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Multiplicity one theorem for the generalized doubling methodSep 18 2019In this work we prove the local multiplicity at most one theorem underlying the definition and theory of local $\gamma$-, $\epsilon$- and $L$-factors, defined by virtue of the generalized doubling method, over any local field of characteristic 0. We also ... More
L^p-Poisson integral representations of the generalized Hua operators on line bundles over SU(n,n)/S(U(n)xU(n))Sep 18 2019Let $\tau_\nu$ ($\nu \in \mathbb{Z}$) be a character of $K=S(U(n)\times U(n))$, and $SU(n,n)\times_K\mathbb{C}$ the associated homogeneous line bundle over $\mathcal{D}=\{Z\in M(n,\mathbb{C}): I-ZZ^* > 0\}$. Let $\mathcal{H}_\nu$ be the Hua operator on ... More
Representations of the orbifold VOAS $L_{\hat{\frak{sl}_2}}(k,0)^{K}$ and the commutant VOAS $C_{{L_{\hat{\mathfrak{so}_m}}(1,0)}^{\otimes 3}}({L_{\hat{\mathfrak{so}_m}}(3,0)})$Sep 18 2019For the Klein group $K$, $k\in\mathbb{Z}_{\geqslant 1}$ and $m\in\mathbb{Z}_{\geqslant 4}$, we study the representations of the orbifold vertex operator algebra $L_{\hat{\mathfrak{sl}_2}}(k,0)^{K}$ and the commutant vertex operator algebra of $L_{\hat{\mathfrak{so}_m}}(3,0)$ ... More
Representation stability for pure braid group Milnor fibersSep 17 2019We prove a representation stability result for the Milnor fiber associated to the pure braid group. Our result connects previous work of Simona Settepenella to representation stability in the sense of Church--Ellenberg--Farb, answering a question of Graham ... More
A Control Theorem for Primitive ideals in Iwasawa algebrasSep 17 2019Let p be a prime, K a p-adic field, G a nilpotent, uniform pro-p group. We prove that all faithful, primitive ideals in the Iwasawa algebra KG are controlled by the centraliser of the second term in the upper central series for G.
Interplay between symmetries of quantum 6-j symbols and the eigenvalue hypothesisSep 17 2019The eigenvalue hypothesis claims that any quantum Racah matrix for finite-dimensional representations of $U_q(sl_N)$ is uniquely determined by eigenvalues of the corresponding quantum $\cal{R}$-matrices. If this hypothesis turns out to be true, then it ... More
Symmetric and Exterior Squares of Hook RepresentationsSep 16 2019We determine the multiplicities of irreducible summands in the symmetric and the exterior squares of hook representations of symmetric groups over an algebraically closed field of characteristic zero.
Unipotent Representations Attached to the Principal Nilpotent OrbitSep 16 2019In this paper, we construct and classify the (special) unipotent representations of a real reductive group attached to the principal nilpotent orbit. Our construction gives rise to a formula for the $\mathbf{K}$-types and associated varieties of all such ... More
Elliptic classes, McKay correspondence and theta identitiesSep 16 2019We revisit the construction of elliptic class given by Borisov and Libgober for singular algebraic varieties. Assuming torus action we adjust the theory to equivariant local situation. We study theta function identities having geometric origin. In the ... More
Distinction for unipotent $p$-adic groupsSep 16 2019Let $F$ be a $p$-adic field and $\mathbf{U}$ be a unipotent group defined over $F$, and set $U=\mathbf{U}(F)$. Let $\sigma$ be an involution of $\mathbf{U}$ defined over $F$. Adapting the arguments of Yves Benoist in the real case, we prove the following ... More
The Kostant invariant and special $ε$-orthogonal representations for $ε$-quadratic colour Lie algebrasSep 16 2019Let k be a field of characteristic not two or three, let $\mathfrak{g}$ be a finite-dimensional colour Lie algebra and let V be a finite-dimensional representation of $\mathfrak{g}$. In this article we give various ways of constructing a colour Lie algebra ... More
James's Conjecture holds for the principal block of the Iwahori-Hecke algebra $\mathcal{H}_{5e}$ for $e\ge5$Sep 16 2019James's Conjecture predicts that the decomposition numbers for blocks of the Iwahori-Hecke algebra of the symmetric group over a field of prime characteristic is equal to that over $\mathbb{C}$ when the weight of the block is strictly less than the characteristic ... More
