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We investigate the Drinfel'd doubles $D(\Lambda_{n,d})$ of a certain family of Hopf algebras. We determine their simple modules and their indecomposable projective modules, and we obtain a presentation by quiver and relations of these Drinfel'd doubles, from which we deduce properties of their representations,... |
We present a unified approach to the study of Radon transforms related to symmetric groups and to general linear groups GL(n,q) regarded as q-analogues of the former. In both cases, we define a sequence of generalized Radon transforms which are intertwining operators for natural representations associated to Gel'f... |
Let ${\mathfrak g}$ be a finite dimensional Lie algebra over a field of characteristic 0, with solvable radical ${\mathfrak r}$ and nilpotent radical ${\mathfrak n}=[{\mathfrak g},{\mathfrak r}]$. Given a finite dimensional ${\mathfrak g}$-module $U$, its nilpotency series $ 0\subset U({\mathfrak n}^1)\subset\cdots\su... |
It is shown how integrability of the generalized Lie equations of a local analytic Moufang loop is related to the reductivity conditions and Sagle-Yamaguti identity. |
In this paper, the rational Ringel-Hall algebras for tame quivers are introduced and are identified with the positive part of the quantum extended Kac-Moody algebras. By using the rational Ringel-Hall algebras, we show that the existence of Hall polynomials for tame quiver algebras. |
In this paper, we introduce the class of Cohen-Macaulay (=CM) dg (=differential graded) modules over Gorenstein dg algebras and study their basic properties. We show that the category of CM dg modules forms a Frobenius extriangulated category, in the sense of Nakaoka and Palu, and it admits almost split extensions. |
We present the general construction of the $U$-projector (the homomorphism of the algebra into its field of $U$-invariants identical on the subalgebra of $U$-invariants). It is shown how to apply $U$-projector to find the systems of free generators of the fields of $U$-invariants for representations of reductive group... |
In this note we compute the centers of the categories of tilting modules for G=SL2 in prime characteristic, of tilting modules for the corresponding quantum group at a complex root of unity, and of projective GgT-modules when g=1,2. |
We study the multiplicities of Young modules as direct summands of permutation modules on cosets of Young subgroups. Such multiplicities have become known as the p-Kostka numbers. |
Let $\pi$ be a Speh module of $\mathbf{GL}(2n,\mathbb{R})$ based on a discrete series of $\textbf{GL}(2,\mathbb{R})$. The aim of this paper is to build a chain complex of $\pi$ by direct sum of auto-duals standard modules, \begin{align}\label{eq:abstract} 0\rightarrow \pi\rightarrow X_{0}\rightarrow \cdots\rightarrow ... |
We study the structure of parabolic inductions of a pro-$p$-Iwahori Hecke algebra. In particular, we give a classification of irreducible modulo $p$ representations of pro-$p$-Iwahori Hecke algebra in terms of supersingular representations. |
It is shown that any generalized Kac-Moody Lie algebra g that has no mutually orthogonal imaginary simple roots can be written as the vector space direct sum of a Kac-Moody subalgebra and subalgebras isomorphic to free Lie algebras over certain modules for the Kac-Moody subalgebra. Also included is a detailed discussi... |
Let $A$ be a finite-dimensional algebra over an algebraically closed field $\Bbbk$. For any finite-dimensional $A$-module $M$ we give a general formula that computes the indecomposable decomposition of $M$ without decomposing it, for which we use the knowledge of AR-quivers that are already computed in many cases. |
We show that string algebras are `homologically tame' in the following sense: First, the syzygies of arbitrary representations of a finite dimensional string algebra $\Lambda$ are direct sums of cyclic representations, and the left finitistic dimensions, both little and big, of $\Lambda$ can be computed from a fini... |
This paper gives a classification of stable vectors in dual Vinberg representations coming from a graded Lie algebra of type $F_4$ in a way that is independent of the field of definition. Relating these gradings to Moy-Prasad filtrations, we obtain the input for Reeder-Yu's construction of epipelagic supercuspidal... |
In previous papers the author introduced a new basis of the Grpthendieck group of unipotent representations of a finite Chevalley group. In type D the definition of this basis was stated without proof. |
