ... More

On the module category of generalized preprojective algebras of Dynkin typesJun 20 2019For a symmetrizable GCM $C$ and its symmetrizer $D$, Geiss-Leclerc-Schr\"oer [Invent. Math. 209 (2017)] has introduced a generalized preprojective algebra $\Pi$ associated to $C$ and $D$, that contains a class of modules, called locally free modules. ... More

Some algebras that are not silting connectedJun 19 2019We give examples of finite-dimensional algebras $A$ for which the silting objects in $K^b(\mbox{proj-}A)$ are not connected by any sequence of (possibly reducible) silting mutations. The argument is based on the fact that silting mutation preserves invariance ... More