$q$-rious and $q$-riouserSep 16 2019Dick Askey is known not just for his beautiful mathematics and his many amazing theorems, but also for posing numerous interesting and important open problems. Dick being Dick, these problems are hardly ever isolated, and often intended to demonstrate ... More
Cyclic sieving phenomenon on dominant maximal weights over affine Kac-Moody algebrasSep 16 2019We construct a (bi)cyclic sieving phenomenon on the union of dominant maximal weights for level $\ell$ highest weight modules over an affine Kac-Moody algebra with exactly one highest weight being taken for each equivalence class, in a way not depending ... More
Springer correspondence for symmetric spacesSep 15 2019This is a survey article on the Springer correspondence for symmetric spaces. We discuss various generalization of the theory of the Springer correspondence for reductive groups to symmetric spaces and exotic symmetric spaces associated to classical groups. ... More
The $A$-fibered Burnside ring as $A$-fibered biset functor in characteristic zeroSep 14 2019Let $A$ be an abelian group and let $\mathbb{K}$ be a field of characteristic zero containing roots of unity of all orders equal to finite element orders in $A$. In this paper we prove foundational properties of the $A$-fibered Burnside ring functor $B_{\mathbb{K}}^A$ ... More
Fusion systems of blocks of finite groups over arbitrary fieldsSep 14 2019To any block idempotent $b$ of a group algebra $kG$ of a finite group $G$ over a field $k$ of characteristic $p>0$, Puig associated a fusion system and proved that it is saturated if the $k$-algebra $kC_G(P)e$ is split, where $(P,e)$ is a maximal $kGb$-Brauer ... More
On Schur multiplier and projective representations of Heisenberg groupsSep 14 2019In this article, we study the Schur mutiplier of the discrete as well as the finite Heisenberg groups and their t-variants. We describe the representation groups of these Heisenberg groups and through these give a construction of their finite dimensional ... More
Bannai-Ito algebras and the universal R-matrix of osp(1|2)Sep 13 2019The Bannai-Ito algebra $BI(n)$ is viewed as the centralizer of the action of $\mathfrak{osp}(1|2)$ in the $n$-fold tensor product of the universal algebra of this Lie superalgebra. The generators of this centralizer are constructed with the help of the ... More
Degeneracy locus formulas for amenable Weyl group elementsSep 13 2019We define a class of amenable Weyl group elements in the Lie types B, C, and D, which we propose as the analogues of vexillary permutations in these Lie types. Our amenable signed permutations index flagged theta and eta polynomials, which generalize ... More
Full Spark Frames in the Orbit of a RepresentationSep 13 2019We present a new infinite family of full spark frames in finite dimensions arising from a unitary group representation, where the underlying group is the semi-direct product of a cyclic group by a group of automorphisms. The only previously known algebraically ... More
The m=2 amplituhedronSep 13 2019The (tree) amplituhedron $\mathcal{A}_{n, k, m}$ is introduced by Arkani-Hamed and Trnka in 2013 in the study of $\mathcal{N}=4$ supersymmetric Yang-Mills theory. It is defined in terms of the totally nonnegative Grassmannians. In this paper, we show ... More
A local Langlands parameterization for generic supercuspidal representations of $p$-adic $G_2$Sep 12 2019We construct a Langlands parameterization of supercuspidal representations of $G_2$ over a $p$-adic field. More precisely, for any finite extension $K / \QQ_p$ we will construct a bijection \[ \CL_g : \CA^0_g(G_2,K) \rightarrow \CG^0(G_2,K) \] from the ... More
Typical representations via fixed point sets in Bruhat--Tits buildingsSep 12 2019We establish two distinct and complementary sufficient conditions for the Mackey components of an essentially tame supercuspidal representation $\pi$ of an arbitrary connected reductive $p$-adic group $G$ to intertwine with representations of $G$ which ... More
Exceptional Sequences and Idempotent FunctionsSep 12 2019We prove that there is one to one correspondence between the following three: idempotent functions on the set of size $n$, exceptional sequences of linear radical square zero Nakayama algebras of rank $n$ and rooted labeled forests with $n$ nodes and ... More