We construct Young wall models for the crystal bases of level $1$ irreducible highest weight representations and Fock space representations of quantum affine algebras in types $E_{6}^{(1)}$, $E_{7}^{(1)}$ and $E_{8}^{(1)}$. In each case, Young walls consist of coloured blocks stacked inside the relevant Young wall pat... |
Inspired by the tropical duality in cluster algebras, we introduce c-vectors for finite-dimensional algebras via $\tau$-tilting theory. Let $A$ be a finite-dimensional algebra over a field $k$. |
Given a classical semisimple complex algebraic group G and a symmetric pair (G, K) of non-Hermitian type, we study the closures of the spherical nilpotent K-orbits in the isotropy representation of K. For all such orbit closures, we study the normality and we describe the K-module structure of the ring of regular funct... |
The aim of this paper is to study bimodule stably Calabi-Yau properties of derivation quotient algebras. We give the definition of a twisted stably Calabi-Yau algebra and show that every twisted derivation quotient algebra $A$ for which the associated bimodule complex gives the beginning of a bimodule resolution for $... |
In this expository note, I present some of the key features of the lattice of torsion classes of a finite-dimensional algebra, focussing in particular on its complete semidistributivity and consequences thereof. This is intended to serve as an introduction to recent work by Barnard-Carroll-Zhu and Demonet-Iyama-Readin... |
We relate the structure of cyclotomic and degenerate cyclotomic BMW algebras, for arbitrary parameter values, to that for admissible parameter values. In particular, we show that these algebras are cellular. |
We determine the unitary dual of the geometric graded Hecke algebras with {unequal} parameters which appear in Lusztig's classification of unipotent representations for {exceptional} $p$-adic groups. The largest such algebra is of type $F_4.$ Via the Barbasch-Moy correspondence of unitarity applied to this setting... |
This paper computes the Dirac index of all the weakly fair $A_{\mathfrak{q}}(\lambda)$ modules of $U(p, q)$. Although counter-examples have been found to a conjecture of Vogan on the unitary dual of $U(p, q)$ phrased by Trapa in 2001, we believe that any irreducible unitary representation of $U(p, q)$ with non-zero Di... |
We examine various consequences of the existence of exceptional representations of an irreducible Weyl group. (These are notes from a talk in the MIT Lie groups seminar.) |
We classify deformations of $\mathfrak{osp}(2|2)-$module structure on the spaces of symbols $\mathfrak{S}_d^2$ of differential operators acting on the space of weighted densities $\mathfrak{F}_{\lambda}^{2}$. |
We define an exact functor $F_{n,k}$ from the category of Harish-Chandra modules for $GL(n,R)$ to the category of finite-dimensional representations for the degenerate affine Hecke algebra for $gl(k)$. Under certain natural hypotheses, we prove that the functor maps standard modules to standard modules (or zero) and i... |
Iwahori-Hecke algebras for the infinite series of complex reflection groups $G(r,p,n)$ were constructed recently in the work of Ariki and Koike, Broué and Malle, and Ariki. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. |
Let an affine Weyl group $\hat W$ act as a group of affine transformations on a real vector space V. We analyze the $\hat W$-orbit of a regular element in V and deduce applications to Kostant's formula for powers of the Euler product and to the representations of $\hat W$ as permutations of the integers. |
We give an explicit combinatorial Chevalley-type formula for the equivariant K-theory of generalized flag varieties G/P which is a direct generalization of the classical Chevalley formula. Our formula implies a simple combinatorial model for the characters of the irreducible representations of G and, more generally, f... |
In this paper, we show that for a Koszul $n$-homogeneous algebra $\Lambda$, the quadratic dual of certain twisted trivial extension is the $(n+1)$-preprojective algebra of its quadratic dual, that is, $ (\Delta_{\nu}\Lambda)^{! ,op} \simeq\Pi( \Lambda^{ !, op })$. |
We settle a long-standing problem in the theory of Hecke algebras of complex reflection groups by constructing many (graded) integral cellular bases of these algebras. As applications, we explicitly construct the simple modules of Ariki's categorification theorem and prove unitriangularity of decomposition matrice... |
We study the representation theory of a quantum symmetric pair $(\mathbf{U},\mathbf{U}^{\jmath})$ with two parameters $p,q$ of type AIII, by using highest weight theory and a variant of Kashiwara's crystal basis theory. Namely, we classify the irreducible $\mathbf{U}^{\jmath}$-modules in a suitable category and as... |
