Representation Theory Of Groups Research Articles

Group Theory and Representation Theory

Group Theory and Representation Theory

Subrepresentations in the homology of finite covers of graphs

AbstractLet $p \;:\; Y \to X$ be a finite, regular cover of finite graphs with associated deck group $G$ , and consider the first homology $H_1(Y;\;{\mathbb{C}})$ of the cover as a $G$ -representation. The main contribution of this article is to broaden the correspondence and dictionary between the representation theory of the deck group $G$ on the one hand and topological properties of homology classes in $H_1(Y;\;{\mathbb{C}})$ on the other hand. We do so by studying certain subrepresentations in the $G$ -representation $H_1(Y;\;{\mathbb{C}})$ .The homology class of a lift of a primitive element in $\pi _1(X)$ spans an induced subrepresentation in $H_1(Y;\;{\mathbb{C}})$ , and we show that this property is never sufficient to characterize such homology classes if $G$ is Abelian. We study $H_1^{\textrm{comm}}(Y;\;{\mathbb{C}}) \leq H_1(Y;\;{\mathbb{C}})$ —the subrepresentation spanned by homology classes of lifts of commutators of primitive elements in $\pi _1(X)$ . Concretely, we prove that the span of such a homology class is isomorphic to the quotient of two induced representations. Furthermore, we construct examples of finite covers with $H_1^{\textrm{comm}}(Y;\;{\mathbb{C}}) \neq \ker\!(p_*)$ .

Character Theory and Categorification

Over a hundred years after the work of Frobenius and Schur, the sheer enormity of what is not known about the character theory of symmetric and alternating groups continues to surprise and awe the uninitiated. How does one decompose the tensor product of a pair of complex characters? Or the restriction of a complex character to a Sylow or wreath product subgroup? Can we understand the vanishing sets of complex characters? What about the asymptotic behaviour of complex characters? What are the dimensions of the modular characters? These questions have been hailed as some of the definitive open problems in representation theory and algebraic combinatorics, they have deep connections with Lie theory, group theory, local-global conjectures in representation theory of finite groups, symplectic geometry, complexity theory, statistical mechanics and quantum information theory. The overarching theme of this proposal is the use of hidden, richer representation theoretic structures arising in modular, local-global, and categorical representation theory in order to prove and disprove conjectures concerning characters of symmetric and alternating groups.

Poisson orders on large quantum groups

We develop a Poisson geometric framework for studying the representation theory of all contragredient quantum super groups at roots of unity. This is done in a uniform fashion by treating the larger class of quantum doubles of bozonizations of all distinguished pre-Nichols algebras [9] belonging to a one-parameter family; we call these algebras large quantum groups. We prove that each of these quantum algebras has a central Hopf subalgebra giving rise to a Poisson order in the sense of [13]. We describe explicitly the underlying Poisson algebraic groups and Poisson homogeneous spaces in terms of Borel subgroups of complex semisimple algebraic groups of adjoint type. The geometry of the Poisson algebraic groups and Poisson homogeneous spaces that are involved and its applications to the irreducible representations of the algebras Uq⊃Uq⩾⊃Uq+ are also described. Besides all (multiparameter) big quantum groups of De Concini–Kac–Procesi and big quantum super groups at roots of unity, our framework also contains the quantizations in characteristic 0 of the 34-dimensional Kac-Weisfeiler Lie algebras in characteristic 2 and the 10-dimensional Brown Lie algebras in characteristic 3. The previous approaches to the above problems relied on reductions to rank two cases and direct calculations of Poisson brackets, which is not possible in the super case since there are 13 kinds of additional Serre relations on up to 4 generators. We use a new approach that relies on perfect pairings between restricted and non-restricted integral forms.

Charge order, frustration relief, and spin-orbit coupling in U3O8

Research efforts on the description of the low-temperature magnetic order and electronic properties of ${\mathrm{U}}_{3}{\mathrm{O}}_{8}$ have been inconclusive so far. Reinterpreting neutron scattering results, we use group representation theory to show that the ground state presents collinear out-of-plane magnetic moments, with antiferromagnetic coupling both in-layer and between layers. Charge order relieves the initial geometric frustration, generating a slightly distorted honeycomb sublattice with N\'eel-type order. The precise knowledge of the characteristics of this magnetic ground state is then used to explain the fine features of the band gap. In this system, spin-orbit coupling (SOC) is of critical importance, as it strongly affects the electronic structure, narrowing the gap by $\ensuremath{\sim}38%$, compared to calculations neglecting SOC. The predicted electronic structure actually explains the salient features of recent optical absorption measurements, further demonstrating the excellent agreement between the calculated ground state properties and experiment.

