Representation Theoretic Approach Research Articles

Intrinsic data quality dimensions: expanding on Wand and Wang’s data quality model

PurposeStudies show that data quality (DQ) issues are extremely costly for companies. To address such issues, as a starting point, there is a need to understand what DQ is. In his context, the 1996 paper “Anchoring data quality dimensions ontological foundations” by Wand and Wang has been highly influential on the understanding of DQ. However, the present study demonstrates that some of the assumptions made in their paper can be challenged. On this basis, this study seeks to develop clearer definitions.Design/methodology/approachThe assumptions behind Wand and Wang’s DQ classification are discussed, on which basis three counter-propositions are formulated. These are investigated through a representation theoretical approach involving analyses of deficient mappings between real-world and information system states. On this basis, an intrinsic DQ classification is derived. A case study is conducted to investigate the value of the developed DQ classification.FindingsThe representation theoretical analysis and the case study support the three propositions. These give rise to the development of a DQ classification that includes three primary intrinsic DQ dimensions (accuracy, completeness and conciseness), which are associated with six primary value-level DQ deficiencies (inaccuracy, incorrectness, meaninglessness, incompleteness, absence and redundancy). The case study supports the value of extending Wand and Wang’s DQ classification with the three additional data deficiencies.Research limitations/implicationsBy improving the conceptual clarity of DQ, this study provides future research with an improved basis for studies and discussions of DQ.Originality/valueThe study advances the understanding of DQ by providing additional clarity.

A Q-Operator for Open Spin Chains II: Boundary Factorization

One of the features of Baxter’s Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang–Baxter equation associated to particular infinite-dimensional representations). To extend such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang–Baxter equation) associated to these representations. In the case of quantum affine sl2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {sl}_2$$\\end{document} and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras.

Towards a classification of multi-faced independence: A representation-theoretic approach

We attack the classification problem of multi-faced independences, the first non-trivial example being Voiculescu's bi-freeness. While the present paper does not achieve a complete classification, it formalizes the idea of lifting an operator on a pre-Hilbert space in a “universal” way to a larger product space, which is key for the construction of (old and new) examples. It will be shown how universal lifts can be used to construct very well-behaved (multi-faced) independences in general. Furthermore, we entirely classify universal lifts to the tensor product and to the free product of pre-Hilbert spaces. Our work brings to light surprising new examples of two-faced independences. Most noteworthy, for many known two-faced independences, we find that they admit continuous deformations within the class of two-faced independences, showing in particular that, in contrast with the single faced case, this class is infinite (and even uncountable).

A Representation-Theoretic Approach to qq-Characters

We raise the question of whether (a slightly generalized notion of) $qq$-characters can be constructed purely representation-theoretically. In the main example of the quantum toroidal $\mathfrak{gl}_1$ algebra, geometric engineering of adjoint matter produces an explicit vertex operator $\mathsf{RR}$ which computes certain $qq$-characters, namely Hirzebruch $\chi_y$-genera, completely analogously to how the R-matrix $\mathsf{R}$ computes $q$-characters. We give a geometric proof of the independence of preferred direction for the refined vertex in this and more general non-toric settings.

The special values of the standard 𝐿-functions for 𝐺𝑆𝑝_{2𝑛}×𝐺𝐿₁

We prove the expected algebraicity property for the critical values of character twists of the standard L L -function associated to vector-valued holomorphic Siegel cusp forms of archimedean type ( k 1 (k_1 , k 2 k_2 , …, k n ) k_n) , where k n ≥ n + 1 k_n \ge n+1 and all k i k_i are of the same parity. For the proof, we use an explicit integral representation to reduce to arithmetic properties of differential operators on vector-valued nearly holomorphic Siegel cusp forms. We establish these properties via a representation-theoretic approach.

