Representation Theory Of Groups Research Articles

Unified Framework for Generalized Statistics: Canonical Partition Function, Maximum Occupation Number, and Permutation Phase of Wave Function

Beyond Bose and Fermi statistics, there still exist various kinds of generalized quantum statistics. Two ways to approach generalized quantum statistics: (1) in quantum mechanics, generalize the permutation symmetry of the wave function and (2) in statistical mechanics, generalize the maximum occupation number of quantum statistics. The connection between these two approaches, however, is obscure. In this paper, we suggest a unified framework to describe various kinds of generalized quantum statistics. We first provide a general formula of canonical partition functions of ideal $N$-particle gases obeying various kinds of generalized quantum statistics. Then we reveal the connection between the permutation phase of the wave function and the maximum occupation number, through constructing a method to obtain the permutation phase and the maximum occupation number from the canonical partition function. In our scheme, the permutation phase of wave functions is generalized to a matrix phase, rather than a number. It is commonly accepted that different kinds of statistics are distinguished by the maximum number. We show that the maximum occupation number is not sufficient to distinguish different kinds of generalized quantum statistics. As examples, we discuss a series of generalized quantum statistics in the unified framework, giving the corresponding canonical partition functions, maximum occupation numbers, and the permutation phase of wave functions. Especially, we propose three new kinds of generalized quantum statistics which seem to be the missing pieces in the puzzle. The mathematical basis of the scheme are the mathematical theory of the invariant matrix, the Schur-Weyl duality, the symmetric function, and the representation theory of the permutation group and the unitary group. The result in this paper builds a bridge between the statistical mechanics and such mathematical theories.

Applying the symmetry groups to study the n body problem

We introduce an algebraic method to study local stability in the Newtonian n-body problem when certain symmetries are present. We use representation theory of groups to simplify the calculations of certain eigenvalue problems. The method should be applicable in many cases, we give two main examples here: the square central configurations with four equal masses, and the equilateral triangular configurations with three equal masses plus an additional mass of arbitrary size at the center. Using representation theory of finite groups, we explicitly found the eigenvalues of certain 8×8 Hessians in these examples, with only some simple calculations of traces. We also studied the local stability properties of corresponding relative equilibria in the four-body problems.

Proof for the electronic band crossing in sliding bilayer graphene

Dirac points are found to emerge due to the crossing of bands in the electronic structure of so-called sliding bilayer graphene. Group representation theory analysis corroborated with a tight-binding model for the ${p}_{z}$ orbitals is employed to demonstrate that the band crossings of energy dispersion curves at generic $\mathbf{k}$ points are guaranteed by the compatibility relations between the symmetries of eigenstates at the high-symmetry $\mathbf{k}$ points in the Brillouin zone. The Lifshitz transition picture and the transport properties of the systems are shown as consequences of the presence of Dirac points in governing the geometrical and topological properties of the Fermi energy surfaces.

On the Galois symmetries for the character table of an integral fusion category

In this paper, we show that integral fusion categories with rational structure constants admit a natural group of symmetries given by the Galois group of their character tables. Based on these symmetries, we generalize a well-known result of Burnside from representation theory of finite groups. More precisely, we show that any row corresponding to a non-invertible object in the character table of a weakly integral fusion category contains a zero entry.

Jordan decomposition for the Alperin–McKay conjecture

Späth showed that the Alperin–McKay conjecture in the representation theory of finite groups holds if the so-called inductive Alperin–McKay condition holds for all finite simple groups. In a previous article, we showed that the Bonnafé–Rouquier equivalence for blocks of finite groups of Lie type can be lifted to include automorphisms of groups of Lie type. We use our results to reduce the verification of the inductive condition for groups of Lie type to quasi-isolated blocks.

The Dipper-Du conjecture revisited

We consider vertices, a notion originating in local representation theory of finite groups, for the categoryO\mathcal {O}of a rational Cherednik algebra and prove the analogue of the Dipper-Du Conjecture for Hecke algebras of symmetric groups in that setting. As a corollary we obtain a new proof of the Dipper-Du Conjecture overC\mathbb {C}.

