Mathematics - Representation Theory Publications (50)


Mathematics - Representation Theory Publications

This article surveys recent advances and future challenges in the $2$-representation theory of finitary $2$-categories with a particular emphasis on problems related to classification of various classes of $2$-representations. Read More

We give a finite presentation for the braid twist group $\operatorname{BT}(\mathbf{S}_\aleph^{g,\mathrm{b}})$ of a decorated surface $\mathbf{S}_\aleph^{g,\mathrm{b}}$. If the surface $\mathbf{S}_\aleph^{g,\mathrm{b}}$ is a marked surface $\mathbf{S}$ and the decoration $\aleph$ is from a triangulation $\mathbf{T}$, we obtain a finite presentation for the spherical twist group of the 3-Calabi-Yau category $\mathcal{D}_{fd}(\Gamma_{\mathbf{T}})$ associated to $\mathbf{S}$, w.r. Read More

We determine the image of the (strongly) parabolic Hitchin map for all parabolics in classical groups and $G_2$. Surprisingly, we find that the image is isomorphic to an affine space in all cases, except for certain "bad parabolics" in type $D$, where the image can be singular. Read More

Given a (complex, smooth) irreducible representation $\pi$ of the general linear group over a non-archimedean local field and an irreducible supercuspidal representation $\sigma$ of a classical group, we show that the (normalized) parabolic induction $\pi\rtimes\sigma$ is reducible if there exists $\rho$ in the supercuspidal support of $\pi$ such that $\rho\rtimes\sigma$ is reducible. In special cases we also give irreducibility criteria for $\pi\rtimes\sigma$ when the above condition is not satisfied. Read More

In this note we show that over uncountable fields a certain class of skew-symmetrizable matrices have a species realization admitting a nondegenerate potential. This gives a partial affirmative answer to a question raised by Daniel Labardini-Fragoso and Jan Geuenich. We also provide an example of a class of skew-symmetrizable $4 \times 4$ integer matrices that have a species realization via field extensions of the rationals. Read More

We introduce the notion of exact tilting objects, which are partial tilting objects $T$ inducing an equivalence between the abelian category generated by $T$ and the category of modules over the endomorphism algebra of $T$. Given a chain of sufficiently negative rational curves on a rational surface, we construct an exceptional sequence whose universal extension is an exact tilting object. For a chain of (-2)-curves, we obtain an equivalence with modules over a well known algebra. Read More

Let Q be an acyclic quiver. The dimension vectors of indecomposable rigid representations are called real Schur roots. We give a conjectural description for real Schur roots of Q using non-self-intersecting paths on Riemann surfaces, and prove it for certain quivers of finite types and for the quivers with three or less vertices and multiple arrows between every pair of vertices. Read More

In this paper we verify Navarro's refinement of the McKay conjecture for quasi-simple groups of Lie type $G$ and the prime $p$, where $p$ is the defining characteristic of $G$. Navarro's refinement takes into account the action of specific Galois automorphisms on the characters present in the McKay conjecture. Our proof of this special case of the conjecture relies on a character correspondence which Maslowski constructed in the context of the inductive McKay conditions by Isaacs, Malle and Navarro. Read More

We give a uniform description of the bijection $\Phi$ from rigged configurations to tensor products of Kirillov--Reshetikhin crystals of the form $\bigotimes_{i=1}^N B^{r_i,1}$ in dual untwisted types: simply-laced types and types $A_{2n-1}^{(2)}$, $D_{n+1}^{(2)}$, $E_6^{(2)}$, and $D_4^{(3)}$. We give a uniform proof that $\Phi$ is a bijection and preserves statistics. We describe $\Phi$ uniformly using virtual crystals for all remaining types, but our proofs are type-specific. Read More

The author's work with Murnaghan on distinguished tame supercuspidal representations is re-examined using a simplified treatment of Jiu-Kang Yu's construction of tame supercuspidal representations of $p$-adic reductive groups. This leads to a unification of aspects of the theories of distinguished cuspidal representations over $p$-adic and finite fields. Read More

We introduce a bt-algebra of type B. We define this algebra doing the natural analogy with the original construction of the bt-algebra. Notably we find a basis for it, a faithful tensorial representation, and we prove that it supports a Markov trace, from which we derive invariants of classical links in the solid torus. Read More

We consider a certain quotient of a polynomial ring categorified by both the isomorphic Green rings of symmetric group and Schur algebra generated by the signed Young permutation modules and mixed powers respectively. They have bases parametrised by pairs of partitions whose second partitions are multiple of the odd prime $p$ the characteristic of the underlying field. We provide an explicit formula rewriting a signed Young permutation module (respectively, mixed power) in terms of signed Young permutation modules (respectively, mixed powers) labelled by those pair of partitions. Read More

