Mathematics - Representation Theory Publications (50)


Mathematics - Representation Theory Publications

Maximal green sequences are important objects in representation theory, cluster algebras, and string theory. It is an open problem to determine what lengths are achieved by the maximal green sequences of a quiver. We combine the combinatorics of surface triangulations and the basics of scattering diagrams to address this problem. Read More

In this short note, we will give an efficient functorial proof of the equivalence of various characterisations of purity in a locally finitely presented additive category $C$. The complications of the proofs for specific choices of $C$ (e.g. Read More

We relate the counting of honeycomb dimer configurations on the cylinder to the counting of certain vertices in Kirillov-Reshetikhin crystal graphs. We show that these dimer configurations yield the quantum Kostka numbers of the small quantum cohomology ring of the Grassmannian, i.e. Read More

In this note we make an attempt to compare a cohomological theory of Hilbert spaces of ground states in the ${\cal N}=(2,2)$ 2d Landau-Ginzburg theory in models describing link embeddings in ${\mathbb{R}}^3$ to Khovanov and Khovanov-Rozansky homologies. To confirm the equivalence we exploit the invariance of Hilbert spaces of ground states for interfaces with respect to homotopy. In this attempt to study solitons and instantons in the Landau-Giznburg theory we apply asymptotic analysis also known in the literature as exact WKB method, spectral networks method, or resurgence. Read More

In this paper, we study divisorial ideals of a Hibi ring which is a toric ring arising from a partially ordered set. We especially characterize the special class of divisorial ideals called conic using the associated partially ordered set. Using our observation of conic divisorial ideals, we also construct a module giving a non-commutative crepant resolution (= NCCR) of the Segre product of polynomial rings. Read More

Let $X$ be a connected scheme, smooth and separated over an algebraically closed field $k$ of characteristic $p\geq 0$, let $f:Y\rightarrow X$ be a smooth proper morphism and $x$ a geometric point on $X$. We prove that the tensor invariants of bounded length $\leq d$ of $\pi_1(X,x)$ acting on the \'etale cohomology groups $H^*(Y_x,F_\ell)$ are the reduction modulo-$\ell$ of those of $\pi_1(X,x)$ acting on $H^*(Y_x,Z_\ell)$ for $\ell$ greater than a constant depending only on $f:Y\rightarrow X$, $d$. We apply this result to show that the geometric variant with $F_\ell$-coefficients of the Grothendieck-Serre semisimplicity conjecture -- namely that $\pi_1(X,x)$ acts semisimply on $H^*(Y_x,F_\ell)$ for $\ell\gg 0$ -- is equivalent to the condition that the image of $\pi_1(X,x)$ acting on $H^*(Y_x,Q_\ell)$ is `almost maximal' (in a precise sense; what we call `almost hyperspecial') with respect to the group of $Q_\ell$-points of its Zariski closure. Read More

We present a study of incidence algebras that are piecewise hereditary, which we denominate Phia algebras. For the Phia algebras $K\Delta$ which are of sheaf type, we study the canonical sincere $K\Delta$-module $M$ and the one-point extension algebra $K\Delta[M]$. We show that if $KQ/I$ is a sincere, quasi-tilted of sheaf type, representation-infinite algebra, then the sincere $KQ/I$-modules are exceptional. Read More

Let $X$ be a set of $4$ generic points in $\mathbb{P}^2$ with homogeneous coordinate ring $R$. We classify indecomposable graded MCM modules over $R$ by reducing the classification to the Four Subspace problem solved by Nazarova and Gel$'$fand-Ponomarev, or equivalently to the representation theory of the $\widetilde{D}_4$ quiver. In particular, the $\mathbb{P}^1$ tubular family of regular representations corresponds to matrix factorizations of the pencil of conics going through $X$, with smooth conics $Q_{t}$ corresponding to rank one tubes and the singular conics $Q_0, Q_1, Q_{\infty}$ giving the remaining rank two tubes. Read More

To every weighting $\theta\in \mathbb{Z}^\ell$, we associate a graded cellular basis of the cyclotomic Hecke algebra. These bases are indexed by the usual (pairs of) standard tableaux but with a grading which is determined by the weighting and multicharge. We show that these distinct cellular bases give rise to non-isomorphic sets of cell-modules and hence provide explicit constructions of many new families of interesting representations for these Hecke algebras. Read More