Shifted Quantum Groups and Matter Multiplets in Supersymmetric Gauge Theories

The notion of shifted quantum groups has recently played an important role in algebraic geometry. This subtle modification of the original definition brings more flexibility in the representation theory of quantum groups. The first part of this paper presents new mathematical results for the shifted quantum toroidal $$\mathfrak {gl}(1)$$ and quantum affine $$\mathfrak {sl}(2)$$ algebras (resp. denoted $${\ddot{U}}_{q_1,q_2}^{\varvec{\mu }}(\mathfrak {gl}(1))$$ and $${\dot{U}}_q^{\varvec{\mu }}(\mathfrak {sl}(2))$$ ). It defines several new representations, including finite dimensional highest $$\ell $$ -weight representations for the toroidal algebra, and a vertex representation of $${\dot{U}}_q^{\varvec{\mu }}(\mathfrak {sl}(2))$$ acting on Hall–Littlewood polynomials. It also explores the relations between representations of $${\dot{U}}_q^{\varvec{\mu }}(\mathfrak {sl}(2))$$ and $${\ddot{U}}_{q_1,q_2}^{\varvec{\mu }}(\mathfrak {gl}(1))$$ in the limit $$q_1\rightarrow \infty $$ ( $$q_2$$ fixed), and present the construction of several new intertwiners. These results are used in the second part to construct BPS observables for 5d $${{\mathcal {N}}}=1$$ and 3d $${{\mathcal {N}}}=2$$ gauge theories. In particular, it is shown that 5d hypermultiplets and 3d chiral multiplets can be introduced in the algebraic engineering framework using shifted representations, and the Higgsing procedure is revisited from this perspective.

On the Notion of Metaplectic Barbasch–Vogan Duality

Abstract In analogy with the Barbasch–Vogan duality for real reductive linear groups, we introduce a duality notion useful for the representation theory of the real metaplectic groups. This is a map on the set of nilpotent orbits in a complex symplectic Lie algebra, whose range consists of the so-called metaplectic special nilpotent orbits. We relate this duality notion with the theory of primitive ideals and extend the notion of special unipotent representations to the real metaplectic groups. We also interpret the duality map in terms of double cells of Weyl group representations.

Polarization singularities in planar electromagnetic resonators with rotation and mirror symmetries

In this work, we apply the group representation theory to systematically study polarization singularities in the in-plane components of the electric fields supported by a planar electromagnetic (EM) resonator with generic rotation and reflection symmetries. We reveal the intrinsic connections between the symmetries and the topological features, i.e., the spatial configuration of the in-plane fields and the associated polarization singularities. The connections are substantiated by a simple relation that links the topological charges of the singularities and the symmetries of the resonator. To verify, a microwave planar resonator with the D8 group symmetries is designed and numerically simulated, which demonstrates the theoretical findings well. Our discussions can be applied to generic EM resonators working in a wide EM spectrum, such as circular antenna arrays, microring resonators, and photonic quasi-crystals, and provide a unique symmetry perspective on many effects in singular optics and topological photonics.

Symmetry analysis of charmonium two-body decay

In the light of SU(3) flavor symmetry, the effective interaction Hamiltonian in tensor form is obtained by virtue of group representation theory. The strong and electromagnetic breaking effects are treated as a spurion octet so that the flavor singlet principle can be utilized as the criterion to determine the form of effective Hamiltonian. Two body decays of both baryonic and mesonic final states are parameterized in the uniform scheme, based on which the relative phase between the strong and electromagnetic amplitudes is studied for various charmonium decay modes, including psi' and/or J/jpsi decay to octet baryon pair, decuplet baryon pair, decuplet-octet baryon final state, and pseudoscalar-pseudoscalar meson final state. In data analysis of samples taken in $e^+e^-$ collider, the details of experimental effects, such as energy spread and initial state radiative correction are taken into consideration in order to make full use of experimental information and acquire the accurate and delicate results.