On a conjecture of Braverman-KazhdanSep 12 2019In this article we prove a conjecture of Braverman and Kazhdan in \cite{BK1} on cohomology vanishing properties of $\gamma$-sheaves on reductive groups in both $\ell$-adic and de Rham settings. We do so by establishing a vanishing conjecture proposed ... More
Continuous cluster categories II: continuous cluster-tilted categoriesSep 11 2019We show that the quotient of the continuous cluster category $\mathcal C_\pi$ modulo the additive subcategory generated by any cluster is an abelian category and we show that it is isomorphic to the category of infinite length modules over the endomorphism ... More
Lines on cubic surfaces and Witt invariantsSep 11 2019The aim of this note is to give a formula expressing the trace form associated with the 27 lines of a cubic surface.
Multiplicity in root components via Geometric SatakeSep 11 2019In this note we explicitly construct top-dimensional components of the cyclic convolution varieties. These components correspond (via the geometric Satake equivalence) to irreducible summands $V(\lambda+\mu-N\beta) \subset V(\lambda) \otimes V(\mu)$ for ... More
Group Representation Theory for Knowledge Graph EmbeddingSep 11 2019Knowledge graph embedding has recently become a popular way to model relations and infer missing links. In this paper, we present a group theoretical perspective of knowledge graph embedding, connecting previous methods with different group actions. Furthermore, ... More
The third Milgram-Priddy class liftsSep 11 2019We show that the third cohomology of the finite general linear group $GL_6(\mathbb{F}_2)$ with trivial mod 2 coefficients is non-zero. The necessarily unique non-trivial element restricts to the third Milgram-Priddy class.
Real orbits of complex spherical homogeneous spaces: the split caseSep 11 2019We identify the $G(\mathbb R)$-orbits of the real locus $X(\mathbb R)$ of any spherical complex variety $X$ defined over $\mathbb R$ and homogeneous under a split connected reductive group $G$ defined also over $\mathbb R$. This is done by introducing ... More
Flag versions of quiver Grassmannians for Dynkin quivers have no odd cohomology over $\mathbb{Z}$Sep 11 2019We prove that flag versions of quiver Grassmannians (also knows as Lusztig's fibers) for Dynkin quivers (types $A$, $D$, $E$) have no odd cohomology over $\mathbb{Z}$. Moreover, for types $A$ and $D$ we prove that these varieties have $\alpha$-partitions ... More
Higher Lie characters and cyclic descent extension on conjugacy classesSep 10 2019A cyclic descent extension of the classical notion of descent set, for permutations as well as standard Young tableaux, was introduced and studied in recent years. The main result of this paper is a full characterization of conjugacy classes in the symmetric ... More
On simple-minded systems and $τ$-periodic modules of self-injective algebrasSep 10 2019Let $A$ be a finite-dimensional self-injective algebra over an algebraically closed field, $\mathcal{C}$ a stably quasi-serial component (i.e. its stable part is a tube) of rank $n$ of the Auslander-Reiten quiver of $A$, and $\mathcal{S}$ be a simple-minded ... More
Noncommutative tensor triangular geometrySep 10 2019We develop a general noncommutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (M$\Delta$Cs). Insight from noncommutative ring theory is used to obtain a framework for prime, semiprime, ... More
A Permutation Module Deligne Category and Stable Patterns of Kronecker CoefficientsSep 09 2019Deligne's category $\underline{{\rm Rep}}(S_t)$ is a tensor category depending on a parameter $t$ "interpolating" the categories of representations of the symmetric groups $S_n$. We construct a family of categories $\mathcal{C}_\lambda$ (depending on ... More
Not even Khovanov homologySep 09 2019We construct a supercategory that can be seen as a skew version of (thickened) KLR algebras for the type $A$ quiver. We use our supercategory to construct homological invariants of tangles and show that for every link our invariant gives a link homology ... More
