Mathematics - Representation Theory Publications (50)


Mathematics - Representation Theory Publications

In this paper, we study the $\mu$-ordinary locus of a Shimura variety with parahoric level structure. Under the Axioms in \cite{HR}, we show that $\mu$-ordinary locus is a union of some maximal Ekedahl-Kottwitz-Oort-Rapoport strata introduced in \cite{HR} and we give criteria on the density of the $\mu$-ordinary locus. Read More

In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus $2$ and $3$ over $\mathbb Q$ whose Jacobians have such maximal adelic Galois representations. Read More

In recent work, we obtained the symmetry algebra for a class of Dirac operators, containing in particular the Dirac-Dunkl operator for arbitrary root system. We now consider the three-dimensional case of the Dirac-Dunkl operator associated to the root system $A_2$, and the associated Dirac equation. The corresponding Weyl group is $\mathrm{S}_3$, the symmetric group on three elements. Read More

This paper studies intersections of principal blocks of a finite group with respect to different primes. We first define the block graph of a finite group G, whose vertices are the prime divisors of |G| and there is an edge between two vertices p \ne q if and only if the principal p- and q-blocks of G have a nontrivial common complex irreducible character of G. Then we determine the block graphs of finite simple groups, which turn out to be complete except those of J_1 and J_4. Read More

Let $n$ and $k$ be natural numbers such that $2^k < n$. We study the restriction to $\mathfrak{S}_{n-2^k}$ of odd-degree irreducible characters of the symmetric group $\mathfrak{S}_n$. This analysis completes the study begun in "Odd partitions in Young's lattice" by A. Read More

These notes are based on a series of lectures given by the author at the Centre Bernoulli (EPFL) in July 2016. They aim at illustrating the importance of the mod-$\ell$ cohomology of Deligne--Lusztig varieties in the modular representation theory of finite reductive groups. Read More

Extending the notion of maximal green sequences to an abelian category, we characterize the stability functions, as defined by Rudakov, that induce a maximal green sequence in an abelian length category. Furthermore, we use $\tau$-tilting theory to give a description of the wall and chamber structure of any finite dimensional algebra. Finally we introduce the notion of green paths in the wall and chamber structure of an algebra and show that green paths serve as geometrical generalization of maximal green sequences in this context. Read More

We use the adelic language to show that any homomorphism between Jacobians of modular curves arises from a linear combination of Hecke modular correspondences. The proof is based on a study of the actions of $\mathrm{GL}_2$ and Galois on the \'etale cohomology of the tower of modular curves. Read More

In this paper we continue to study the degrees of matrix coefficients of intertwining operators associated to reductive groups over $p$-adic local fields. Together with previous analysis of global normalizing factors we can control the analytic properties of global intertwining operators for a large class of reductive groups over number fields, in particular for inner forms of $GL(n)$ and $SL(n)$ and quasi-split classical groups. This has a direct application to the limit multiplicity problem for these groups. Read More

It is well-known that the Gauss decomposition of the generator matrix in the $R$-matrix presentation of the Yangian in type $A$ yields generators of its Drinfeld presentation. Defining relations between these generators are known in an explicit form thus providing an isomorphism between the presentations. It has been an open problem since the pioneering work of Drinfeld to extend this result to the remaining types. Read More

We classify Morita equivalence classes of indecomposable self-injective cellular algebras which have polynomial growth representation type, assuming that the base field has an odd characteristic. This assumption on the characteristic is for the cellularity to be a Morita invariant property. Read More

We show that any finite subgroup of automorphisms of the universal enveloping algebra of a semi-simple Lie algebra $\mathfrak{g}$ is isomorphic to a subgroup of Lie algebra automorphisms of $\mathfrak{g}$. Read More

We show that the category of graded modules over a finite-dimensional algebra admitting a triangular decomposition can be endowed with the structure of a highest weight category. When the algebra is self-injective, we show that this highest weight category has a tilting theory, in the sense of Ringel. As a consequence we are able to show that the degree zero part of the algebra is cellular, whereas, in all cases of interest, the algebra itself is not cellular. Read More