Finite group p-modular representation theory of a finite group with a non-trivial central $$p'$$-subgroup

Finite group p-modular representation theory of a finite group with a non-trivial central $$p'$$-subgroup

Kähler-Ricci flow on rational homogeneous varieties

In this work, we study the Kähler-Ricci flow on rational homogeneous varieties exploring the interplay between projective algebraic geometry and representation theory which underlies the classical Borel-Weil theorem. By using elements of representation theory of semisimple Lie groups and Lie algebras, we give an explicit description for all solutions of the Kähler-Ricci flow with homogeneous initial condition. This description enables us to compute explicitly the maximal existence time for any solution starting at a homogeneous Kähler metric and obtain explicit upper and lower bounds for several geometric quantities along the flow, including curvatures, volume, diameter, and the first non-zero eigenvalue of the Laplacian. As an application of our main result, we investigate the relationship between certain numerical invariants associated with ample divisors and numerical invariants arising from solutions of the Kähler-Ricci flow in the homogeneous setting.

The Gini index in the representation theory of the general linear group

The Gini index in the representation theory of the general linear group

Factorial multiparameter Hecke von Neumann algebras and representations of groups acting on right-angled buildings

We obtain a complete characterisation of factorial multiparameter Hecke von Neumann algebras associated with right-angled Coxeter groups. Considering their ℓp-convolution algebra analogues, we exhibit an interesting parameter dependence, contrasting phenomena observed earlier for group Banach algebras. Translated to Iwahori-Hecke von Neumann algebras, these results allow us to draw conclusions on spherical representation theory of groups acting on right-angled buildings, which are in strong contrast to behaviour of spherical representations in the affine case. We also investigate certain graph product representations of right-angled Coxeter groups and note that our von Neumann algebraic structure results show that these are finite factor representations. Further classifying a suitable family of them up to unitary equivalence allows us to reveal high-dimensional Euclidean subspaces of the space of extremal characters of right-angled Coxeter groups.

Cross-Toeplitz operators on the Fock–Segal–Bargmann spaces and two-sided convolutions on the Heisenberg group

We introduce an extended class of cross-Toeplitz operators which act between Fock–Segal–Bargmann spaces with different weights. It is natural to consider these operators in the framework of representation theory of the Heisenberg group. Our main technique is representation of cross-Toeplitz by two-sided relative convolutions from the Heisenberg group. In turn, two-sided convolutions are reduced to usual (one-sided) convolutions on the Heisenberg group of the doubled dimensionality. This allows us to utilise the powerful group-representation technique of coherent states, co- and contra-variant transforms, twisted convolutions, symplectic Fourier transform, etc. We discuss connections of (cross-)Toeplitz operators with pseudo-differential operators, localisation operators in time–frequency analysis, and characterisation of kernels in terms of ladder operators. The paper is written in a detailed and reasonably self-contained manner to be suitable as an introduction into group-theoretical methods in phase space and time–frequency operator theory.

A Tour of p-Permutation Modules and Related Classes of Modules

This survey provides an overview of numerous results on p-permutation modules and the closely related classes of endo-trivial, endo-permutation and endo-p-permutation modules. These classes of modules play an important role in the representation theory of finite groups. For example, they are important building blocks used to understand and parametrise several kinds of categorical equivalences between blocks of finite group algebras. For this reason, there has been, since the late 1990’s, much interest in classifying such modules. The aim of this manuscript is to review classical results as well as all the major recent advances in the area. The first part of this survey serves as an introduction to the topic for non-experts in modular representation theory of finite groups, outlining proof ideas of the most important results at the foundations of the theory. Simultaneously, the connections between the aforementioned classes of modules are emphasised. In this respect, results, which are dispersed in the literature, are brought together, and emphasis is put on common properties and the role played by the p-permutation modules throughout the theory. Finally, in the last part of the manuscript, lifting results from positive characteristic to characteristic zero are collected and their proofs sketched.