This note announces results on the relations between the approach of Beilinson and Drinfeld to the geometric Langlands correspondence based on conformal field theory, the approach of Kapustin and Witten based on $N=4$ SYM, and the AGT-correspondence. The geometric Langlands correspondence is described as the Nekrasov-Shatashvili limit of a generalisation of the AGT-correspondence in the presence of surface operators. Following the approaches of Kapustin - Witten and Nekrasov - Witten we interpret some aspects of the resulting picture using an effective description in terms of two-dimensional sigma models having Hitchin's moduli spaces as target-manifold. Read More

We examine situations, where representations of a finite-dimensional $F$-algebra $A$ defined over a separable extension field $K/F$, have a unique minimal field of definition. Here the base field $F$ is assumed to be a $C_1$-field. In particular, $F$ could be a finite field or $k(t)$ or $k((t))$,where $k$ is algebraically closed. Read More

We are going to study some conditions on which the Radon transform and its dual are invertible. Two function spaces are introduced that the Radon transform on which is bijective linear operator. In this regards, a reconstruction formula is constructed. Read More

A generalization of the quotient integral formula is presented and some of its properties are investigated. Also the relations between two function spaces related to the spacial homogeneous spaces are derived by using general quotient integral formula. Finally our results are supported by some examples. Read More

We construct explicit families of right coideal subalgebras of quantum groups, where all irreducible representations are one-dimensional and which are maximal with this property. We have previously called such a right coideal subalgebra a Borel subalgebra. Conversely we can prove that any tringular Borel subalgebra fulfilling a certain non-degeneracy property is of the form we construct; this classification requires a key assertion about Weyl groups which we could only prove in type $A_n$. Read More

Let $ H $ be a compact subgroup of a locally compact group $G$. In this paper we define a convolution on $ M(G/H) $, the space of all complex bounded Radon measures on the homogeneous space G/H. Then we prove that the measure space $ M(G/H, *) $ is a non-unital Banach algebra that possesses an approximate identity. Read More

We complete the classification of conformal embeddings of a maximally reductive subalgebra $\mathfrak k$ into a simple Lie algebra $\mathfrak g$ at non-integrable non-critical levels $k$ by dealing with the case when $\mathfrak k$ has rank less than that of $\mathfrak g$. We describe some remarkable instances of decomposition of the vertex algebra $V_{k}(\mathfrak g)$ as a module for the vertex subalgebra generated by $\mathfrak k$. We discuss decompositions of conformal embeddings and constructions of new affine Howe dual pairs at negative levels. Read More

We prove that certain crystabelline deformation rings of two dimensional residual representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ are Cohen-Macaulay. As a consequence, this allows to improve Kisin's $R[1/p]=\mathbb{T}[1/p]$ theorem to an $R=\mathbb{T}$ theorem. Read More

We estimate the dimension of the variety of homomorphisms from $S_{2g}$ to any semisimple real algebraic group, where $S_{2g}$ is a surface group of genus $g \geq 2$. Read More

Let $\Lambda$ be a finite-dimensional algebra over an algebraically closed field $k$ and $G$ a finite group whose order is invertible in $\Lambda$ acting on $\Lambda$. We prove that $\Lambda$ satisfies the Gorenstein symmetry conjecture (resp. finitistic dimension conjecture, Auslander-Gorenstein conjecture, Nakayama conjecture) if and only if so does the skew group algebra $\Lambda G$. Read More

We study the structure of the $(\mathfrak{g},K)$-modules of the principal series representations of $SL(3,\mathbb{R})$ and $Sp(2,\mathbb{R})$ induced from minimal parabolic subgroups, in the case when the infinitesimal character is nonsingular. The composition factors of these modules are known by Kazhdan-Lusztig-Vogan conjecture. In this paper, we give complete descriptions of the socle filtrations of these modules. Read More

We show that for any dominant integral weight $\lambda$ of a complex semisimple Lie algebra $\mathfrak{g}$, the endomorphism algebra of any projective-injective module in the parabolic BGG category $\mathcal{O}_\lambda^{\mathfrak{p}}$ is a symmetric algebra (as conjectured by Khovanov) extending the results of Mazorchuk and Stroppel for the regular dominant integral weight. Moreover, all the indecomposable projective-injective modules in each block of $\mathcal{O}_\lambda^{\mathfrak{p}}$ have the same Loewy length and the endomorphism algebra of any projective-injective module in the block equips with a homogeneous non-degenerate symmetrizing form. Read More