Revisiting the Ramond sector of the N=1 superconformal minimal models

Key to the exact solubility of the unitary minimal models in two-dimensional conformal field theory is the organization of their Hilbert space into Verma modules, whereby all eigenstates of the Hamiltonian are obtained by the repeated action of Virasoro lowering operators onto a finite set of highest-weight states. The usual representation-theoretic approach to removing from all modules zero-norm descendant states generated in such a way is based on the assumption that those states form a nested sequence of Verma submodules built upon singular vectors, i.e., descendant highest-weight states. We show that this fundamental assumption breaks down for the Ramond-sector Verma module with highest weight $c/24$ in the even series of $\mathcal{N}\!=\!1$ superconformal minimal models with central charge $c$. To resolve this impasse, we conjecture, and prove at low orders, the existence of a nested sequence of linear-dependence relations that enables us to compute the character of the irreducible $c/24$ module. Based on this character formula, we argue that imposing modular invariance of the torus partition function requires the introduction of a non-null odd-parity Ramond-sector ground state. This symmetrization of the ground-state manifold allows us to uncover a set of conformally invariant boundary conditions not previously discussed and absent in the odd series of superconformal minimal models, and to derive for the first time a complete set of fusion rules for the even series of those models.

A Note on Critical p-adic L-functions

We study the adjunction property of the Jacquet-Emerton functor in certain neighborhoods of critical points in the eigencurve. As an application, we construct two-variable p-adic L-functions around critical points via Emerton’s representation theoretic approach.

A comparison of classes in the Johnson cokernels of the mapping class groups of surfaces

In [8], the first and the third authors introduced new classes in the Johnson cokernels of the mapping class groups of surfaces by a representation theoretic approach based on some previous results for the Johnson cokernels of the automorphism groups of free groups. On the other hand, in [15], Kawazumi and the second author introduced another type of classes by a topological consideration of self-intersections of curves on a surface.In this paper, we show that the classes found in [15] are contained in the classes found in [8] in a stable range. Furthermore, we prove that the anti-Morita obstructions [14m+1] for m≥1 obtained in [8] and a hook-type component [3,15] detected in [6] appear in their gap.

Homological classification of 4d mathcal{N} = 2 QFT. Rank-1 revisited

Argyres and co-workers started a program to classify all 4d mathcal{N} = 2 QFT by classifying Special Geometries with appropriate properties. They completed the program in rank-1. Rank-1 mathcal{N} = 2 QFT are equivalently classified by the Mordell-Weil groups of certain rational elliptic surfaces. The classification of 4d mathcal{N} = 2 QFT is also conjectured to be equivalent to the representation theoretic (RT) classification of all 2-Calabi-Yau categories with suitable properties. Since the RT approach smells to be much simpler than the Special-Geometric one, it is worthwhile to check this expectation by reproducing the rank-1 result from the RT side. This is the main purpose of the present paper. Along the route we clarify several issues and learn new details about the rank-1 SCFT. In particular, we relate the rank-1 classification to mirror symmetry for Fano surfaces. In the follow-up paper we apply the RT methods to higher rank 4d mathcal{N} = 2 SCFT.

Permutation invariant Gaussian matrix models

Permutation invariant Gaussian matrix models were recently developed for applications in computational linguistics. A 5-parameter family of models was solved. In this paper, we use a representation theoretic approach to solve the general 13-parameter Gaussian model, which can be viewed as a zero-dimensional quantum field theory. We express the two linear and eleven quadratic terms in the action in terms of representation theoretic parameters. These parameters are coefficients of simple quadratic expressions in terms of appropriate linear combinations of the matrix variables transforming in specific irreducible representations of the symmetric group SD where D is the size of the matrices. They allow the identification of constraints which ensure a convergent Gaussian measure and well-defined expectation values for polynomial functions of the random matrix at all orders. A graph-theoretic interpretation is known to allow the enumeration of permutation invariants of matrices at linear, quadratic and higher orders. We express the expectation values of all the quadratic graph-basis invariants and a selection of cubic and quartic invariants in terms of the representation theoretic parameters of the model.

A Representation Theory Perspective on Simultaneous Alignment and Classification.

Single particle cryo-electron microscopy (EM) is a method for determining the 3-D structure of macromolecules from many noisy 2-D projection images of individual macromolecules whose orientations and positions are random and unknown. The problem of orientation assignment for the images motivated work on general multireference alignment. The recently introduced non-unique games framework provides a representation theoretic approach to alignment over compact groups, and offers a convex relaxation which is formulated as semidefinite programs with certificates of global optimality under certain circumstances. One of the great opportunities in cryo-EM is studying heterogeneous samples, containing two or more distinct classes or conformations of molecules. Taking advantage of this opportunity presents an algorithmic challenge: determining both the class and orientation of each particle. We generalize multireference alignment to a problem of alignment and classification, and we propose to extend non-unique games to the problem of simultaneous alignment and classification with the goal of simultaneously classifying cryo-EM images and aligning them within their respective classes.