Representations of code loops by binary codesSep 09 2019Code loops are Moufang loops constructed from doubly even binary codes. Then, given a code loop L, we ask which doubly even binary code V produces L. In this sense, V is called a representation of L. In this article we define and determine all minimal ... More
Two Generation of Finite Simple GroupsSep 09 2019This expository article revolves around the question to find short presentations of finite simple groups. This subject is one of the most active research areas of group theory in recent times. We bring together several known results on two-generation ... More
The heat kernel on $SL(2,\mathbb{R})$Sep 09 2019Let $G$ be a noncompact semisimple Lie group equipped with a certain invariant Riemannian metric. Then, we can consider a heat kernel function on $G$ associated to the Riemannian metric. We give an explicit formula for the heat kernel when $G = SL(2,\mathbb{R})$. ... More
The heat kernel on $SL(2,\mathbb{R})$Sep 09 2019Sep 11 2019Let $G$ be a noncompact semisimple Lie group equipped with a certain invariant Riemannian metric. Then, we can consider a heat kernel function on $G$ associated to the Riemannian metric. We give an explicit formula for the heat kernel when $G = SL(2,\mathbb{R})$. ... More
(GL_k x S_n)-modules and nabla of hook-indexed Schur functionsSep 08 2019Sep 10 2019The aim of this paper is to describe structural properties of spaces of diagonal rectangular harmonic polynomials in several sets (say $k$) of $n$ variables, both as $GL_k$-modules and $S_n$-modules. We construct explicit such modules associated to any ... More
Lecture Notes on Quiver Representations and Moduli Problems in Algebraic GeometrySep 08 2019These lecture notes consist of an introduction to moduli spaces in algebraic geometry, with a strong emphasis placed on examples related to the theory of quiver representations. The goal is to provide the background necessary to understand the construction ... More
The representation theory of seam algebrasSep 08 2019The boundary seam algebras $\mathsf{b}_{n,k}(\beta=q+q^{-1})$ were introduced by Morin-Duchesne, Ridout and Rasmussen to formulate algebraically a large class of boundary conditions for two-dimensional statistical loop models. The representation theory ... More
Period Relations for Standard $L$-functions of Symplectic TypeSep 08 2019This article is to understand the critical values of $L$-functions $L(s,\Pi\otimes \chi)$ and to establish the relation of the relevant global periods at the critical places. Here $\Pi$ is an irreducible regular algebraic cuspidal automorphic representation ... More
p-reduced multicomponent KP hierarchy and classical W-algebras W(gl_N,p)Sep 07 2019For each partition p of an integer N \geq 2, consisting of r parts, an integrable hierarchy of Lax type Hamiltonian PDE has been constructed recently by some of us. In the present paper we show that any tau-function of the p-reduced r-component KP hierarchy ... More
Character correspondeces in solvable groups with a self-normalizing Sylow subgroupSep 07 2019In 1973, I. M. Isaacs described a correspondence between characters of degree not divisible by a fixed prime $p$ of a finite solvable group $G$ and those of the normalizer of Sylow $p$-subgroup of $G$, whenever the index of the normalizer in $G$ is odd. ... More
Computing fusion rules for spherical G-extensions of fusion categoriesSep 06 2019A $G$-graded extension of a fusion category $\mathcal{C}$ yields a categorical action $\rho\colon \underline{G}\rightarrow \underline{\operatorname{Aut}}^{\mathrm{br}}_{\otimes}( Z(\mathcal C))$. If the extension admits a spherical structure, we provide ... More
Representations for three-point Lie algebras of genus zeroSep 06 2019In this paper, we study representations for three-point Lie algebras of genus zero based on the Cox-Jurisich's presentations. We construct two functors which transform simple restricted modules with nonzero levels over the standard affine algebras into ... More
Weil representation and Arithmetic Fundamental LemmaSep 06 2019By a global approach, we prove the arithmetic fundamental lemma conjecture for unitary groups in $n$ variables over $\mathbb{Q}_p$ when $p\geq n$.