In this paper we extend classical Titchmarsh theorems on the Fourier transform of H\"older-Lipschitz functions to the setting of compact homogeneous manifolds. As an application, we derive a Fourier multiplier theorem for $L^2$-H\"older-Lipschitz spaces on compact Lie groups. We also derive conditions and a characterisation for Dini-Lipschitz classes on compact homogeneous manifolds in terms of the behaviour of their Fourier coefficients. Read More

Let ${\bf G}$ be a simply connected semisimple algebraic group over $\overline{\mathbb{F}}_q$, the algebraically closure of $\mathbb{F}_q$ (the finite field with $q=p^e$ elements), and $F$ be the standard Frobenius map. Let ${\bf B}$ be an $F$-stable Borel subgroup and ${\bf T}$ an $F$-stable maximal torus contained in ${\bf B}$. Set $G_{q^r}={\bf G}^{F^r}$ and $B_{q^r}={\bf B}^{F^r}$ for any $r>0$. Read More

The cutoff phenomenon was recently confirmed for random walks on Ramanujan graphs by the first author and Peres (2016). In this work, we obtain analogs in higher dimensions, for random walk operators on any Ramanujan complex associated with a simple group $G$ over a local field $F$. We show that if $T$ is any $k$-regular $G$-equivariant operator on the Bruhat-Tits building with a simple combinatorial property (collision-free), the associated random walk on the $n$-vertex Ramanujan complex has cutoff at time $\log_k n$. Read More

Given an odd prime $p$, we investigate the position of simple modules in the stable Auslander-Reiten quiver of the principal block of a finite group with non-cyclic abelian Sylow $p$-subgroups. In particular, we prove a reduction to finite simple groups. In the case that the characteristic is $3$, we prove that simple modules in the principal block all lie at the end of their components Read More

For abelian length categories the borderline between finite and infinite representation type is discussed. Characterisations of finite representation type are extended to length categories of infinite height, and the minimal length categories of infinite height are described. Read More

We compute toric degenerations arising from the tropicalization of the full flag varieties $\mathcal{F}\ell_4$ and $\mathcal{F}\ell_5$ embedded in a product of Grassmannians. For $\mathcal{F}\ell_4$ and $\mathcal{F}\ell_5$ we compare toric degenerations arising from string polytopes and the FFLV polytope with those obtained from the tropicalization of the flag varieties. We also present a general procedure to find toric degenerations in the cases where the initial ideal arising from a cone of the tropicalization of a variety is not prime. Read More

In earlier work, the author introduced a method for constructing a Frobenius categorification of a cluster algebra with frozen variables, requiring as input a suitable candidate for the endomorphism algebra of a cluster-tilting object in such a category. In this paper, we construct such candidates in the case of acyclic cluster algebras with 'polarised' principal coefficients, and study the resulting Frobenius categorifications. Since cluster algebras with principal coefficients are obtained from those with polarised principal coefficients by setting half of the frozen variables to 1, our categories also indirectly model cluster algebras with principal coefficients, for which no Frobenius categorification can exist. Read More

We consider non-trivial irreducible tensor products of modular representations of a symmetric group $S_n$ in characteristic 2 for even $n$ completing the proof of a classification conjecture of Gow and Kleshchev about such products. Read More

We prove that a finite coprime linear group G in characteristic p>=(|G|-1)/2 has a regular orbit. This bound on p is best possible. We also give an application to blocks with abelian defect groups. Read More

We show that the blocks of category O for the Lie superalgebra q_n associated to half-integral weights carry the structure of a tensor product categorification for the infinite rank Kac-Moody algebra of type C. This allows us to prove two conjectures formulated by Cheng, Kwon and Lam. We then focus on the full subcategory consisting of finite-dimensional representations, which we show is a highest weight category with blocks that are Morita equivalent to certain generalized Khovanov arc algebras. Read More