We investigate the relationship between the delooping level (dell) and the finitistic dimension of left and right serial path algebras. These 2-syzygy finite algebras have finite delooping level, and it can be calculated with an easy and finite algorithm. |
The additive monoid $R_+(x)$ is defined as the set of all nonnegative integer linear combinations of binomial coefficients $\binom{x}{n}$ for $n \in \mathbb Z_+$. This paper is concerned with the inquiry into the structure of $R_+(\alpha)$ for complex numbers $\alpha. |
Let $\Gamma^{n}$ be the cone of an $(n-1)$-complete algebra over an algebraically closed field $k$. In this paper, we prove that if the bound quiver $(Q_{n},\rho_{n})$ of $\Gamma^{n}$ is a truncation from the bound McKay quiver $(Q_{G},\rho_{G})$ of a finite subgroup $G$ of $GL(n,k)$, the bound quiver $(Q_{n+1}, \rho_... |
Degenerating the action of the elliptic Hall algebra on the Fock space, we give a combinatorial formula for the Shalika germs of tamely ramified regular semisimple elements $\gamma$ of $GL_n$ over a nonarchimedean local field. As a byproduct, we compute the weight polynomials of affine Springer fibers in type A and or... |
First and second fundamental theorems are given for polynomial invariants of a class of pseudo-reflection groups (including the Weyl groups of type $B_n$), under the assumption that the order of the group is invertible in the base field. Special case of the result is a finite presentation of the algebra of multisymmet... |
We present a novel approach to the classification of conformally equivariant differential operators on spinors in the case of homogeneous conformal geometry. It is based on the classification of solutions for a vector-valued system of partial differential equations, associated to $\mathcal{D}$-modules for the homogene... |
The symmetries provided by representations of the centrally extended Lie superalgebra $\mathfrak{psl}(2|2)$ are known to play an important role in the spin chain models originated in the planar anti-de Sitter/conformal field theory correspondence and one-dimensional Hubbard model. We give a complete description of fin... |
We deal with finitely generated modules over an artin algebra. In his Philadelphia Notes, <a href="http://M.Auslander" rel="external noopener nofollow" class="link-external link-http">this http URL</a> showed that any homomorphism is right determined by a module C, but a formula for C which he wrote down has to be mod... |
The goal of this paper is to construct an action of the cactus group of a Weyl group W on W that is nicely compatible with Kazhdan-Lusztig cells. The action is realized by the wall-crossing bijections that are combinatorial shadows of wall-crossing functors on the category O. |
Like the RSK correspondence for symmetric groups, Garfinkle defined a domino correspondence for type $\mathrm{B}$ and $\mathrm{D}$ Coxeter groups. Similar to the Knuth relations, Taskin and Pietraho give the plactic relations for the domino correspondence and Bonnafé use them to classify the cells for type $\mathrm{B}... |
The weighted triangulation algebras associated to triangulation quivers and their socle deformations were recently introduced and studied in [15]-[20] and [2]. These algebras, based on surface triangulations and originated from the theory of cluster algebras, were also proved to be (with some minor exceptions) finite-... |
In a recent preprint, Sakellaridis and Venkatesh considered the spectral decomposition of the space $L^2(X)$, where $X = H\G$ is a spherical variety and $G$ is a real or $p$-adic group, and stated a conjecture describing this decomposition in terms of a dual group $\check{G}_X$ associated to $X$. The main purpose of t... |
We give a functorial construction of the genus zero chiral algebras of class $\mathcal{S}$, that is, the vertex algebras corresponding to the theory of class $\mathcal{S}$ associated with genus zero pointed Riemann surfaces via the 4d/2d duality discovered by Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees in phys... |
In this paper we express certain multiplicities in modular representation-theoretic categories of type A in terms of affine p-Kazhdan-Lusztig polynomials. The representation-theoretic categories we deal with include the categories of rational representations of GL(n), representations of the quantum group for gl(n), an... |
We prove some basic facts on parahoric subgroups and on Iwahori-Weyl groups. |
Let G be a reductive algebraic group over the algebraic closure of a finite field F_q of good characteristic. In this paper, we demonstrate a remarkable compatibility between the Springer correspondence for G and the parametrization of unipotent characters of G(F_q). |