The spectrum of $L^2$ on a pseudo-unitary group $U(p,q)$ (we assume $p\ge q$ naturally splits into $q+1$ types. We write explicitly orthogonal projectors in $L^2$ to subspaces with uniform spectra (this is an old question formulated by Gelfand and Gindikin). We also write two finer separations of $L^2$. Read More

A famous result of Zimmermann-Huisgen, Hille and Reineke asserts that any projective variety occurs as a quiver Grassmannian for a suitable representation of some wild acyclic quiver. We show that this happens for any wild acyclic quiver. Read More

Let $A$ be a (left and right) Noetherian ring that is semiperfect. Let $e$ be an idempotent of $A$ and consider the algebra $\Gamma:=(1-e)A(1-e)$ and the semi-simple right $A$-module $S_e : = eA/e{\rm rad}A$. In this paper, we investigate the relationship between the global dimensions of $A$ and $\Gamma$, by using the homological properties of $S_e$. Read More

Let $G$ be an unramified $p$-adic group. We decompose $Rep_{\Lambda}^{0}(G)$, the abelian category of smooth level $0$ representations of $G$ with coefficients in $\Lambda=\overline{\mathbb{Q}}_{l}$ or $\overline{\mathbb{Z}}_{l}$, into a product of subcategories indexed by inertial Langlands parameters. We construct these categories via systems of idempotents on the Bruhat-Tits building and Deligne-Lusztig theory. Read More

We give the triangulated categories version of our general homotopy theory of additive categories with suspensions developed in [11,12]. We show that the Verdier quotients are just the corresponding homotopy categories of our theory under suitable conditions. From which we can get the recent work of Iyama-Yang, Nakaoka and Wei. Read More

The $K$-homology ring of the affine Grassmannian of $SL_n(C)$ was studied by Lam, Schilling, and Shimozono. It is realized as a certain concrete Hopf subring of the ring of symmetric functions. On the other hand, for the quantum $K$-theory of the flag variety $Fl_n$, Kirillov and Maeno provided a conjectural presentation based on the results obtained by Givental and Lee. Read More

We studied an enhanced adjoint action of the general linear group on a product of its Lie algebra and a vector space consisting of several copies of defining representations and its duals. We determined regular semisimple orbits (i.e. Read More

Let $G:=\widehat{SL_2}$ denote the affine Kac-Moody group associated to $SL_2$ and $\bar{\mathcal{X}}$ the associated affine Grassmanian. We determine an inductive formula for the Schubert basis structure constants in the torus-equivariant Grothendieck group of $\bar{\mathcal{X}}$. In the case of ordinary (non-equivariant) $K$-theory we find an explicit closed form for the structure constants. Read More

Let $\mathbf{k}$ be field of arbitrary characteristic and let $\Lambda$ be a finite dimensional $\mathbf{k}$-algebra. From results previously obtained by F.M Bleher and the author, it follows that if $V^\bullet$ is an object of the bounded derived category $\mathcal{D}^b(\Lambda\textup{-mod})$ of $\Lambda$, then $V^\bullet$ has a well-defined versal deformation ring $R(\Lambda, V^\bullet)$, which is complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$, and which is universal provided that $\textup{Hom}_{\mathcal{D}^b(\Lambda\textup{-mod})}(V^\bullet, V^\bullet)=\mathbf{k}$. Read More

We complete the classification of maximal representations of uniform complex hyperbolic lattices in Hermitian Lie groups by dealing with the exceptional groups ${\rm E}_6$ and ${\rm E}_7$. We prove that if $\rho$ is a maximal representation of a uniform complex hyperbolic lattice $\Gamma\subset{\rm SU}(1,n)$, $n>1$, in an exceptional Hermitian group $G$, then $n=2$ and $G={\rm E}_6$, and we describe completely the representation $\rho$. The case of classical Hermitian target groups was treated by Vincent Koziarz and the second named author (arxiv:1506. Read More

We start with observing that the only connected finite dimensional algebras with finitely many isomorphism classes of indecomposable bimodules are the quotients of the path algebras of uniformly oriented $A_n$-quivers modulo the radical square zero relations. For such algebras we study the (finitary) tensor category of bimodules. We describe the cell structure of this tensor category, determine existing adjunctions between its $1$-morphisms and find a minimal generating set with respect to the tensor structure. Read More