Role of Symmetry in Quantum Search via Continuous-Time Quantum Walk

For quantum search via the continuous-time quantum walk, the evolution of the whole system is usually limited in a small subspace. In this paper, we discuss how the symmetries of the graphs are related to the existence of such an invariant subspace, which also suggests a dimensionality reduction method based on group representation theory. We observe that in the one-dimensional subspace spanned by each desired basis state which assembles the identically evolving original basis states, we always get a trivial representation of the symmetry group. So, we could find the desired basis by exploiting the projection operator of the trivial representation. Besides being technical guidance in this type of problem, this discussion also suggests that all the symmetries are used up in the invariant subspace and the asymmetric part of the Hamiltonian is very important for the purpose of quantum search.

Self-equilibrium and super-stability of rhombic truncated regular tetrahedral and cubic tensegrities using symmetry-adapted force-density matrix method

Tensegrities consisting of axially loaded strings and bars have various technologically important applications, for which symmetric configurations are preferred as prototypes. In this study, we investigate the self-equilibrated states of rhombic truncated regular tetrahedral and cubic tensegrities by combining group representation theory and force-density matrix method. The force-density matrices of these tensegrities are analytically block-diagonalized using the irreducible representation matrices of tetrahedral and cubic groups. By making the symmetry-adapted force-density matrix satisfy the necessary number of nullity and positive semi-definiteness, we derive the analytical expressions for self-equilibrium and super-stability of rhombic truncated regular tetrahedral and cubic tensegrities. This work helps to design highly symmetric tensegrities for developing biomechanical models, mechanical metamaterials, and flexible robotics.

Quantum Spectrum Testing

In this work, we study the problem of testing properties of the spectrum of a mixed quantum state. Here one is given n copies of a mixed state $$\rho \in \mathbb {C}^{d\times d}$$ ρ ∈ C d × d and the goal is to distinguish (with high probability) whether $$\rho $$ ρ ’s spectrum satisfies some property $${\mathcal {P}}$$ P or whether it is at least $$\epsilon $$ ϵ -far in $$\ell _1$$ ℓ 1 -distance from satisfying $${\mathcal {P}}$$ P . This problem was promoted in the survey of Montanaro and de Wolf (A survey of quantum property testing. Technical report, arXiv:1310.2035 , 2013) under the name of testing unitarily invariant properties of mixed states. It is the natural quantum analogue of the classical problem of testing symmetric properties of probability distributions. Unlike property testing probability distributions—where one generally hopes for algorithms with sample complexity that is sublinear in the domain size—here the hope is for algorithms with subquadratic copy complexity in the dimension d. This is because the (frequently rediscovered) “empirical Young diagram (EYD) algorithm” (Alicki et al. in J Math Phys 29(5):1158–1162, 1988; Keyl and Werner in Phys Rev A 64(5):052311, 2011; Hayashi and Matsumoto in Phys Rev A 66(2):022311, 2002; Christandl and Mitchison in Commun. Math. Phys. 261(3):789–797, 2006) can estimate the spectrum of any mixed state up to $$\epsilon $$ ϵ -accuracy using only $${\widetilde{O}}(d^2/\epsilon ^2)$$ O ~ ( d 2 / ϵ 2 ) copies. In this work, we show that given a mixed state $$\rho \in \mathbb {C}^{d \times d}$$ ρ ∈ C d × d : $$\Theta (d/\epsilon ^2)$$ Θ ( d / ϵ 2 ) copies are necessary and sufficient to test whether $$\rho $$ ρ is the maximally mixed state, i.e., has spectrum $$(\frac{1}{d}, \dots , \frac{1}{d})$$ ( 1 d , ⋯ , 1 d ) . This can be viewed as the quantum analogue of Paninski (IEEE Trans Inf Theory 54(10):4750–4755, 2008) ’s sharp bounds for classical uniformity-testing. $$\Theta (r^2/\epsilon )$$ Θ ( r 2 / ϵ ) copies are necessary and sufficient to test with one-sided error whether $$\rho $$ ρ has rank r, i.e., has at most r nonzero eigenvalues. For two-sided error, a lower bound of $$\Omega (r/\epsilon )$$ Ω ( r / ϵ ) copies holds. $${\widetilde{\Theta }}(r^2)$$ Θ ~ ( r 2 ) copies are necessary and sufficient to distinguish whether $$\rho $$ ρ is maximally mixed on an r-dimensional or an $$(r+1)$$ ( r + 1 ) -dimensional subspace. More generally, for r vs. $$r+\Delta $$ r + Δ (with $$1 \le \Delta \le r$$ 1 ≤ Δ ≤ r ), $${\widetilde{\Theta }}(r^2/\Delta )$$ Θ ~ ( r 2 / Δ ) copies are necessary and sufficient. The EYD algorithm requires $$\Omega (d^2/\epsilon ^2)$$ Ω ( d 2 / ϵ 2 ) copies to estimate the spectrum of $$\rho $$ ρ up to $$\epsilon $$ ϵ -accuracy, nearly matching the known upper bound. In addition, we simplify part of the proof of the $${\widetilde{O}}(d^2/\epsilon ^2)$$ O ~ ( d 2 / ϵ 2 ) upper bound. Our techniques involve the asymptotic representation theory of the symmetric group; in particular Kerov’s algebra of polynomial functions on Young diagrams.