The theory of total positivity for reductive groups is here extended to the case of symmetric spaces. |
For a finite dimensional semisimple Lie algebra and a root of unity, Lusztig defined an infinite dimensional quantum group of divided powers. Under certain restrictions on the order of the root of unity, he constructed a Frobenius homomorphism with finite dimensional Hopf kernel and with image the universal enveloping... |
We establish the analog for real spherical varieties of the Scattering Theorem of Sakellaridis and Venkatesh (\cite{SV}, Theorem 7.3.1) for p-adic spherical varieties. We use properties of the Harish-Chandra homomorphism of Knop for invariant differential operators of the variety, special coverings of the variety and ... |
We show the existence of a large family of representations supported by the orbit closure of the determinant. However, the validity of our result is based on the validity of the celebrated `Latin Square Conjecture' due to Alon-Tarsi or more precisely on the validity of an equivalent `column Latin Square Conjecture... |
We extend our previous study of quaternionic analysis based on representation theory to the case of split quaternions H_R. The special role of the unit sphere in the classical quaternions H identified with the group SU(2) is now played by the group SL(2,R) realized by the unit quaternions in H_R. |
Let Q be a Dynkin quiver. The bounded derived category of the path algebra of Q has an autoequivalence given by the composition of the Auslander-Reiten translate and the square of the shift functor. |
For a simple Lie algebra $\mathfrak{g}$ of type $A_n,B_n,C_n$ or $D_n$, we give a characterization of the set of dominant integral weights $\lambda$ such that for any rational point $\mu$ in the fundamental Weyl chamber, $2\lambda-\mu$ is a non-negative rational combination of the simple roots if and only if $V_{m\mu}\... |
We classify the irreducible projective representations of symmetric and alternating groups of minimal possible and second minimal possible dimensions, and get a lower bound for the third minimal dimension. On the way we obtain some new results on branching which might be of independent interest. |
We prove that the conjectural depth zero local Langlands correspondence of DeBacker/Reeder agrees with the depth zero local Langlands correspondence as described by Moy, for the group $GL(\ell,F)$, where $\ell$ is prime and F is a local non-archimedean field of characteristic 0. We also prove that if one assumes a cer... |
We classify the simple integrable modules of double affine Hecke algebras via perverse sheaves. We get also some estimate for the Jordan-Holder multiplicities of induced modules. |
In this paper, we give a refinement of a generalized Dedekind's theorem. In addition, we show that all possible values of integer group determinants of any group are also possible values of integer group determinants of its any abelian subgroup. |
We give an explicit formula for the exchange matrix correponding to the tensor product of two copies of the natural (standard) evaluation representation of the quantum group associated to the affine Lie algebra of sl(n+1). Then we calculate the gauge trasformations that transforms it into Felder's elliptic solutio... |
Let $(\mathcal{A,B})$ be a GP-admissible pair and $(\mathcal{Z,W})$ be a GI-admissible pair of classes of objects in an abelian category $\mathcal{C}$, and consider the class $\pi\mathcal{GP}_{(\omega,\mathcal{B},1)}$ of $1$-periodic $(\omega,\mathcal{B})$-Gorenstein projective objects, where $\omega := \mathcal{A} \ca... |
For a Hermitian Lie group $G$, we study the family of representations induced from a character of the maximal parabolic subgroup $P=MAN$ whose unipotent radical $N$ is a Heisenberg group. Realizing these representations in the non-compact picture on a space $I(\nu)$ of functions on the opposite unipotent radical $\bar... |
We construct a faithful categorical representation of an infinite Temperley-Lieb algebra on the periplectic analogue of Deligne's category. We use the corresponding combinatorics to classify thick tensor ideals in this periplectic Deligne category. |
Baer's Criterion for Injectivity is a basic tool of the theory of modules and complexes of modules. Its dual version (DBC) is known to hold for all right perfect rings, but its validity for non-right perfect rings is a complex problem (first formulated by Faith in 1976 \cite{F}). |
We show that the totally nonnegative part of a partial flag variety (in the sense of Lusztig) is homeomorphic to a closed ball. |