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$, $B$ and $B_-$ be its two opposite Borel subgroups. For two elements $u$, $v$ of the Weyl group $W$, it is known that the coordinate ring ${\mathbb C}[G^{u,v}]$ of the double Bruhat cell $G^{u,v}=BuB\cup B_-vB_-$ is isomorphic to a cluster algebra $\mathcal{A}(\textbf{i})_{{\mathbb C}}$ [arXiv:math/0305434, arXiv:1602.00498]. Read More

Let X(F, G) be the G-character variety of F where G is a rank 1 complex affine algebraic group and F is a finitely presentable discrete group. We describe an algorithm, which we implement in "Mathematica," that takes a finite presentation for F and produces a finite presentation of the coordinate ring of X(F,G). The main results in this paper are not new, although we hope that as a well-referenced exposition with a companion computer program, it will be useful. Read More

If ${\mathfrak g}$ is a real reductive Lie algebra and ${\mathfrak h} < {\mathfrak g}$ is a subalgebra, then $({\mathfrak g}, {\mathfrak h})$ is called real spherical provided that ${\mathfrak g} = {\mathfrak h} + {\mathfrak p}$ for some choice of a minimal parabolic subalgebra ${\mathfrak p} \subset {\mathfrak g}$. In this paper we classify all real spherical pairs $({\mathfrak g}, {\mathfrak h})$ where ${\mathfrak g}$ is semi-simple but not simple and ${\mathfrak h}$ is a reductive real algebraic subalgebra. The paper is based on the classification of the case where ${\mathfrak g}$ is simple (see arXiv:1609. Read More

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. The proof of the formula here is much simpler than that in a prior literature by Dowbor and Mr\'oz. Read More

It is well known that a dense subgroup $G$ of the complex unitary group $U(d)$ cannot be amenable as a discrete group when $d>1$. When $d$ is large enough we give quantitative versions of this phenomenon in connection with certain estimates of random Fourier series on the compact group $\bar G$ that is the closure of $G$. Roughly, we show that if $\bar G$ covers a large enough part of $U(d)$ in the sense of metric entropy then $G$ cannot be amenable. Read More

We show that a separable equivalence between symmetric algebras preserves the dominant dimensions of certain endomorphism algebras of modules. We apply this to show that the dominant dimension of the category coMack(B) of cohomological Mackey functors of a p-block B of a finite group with a nontrivial defect group is 2. Read More

We study a one dimensional directed polymer model in an inverse-gamma random environment, known as the log-gamma polymer, in three different geometries: point-to-line, point-to-half line and when the polymer is restricted to a half space with end point lying free on the corresponding half line.Via the use of A.N. Read More

Introductory and pedagogical treatmeant of the article : P. Broussous "Distinction of the Steinberg representation", with an appendix by Fran\c{c}ois Court\`es, IMRN 2014, no 11, 3140-3157. To appear in Proceedings of Chaire Jean Morlet, Dipendra Prasad, Volker Heiermann Ed. Read More

These notes are devoted to a detailed exposition of the proof of the Geometric Satake Equivalence in the case the coefficients are a field of characteristic 0, following Mirkovic-Vilonen. We follow their arguments closely, adding only a few details where this might be useful. Read More

We present a simplified way to construct the Gelfand-Tsetlin modules over $\mathfrak{gl}(n,\mathbb C)$ related to a $1$-singular GT-tableau defined by Futorny, Grantcharov and Ramirez. We begin by reframing the classical construction of generic Gelfand-Tsetlin modules found by Drozd, Futorny and Ovsienki, showing that they form a flat family over generic points of $\mathbb C^{\binom{n}{2}}$. We then show that this family can be extended to a flat family over a variety including generic points and $1$-singular points for a fixed singular pair of entries. Read More

We extend the Weil representation of infinite-dimensional symplectic group to a representation a certain category of linear relations. Read More

This article is part of a project which consists in investigating Arthur packets for real classical groups. Our goal is to give an explicit description of these packets and to establish the multiplicity one property (which is known to hold for $p$-adic and complex groups). The main result in this paper is a construction of packets from unipotent packets on $c$-Levi factors using cohomological induction. Read More

In this paper we study a key example of a Hermitian symmetric space and a natural associated double flag variety, namely for the real symplectic group $G$ and the symmetric subgroup $L$, the Levi part of the Siegel parabolic $P_S$. We give a detailed treatment of the case of the maximal parabolic subgroups $Q$ of $L$ corresponding to Grassmannians and the product variety of $G/P_S$ and $L/Q$; in particular we classify the $L$-orbits here, and find natural explicit integral transforms between degenerate principal series of $L$ and $G$. Read More

