Finite-dimensional Representations Research Articles

Non virtually solvable subgroups of mapping class groups have non virtually solvable representations

Let \Sigma be a compact orientable surface of finite type with at least one boundary component. Let \Gamma \leq \mathrm{Mod}(\Sigma) be a non virtually solvable subgroup. We answer a question of Lubotzky by showing that there exists a finite dimensional homological representation \rho of \mathrm{Mod}(\Sigma) such that \rho(\Gamma) is not virtually solvable. We then apply results of Lubotzky and Meiri to show that for any random walk on such a group the probability of landing on a power, or on an element with topological entropy 0 both decrease exponentially in the length of the walk.

More on Wilson toroidal networks and torus blocks

We consider the Wilson line networks of the Chern-Simons 3d gravity theory with toroidal boundary conditions which calculate global conformal blocks of degenerate quasi-primary operators in torus 2d CFT. After general discussion that summarizes and further extends results known in the literature we explicitly obtain the one-point torus block and two-point torus blocks through particular matrix elements of toroidal Wilson network operators in irreducible finite-dimensional representations of sl(2, ℝ) algebra. The resulting expressions are given in two alternative forms using different ways to treat multiple tensor products of sl(2, ℝ) representations: (1) 3mj Wigner symbols and intertwiners of higher valence, (2) totally symmetric tensor products of the fundamental sl(2, ℝ) representation.

English

Let H denote a finite index subgroup of the modular group Γ and let ϱ denote a finite-dimensional complex representation of H. Let M(ϱ) denote the collection of holomorphic vector-valued modular forms for ϱ and let M(H) denote the collection of modular forms on H. Then M(ϱ) is a ℤ-graded M(H)-module. It has been proven that M(ϱ) may not be projective as a M(H)-module. We prove that M(ϱ) is Cohen-Macaulay as a M(H)-module. We also explain how to apply this result to prove that if M(H) is a polynomial ring, then M(ϱ) is a free M(H)-module of rank dim ϱ.

Etingof’s conjecture for quantized quiver varieties

We compute the number of finite dimensional irreducible modules for the algebras quantizing Nakajima quiver varieties. We get a lower bound for all quivers and vectors of framing. We provide an exact count in the case when the quiver is of finite type or is of affine type and the framing is the coordinate vector at the extending vertex. The latter case precisely covers Etingof’s conjecture on the number of finite dimensional irreducible representations for Symplectic reflection algebras associated to wreath-product groups. We use several different techniques, the two principal ones are categorical Kac–Moody actions and wall-crossing functors.

Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type

The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups H2, H3 and H4. Using a representation-orbit replacement, the definitions and properties of the indices are formulated for individual orbits of the examined groups. The indices of orders two and four of the tensor product of k orbits are determined. Using the branching rules for the non-crystallographic Coxeter groups, the embedding index is defined similarly to the Dynkin index of a representation. Moreover, since the definition of the indices can be applied to any orbit of non-crystallographic type, the algorithm allowing to search for the orbits of smaller radii contained within any considered one is presented for the Coxeter groups H2 and H3. The geometrical structures of nested polytopes are exemplified.

Jacobi–Trudi type formula for a class of irreducible representations of 𝔤𝔩(m|n)

We prove a determinantal type formula to compute the characters of a class of finite-dimensional irreducible representations of the general Lie super-algebra [Formula: see text] in terms of the characters of the symmetric powers of the fundamental representation and their duals. This formula, originally conjectured by van der Jeugt and Moens, can be regarded as a generalization of the well-known Jacobi–Trudi formula.

Degenerate Sklyanin algebras, Askey–Wilson polynomials and Heun operators

The q-difference equation, the shift and the contiguity relations of the Askey–Wilson polynomials are cast in the framework of the three and four-dimensional degenerate Sklyanin algebras and . It is shown that the q-para Racah polynomials corresponding to a non-conventional truncation of the Askey–Wilson polynomials form a basis for a finite-dimensional representation of . The first order Heun operators defined by a degree raising condition on polynomials are shown to form a five-dimensional vector space that encompasses . The most general quadratic expression in the five basis operators and such that it raises degrees by no more than one is identified with the Heun–Askey–Wilson operator.

CONNECTIVITY PROPERTIES OF MCKAY QUIVERS

AbstractWe present several results on the connectivity of McKay quivers of finite-dimensional complex representations of finite groups, with no restriction on the faithfulness or self-duality of the representations. We give examples of McKay quivers, as well as quivers that cannot arise as McKay quivers, and discuss a necessary and sufficient condition for two finite groups to share a connected McKay quiver.