Affinoid Dixmier Modules and the Deformed Dixmier-Moeglin Equivalence

The affinoid enveloping algebra widehat {U({mathscr{L}})}_{K} of a free, finitely generated mathbb {Z}_{p}-Lie algebra {mathscr{L}} has proven to be useful within the representation theory of compact p-adic Lie groups, and we aim to further understand its algebraic structure. To this end, we define the notion of a Dixmier module over widehat {U({mathscr{L}})}_{K}, a generalisation of the Verma module, and we prove that when {mathscr{L}} is nilpotent, all primitive ideals of widehat {U({mathscr{L}})}_{K} can be described in terms of annihilator ideals of Dixmier modules. Using this, we take steps towards proving that this algebra satisfies a version of the classical Dixmier-Moeglin equivalence.

A unified FFT-based approach to maximum assignment problems related to transitive finite group actions

This paper studies the cross-correlation of two real-valued functions whose common domain is a set on which a finite group acts transitively. Such a group action generalizes, e.g., circular time shifts for discrete periodic time signals or spatial translations in the context of image processing. Cross-correlations can be reformulated as convolutions in group algebras and are thus transformable into the spectral domain via Fast Fourier Transforms. The resulting general discrete cross-correlation theorem will serve as the basis for a unified FFT-based approach to maximum assignment problems related to transitive finite group actions. Our main focus is on those assignment problems arising from the actions of symmetric groups on the cosets of Young subgroups. The “simplest” nontrivial representatives of the latter class of problems are the NP-hard symmetric and asymmetric maximum quadratic assignment problem. We present a systematic spectral approach in terms of the representation theory of symmetric groups to solve such assignment problems. This generalizes and algorithmically improves Kondor's spectral branch-and-bound approach (SODA, 2010) to the exact solution of the asymmetric maximum quadratic assignment problem.

Robinson–Schensted–Knuth correspondence in the representation theory of the general linear group over a non-archimedean local field

We construct new “standard modules” for the representations of general linear groups over a local non-archimedean field. The construction uses a modified Robinson–Schensted–Knuth correspondence for Zelevinsky’s multisegments. Typically, the new class categorifies the basis of Doubilet, Rota, and Stein (DRS) for matrix polynomial rings, indexed by bitableaux. Hence, our main result provides a link between the dual canonical basis (coming from quantum groups) and the DRS basis.

Folded and contracted solutions of coupled classical dynamical Yang–Baxter and reflection equations

In this paper we give a concrete recipe how to construct triples of algebra-valued meromorphic functions on a complex vector space a satisfying three coupled classical dynamical Yang–Baxter equations and an associated classical dynamical reflection equation. Such triples provide the local factors of a consistent system of first order differential operators on a, generalising asymptotic boundary Knizhnik–Zamolodchikov–Bernard (KZB) equations.The recipe involves folding and contracting a-invariant and θ-twisted symmetric classical dynamical r-matrices along an involutive automorphism θ. In case of the universal enveloping algebra of a simple Lie algebra g we determine the subclass of Schiffmann’s classical dynamical r-matrices which are a-invariant and θ-twisted.The paper starts with a section highlighting the connections between asymptotic (boundary) KZB equations, representation theory of semisimple Lie groups, and integrable quantum field theories.