On representations of semidirect products of a compact quantum group with a finite groupSep 05 2019We study unitary representations of semidirect products of a compact quantum group with a finite group. We give a classification of all irreducible unitary representations, a description of the conjugate representation of irreducible unitary representations ... More
Convolution morphisms and Kottwitz conjectureSep 05 2019We introduce convolution morphisms, duality morphisms and twist morphisms between moduli spaces of mixed characteristic local shtukas. Using these morphisms, we relate the etale cohomology of different moduli spaces of mixed characteristic local shtukas. ... More
Exact Hochschild extensions and deformed Calabi-Yau completionsSep 05 2019We introduce the Hochschild extensions of dg algebras, which are $A_\infty$-algebras. We show that all exact Hochschild extensions are symmetric Hochschild extensions, more precisely, every exact Hochschild extension of a finite dimensional complete typical ... More
Gerstenhaber brackets on Hochschild cohomology of general twisted tensor productsSep 05 2019We present techniques for computing Gerstenhaber brackets on Hochschild cohomology of general twisted tensor product algebras. These techniques involve twisted tensor product resolutions and are based on recent results on Gerstenhaber brackets expressed ... More
Gerstenhaber brackets on Hochschild cohomology of general twisted tensor productsSep 05 2019Sep 06 2019We present techniques for computing Gerstenhaber brackets on Hochschild cohomology of general twisted tensor product algebras. These techniques involve twisted tensor product resolutions and are based on recent results on Gerstenhaber brackets expressed ... More
Equivariant decomposition of polynomial vector fieldsSep 04 2019To compute the unique formal normal form of families of vector fields with nilpotent linear part, we choose a basis of the Lie algebra consisting of orbits under the linear nilpotent. This creates a new problem: to find explicit formulas for the structure ... More
Finding the dimension of a non-empty orthogonal array polytopeSep 04 2019By using representation theory, we reduce the size of the set of possible values for the dimension of the convex hull of all feasible points polytope of an orthogonal array defining integer linear program (ILP). Our results address the conjecture that ... More
On universal quantum dimensions of certain two-parameter series of representationsSep 04 2019We present the universal, in Vogel's sense, expression for the quantum dimension of Cartan product of an arbitrary number of adjoint and $X_2$ representations of simple Lie algebras. The same formula mysteriously gives quantum dimensions of some other ... More
On total Springer representations for the symplectic Lie algebra in characteristic 2 and the exotic caseSep 04 2019Let $W$ be the Weyl group of type $BC_n$. We first provide restriction formulas of the total Springer representations for the symplectic Lie algebra in characteristic 2 and the exotic case to the maximal parabolic subgroup of $W$ which is of type $BC_{n-1}$. ... More
Multiplicities under basechange: finite field caseSep 04 2019A general proposition is proved relating multiplicities (of restriction of a representation of a group to a subgroup) under basechange, and used to calculate some multiplicities for cuspidal representations which become principal series representations ... More
Variants on a conjecture relating block source algebras to characteristic bisetsSep 04 2019Given a block of a finite group, any source algebra has a basis invariant under the multiplicative actions of the defect group. Is such a basis a characteristic biset of the block fusion system? If the basis can be chosen to consist entirely of units, ... More
Intertwining Operator for $Sp(4,\mathbb{R})$ and Orthogonal PolynomialsSep 04 2019We calculate the $(\mathfrak{g},K)$ module structure for the principal series representation of $Sp(4,\mathbb{R})$. Furthermore, we introduced a hypergeometric generating function together with an inverse Mellin transform technique as an improvement to ... More