A Poisson transform adapted to the Rumin complex

Let G be a semisimple Lie group with finite center, [Formula: see text] a maximal compact subgroup, and [Formula: see text] a parabolic subgroup. Following ideas of P. Y. Gaillard, one may use G-invariant differential forms on [Formula: see text] to construct G-equivariant Poisson transforms mapping differential forms on G/P to differential forms on G/K. Such invariant forms can be constructed using finite-dimensional representation theory. In this general setting, we first prove that the transforms that always produce harmonic forms are exactly those that descend from the de Rham complex on G/P to the associated Bernstein–Gelfand–Gelfand (or BGG) complex in a well defined sense. The main part of this paper is devoted to an explicit construction of such transforms with additional favorable properties in the case that [Formula: see text]. Thus, G/P is [Formula: see text] with its natural CR structure and the relevant BGG complex is the Rumin complex, while G/K is complex hyperbolic space of complex dimension [Formula: see text]. The construction is carried out both for complex and for real differential forms and the compatibility of the transforms with the natural operators that are available on their sources and targets are analyzed in detail.

On the representation theory of the infinite Temperley–Lieb algebra

We begin the study of the representation theory of the infinite Temperley–Lieb algebra. We fully classify its finite-dimensional representations, then introduce infinite link state representations and classify when they are irreducible or indecomposable. We also define a construction of projective indecomposable representations for TL[Formula: see text] that generalizes to give extensions of TL[Formula: see text] representations. Finally, we define a generalization of the spin chain representation and conjecture a generalization of Schur–Weyl duality.

Spin and flavor projection operators in theSU(2Nf)spin-flavor group

The quadratic Casimir operator of the special unitary $SU(N)$ group is used to construct projection operators, which can decompose any of its reducible finite-dimensional representation spaces contained in the tensor product of two and three adjoint spaces into irreducible components. Although the method is general enough, it is specialized to the $SU(2N_f) \to SU(2)\otimes SU(N_f)$ spin-flavor symmetry group, which emerges in the baryon sector of QCD in the large-$N_c$ limit, where $N_f$ and $N_c$ are the numbers of light quark flavors and color charges, respectively. The approach leads to the construction of spin and flavor projection operators that can be implemented in the analysis of the $1/N_c$ operator expansion. The use of projection operators allows one to successfully project out the desired components of a given operator and subtract off those that are not needed. Some explicit examples in $SU(2)$ and $SU(3)$ are detailed.

A semi-supervised learning framework for quantitative structure-activity regression modelling.

MotivationQuantitative structure–activity relationship (QSAR) methods are increasingly used in assisting the process of preclinical, small molecule drug discovery. Regression models are trained on data consisting of a finite-dimensional representation of molecular structures and their corresponding target-specific activities. These supervised learning models can then be used to predict the activity of previously unmeasured novel compounds.ResultsThis work provides methods that solve three problems in QSAR modelling: (i) a method for comparing the information content between finite-dimensional representations of molecular structures (fingerprints) with respect to the target of interest, (ii) a method that quantifies how the accuracy of the model prediction degrades as a function of the distance between the testing and training data and (iii) a method to adjust for screening dependent selection bias inherent in many training datasets. For example, in the most extreme cases, only compounds which pass an activity-dependent screening threshold are reported. A semi-supervised learning framework combines (ii) and (iii) and can make predictions, which take into account the similarity of the testing compounds to those in the training data and adjust for the reporting selection bias. We illustrate the three methods using publicly available structure–activity data for a large set of compounds reported by GlaxoSmithKline (the Tres Cantos AntiMalarial Set, TCAMS) to inhibit asexual in vitro Plasmodium falciparum growth.Availabilityand implementation https://github.com/owatson/PenalizedPrediction.Supplementary information Supplementary data are available at Bioinformatics online.

The Limit of Stationary Distributions of Many-Server Queues in the Halfin–Whitt Regime

We consider the so-called GI/GI/N queue, in which a stream of jobs with independent and identically distributed service times arrive as a renewal process to a common queue that is served by [Formula: see text] identical parallel servers in a first-come, first-served manner. We introduce a new representation for the state of the system and, under suitable conditions on the service and interarrival distributions, establish convergence of the corresponding sequence of centered and scaled stationary distributions in the so-called Halfin–Whitt asymptotic regime. In particular, this resolves an open question posed by Halfin and Whitt in 1981. We also characterize the limit as the stationary distribution of an infinite-dimensional, two-component Markov process that is the unique solution to a certain stochastic partial differential equation. Previous results were essentially restricted to exponential service distributions or service distributions with finite support, for which the corresponding limit process admits a reduced finite-dimensional Markovian representation. We develop a different approach to deal with the general case when the Markovian representation of the limit is truly infinite dimensional. This approach is more broadly applicable to a larger class of networks.