On Automorphic Forms of Small Weight for Fake Projective Planes

On the projective plane there is a unique cubic root of the canonical bundle and this root is acyclic. On fake projective planes such root exists and is unique if there are no 3-torsion divisors (and usually exists, but not unique, otherwise). Earlier we conjectured that any such cubic root must be acyclic. In the present note we give two short proofs of this statement and show acyclicity of some other line bundles on the fake projective planes with at least $9$ automorphisms. Similarly to our earlier work we employ simple representation theory for non-abelian finite groups. The first proof is based on the observation that if some line bundle is non-linearizable with respect to a finite abelian group, then it should be linearized by a finite, \emph{non-abelian}, Heisenberg group. For the second proof, we also demonstrate vanishing of odd Betti numbers for a class of abelian covers, and use linearization of an auxiliary line bundle as well.

On certain McKay numbers of symmetric groups

For primes $$\ell $$ and nonnegative integers a, we study the partition functions $$\begin{aligned} p_\ell (a;n):= \#\{\lambda \vdash n : \textrm{ord}_\ell (H(\lambda ))=a\}, \end{aligned}$$ where $$H(\lambda )$$ denotes the product of hook lengths of a partition $$\lambda $$ . These partition values arise as the McKay numbers $$m_\ell (\textrm{ord}_\ell (n!) - a; S_n)$$ in the representation theory of the symmetric group. We determine the generating functions for $$p_\ell (a;n)$$ in terms of $$p_\ell (0;n)$$ and specializations of specific D’Arcais polynomials. For $$\ell = 2$$ and 3, we give an exact formula for the $$p_\ell (a;n)$$ and prove that these values are zero for almost all n. For larger primes $$\ell $$ , the $$p_\ell (a;n)$$ are positive for sufficiently large n. Despite this positivity, we prove that $$p_\ell (a;n)$$ is almost always divisible by m for any integer m. Furthermore, with these results we prove several Ramanujan-type congruences. These include the congruences $$\begin{aligned}p_\ell (a;\ell ^k n - \delta (a,\ell )) \equiv 0 \,\,(\textrm{mod}\,\,{\ell ^{k+1}}), \end{aligned}$$ for $$0<a< \ell $$ , where $$\ell = 5, 7, 11$$ and $$\delta (a,\ell ) := (\ell ^2 - 1)/24 + a\ell $$ , which answer a question of Ono.

Representation Varieties of Twisted Hopf Links

In this paper, we study the representation theory of the fundamental group of the complement of a Hopf link with n twists. A general framework is described to analyze the {{,textrm{SL},}}_r({mathbb {C}})-representation varieties of these twisted Hopf links as byproduct of a combinatorial problem and equivariant Hodge theory. As application, close formulas of their E-polynomials are provided for ranks 2 and 3, both for the representation and character varieties.

On the intersection density of the symmetric group acting on uniform subsets of small size

Given a finite transitive group G≤Sym(Ω), a subset F of G is intersecting if any two elements of F agree on some elements of Ω. The intersection density of G, denoted by ρ(G), is the maximum of the rational number |F|(|G||Ω|)−1 when F runs through all intersecting sets in G. In this paper, we prove that if G is the group Sym(n) or Alt(n) acting on the k-subsets of {1,2,3…,n}, for k∈{3,4,5}, then ρ(G)=1. Our proof relies on the representation theory of the symmetric group and the ratio bound.

Surjektifitas Pemetaan Eksponensial untuk Grup Lie Heisenberg yang Diperumum

The Heisenberg Lie Group is the most frequently used model for studying the representation theory of Lie groups. This Lie group is modular-noncompact and its Lie algebra is nilpotent. The elements of Heisenberg Lie group and algebra can be expressed in the form of matrices of size 3×3. Another specialty is also inherited by its three-dimensional Lie algebra and is called the Lie Heisenberg algebra. The Heisenberg Lie Group whose Lie Algebra is extended to the dimension 2n+1 is called the generalized Heisenberg Lie group and it is denoted by H whose Lie algebra is h_n. In this study, the surjectiveness of exponential mapping for H was studied with respect to h_n=⟨x ̅,y ̅,z ̅⟩ whose Lie bracket is given by [X_i,Y_i ]=Z. The purpose of this research is to prove the characterization of the Lie subgroup with respect to h_n. In this study, the results were obtained that if ⟨x ̅,y ̅ ⟩=:V⊆h_n a subspace and a set {e^(x_i ) e^(x_j ) ┤| x_i,x_j∈V }=:L⊆H then L=H and consequently Lie(L)≠V.