Mathematics - Representation Theory Publications (50)


Mathematics - Representation Theory Publications

We consider a nonlinear representation of a Lie algebra which is regular on an abelian ideal, we define a normal form which generalizes that defined in [D. Arnal, M. Ben Ammar, M. Read More

We introduce a Schr\"odinger model for the unitary irreducible representations of a Heisenberg motion group and we show that the usual Weyl quantization then provides a Stratonovich-Weyl correspondence. Read More

The integral group ring $\mathbb{Z} G$ of a group $G$ has only trivial central units, if the only central units of $\mathbb{Z} G$ are $\pm z$ for $z$ in the center of $G$. We show that the order of a finite solvable group $G$ with this property, can only be divisible by $2$, $3$, $5$ and $7$, by linking this to inverse semi-rational groups and extending one result on this class of groups. We also classify the Frobenius groups whose integral group rings have only trivial central units. Read More

We show that the idempotent completion of an n-angulated category admits a unique n-angulated structure such that the inclusion is an n-angulated functor, which satisfies a universal property. Read More

For a finite free and projective EI category, we prove that Gorenstein-projective modules over its category algebra are closed under the tensor product if and only if each morphism in the given category is a monomorphism. Read More

We introduce generalizations of Kac-Moody 2-categories in which the quiver Hecke algebras of Khovanov, Lauda and Rouquier are replaced by the quiver Hecke superalgebras of Kang, Kashiwara and Tsuchioka. Read More

For a non-abelian Lie algebra $L$ of dimension $n$ with the derived subalgebra of dimension $m$ , the first author earlier proved that the dimension of its Schur multiplier is bounded by $\frac{1}{2}(n+m-2)(n-m-1)+1$. In the current work, we give some new inequalities on the exterior square and the Schur multiplier of Lie algebras and then we obtain the class of all nilpotent Lie algebras which attains the above bound. Moreover, we also improve this bound as much as possible. Read More

This paper first gives a necessary and sufficient condition that a lattice $L$ can be represented as the collection of all up-sets of a poset. Applying the condition, it obtains a necessary and sufficient condition that a lattice can be embedded into the lattice $L$ such that all infima, suprema, the top and bottom elements are preserved under the embedding by defining a monotonic operator on a poset. This paper finally shows that the quotient of the set of the monotonic operators under an equivalence relation can be naturally ordered and it is a lattice if $L$ is a finite distributive lattice. Read More

From Morita theoretic viewpoint, computing Morita invariants is important. We prove that the intersection of the center and the $n$th right socle $ZS^n(A) := Z(A) \cap \operatorname{Soc}^n(A)$ of a finite dimensional algebra $A$ is a Morita invariant; This is a generalization of important Morita invariants, the center $Z(A)$ and the Reynolds ideal $ZS^1(A)$. As an example, we also studied $ZS^n(FP)$ for the group algebra $FP$ of a finite $p$-group $P$ over a field $F$ of positive characteristic $p$. Read More

We study two question concerning irreducible matrices. The first problem is a counting problem. We count the number of irreducible integral matrices which are annihilated by $X^2 -nX$, for $n>0$. Read More

It is well-known that affine Hecke algebras are very useful to describe the smooth representations of any connected reductive p-adic group G, in terms of the supercuspidal representations of its Levi subgroups. The goal of this paper is to create a similar role for affine Hecke algebras on the Galois side of the local Langlands correspondence. To every Bernstein component of enhanced Langlands parameters for G we canonically associate an affine Hecke algebra (possibly extended with a finite R-group). Read More

Let $\mathfrak l:= \mathfrak q(n)\times\mathfrak q(n)$, where $\mathfrak q(n)$ denotes the queer Lie superalgebra. The associative superalgebra $V$ of type $Q(n)$ has a left and right action of $\mathfrak q(n)$, and hence is equipped with a canonical $\mathfrak l$-module structure. We consider a distinguished basis $\{D_\lambda\}$ of the algebra of $\mathfrak l$-invariant super-polynomial differential operators on $V$, which is indexed by strict partitions of length at most $n$. Read More

It is well known that the normaized characters of integrable highest weight modules of given level over an affine Lie algebra $\hat{\frak{g}}$ span an $SL_2(\mathbf{Z})$-invariant space. This result extends to admissible $\hat{\frak{g}}$-modules, where $\frak{g}$ is a simple Lie algebra or $osp_{1|n}$. Applying the quantum Hamiltonian reduction (QHR) to admissible $\hat{\frak{g}}$-modules when $\frak{g} =sl_2$ (resp. Read More