In this article, we prove that induced modules of support $\tau$-tilting modules over blocks of finite groups satisfying inertial-invariant condition are also support $\tau$-tilting modules. |
Williamson defined the category of singular Soergel bimodules attached to a reflection faithful representation of a Coxeter group. We generalize this construction to more general realizations of Coxeter groups. |
Let $Z$ be a unimodular real spherical space. We develop a theory of constant terms for tempered functions on $Z$ which parallels the work of Harish-Chandra. |
We introduce mutation along infinite admissible sequences for infinitely marked surfaces, that is surfaces with infinitely many marked points on the boundary. We show that mutation along such admissible sequences produces a preorder on the set of triangulations of a fixed infinitely marked surface. |
We study the structure and representation theory of affine wreath product algebras and their cyclotomic quotients. These algebras, which appear naturally in Heisenberg categorification, simultaneously unify and generalize many important algebras appearing in the literature. |
Motivated by construction in Algebraic Quantum Field Theory we introduce wedge domains in compactly causal symmetric spaces M=G/H, which includes in particular anti de Sitter space in all dimensions and its coverings. Our wedge domains generalize Rindler wedges in Minkowski space. |
We obtain two explicit formulas for the full local character expansion of any irreducible representation of a p-adic general linear group in principal blocks. The first, generalizing previous work of the author on the Iwahori-spherical case, expresses the expansion in terms of dimensions of degenerate Whittaker models... |
Given any symmetric Cartan datum, Lusztig has provided a pair of key lemmas to construct the perverse sheaves over the corresponding quiver and the functions of irreducible components over the corresponding preprojective algebra respectively. In the present article, we prove that these two inductive algorithms of Lusz... |
We consider generic bricks and use them in the study of arbitrary biserial algebras over algebraically closed fields. For a biserial algebra $\Lambda$, we show that $\Lambda$ is brick-infinite if and only if it admits a generic brick, that is, there exists a generic $\Lambda$-module $G$ with $End_{\Lambda}(G)=k(x)$. |
In this paper, we give a characterization of tau-rigid-finite algebras with radical square zero in terms of the separated quivers, which is an analog of a famous characterization of representation-finite algebras with radical square zero. |
We show that the space spanned by the class of simple perverse sheaves in [Zh08] without localization is isomorphic to the tensor product of a Verma module with a tensor product of irreducible integrable modules of the quantum enveloping algebra associated with a graph. Under the isomorphism, the simple perverse sheav... |
In this paper we give the description of generic representations of metaplectic groups over p-adic fields in terms of their Langlands parameters and calculate their theta lifts on all levels for any tower of odd orthogonal groups. We also describe precisely all the occurrences of the failure of the standard module con... |
Let F be a non-Archimedean locally compact field with residual characteristic p, let G be an inner form of GL(n,F) for a positive integer n and let R be an algebraically closed field of characteristic different from p. When R has characteristic $\ell>0$, the image of an irreducible smooth R-representation $\pi$ of ... |
In this short note, we refine a result of Schiffmann-Vasserot, by showing that the localized preprojective cohomological Hall algebra of any quiver is spherical, i.e. generated by elements of minimal dimension. |
Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field $k$. We survey some results on algebras of finite global dimension and address some open problems. |
We show that tilting modules for quantum groups over local Noetherian domains exist and that the indecomposable tilting modules are parametrized by their highest weight. For this we introduce a model category ${\mathcal X}={\mathcal X}_{\mathscr A}(R)$ associated with a Noetherian ${\mathbb Z}[v,v^{-1}]$-domain ${\mat... |
We produce examples in the cohomology of algebraic groups which answer two questions of Parshall and Scott. Specifically, if $G=SL_2$, then we show: |
We give a complete description of the bounded (i.e. norm continuous) unitary representations of the Fréchet-Lie algebra of all smooth sections, as well as of the LF-Lie algebra of compactly supported smooth sections, of a smooth Lie algebra bundle whose typical fiber is a compact Lie algebra. For the Lie algebra of al... |