Based on recent work of Kaletha, we aim to apply Hakim-Murnaghan theory to study distinguished regular supercuspidal representations of tamely ramified p-adic reductive groups, and investigate the relation between distinction and Langlands functoriality. Assuming p is sufficiently large, we obtain a necessary condition in general case, and sufficient and necessary conditions in depth-zero case, for regular supercuspidal representations to be distinguished. Read More

We study the hyperplane arrangements associated, via the minimal model programme, to symplectic quotient singularities. It is expected that this hyperplane arrangement equals the arrangement of "essential hyperplanes" coming from the representation theory of restricted rational Cherednik algebras. We show that this statement would follow from a rather innocuous conjecture about the number of torus fixed points on the universal Poisson deformation of a $\mathbb{Q}$-factorial terminalization. Read More

The results in this paper provide a comparison between the $K$-structure of unipotent representations and regular sections of bundles on nilpotent orbits. Precisely, let $\tilde{G}=\tilde{Spin}(a,b)$ with $a+b=2n$, the nonlinear double cover of $Spin(a,b)$, and let $\tilde{K}=Spin(a)\times Spin(b)$ be the maximal compact subgroup of $\tilde{G}$. We consider the nilpotent orbit $\mathcal O_c$ parametrized by $[3 \ 2^{2k} \ 1^{2n-4k-3}]$ with $k>0$. Read More

We investigate the rational cohomology algebra of various unipotent group schemes defined over an algebraically closed field $k$ of characteristic $p > 0$. For the $r$-th Fobenius kernel $U_{(r)}$ of many unipotent algebraic groups $U$, we construct an algebra $\overline{S}^*(U_{(r)})$ given by explicit generators and relations together with a map $\overline {\eta}_{U,r}: \overline{S}^*(U_{(r)}) \to H^*(U_{(r)},k)$ of graded $k$-algebras which is an isomorphism modulo nilpotents. To achieve our computations, we refine more qualitative and more generally applicable results of A. Read More

The famous pentagon identity for quantum dilogarithms has a generalization for every Dynkin quiver, due to Reineke. A more advanced generalization is associated with a pair of alternating Dynkin quivers, due to Keller. The description and proof of Keller's identities involves cluster algebras and cluster categories, and the statement of the identity is implicit. Read More

In this note we compute several invariants (e.g. algebraic K-theory, cyclic homology and topological Hochschild homology) of the noncommutative projective schemes associated to Koszul algebras of finite global dimension. Read More

Let $Z$ be a unimodular real spherical space which is assumed of wave-front type. Generalizing some ideas of Harish-Chandra, we show the existence of the constant term for smooth tempered functions on $Z$, while Harish-Chandra dealt with $K$-finite functions on the group (see also the work of Wallach, dealing with smooth functions on the group and using asymptotic expansions). By applying this theory, we get a characterization of the relative discrete series for $Z$. Read More

We describe the combinatorics of the multisemigroup with multiplicities for the tensor category of subbimodules of the identity bimodule, for an arbitrary non-uniform orientation of a finite cyclic quiver. Read More

We provide a framework connecting several well known theories related to the linearity of graded modules over graded algebras. In the first part, we pay a particular attention to the tensor products of graded bimodules over graded algebras. Finally, we provide a tool to evaluate the possible degrees of a module appearing in a graded projective resolution once the generating degrees for the first term of some particular projective resolution are known. Read More

We show that for the path algebra $A$ of an acyclic quiver, the singularity category of the derived category $\mathsf{D}^{\rm b}(\mathsf{mod}\,A)$ is triangle equivalent to the derived category of the functor category of $\underline{\mathsf{mod}}\,A$, that is, $\mathsf{D}_{\rm sg}(\mathsf{D}^{\rm b}(\mathsf{mod}\,A))\simeq \mathsf{D}^{\rm b}(\mathsf{mod}(\underline{\mathsf{mod}}\,A))$. This extends a result of Iyama-Oppermann for the path algebra $A$ of a Dynkin quiver. An important step is to establish a functor category analog of Happel's triangle equivalence for repetitive algebras. Read More

Local categories of a finite group $G$ are considered as generalized subquotient groups, and their representations contribute to group representations. To show this, we investigate the (stable) Grothendicek groups of the relevant local category algebras, as well as several natural maps among them. In this way, we may realize homology representations of $G$ via a generalized induction process. Read More