The relationship between bulk and boundary properties is one of the founding features of (Rational) Conformal Field Theory. Our goal in this paper is to explore the possibility of having an equivalent relationship in the context of lattice models. We focus on models based on the Temperley-Lieb algebra, and use the concept of braid translation, which is a natural way to close an open spin chain by adding an interaction between the first and last spins using braiding to bring them next to each other. Read More

For an arbitrary finite dimensional algebra $\Lambda$, we prove that any wide subcategory of $\mathsf{mod} \Lambda$ satisfying a certain finiteness condition is $\theta$-semistable for some stability condition $\theta$. More generally, we show that wide subcategories of $\mathsf{mod} \Lambda$ associated with two-term presilting complexes of $\Lambda$ are semistable. This provides a complement for Ingalls-Thomas-type bijections for finite dimensional algebras. Read More

We consider the decomposition into irreducible components of the external power $\Lambda^p(\mathbb{C}^m\otimes \mathbb{C}^n)$ regarded as a $\operatorname{GL}_m\times\operatorname{GL}_n$-module. The Young diagrams from each pair $(\lambda,\mu)$ which contributes to this decomposition turn out to be conjugate one to the other, i.e. Read More

In this joint introduction to an Asterisque volume, we give a short discussion of the historical developments in the study of nonlinear covering groups, touching on their structure theory, representation theory and the theory of automorphic forms. This serves as a historical motivation and sets the scene for the papers in the volume. Our discussion is necessarily subjective and will undoubtedly leave out the contributions of many authors, to whom we apologize in earnest. Read More

We study the dimension of the space of Whittaker functionals for depth zero representations of covering groups. In particular, we determine such dimensions for arbitrary Brylinski-Deligne coverings of the general linear group. The results in the paper are motivated by and compatible with the work of Howard and the second author, and earlier work by Blondel. Read More

We prove that the Grothendieck rings of category $\mathcal{C}^{(t)}_Q$ over quantum affine algebras $U_q'(\g^{(t)})$ $(t=1,2)$ associated to each Dynkin quiver $Q$ of finite type $A_{2n-1}$ (resp. $D_{n+1}$) is isomorphic to one of category $\mathcal{C}_{\mQ}$ over the Langlands dual $U_q'({^L}\g^{(2)})$ of $U_q'(\g^{(2)})$ associated to any twisted adapted class $[\mQ]$ of $A_{2n-1}$ (resp. $D_{n+1}$). Read More

A group K is said to be a B-group if every permutation group containing K as a regular subgroup is either imprimitive or 2-transitive. In the second edition of his influential textbook on finite groups, Burnside published a proof that cyclic groups of composite prime-power degree are B-groups. Ten years later in 1921 he published a proof that every abelian group of composite degree is a B-group. Read More

We consider the cohomological Hall algebra Y of a Lagrangian substack of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and its actions on the cohomology of quiver varieties. We conjecture that Y is equal, after a suitable extension of scalars, to the Yangian introduced by Maulik and Okounkov, and we construct an embedding of Y in the Yangian, intertwining the respective actions of both algebras on the cohomology of quiver varieties. Read More

We study the cohomological Hall algebra Y of a lagrangian substack of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and their actions on the cohomology of Nakajima quiver varieties. We prove that Y is pure and we compute its Poincare polynomials in terms of (nilpotent) Kac polynomials. We also provide a family of algebra generators. Read More

Let $A$ be the path algebra of a Dynkin quiver $Q$ over a finite field, and $\mathscr{P}$ be the category of projective $A$-modules. Denote by $C^1(\mathscr{P})$ the category of 1-cyclic complexes over $\mathscr{P}$, and $\tilde{\mathfrak{n}}^+$ the vector space spanned by the isomorphism classes of indecomposable and non-acyclic objects in $C^1(\mathscr{P})$. In this paper, we prove the existence of Hall polynomials in $C^1(\mathscr{P})$, and then establish a relationship between the Hall numbers for indecomposable objects therein and those for $A$-modules. Read More