If $A$ and $B$ are $n$- and $m$-representation finite $k$-algebras, then their tensor product $\Lambda = A\otimes_k B$ is not in general $(n+m)$-representation finite. However, we prove that if $A$ and $B$ are acyclic and satisfy the weaker assumption of $n$- and $m$-completeness, then $\Lambda$ is $(n+m)$-complete. This mirrors the fact that taking higher Auslander algebra does not preserve $d$-representation finiteness in general, but it does preserve $d$-completeness. Read More

We prove that up to scaling there are only finitely many integral lattices L of signature (2,n) with n>20 or n=17 such that the modular variety defined by the orthogonal group of L is not of general type. In particular, when n>107, every modular variety defined by an arithmetic group for a rational quadratic form of signature (2,n) is of general type. We also obtain similar finiteness in n>8 for the stable orthogonal groups. Read More

The Heisenberg product is an associative product defined on symmetric functions which interpolates between the usual product and the Kronecker product. In 1938, Murnaghan discovered that the Kronecker product of two Schur functions stabilizes. We prove an analogous result for the Heisenberg product of Schur functions. Read More

We describe the modules in the Ziegler closure of ray and coray tubes in module categories over finite-dimensional algebras. We improve slightly on Krause's result for stable tubes by showing that the inverse limit along a coray in a ray or coray tube is indecomposable, so in particular, the inverse limit along a coray in a stable tube is indecomposable. In order to do all this, we first describe the finitely presented modules over and the Ziegler spectra of iterated one-point extensions of valuation domains. Read More

We formulate a two-parameter generalization of the geometric Langlands correspondence, which we prove for all simply-laced Lie algebras. It identifies the q-conformal blocks of the quantum affine algebra and the deformed W-algebra associated to two Langlands dual Lie algebras. Our proof relies on recent results in quantum K-theory of the Nakajima quiver varieties. Read More

We describe the structure of bimodules (over finite dimensional algebras) which have the property that the functor of tensoring with such a bimodule sends any module to a projective module. The main result is that all such bimodules are $\Bbbk$-split in the sense that they factor (inside the tensor category of bimodules) over $\Bbbk$-vector spaces. As one application, we show that any simple $2$-category has a faithful $2$-representation inside the $2$-category of $\Bbbk$-split bimodules. Read More

We investigate the saturation rank of a finite group scheme, defined over an algebraically closed field $\Bk$ of positive characteristic $p$. We begin by exploring the saturation rank for finite groups and infinitesimal group schemes. Special attention is given to reductive Lie algebras and the second Frobenius kernel of the algebraic group $\SL_{n}$. Read More

We define web categories describing intertwiners for the orthogonal and symplectic Lie algebras, and, in the quantized setup, for certain orthogonal and symplectic coideal subalgebras. They generalize the Brauer category, and allow us to prove quantum versions of some classical type $\mathbf{B}\mathbf{C}\mathbf{D}$ Howe dualities. Read More

For a fixed Coxeter element of a Coxeter group, we introduce the notion of preprojective root that is analogous to the notion of indecomposable preprojective representation of a finite acyclic quiver. We prove that a Coxeter group is finite if and only if each positive root is preprojective. Read More

The (fixed-point-free) involution Stanley symmetric functions $\hat{F}_y$ and $\hat{F}^{\tt{FPF}}_z$ are the stable limits of the analogues of Schubert polynomials for the orbits of the orthogonal and symplectic groups in the flag variety. These symmetric functions are also generating functions for involution words, and are indexed by the (fixed-point-free) involutions in the symmetric group. It holds by construction that both $\hat{F}_y$ and $\hat{F}^{\tt{FPF}}_z$ are sums of Stanley symmetric functions and therefore Schur positive. Read More

We study Auslander-Reiten components of an artin algebra with bounded short cycles, namely, there exists a bound for the depths of maps appearing on short cycles of non-zero non-invertible maps between modules in the given component. First, we give a number of combinatorial characteri\-zations of almost acyclic Auslander-Reiten components. Then, we show that an Auslander-Reiten component with bounded short cycles is closely related to the connec\-ting component of a tilted quotient algebra. Read More

In this paper we study intertwining functors (Radon transforms) for twisted D-modules on partial flag varieties and their relation to the representations of semisimple Lie algebras. We show that certain intertwining functors give equivalences of derived categories of twisted D-modules. This is a generalization of a result by Marastoni. Read More