Asymptotic representation theory and the spectrum of a random geometric graph on a compact Lie group

Let $G$ be a compact Lie group, $N\geq 1$ and $L>0$. The random geometric graph on $G$ is the random graph $\Gamma _{\mathrm{geom} }(N,L)$ whose vertices are $N$ random points $g_1,\ldots ,g_N$ chosen under the Haar measure of $G$, and whose edges are the pairs $\{g_i,g_j\}$ with $d(g_i,g_j)\leq L$, $d$ being the distance associated to the standard Riemannian structure on $G$. In this paper, we describe the asymptotic behavior of the spectrum of the adjacency matrix of $\Gamma _{\mathrm{geom} }(N,L)$, when $N$ goes to infinity. 1. If $L$ is fixed and $N \to + \infty $ (Gaussian regime), then the largest eigenvalues of $\Gamma _{\mathrm{geom} }(N,L)$ converge after an appropriate renormalisation towards certain explicit linear combinations of values of Bessel functions. 2. If $L = O(N^{-\frac{1} {\dim G}})$ and $N \to +\infty $ (Poissonian regime), then the geometric graph $\Gamma _{\mathrm{geom} }(N,L)$ converges in the local Benjamini–Schramm sense, which implies the weak convergence in probability of the spectral measure of $\Gamma _{\mathrm{geom} }(N,L)$. In both situations, the representation theory of the group $G$ provides us with informations on the limit of the spectrum, and conversely, the computation of this limiting spectrum involves many classical tools from representation theory: Weyl’s character formula and the weight lattice in the Gaussian regime, and a degeneration of these objects in the Poissonian regime. The representation theoretic approach allows one to understand precisely how the degeneration from the Gaussian to the Poissonian regime occurs, and the article is written so as to highlight this degeneration phenomenon. In the Poissonian regime, this approach leads us to an algebraic conjecture on certain functionals of the irreducible representations of $G$.

Discriminating distinguishability

Particle distinguishability is a significant challenge for quantum technologies, in particular photonics where the Hong-Ou-Mandel (HOM) effect clearly demonstrates it is detrimental to quantum interference. We take a representation theoretic approach in first quantisation, separating particles' Hilbert spaces into degrees of freedom that we control and those we do not, yielding a quantum information inspired bipartite model where distinguishability can arise as correlation with an environment carried by the particles themselves. This makes clear that the HOM experiment is an instance of a (mixed) state discrimination protocol, which can be generalised to interferometers that discriminate unambiguously between ideal indistinguishable states and interesting distinguishable states, leading to bounds on the success probability of an arbitrary HOM generalisation for multiple particles and modes. After setting out the first quantised formalism in detail, we consider several scenarios and provide a combination of analytical and numerical results for up to nine photons in nine modes. Although the Quantum Fourier Transform features prominently, we see that it is suboptimal for discriminating completely distinguishable states.

Hyper-Algebras of Vector-Valued Modular Forms

We define graded hyper-algebras of vector-valued Siegel modular forms, which allow us to study tensor products of the latter. We also define vector-valued Hecke operators for Siegel modular forms at all places of ${\mathbb Q}$, acting on these hyper-algebras. These definitions bridge the classical and representation theoretic approach to Siegel modular forms. Combining both the product structure and the action of Hecke operators, we prove in the case of elliptic modular forms that all cusp forms of sufficiently large weight can be obtained from products involving only two fixed Eisenstein series. As a byproduct, we obtain inclusions of cuspidal automorphic representations into the tensor product of global principal series.