We will construct a family of irreducible generic supercuspidal representations of the symplectic groups over non-archimedian local field $F$ of odd residual characteristic. The supercuspidal representations are compactly induced from irreducible representations of the hyperspecial compact subgroup which are inflated from irreducible representations of finite symplectic groups over the finite quotient ring of the integer ring of $F$ modulo high powers of the prime element. Read More

Let $A$ be the one point extension of an algebra $B$ by a projective $B$-module. We prove that the extension of a given support $\tau$-tilting $B$-module is a support $\tau$-tilting $A$-module; and, conversely, the restriction of a given support $\tau$-tilting $A$-module is a support $\tau$-tilting $B$-module. Moreover, we prove that there exists a full embedding of quivers between the corresponding poset of support $\tau$-tilting modules. Read More

Categorical equivalences between block algebras of finite groups - such as Morita and derived equivalences - are well-known to induce character bijections which commute with the Galois groups of field extensions. This is the motivation for attempting to realise known Morita and derived equivalences over non splitting fields. This article presents various result on the theme of descent. Read More

We study the exceptional theta correspondence for real groups obtained by restricting the minimal representation of the split exceptional group of the type E_n, to a split dual pair where one member is the exceptional group of the type G_2. We prove that the correspondence gives a bijection between spherical representations if n=6,7, and a slightly weaker statement if n=8. Read More

We give a complete picture of when the tensor product of an induced module and a Weyl module is a tilting module for the algebraic group $SL_2$ over an algebraically closed field of characteristic $p$. Whilst the result is recursive by nature, we give an explicit statement in terms of the $p$-adic expansions of the highest weight of each module. Read More

Recently de Thanhoffer de V\"olcsey and Van den Bergh classified the Euler forms on a free abelian group of rank 4 having the properties of the Euler form of a smooth projective surface. There are two types of solutions: one corresponding to $\mathbb{P}^1\times\mathbb{P}^1$ (and noncommutative quadrics), and an infinite family indexed by the natural numbers. For $m=0,1$ there are commutative and noncommutative surfaces having this Euler form, whilst for $m\geq 2$ there are no commutative surfaces. Read More

van den Ban and Kluit have found a serious error in a key lemma in the proof of the Whittaker Plancherel theorem. One purpose of this article is to fix the aspects of the proof of the theorem that are affected by the error, thereby giving the first full proof of the theorem. The other is to give an exposition of the structure of proof of the Whittaker Plancherel theorem and, thereby, of the Harish-Chandra Plancherel theorem. Read More

Counting formulae for general primary fields in free four dimensional conformal field theories of scalars, vectors and matrices are derived. These are specialised to count primaries which obey extremality conditions defined in terms of the dimensions and left or right spins (i.e. Read More

For an affine algebraic variety $X$ we study a category of modules that admit compatible actions of both the algebra of functions on $X$ and the Lie algebra of vector fields on $X$. In particular, for the case when $X$ is the sphere $\mathbb{S}^2$, we construct a set of simple modules that are finitely generated over $A$. In addition, we prove that the monoidal category that these modules generate is equivalent to the category of finite-dimensional rational $\mathrm{GL}_2$-modules. Read More

Let $\CX$ be a contravariantly finite subcategory of an abelian category $\CA$ with enough projective objects. Assume that $\CX$ contains all projective objects. In this paper, we investigate $\mmod \underline{\CX}$, the category of finitely presented functors on the stable category $\underline{\CX}$ of $\CX$. Read More