We prove the conjecture that flag versions of quiver Grassmannians (also known as Lusztig's fibers) for Dynkin quivers (types $A$, $D$, $E$) have no odd cohomology groups over an arbitrary ring. Moreover, for types A and D we prove that these varieties have affine pavings. |
In 2005, the second author and Todorov introduced an upper bound on the finitistic dimension of an Artin algebra, now known as the {\phi}-dimension. The {\phi}-dimension conjecture states that this upper bound is always finite, a fact that would imply the finitistic dimension conjecture. |
Let $\cal C$ be a $(d+2)$-angulated category. In this note, we show that if $\cal C$ is a locally finite, then $\cal C$ has Auslander-Reiten $(d+2)$-angles. |
We show that rectangular Kronecker coefficients stabilize when the lengths of the sides of the rectangle grow, and we give an explicit formula for the limit values in terms of invariants of $\fsl_n$. |
We introduce Sylow subgroups and $0$-groups to the theory of complex algebraic supergroups, which mimic Sylow subgroups and $p$-groups in the theory of finite groups. We prove that Sylow subgroups are always $0$-groups, and show that they are unique up to conjugacy. |
The traditional construction of Chevalley groups relies on the choice of certain signs for a Chevalley basis of the underlying Lie algebra~$\mathfrak{g}$. Recently, Lusztig simplified this construction for groups of adjoint type by using the "canonical basis" of the adjoint representation of~$\mathfrak{g}$, in... |
Bloch and Okounkov's correlation function on the infinite wedge space has connections to Gromov-Witten theory, Hilbert schemes, symmetric groups, and certain character functions of $\hgl_\infty$-modules of level one. Recent works have calculated these character functions for higher levels for $\hgl_\infty$ and its... |
We generalize the work of DeBacker and Reeder to the case of unitary groups split by a tame extension. The approach is broadly similar and the restrictions on the parameter the same, but many of the details of the arguments differ. |
We prove for finite reductive groups $G$ of classical type, that every irreducible character of $L$ extends to its inertia group in $N$, where $L$ is an abelian centraliser of a Sylow $d$-torus $\mathbf S$ of $G$ and $N:=N_G(\mathbf S)$. This gives a precise description of the irreducible characters of $N$. |
The Foulkes module H^(a^b) is the permutation module for the symmetric group S_ab given by the action of S_ab on the collection of set partitions of a set of size ab into b sets each of size a. The main result of this paper is a sufficient condition for a simple CS_{ab}-module to have zero multiplicity in H^(a^b). |
Let $K$ be an infinite field of characteristic $p>0$ and let $\lambda, \mu$ be partitions, where $\mu$ has two parts. We find sufficient arithmetic conditions on $p, \lambda, \mu$ for the existence of a nonzero homomorphism $\Delta(\lambda) \to \Delta (\mu)$ of Weyl modules for the general linear group $GL_n(K)$. |
The classification of unitary representations for the non-compact real form E6(-14) of the exceptional Lie group E6 has long been hindered by computational bottlenecks due to its complex root system (72 roots) and large Weyl group (order 51840). This paper proposes a dimensional reduction method integrating root subsy... |
We consider integrable category $\mathcal{O}$ representations of Borcherds--Kac--Moody algebras whose Cartan matrix is finite dimensional, and determine the necessary and sufficient conditions for which the tensor product of irreducible representations from this category is isomorphic to another. This result generaliz... |
In his article "Unitary Representations and Complex Analysis", David Vogan gives a characterization of the continuous invariant Hermitian forms defined on the compactly supported sheaf cohomology groups of certain homogeneous analytic sheaves defined on open orbits in generalized flag manifolds. In the last se... |
The Weil representation of the symplectic group associated to a finite abelian group of odd order is shown to have a multiplicity-free decomposition. When the abelian group is p-primary, the irreducible representations occurring in the Weil representation are parametrized by a partially ordered set which is independen... |
A Lie group G is called a trace class group if for every irreducible unitary representation R of G and every C-infinity function f with compact support the operator R(f) is of trace class. In this note we prove that the semidirect product of R^n and a real semisimple algebraic subgroup G of GL(n;R) is a trace class gr... |
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