Let $q\geq 2$ be an integer, and $\Bbb F_q^d$, $d\geq 1$, be the vector space over the cyclic space $\Bbb F_q$. The purpose of this paper is two-fold. First, we obtain sufficient conditions on $E \subset \Bbb F_q^d$ such that the inverse Fourier transform of $1_E$ generates a tight wavelet frame in $L^2(\Bbb F_q^d)$. Read More

We resolve a conjecture of Li and Ramos that relates the regularity of an FI-module to its local cohomology groups. This is an analogue of the familiar relationship between regularity and local cohomology in commutative algebra. Read More

We use Littlewood-Paley-Stein type g-functions (also called generalized square functions) associated to symmetric diffusion semigroups to obtain a characterization of inhomogeneous abstract Besov spaces on the abstract Hilbert spaces. Then we apply our results for the abstract Besov spaces defined through the Poisson and Gauss-Weierstrass semigroups. Read More

This paper is devoted to the study of geometry properties of wavelet and Riesz wavelet sets on locally compact abelian groups. The catalyst for our research is a result by Wang ([32], Theorem 1.1) in the Euclidean wavelet theory. Read More

We compute the group of braided tensor autoequivalences and the Brauer-Picard group of the representation category of the small quantum group $\mathfrak{u}_q(\mathfrak{g})$, where $q$ is a root of unity. Read More

We investigate the local descents for special orthogonal groups over p-adic local fields of characteristic zero, and obtain explicit spectral decomposition of the local descents at the first occurrence index in terms of the local Langlands data via the explicit local Langlands correspondence. The main result can be regarded as a refinement of the local Gan-Gross-Prasad conjecture. Read More

A large class of N=2 quantum field theories admits a BPS quiver description and the study of their BPS spectra is then reduced to a representation theory problem. In such theories the coupling to a line defect can be modelled by framed quivers. The associated spectral problem characterises the line defect completely. Read More

We construct and investigate Specht modules $\mathcal{S}^\lambda$ for cyclotomic quiver Hecke algebras in type $C^{(1)}_\ell$ and $C_\infty$, which are labelled by multipartitions $\lambda$. It is shown that in type $C_\infty$, the Specht module $\mathcal{S}^\lambda$ has a homogeneous basis indexed by standard tableaux of shape $\lambda$, which yields a graded character formula and good properties with the exact functors $E_i^\Lambda$ and $F_i^\Lambda$. For type $C^{(1)}_\ell$, we propose a conjecture. Read More

Let $\Lambda$ be a finite dimensional string algebra over a field with the quiver $Q$ such that the underlying graph of $Q$ is a tree, and let $|\Det(\Lambda)|$ be the number of the minimal right determiners of all irreducible morphisms between indecomposable left $\Lambda$-modules. Then we have $$|\Det(\Lambda)|=2n-p-q-1,$$ where $n$ is the number of vertices in $Q$, $p=|\{i\mid i$ is a source in $Q$ with two neighbours$\}|$ and $q$ is the number of non-zero vertex ideals of $\Lambda$. Read More

In this paper we explore a relationship between the topology of the complex hyperplane complements $\mathcal{M}_{BC_n} (\mathbb{C})$ in type B/C and the combinatorics of certain spaces of degree-$n$ polynomials over a finite field $\mathbb{F}_q$. This relationship is a consequence of the Grothendieck trace formula and work of Lehrer and Kim. We use it to prove a correspondence between a representation-theoretic convergence result on the cohomology algebras $H^*(\mathcal{M}_{BC_n} (\mathbb{C});\mathbb{C})$, and an asymptotic stability result for certain polynomial statistics on monic squarefree polynomials over $\mathbb{F}_q$ with nonzero constant term. Read More

In this paper, we give a BLM realization of the positive part of the quantum group of $U_v(gl_n)$ with respect to RTT relations. Read More

In this paper, we define a set which has a finite group action and is generated by a finite color set, a set which has a finite group action, and a subset of the set of non negative integers. we state its properties to apply one of solution of the following two problems, respectively. First, we calculate the generating function of the character of symmetric powers of permutation representation associated with a set which has a finite group action. Read More

Let $\rk$ be a local field of characteristic zero. Let $\pi$ be an irreducible admissible smooth representation of $\GL_{2n}(\rk)$. We prove that for all but countably many characters $\chi$ of $\GL_n(\rk)\times \GL_n(\rk)$, the space of $\chi$-equivariant (continuous in the archimedean case) linear functionals on $\pi$ is at most one dimensional. Read More

Given the pair of a dualizing $k$-variety and its functorially finite subcategory, we show that there exists a recollement consisting of their functor categories of finitely presented objects. We provide several applications for Auslander's formulas: The first one realizes a module category as a Serre quotient of a suitable functor category. The second one shows a close connection between Auslander-Bridger sequences and recollements. Read More