We introduce the notion of the relative singular locus Sing$(T/S)$ of a closed subscheme $T$ of a Noetherian scheme $S$, and for a separated Noetherian scheme $X$ with an ample family of line bundles and a non-zero-divisor $W\in\Gamma(X,L)$ of a line bundle $L$ on $X$, we classify certain thick subcategories of the derived matrix factorization category DMF$(X,L,W)$ by means of specialization-closed subsets of the relative singular locus Sing$(X_0/X)$ of the zero scheme $X_0:=W^{-1}(0)\subset X$. Furthermore, we show that the spectrum of the tensor triangulated category (DMF$(X,L,W)$, $\otimes^{\frac{1}{2}}$) is homeomorphic to the relative singular locus Sing$(X_0/X)$ by using the classification result and the theory of Balmer's tensor triangular geometry. Read More

Let $F$ be a non-Archimedean local field. We study the restriction of an irreducible admissible genuine representations of the two fold metaplectic cover $\widetilde{GL}_{2}(F)$ of $GL_{2}(F)$ to the inverse image in $\widetilde{GL}_{2}(F)$ of a maximal torus in $GL_{2}(F)$. Read More

We combine the ideas of a Harish-Chandra--Howe local character expansion, which can be centred at an arbitrary semisimple element, and a Kim--Murnaghan asymptotic expansion, which so far has been considered only around the identity. We show that, for most smooth, irreducible representations (those containing a good, minimal K-type), Kim--Murnaghan-type asymptotic expansions are valid on explicitly defined neighbourhoods of nearly arbitrary semisimple elements. We then give an explicit, inductive recipe for computing the coefficients in an asymptotic expansion for a tame supercuspidal representation. Read More

Lie-theoretic structures of type $E_8$ (e.g., Lie groups and algebras, Hecke algebras and Kazhdan-Lusztig cells, . Read More

We introduce a quotient of the affine Temperley-Lieb category that encodes all weight-preserving linear maps between finite-dimensional sl(2)-representations. We study the diagrammatic idempotents that correspond to projections onto extremal weight spaces and find that they satisfy similar properties as Jones-Wenzl projectors, and that they categorify the Chebyshev polynomials of the first kind. This gives a categorification of the Kauffman bracket skein algebra of the annulus, which is well adapted to the task of categorifying the multiplication on the Kauffman bracket skein module of the torus. Read More

In this paper, we construct quantum twist automorphisms on quantum unipotent cells, which are quantum analogues of the Berenstein-Fomin-Zelevinsky twist automorphisms on unipotent cells. We show that those quantum twist automorphisms preserve the dual canonical bases of quantum unipotent cells. We also prove that quantum twist automorphisms admit the quantum analogues of Gei{\ss}-Leclerc-Schr\"oer's description of (non-quantum) twist automorphisms. Read More

In this paper, we are interested in the decomposition of the tensor product of two representations of a symmetrizable Kac-Moody Lie algebra $\mathfrak g$. Let $P\_+$ be the set of dominant integral weights. For $\lambda\in P\_+$ , $L(\lambda)$ denotes the irreducible, integrable, highest weight representation of g with highest weight $\lambda$. Read More

Let $W$ be a Coxeter group. The goal of the paper is to construct new Hopf algebras that contain Hecke algebras $H_{\bf q}(W)$ as (left) coideal subalgebras. Our Hecke-Hopf algebras ${\bf H}(W)$ have a number of applications. Read More

We prove that the category of solitons of a finite index conformal net is a bicommutant category, and that its Drinfel'd center is the category of representations of the conformal net. In the special case of a chiral WZW conformal net with finite index, the second result specializes to the statement that the Drinfel'd center of the category of positive energy representations of the based loop group is equivalent to the category of positive energy representations of the free loop group. These results were announced in [arXiv:1503. Read More

Classifying Hopf algebras of a given dimension is a hard and open question. Using the generalized lifting method, we determine all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose coradical generates a Hopf algebra $H$ of dimension $16$ without the Chevalley property and the corresponding infinitesimal braidings are simple objects in $\HYD$. In particular, we figure out $8$ classes of new Hopf algebras of dimension $128$ without the Chevalley property. Read More

We study the geometry and the singularities of the principal direction of the Drinfeld-Lafforgue-Vinberg degeneration of the moduli space of G-bundles Bun_G for an arbitrary reductive group G, and their relationship to the Langlands dual group of G. The article consists of two parts. In the first and main part, we study the monodromy action on the nearby cycles sheaf along the principal degeneration of Bun_G and relate it to the Langlands dual group of G. Read More

We compare two important bases of an irreducible representation of the symmetric group: the web basis and the Specht basis. The web basis has its roots in the Temperley-Lieb algebra and knot-theoretic considerations. The Specht basis is a classic algebraic and combinatorial construction of symmetric group representations which arises in this context through the geometry of varieties called Springer fibers. Read More