We give a sufficient condition for a Verdier quotient $\ct/\cs$ of a triangulated category $\ct$ by a thick subcategory $\cs$ to be realized inside of $\ct$ as an ideal quotient. As applications, we deduce three significant results by Buchweitz, Orlov and Amiot--Guo--Keller. Read More

In the present paper, a class of irreducible twisted Heisenberg-Virasoro modules by taking tensor products of irreducible twisted Heisenberg-Virasoro modules $\Omega(\lambda,a,b)$ defined in \cite{CG2} with irreducible twisted Heisenberg-Virasoro modules $\mathrm{Ind}(M)$ defined in \cite{CG} are obtained. We determine the necessary and sufficient conditions for two such irreducible tensor product modules to be isomorphic. Moreover, two classes of non-weight modules $\mathrm{Ind}_{\underline y,\lambda}(\C_{RS})$ and $\mathrm{Ind}_{\underline z,\lambda}(\C_{PQ})$ over the twisted Heisenberg-Virasoro algebra are constructed. Read More

In this article we study the perverse sheaf on Drinfeld's compactification obtained by applying the geometric Jacquet functor (alias nearby cycles) to a nondegenerate Whittaker sheaf. Namely, we describe its restrictions along the defect stratification in terms of the Langlands dual Lie algebra, in particular showing that this nearby cycles sheaf is tilting. We also describe the subquotients of the monodromy filtration using the Picard-Lefschetz oscillators introduced by S. Read More

A simple geometric algebra is shown to contain automatically the leptons and quarks of a generation of the Standard Model, and the electroweak and color gauge symmetries, without predicting extra particles and symmetries. The algebra is already naturally present in the Standard Model, in two instances of the Clifford algebra $\mathbb{C}\ell_6$, one being algebraically generated by the Dirac algebra and the weak symmetry generators, and the other by a complex three-dimensional representation of the color symmetry, which generates a Witt decomposition which leads to the decomposition of the algebra into ideals representing leptons and quarks. The two instances being isomorphic, the minimal approach is to identify them, resulting in the model proposed here. Read More

We prove that under mild assumptions the zero set of the discriminant ideal of a prime PI algebra R coincides with the zero set of the modified discriminant ideal of R. Furthermore, we prove that this set is precisely the complement of the Azumaya locus of R. This is used to classify the Azumaya loci of the mutiparameter quantized Weyl algebras at roots of unity. Read More

We investigate the generalized Kronecker algebra $\mathcal{K}_r = k\Gamma_r$ with $r \geq 3$ arrows. Given a regular component $\mathcal{C}$ of the Auslander-Reiten quiver of $\mathcal{K}_r$, we show that the quasi-rank $rk(\mathcal{C}) \in \mathbb{Z}_{\leq 1}$ can be described almost exactly as the distance $\mathcal{W}(\mathcal{C}) \in \mathbb{N}_0$ between two non-intersecting cones in $\mathcal{C}$, given by modules with the equal images and the equal kernels property; more precisley, we show that the two numbers are linked by the inequality \[ -\mathcal{W}(\mathcal{C}) \leq rk(\mathcal{C}) \leq - \mathcal{W}(\mathcal{C}) + 3.\] Utilizing covering theory, we construct for each $n \in \mathbb{N}_0$ a bijection $\varphi_n$ between the field $k$ and $\{ \mathcal{C} \mid \mathcal{C} \ \text{regular component}, \ \mathcal{W}(\mathcal{C}) = n \}$. Read More

In this paper, we introduce two new classes of representations of the framed braid groups. One is the homological representation constructed as the action of a mapping class group on a certain homology group. The other is the monodromy representation of the confluent KZ equation, which is a generalization of the KZ equation to have irregular singularities. Read More

Let $G_n=U(n)\ltimes {\mathbb H}_n $ be the semi-direct product of the unitary group acting by automorphisms on the Heisenberg group ${\mathbb H}_n$. According to Lipsman, the unitary dual $\widehat {G_n} $ of $G_n $ is in one to one correspondence with the space of admissible coadjoint orbits $\mathfrak g_n^\ddagger /G_n $ of $G_n $. In this paper, we determine the topology of the space $\mathfrak g_n^\ddagger /G_n $ and we show that the correspondence with $\widehat {G_n} $ is a homeomorphism. Read More