Given a Hopf subalgebra $R$ of a finite-dimensional Hopf algebra $H$, we continue a study of certain properties of the quotient $H$-module coalgebra $Q = H/R^+H$. We prove that $Q$ has a nonzero right $H$/$R$-integral if and only if $H$ is a Frobenius extension of $R$. We begin a study of a Mackey theory of quotient modules with labels for varying Hopf subalgebras: we show that when $H$ is a group algebra and $R$ is a separable subgroup algebra, their h-depth has an upper bound in terms of the combinatorial depth or the number of conjugate subgroups which intersect to equal the core of a subgroup. Read More

The main result of the paper is a construction of a five-parameter family of new bases in the algebra of symmetric functions. These bases are inhomogeneous and share many properties of systems of orthogonal polynomials on an interval of the real line. This means, in particular, that the algebra of symmetric functions is embedded into the algebra of continuous functions on a certain compact space Omega, and under this realization, our bases turn into orthogonal bases of weighted Hilbert spaces corresponding to certain probability measures on Omega. Read More

Irreducibility results for parabolic induction of representations of the general linear group over a local non-archimedean field can be formulated in terms of Kazhdan--Lusztig polynomials of type $A_n$. Spurred by these results, we hypothesize a simple identity for certain alternating sums of $2^n$ Kazhdan-Lusztig polynomials with respect to $S_{2n}$. Read More

In this paper we will derive an explicit description of the genuine projective representations of the symmetric group $S_n$ using Dirac cohomology and the branching graph for the irreducible genuine projective representations of $S_n$. In 2015 Ciubotaru and He, using the extended Dirac index, showed that the characters of the projective representations of $S_n$ are related to the characters of elliptic graded modules. We derived the branching graph using Dirac theory and combinatorics relating to the cohomology of Borel varieties $\mathcal{B}_e$ of $\mathfrak{g}$ and were able to use Dirac cohomology to construct an explicit model for the projective representations. Read More

The aim of this work is to study the representation dimension of cluster tilted algebras. We prove that the weak representation dimension of tame cluster tilted algebras is equal to three. We construct a generator module that reaches the weak representation dimension, unfortunately this module is not always a cogenerator. Read More

Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $G$ be a connected reductive algebraic group over $k$. Under some standard hypothesis on $G$, we give a direct approach to the finite $W$-algebra $U(\mathfrak g,e)$ associated to a nilpotent element $e \in \mathfrak g = \operatorname{Lie} G$. We prove a PBW theorem and deduce a number of consequences, then move on to define and study the $p$-centre of $U(\mathfrak g,e)$, which allows us to define reduced finite $W$-algebras $U_\eta(\mathfrak g,e)$ and we verify that they coincide with those previously appearing in the work of Premet. Read More

A QSIN group is a locally compact group $G$ whose group algebra $L^1(G)$ admits a quasi-central bounded approximate identity. Examples of QSIN groups include every amenable group and every discrete group. It is shown that if $G$ is a QSIN group, $H$ is a closed subgroup of $G$, and $\pi$ is a unitary representation of $H$, then $\pi$ is weakly contained in $(\mathrm{Ind}_H^G\pi)|_H$. Read More

A real reductive pair $(G,H)$ is called strongly spherical if the homogeneous space $(G\times H)/{\rm diag}(H)$ is real spherical. This geometric condition is equivalent to the representation theoretic property that ${\rm dim\,Hom}_H(\pi|_H,\tau)<\infty$ for all smooth admissible representations $\pi$ of $G$ and $\tau$ of $H$. In this paper we explicitly construct for all strongly spherical pairs $(G,H)$ intertwining operators in ${\rm Hom}_H(\pi|_H,\tau)$ for $\pi$ and $\tau$ spherical principal series representations of $G$ and $H$. Read More

In \cite{JS} Jensen and Su constructed 0-Schur algebras on double flag varieties. The construction leads to a presentation of 0-Schur algebras using quivers with relations and the quiver approach naturally gives rise to a new class of algebras. That is, the path algebras defined on the quivers of 0-Schur algebras with relations modified from the defining relations of 0-Schur algebras by a tuple of parameters $\ut$. Read More