Quantum Grothendieck rings as quantum cluster algebras.

We define and construct a quantum Grothendieck ring for a certain monoidal subcategory of the category O of representations of the quantum loop algebra introduced by Hernandez–Jimbo. We use the cluster algebra structure of the Grothendieck ring of this category to define the quantum Grothendieck ring as a quantum cluster algebra. When the underlying simple Lie algebra is of type A, we prove that this quantum Grothendieck ring contains the quantum Grothendieck ring of the category of finite‐dimensional representations of the associated quantum affine algebra. In type A1, we identify remarkable relations in this quantum Grothendieck ring.

Bogomolov multipliers of some groups of order p 6

Let G be a finite group, V a faithful finite-dimensional representation of G over the complex field and be the corresponding invariant field. The Bogomolov multiplier of G is canonically isomorphic to the unramified cohomological group which has been used by Saltman in 1984 and Bogomolov in 1988 to provide counterexamples to the rationality problem of for finite p-groups P over In this paper, we investigate the vanishing property of where P denotes a p-group of order p 6 for

Codimension growth for weak polynomial identities, and non-integrality of the PI exponent

Let K be a field of characteristic zero. In this paper, we study the polynomial identities of representations of Lie algebras, also called weak identities, or identities of pairs. These identities are determined by pairs of the form (A, L) where A is an associative enveloping algebra for the Lie algebra L. Then a weak identity of (A, L) (or an identity for the representation of L associated to A) is an associative polynomial which vanishes when evaluated on elements of L⊆ A. One of the most influential results in the area of PI algebras was the theory developed by Kemer. A crucial role in it was played by the construction of the Grassmann envelope of an associative algebra and the close relation of the identities of the algebra and its Grassmann envelope. Here we consider varieties of pairs. We prove that under some restrictions one can develop a theory similar to that of Kemer's in the study of identities of representations of Lie algebras. As a consequence, we establish that in the case when K is algebraically closed, if a variety of pairs does not contain pairs corresponding to representations of sl2(K), and if the variety is generated by a pair where the associative algebra is PI then it is soluble. As another consequence of the methods used to obtain the above result, and applying ideas from papers by Giambruno and Zaicev, we were able to construct a pair (A, L) such that its PI exponent (if it exists) cannot be an integer. We recall that the PI exponent exists and is an integer whenever A is an associative (a theorem by Giambruno and Zaicev), or a finite-dimensional Lie algebra (Zaicev). Gordienko also proved that the PI exponent exists and is an integer for finite-dimensional representations of Lie algebras.

Twisted Burnside–Frobenius theorem and $R_infty$-property for lamplighter-type groups

We prove that the restricted wreath product ${\mathbb{Z}_n \mathbin{\mathrm{wr}} \mathbb{Z}^k}$ has the $R_\infty$-property, i. e. every its automorphism $\varphi$ has infinite Reidemeister number $R(\varphi)$, in exactly two cases: (1) for any $k$ and even $n$; (2) for odd $k$ and $n$ divisible by 3. In the remaining cases there are automorphisms with finite Reidemeister number, for which we prove the finite-dimensional twisted Burnside--Frobenius theorem (TBFT): $R(\varphi)$ is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations fixed by the action ${[\rho]\mapsto[\rho\circ\varphi]}$.

A constructive method for decomposing real representations

A constructive method for decomposing finite dimensional representations of semisimple real Lie algebras is developed. The method is illustrated by an example. We also discuss an implementation of the algorithm in the language of the computer algebra system GAP4.

Representation Theory of the Yangian of the Lie Superalgebra and the Quantum Loop Superalgebra

Based on the description of irreducible finite-dimensional representations of the Yangian of special linear Lie superalgebra and quantum loop special linear superalgebra we introduce the counterparts of the categories $$\mathfrak{O}$$ for representations of these quantum superalgebras. The theorem of equivalence of these representation categories is announced.

Two Examples Related to the Twisted Burnside–Frobenius Theory for Infinitely Generated Groups

TheTBFTf conjecture, which is a modification of a conjecture by Fel’shtyn and Hill, says that if the Reidemeister number R(𝜙) of an automorphism 𝜙 of a (countable discrete) group G is finite, then it coincides with the number of fixed points of the corresponding homeomorphism $$ \hat{\phi} $$ of $$ {\hat{G}}_f $$ (the part of the unitary dual formed by finite-dimensional representations). The study of this problem for residually finite groups has been the subject of some recent activity. We prove here that for infinitely generated residually finite groups there are positive and negative examples for this conjecture. It is detected that the finiteness properties of the number of fixed points of 𝜙 itself also differ from the finitely generated case.