Correction to: Homological multiplicities in representation theory of p-adic groups

Regrettably, the paper [1] had two incorrect lemmas. Nevertheless, the statements of the theorems in [1] are correct with no modification required. Specifically, [1, Lemma 6.4] does not hold for spherical spaces, and also the conclusion of [1, Lemma B.2] is not valid in the generality stated there. The effect of the error in [1, Lemma 6.4] on the proof of [1, Theorem 6.1] is significant. In § 1 one we provide a proof of [1, Theorem 6.1] that does not relay upon [1, Lemma 6.4]. The effect of the error in [1, Lemma B.2] is minor and we explain the nessecary modifications in §2.

A useful tool for constructing linear codes

We introduce and discuss an elementary tool from representation theory of finite groups for constructing linear codes invariant under a given permutation group G. The tool gives theoretical insight as well as a recipe for computations of generator matrices and weight distributions. In some interesting cases a classification of code vectors under the action of G can be obtained. As an explicit example a class of binary codes is studied extensively which is closely related to the class of binary codes associated to triangular graphs. A second explicit application is related to the action of the Mathieu simple group M24 on the set of octads giving many binary codes of length 759 with interesting properties. We also obtain new alternative proofs for several other theorems and construct several new codes invariant under various subgroups of the Conway simple group Co1.

Sinkhorn Algorithm for Quantum Permutation Groups

We introduce a Sinkhorn-type algorithm for producing quantum permutation matrices encoding symmetries of graphs. Our algorithm generates square matrices whose entries are orthogonal projections onto one-dimensional subspaces satisfying a set of linear relations. We use it for experiments on the representation theory of the quantum permutation group and quantum subgroups of it. We apply it to the question whether a given finite graph (without multiple edges) has quantum symmetries in the sense of Banica. In order to do so, we run our Sinkhorn algorithm and check whether or not the resulting projections commute. We discuss the produced data and some questions for future research arising from it.

Spontaneous symmetry breaking of dislocation core in SrTiO3

Dislocation endows intriguing properties in materials and attracted extensive interest, while its potential is far from being fully explored. Here, we use first-principles simulations to explore the property of an edge dislocation core in SrTiO3, an incipient ferroelectricity with a prototype perovskite structure. We discovered the spontaneous symmetry breaking in the dislocation, which is introduced by the atomic movement in the core and the Ti–O octahedron's distortion in the nearby unit cells. Next, we systematically investigate the coupling relationship between strain modes and ferroelectric atomic displacement modes based on group representation theory, and further proposed a model to quantificationally estimate the atomic configuration of SrTiO3 under complex strain conditions, which predicts the polarization near the dislocation decently.

Universal p‐ary designs

AbstractWe investigate ‐ary ‐designs which are simultaneously designs for all , which we call universal ‐ary designs. Null universal designs are well understood due to Gordon James via the representation theory of the symmetric group. We study non‐null designs and determine necessary and sufficient conditions on the coefficients for such a design to exist. This allows us to classify all universal designs, up to similarity.

Understanding star-fundamental algebras

Star-fundamental algebras are special finite dimensional algebras with involution ∗ * over an algebraically closed field of characteristic zero defined in terms of multialternating ∗ * -polynomials. We prove that the upper-block matrix algebras with involution introduced in Di Vincenzo and La Scala [J. Algebra 317 (2007), pp. 642–657] are star-fundamental. Moreover, any finite dimensional algebra with involution contains a subalgebra mapping homomorphically onto one of such algebras. We also give a characterization of star-fundamental algebras through the representation theory of the symmetric group.

Computing homological residue fields in algebra and topology

We determine the homological residue fields, in the sense of tensor-triangular geometry, in a series of concrete examples ranging from topological stable homotopy theory to modular representation theory of finite groups.