We study ${\mathbb Z}$-graded thread $W^+$-modules $$V=\oplus_i V_i, \; \dim{V_i}=1, -\infty \le k< i < N\le +\infty, \; \dim{V_i}=0, \; {\rm \; otherwise},$$ over the positive part $W^+$ of the Witt (Virasoro) algebra $W$. There is well-known example of infinite-dimensional ($k=-\infty, N=\infty$) two-parametric family $V_{\lambda, \mu}$ of $W^+$-modules induced by the twisted $W$-action on tensor densities $P(x)x^{\mu}(dx)^{-\lambda}, \mu, \lambda \in {\mathbb K}, P(x) \in {\mathbb K}[t]$. Another family $C_{\alpha, \beta}$ of $W^+$-modules is defined by the action of two multiplicative generators $e_1, e_2$ of $W^+$ as $e_1f_i=\alpha f_{i+1}$ and $e_2f_j=\beta f_{j+2}$ for $i,j \in {\mathbb Z}$ and $\alpha, \beta$ are two arbitrary constants ($e_if_j=0, i \ge 3$). Read More

It follows from the work of Burban and Drozd arXiv:0905.1231 that for nodal curves $C$, the derived category of modules over the Auslander order $\mathcal{A}_C$ provides a categorical (smooth and proper) resolution of the category of perfect complexes $\mathrm{Perf}(C)$. On the A-side, it follows from the work of Haiden-Katzarkov-Kontsevich arXiv:1409. Read More

In this paper, we give some low-dimensional examples of local cocycle 3-Lie bialgebras and double construction 3-Lie bialgebras which were introduced in the study of the classical Yang-Baxter equation and Manin triples for 3-Lie algebras. We give an explicit and practical formula to compute the skew-symmetric solutions of the 3-Lie classical Yang-Baxter equation (CYBE). As an illustration, we obtain all skew-symmetric solutions of the 3-Lie CYBE in complex 3-Lie algebras of dimension 3 and 4 and then the induced local cocycle 3-Lie bialgebras. Read More

There is renewed interest in the coideal subalgebras used to form quantum symmetric pairs because of recent discoveries showing that they play a fundamental role in the representation theory of quantized enveloping algebras. However, there is still no general theory of finite-dimensional modules for these coideals. In this paper, we establish an important step in this direction: we show that every quantum symmetric pair coideal subalgebra admits a quantum Cartan subalgebra which is a polynomial ring that specializes to its classical counterpart. Read More

We show that the Lie algebra of polynomial vector fields on an irreducible affine variety X is simple if and only if X is a smooth variety. This completes the result of Jordan on the simplicity of the derivation algebra \cite{Jo}. Given proof is self-contained and does not depend on the results of Jordan. Read More

We give an elementary construction of a $p\geq 1$-singular Gelfand-Tsetlin $\mathfrak{gl}_n(\mathbb C)$-module in terms of local distributions. This is a generalization of the universal $1$-singular Gelfand-Tsetlin $\mathfrak{gl}_n(\mathbb C)$-module obtained in [FGR1]. We expect that the family of new Gelfand-Tsetlin modules that we obtained will lead to a classification of all irreducible $p>1$-singular Gelfand-Tsetlin modules. Read More

A randomisation of the Berele insertion algorithm is proposed, where the insertion of a letter to a symplectic Young tableau leads to a distribution over the set of symplectic Young tableaux. Berele's algorithm provides a bijection between words from an alphabet and a symplectic Young tableau along with a recording oscillating tableau. The randomised version of the algorithm is achieved by introducing a parameter $0 < q < 1$. Read More

The possible spectra of one-particle reduced density matrices that are compatible with a pure multipartite quantum system of finite dimension form a convex polytope. We introduce a new construction of inner- and outer-bounding polytopes that constrain the polytope for the entire quantum system. The outer bound is sharp. Read More