Quantum simulation of partially distinguishable boson sampling

Boson Sampling is the problem of sampling from the same output probability distribution as a collection of indistinguishable single photons input into a linear interferometer. It has been shown that, subject to certain computational complexity conjectures, in general the problem is difficult to solve classically, motivating optical experiments aimed at demonstrating quantum computational ``supremacy.'' There are a number of challenges faced by such experiments, including the generation of indistinguishable single photons. We provide a quantum circuit that simulates Boson Sampling with arbitrarily distinguishable particles. This makes clear how distinguishabililty leads to decoherence in the standard quantum circuit model, allowing insight to be gained. At the heart of the circuit is the quantum Schur transform, which follows from a representation theoretic approach to the physics of distinguishable particles in first quantization. The techniques are quite general and have application beyond Boson Sampling.

Refined cyclic sieving on words for the major index statistic

Reiner–Stanton–White (Reiner et al., 2004) defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action and a polynomial. A key example arises from the length generating function for minimal length coset representatives of a parabolic quotient of a finite Coxeter group. In type A, this result can be phrased in terms of the natural cyclic action on words of fixed content.There is a natural notion of refinement for many CSP’s. We formulate and prove a refinement, with respect to the major index statistic, of this CSP on words of fixed content by also fixing the cyclic descent type. The argument presented is completely different from Reiner–Stanton–White’s representation-theoretic approach. It is combinatorial and largely, though not entirely, bijective in a sense we make precise with a “universal” sieving statistic on words, flex.A building block of our argument involves cyclic sieving for shifted subset sums, which also appeared in Reiner–Stanton–White. We give an alternate, largely bijective proof of a refinement of this result by extending some ideas of Wagon–Wilf (Wagon and Wilf, 1994).

A representation-theoretic approach to the calculation of evolutionary distance in bacteria

In the context of bacteria and models of their evolution under genome rearrangement, we explore a novel application of group representation theory to the inference of evolutionary history. Our contribution is to show, in a very general maximum likelihood setting, how to use elementary matrix algebra to sidestep intractable combinatorial computations and convert the problem into one of eigenvalue estimation amenable to standard numerical approximation techniques.

The cohomological crepant resolution conjecture for the Hilbert–Chow morphisms

We prove that Ruan’s Cohomological Crepant Resolution Conjecture holds for the Hilbert–Chow morphisms. There are two main ideas in the proof. The first one is to use the representation theoretic approach proposed in [QW] which involves vertex operator techniques. The second is to prove certain universality structures about the $3$-pointed genus-$0$ extremal Gromov–Witten invariants of the Hilbert schemes by using the indexing techniques from [LiJ], the product formula from [Beh2] and the co-section localization from [KL1, KL2, LL]. We then reduce Ruan’s Conjecture from the case of an arbitrary surface to the case of smooth projective toric surfaces which has already been proved in [Che].

Irreducible representations of Bost–Connes systems

The classification problem of Bost–Connes systems was studied by Cornelissen and Marcolli partially, but still remains unsolved. In this paper, we give a representation-theoretic approach to this problem. We generalize the result of Laca and Raeburn, which is concerned with the primitive ideal space of the Bost–Connes system for \mathbb Q . As a consequence, the Bost–Connes C^* -algebra for a number field K has h^1_K -dimensional irreducible representations and does not have finite-dimensional irreducible representations for the other dimensions, where h^1_K is the narrow class number of K . In particular, the narrow class number is an invariant of Bost–Connes C^* -algebras.

Confluences of the Painlevé equations, Cherednik algebras and q-Askey scheme

In this paper we produce seven new algebras as confluences of the Cherednik algebra of type and we characterise their spherical-sub-algebras.The limit of the spherical sub-algebra of the Cherednik algebra of type is the monodromy manifold of the Painlevé VI equation (Oblomkov 2004 Int. Math. Res. Not. 2004 877–912). Here we prove that by considering the limits of the spherical sub-algebras of our new confluent algebras, one obtains the monodromy manifolds of all other Painlevé differential equations. Moreover, we introduce confluent versions of the Zhedanov algebra and prove that each of them (quotiented by their Casimir) is isomorphic to the corresponding spherical sub-algebra of our new confluent Cherednik algebras. We show that in the basic representation our confluent Zhedanov algebras act as symmetries of certain elements of the q-Askey scheme, thus setting a stepping stone towards the solution of the open problem of finding the corresponding quantum algebra for each element of the q-Askey scheme.These results establish a new link between the theory of the Painlevé equations and the theory of the q-Askey scheme making a step towards the construction of a representation theoretic approach for the Painlevé theory.