A categorification of the Malvenuto--Reutenauer algebra via a tower of groupsSep 03 2019There is a long tradition of categorifying combinatorial Hopf algebras by the modules of a tower of algebras (or even better via the representation theory of a tower of groups). From the point of view of combinatorics, such a categorification supplies ... More
On ordered $k$-paths and rims for certain families of Kazhdan-Lusztig cells of $S_n$Sep 03 2019For a composition $\lambda$ of $n$ we consider the Kazhdan-Lusztig cell in the symmetric group $S_n$ containing the longest element of the standard parabolic subgroup of $S_n$ associated to $\lambda$. In this paper we extend some of the ideas and results ... More
An inductive method for $\mathrm{OI}$-modulesSep 03 2019In this paper we introduce an inductive method to study $\mathrm{OI}$-modules presented in finite degrees, where $\mathrm{OI}$ is a skeleton of the category of finitely totally ordered sets and strictly increasing maps. As an application, we obtain an ... More
Quasi-split symmetric pairs of $U(\mathfrak{gl}_N)$ and their Schur algebrasSep 03 2019We establish explicit isomorphisms of two seemingly-different algebras, and their Schur algebras, arising from the centralizers of two different type B Weyl group actions in Schur-like dualities. We provide a presentation of the geometric counterpart ... More
Many zeros of many characters of GL(n,q)Sep 03 2019For $G={\rm GL}(n,q)$, the proportion $P_{n,q}$ of pairs $(\chi,g)$ in ${\rm Irr}(G)\times G$ with $\chi(g)\neq 0$ satisfies $P_{n,q}\to 0$ as $n\to\infty$.
Principal Series Representation of $SU(2,1)$ and Its Intertwining OperatorSep 03 2019In this paper, following a similar procedure developed by Buttcane and Miller in \cite{MillerButtcane} for $SL(3,\RR)$, the $(\frakg,K)$-module structure of the minimal principal series of real reductive Lie groups $SU(2,1)$ is described explicitly by ... More
Kazhdan-Lusztig right cells and associated varieties of highest weight modulesSep 03 2019Let $\mathfrak{g}$ be a simple Lie algebra with a Weyl group $W$. Let $L_w$ be a simple module with highest weight $-w\rho-\rho$. By using a conjecture of Tanisaki, we show that there is a bijection between the right cells and associated varieties of ... More
Total positivity in Springer fibresSep 03 2019Let u be a unipotent element in the totally positive part of a complex reductive group. We consider the intersection of the Springer fibre at u with the totally positive part of the flag manifold. We show that this intersection has a natural cell decomposition ... More
Total positivity in Springer fibresSep 03 2019Sep 08 2019Let u be a unipotent element in the totally positive part of a complex reductive group. We consider the intersection of the Springer fibre at u with the totally positive part of the flag manifold. We show that this intersection has a natural cell decomposition ... More
Singular tuples of matrices is not a null cone (and, the symmetries of algebraic varieties)Sep 02 2019The following multi-determinantal algebraic variety plays a central role in algebra, algebraic geometry and computational complexity theory: ${\rm SING}_{n,m}$, consisting of all $m$-tuples of $n\times n$ complex matrices which span only singular matrices. ... More
Convex topological algebras via linear vector fields and Cuntz algebrasSep 02 2019Realization by linear vector fields is constructed for any Lie algebra which admits a biorthogonal system and for its any suitable representation. The embedding into Lie algebras of linear vector fields is analogous to the classical Jordan-Schwinger map. ... More
A characterization of some highest weight modules for type A with irreducible associated varietiesSep 02 2019For $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$, let $w$ be an element in the Weyl group $W=S_n$. We use $L_w$ to denote a highest weight module of $\mathfrak{sl}(n,\mathbb{C})$ with highest weight $-w\rho-\rho$. In this paper we will give a characterization ... More
Invariant theory and wheeled PROPsSep 01 2019We study the category of wheeled PROPs using tools from Invariant Theory. A typical example of a wheeled PROP is the mixed tensor algebra ${\mathcal V}=T(V)\otimes T(V^\star)$, where $T(V)$ is the tensor algebra on an $n$-dimensional vector space over ... More