We give a closed expression for the number of points over finite fields (or the motive) of the Lusztig nilpotent variety associated to any quiver, in terms of Kac's A-polynomials. When the quiver has 1-loops or oriented cycles, there are several possible variants of the Lusztig nilpotent variety, and we provide formulas for the point count of each. This involves nilpotent versions of the Kac A-polynomial, which we introduce and for which we give a closed formula similar to Hua's formula for the usual Kac A-polynomial. Read More

Let $G$ be a simply connected semisimple algebraic group over $\mathbb{C}$ and let $\rho :G\rightarrow GL(V_\lambda)$ be an irreducible representation of highest weight $\lambda$. Suppose that $\rho$ has finite kernel. Springer defined adjoint-invariant regular map with Zariski dense image from the group to its Lie algebra, $\theta_\lambda:G\rightarrow\mathfrak{g}$, which depends on $\lambda$ [Kumar]. Read More

This paper is a survey of work done on $\mathbb{N}$-graded Clifford algebras (GCAs) and $\mathbb{N}$-graded \textit{skew} Clifford algebras (GSCAs) \cite{VVW, SV, CaV, NVZ, VVe1, VVe2}. In particular, we discuss the hypotheses necessary for these algebras to be Artin Schelter-regular \cite{AS, ATV1} and show how certain `points' called, point modules, can be associated to them. We may view an AS-regular algebra as a noncommutative analog of the polynomial ring. Read More

We give an explicit construction of global Galois gerbes constructed more abstractly by Kaletha to define global rigid inner forms. This notion is crucial to formulate Arthur's multiplicity formula for inner forms of quasi-split reductive groups. As a corollary, we show that any global rigid inner form is almost everywhere unramified, and we give an algorithm to compute the resulting local rigid inner forms at all places in a given finite set. Read More

We use the progenerator constructed in our previous paper to give a necessary condition for a simple module of a finite reductive group to be cuspidal, or more generally to obtain information on which Harish-Chandra series it can lie in. As a first application we show the irreducibility of the smallest unipotent character in any Harish-Chandra series. Secondly, we determine a unitriangular approximation to part of the unipotent decomposition matrix of finite orthogonal groups and prove a gap result on certain Brauer character degrees. Read More

In this note, let $\A$ be a finitary hereditary abelian category with enough projectives. By using the associativity formula of Hall algebras, we give a new and simple proof of the main theorem in \cite{Yan}, which states that the Bridgeland's Hall algebra of 2-cyclic complexes of projective objects in $\A$ is isomorphic to the Drinfeld double Hall algebra of $\A$. In a similar way, we give a simplification of the key step in the proof of Theorem 4. Read More

We realize the non-split Bessel model of Novodvorsky and Piatetski-Shapiro as a generalized Gelfand-Graev representation of GSp(4), as suggested by Kawanaka. With uniqueness of the model already established by Novodvorsky and Piatetski-Shapiro, we establish existence of a Bessel model for unramified principal series representations. We then connect the Iwahori-fixed vectors in the Bessel model to a linear character of the Hecke algebra of GSp(4) following the method outlined more generally by Brubaker, Bump, and Friedberg. Read More

We show that the Poisson centre of truncated maximal parabolic subalgebras of a simple Lie algebra of type B, D and E_6 is a polynomial algebra. This allows us to answer positively for these algebras Dixmier's fourth problem, namely whether the field of invariant fractions of the enveloping algebra of a Lie algebra is a purely transcendental extension of the base field. In roughly half of the cases the polynomiality of the Poisson centre was already known by a completely different method. Read More

We study thick subcategories defined by modules of complexity one in $\underline{\md}R$, where $R$ is the exterior algebra in $n+1$ indeterminates. Read More

Recently, Chen and Koenig in \cite{CheKoe} and Iyama and Solberg in \cite{IyaSol} independently introduced and characterised algebras with dominant dimension coinciding with the Gorenstein dimension and both dimensions being larger than or equal to two. In \cite{IyaSol}, such algebras are named Auslander-Gorenstein algebras. Those classes of algebras clearly generalise the well known class of higher Auslander algebras, where the dominant dimension additionally coincides with the global dimension. Read More

Basis functions which are invariant under the operations of a rotational polyhedral group $G$ are able to describe any 3-D object which exhibits the rotational symmetry of the corresponding Platonic solid. However, in order to characterize the spatial statistics of an ensemble of objects in which each object is different but the statistics exhibit the symmetry, a larger set of basis functions is required. In particular, for each irreducible representation (irrep) of $G$, it is necessary to include basis functions that transform according to that irrep. Read More

Necessary and sufficient conditions for finite semihypergroups to be built from abelian groups of the same order are established Read More