Cohomology of bimultiplicative local systems on unipotent groupsSep 01 2019Let $U_1, U_2$ be connected commutative unipotent algebraic groups defined over an algebraically closed field $k$ of characteristic $p>0$ and let $\mathcal{L}$ be a bimultiplicative $\overline{\mathbb{Q}}_\ell$-local system on $U_1\times U_2$. In this ... More
Exceptional Periodicity and Magic Star Algebras. I : FoundationsSep 01 2019We introduce and start investigating the properties of a periodic countably infinite chain of finite-dimensional generalizations of the exceptional Lie algebras: each exceptional Lie algebra (but $\mathbf{g}_{2}$) is part of an infinite family of finite-dimensional ... More
Alcove paths and Gelfand-Tsetlin patternsSep 01 2019In their study of the equivariant K-theory of the generalized flag varieties $G/P$, where $G$ is a complex semisimple Lie group, and $P$ is a parabolic subgroup of $G$, Lenart and Postnikov introduced a combinatorial tool, called the alcove paths model. ... More
Thick morphisms of supermanifolds, quantum mechanics, and spinor representationAug 31 2019Sep 04 2019"Thick" or "microformal" morphisms of supermanifolds generalize ordinary maps. They were discovered as a tool for homotopy algebras. Namely, the corresponding pullbacks provide $L_{\infty}$-morphisms for $S_{\infty}$ or Batalin--Vilkovisky algebras. It ... More
Singular Nonsymmetric Macdonald Polynomials and QuasistaircasesAug 30 2019Singular nonsymmetric Macdonald polynomials are constructed by use of the representation theory of the Hecke algebras of the symmetric groups. These polynomials are labeled by quasistaircase partitions and are associated to special parameter values $(q,t)$. ... More
Rigidity of flag supermanifoldsAug 30 2019We prove that under certain assumptions a supermanifold of flags is rigid, this is its complex structure does not admit any non-trivial small deformation. Moreover under the same assumptions we show that a supermanifold of flags is unique non-split supermanifold ... More
Bases of the Intersection Cohomology of Grassmannian Schubert VarietiesAug 30 2019Shigechi and Zinn-Justin (arXiv:1001.1080) described the parabolic Kazhdan-Lusztig polynomials for Grassmannians by counting certain Dyck partitions. We "lift" their combinatorial formula to the intersection cohomology of Schubert varieties in Grassmannians ... More
Tropical Duality in $(d+2)$-angulated categoriesAug 30 2019Let $\mathscr{C}$ be a $2$-Calabi-Yau triangulated category with two cluster tilting subcategories $\mathscr{T}$ and $\mathscr{U}$. Results by Demonet-Iyama-Jasso and J{\o}rgensen-Yakimov known as tropical duality says that the index with respect to $\mathscr{T}$ ... More
Geometric MultiplicitiesAug 30 2019In this paper, we introduce geometric multiplicities, which are positive varieties with potential fibered over the Cartan subgroup $H$ of a reductive group $G$. They form a monoidal category and we construct a monoidal functor from this category to the ... More
Tensor product of the Fock representation with its dual and the Deligne categoryAug 30 2019We describe the structure of the tensor product of the basic Fock representation of sl(\infty) with its shifted dual. More precisely we prove that this tensor product has a unique decreasing filtration with simple quotients. We use the categorification ... More
Selfextensions of modules for Nakayama and Brauer tree algebrasAug 29 2019For Nakayama algebras $A$, we prove that in case $Ext_A^1(M,M) \neq 0$ for an indecomposable $A$-module $M$, we have that the projective dimension of $M$ is infinite. As an application we give a new proof of a classical result from \cite{Gus} on bounds ... More
Some martingales associated with multivariate Jacobi processes and Aomoto's Selberg integralAug 29 2019We study $\beta$-Jacobi diffusion processes on alcoves in $\mathbb R^N$, depending on 3 parameters. Using elementary symmetric functions, we present space-time-harmonic functions and martingales for these processes $(X_t)_{t\ge0}$ which are independent ... More
Split bounded extension algebras and Han's conjectureAug 29 2019A main purpose of this paper is to prove that the class of finite dimensional algebras which verify Han's conjecture is closed under split bounded extensions.
Finite Gelfand pairs and cracking points of the symmetric groupsAug 29 2019Let $\Gamma$ be a finite group. Consider the wreath product $G_n := \Gamma^n \rtimes S_n$ and the subgroup $K_n := \Delta_n \times S_n\subseteq G_n$, where $S_n$ is the symmetric group and $\Delta_n$ is the diagonal subgroup of $\Gamma^n$. For certain ... More
Flagged Littlewood-Richardson tableaux and branching rule for classical groupsAug 29 2019We give a new formula for the branching rule from ${\rm GL}_n$ to ${\rm O}_n$ generalizing the Littlewood's restriction formula. The formula is given in terms of Littlewood-Richardson tableaux with certain flag conditions which vanish in a stable range. ... More
The Mackey bijection for complex reductive groups and continuous fields of reduced group C*-algebrasAug 29 2019The purpose of this paper is to make a further contribution to the Mackey bijection for a complex reductive group G, between the tempered dual of G and the unitary dual of the associated Cartan motion group. We shall construct an embedding of the C*-algebra ... More
Singular Rouquier ComplexesAug 28 2019We generalise the construction of Rouquier complexes to the setting of singular Soergel bimodules by taking minimal complexes of the restriction of Rouquier complexes. We show that they retain many of the properties of ordinary Rouquier complexes: they ... More
On the Gan-Gross-Prasad problem for finite unitary groupsAug 28 2019In this paper we study the Gan-Gross-Prasad problem for unitary groups over finite fields. Our results provide complete answers for unipotent representations, and we obtain the explicit branching of these representations.
Descents of unipotent cuspidal representations of finite classical groupsAug 28 2019Inspired by the Gan-Gross-Prasad conjecture and the descent problem for classical groups, in this paper we study the descents of unipotent cuspidal representations of orthogonal and symplectic groups over finite fields.
The pro-étale cohomology of Drinfeld's upper half spaceAug 28 2019We determine the geometric pro-\'etale cohomology of Drinfeld's upper half space ${\mathcal X}$ over a p-adic field. The strategy is different from the one given by Colmez, Dospinescu and Niziol. It uses the approach developed in a former paper of the ... More
The Siegel-Weil formula for unitary groups: the second term rangeAug 28 2019We study the Siegel-Weil formula in the second term range ($n+1\leq m\leq n+r$) for unitary groups of hermitian forms over a skew-field $D$ with involution of the second kind.
Exterior square gamma factors for cuspidal representations of $\mathrm{GL}_n$: simple supercuspidal representationsAug 27 2019We compute the local twisted exterior square gamma factors for simple supercuspidal representations, using which we prove a local converse theorem for simple supercuspidal representations.
Singular chains on Lie groups and the Cartan relations IAug 27 2019Let $G$ be a simply connected Lie group with Lie algebra $\mathfrak{g}$. We show that the following categories are naturally equivalent. The category $\mathsf{Mod}(C(G))$, of sufficiently smooth modules over the DG-algebra of singular chains on $G$. The ... More
Twisted doubling integrals for classical groupsAug 27 2019We describe the twisted doubling integrals of Cai-Friedberg-Ginzburg-Kaplan in a conceptual way. This also extends the construction to the quaternionic unitary groups. We carry out the unfolding argument uniformly in this article. To do so, we define ... More