Rm Rep Research Articles

Classification of irreducible based modules over the complex representation ring of $ S_4 $

<abstract><p>The complex representation rings of finite groups are the fundamental class of fusion rings, categorified by the corresponding fusion categories of complex representations. The category of $ \mathbb{Z}_+ $-modules of finite rank over such a representation ring is also semisimple. In this paper, we classify the irreducible based modules of rank up to 5 over the complex representation ring $ r(S_4) $ of the symmetric group $ S_4 $. In total, 16 inequivalent irreducible based modules were obtained. In this process, the MATLAB program was used in order to obtain some representation matrices. Based on such a classification result, we further discuss the categorification of based modules over $ r(S_4) $ by module categories over the complex representation category $ {\rm Rep}(S_4) $ of $ S_4 $ arisen from projective representations of certain subgroups of $ S_4 $.</p></abstract>

Variational methods for characterizing matrix product operator symmetries

We present a method of extracting information about topological order from the ground state of a strongly correlated two-dimensional system represented by an infinite projected entangled pair state (iPEPS). As in Phys. Rev. B 101, 041108 (2020) and 102, 235112 (2020) we begin by determining symmetries of the iPEPS represented by infinite matrix product operators (iMPO) that map between the different iPEPS transfer matrix fixed points, to which we apply the fundamental theorem of MPS to find zipper tensors between products of iMPO's that encode fusion properties of the anyons. The zippers can be combined to extract topological $F$-symbols of the underlying fusion category, which unequivocally identify the topological order of the ground state. We bring the $F$-symbols to the canonical gauge, as well as compute the Drinfeld center of this unitary fusion category to extract the topological $S$ and $T$ matrices encoding mutual- and self-statistics of the emergent anyons. The algorithm is applied to Abelian toric code, double semion and twisted quantum double of $Z_3$, as well as to non-Abelian double Fibonacci, double Ising, and quantum double of $S_3$ and ${\rm Rep}(S_3)$ string net models.

A gap of the exponents of repetitions of Sturmian words

By measuring second occurring times of factors of an infinite word $x$, Bugeaud and Kim introduced a new quantity ${\rm rep}(x)$ called the exponent of repetition of $x$. It was proved by Bugeaud and Kim that $1 \leq {\rm rep}(x) \leq r_{\max} = \sqrt{10} - 3/2$ if $x$ is a Sturmian words. In this paper, we determine the value $r_1$ such that there is no Sturmian word $x$ satisfying $r_1 < {\rm rep}(x) < r_{\max}$ and $r_1$ is an accumulation point of the set of ${\rm rep}(x)$ when $x$ runs over the Sturmian words.

Unitary versus pseudounitary time evolution and statistical effects in the dynamical Sauter-Schwinger process

Dynamical Sauter-Schwinger mechanism of pair creation by a time-dependent electric field comprising of $N_{\rm rep}$ identical pulses is analyzed within the framework of the spinor and scalar quantum electrodynamics. For linearly polarized pulses, both theories predict that a single eigenmode of the matter wave follows the dynamics of a two-level system. This dynamics, however, is either governed by a Hermitian (for spin 1/2 particles) or pseudo-Hermitian (for spin 0 particles) Hamiltonian. Essentially, both theories lead to a Fraunhofer-type enhancement of the momentum distributions of created pairs. While in the fermionic case the enhancement is never perfect and it deteriorates with increasing the number of pulses in a train $N_{\rm rep}$, in the bosonic case we observe the opposite. More specifically, it is at exceptional points where the spectra of bosonic pairs scale exactly as $N_{\rm rep}^2$, and this scaling is even enhanced with increasing the number of pulses in a train.

Vanishing theorems for representation homology and the derived cotangent complex

Let $G$ be a reductive affine algebraic group defined over a field $k$ of characteristic zero. In this paper, we study the cotangent complex of the derived $G$-representation scheme $ {\rm DRep}_G(X)$ of a pointed connected topological space $X$. We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of $ {\rm DRep}_G(X) $ to the representation homology $ {\rm HR}_*(X,G) := \pi_*{\mathcal O}[{\rm DRep}_G(X)] $ to prove some vanishing theorems for groups and geometrically interesting spaces. Our examples include virtually free groups, Riemann surfaces, link complements in $ {\mathbb R}^3 $ and generalized lens spaces. In particular, for any f.g. virtually free group $ \Gamma $, we show that $\, {\rm HR}_i({\rm B}\Gamma, G) = 0 \,$ for all $ i > 0 $. For a closed Riemann surface $\Sigma_g $ of genus $ g \ge 1 $, we have $\, {\rm HR}_i(\Sigma_g, G) = 0 \,$ for all $ i > \dim G $. The sharp vanishing bounds for $ \Sigma_g $ depend actually on the genus: we conjecture that if $ g = 1 $, then $\, {\rm HR}_i(\Sigma_g, G) = 0 \,$ for $ i > {\rm rank}\,G $, and if $ g \ge 2 $, then $\, {\rm HR}_i(\Sigma_g, G) = 0 \,$ for $ i > \dim\,{\mathcal Z}(G) \,$, where $ {\mathcal Z}(G) $ is the center of $G$. We prove these bounds locally on the smooth locus of the representation scheme $ {\rm Rep}_G[\pi_1(\Sigma_g)]\,$ in the case of complex connected reductive groups. One important consequence of our results is the existence of a well-defined $K$-theoretic virtual fundamental class for $ {\rm DRep}_G(X)$ in the sense of Ciocan-Fontanine and Kapranov. We give a new `Tor formula' for this class in terms of functor homology.

Poincaré series of character varieties for nilpotent groups

Abstract For any compact, connected Lie group G and any finitely generated nilpotent group Γ, we determine the cohomology of the path component of the trivial representation of the group character variety (representation space) Rep ⁢ ( Γ , G ) 1 {{\rm Rep}(\Gamma,G)_{1}} , with coefficients in a field 𝔽 {{\mathbb{F}}} with characteristic 0 or relatively prime to the order of the Weyl group W. We give explicit formulas for the Poincaré series. In addition, we study G-equivariant stable decompositions of subspaces X ⁢ ( q , G ) {{\rm X}(q,G)} of the free monoid J ⁢ ( G ) {J(G)} generated by the Lie group G, obtained from representations of finitely generated free nilpotent groups.

Performance of the constrained minimization of the total energy in density functional approximations: the electron repulsion density and potential

In the constrained minimization method of Gidopoulos and Lathiotakis (J. Chem. Phys. 136, 224109), the Hartree exchange and correlation Kohn-Sham potential of a finite $N$-electron system is replaced by the electrostatic potential of an effective charge density that is everywhere positive and integrates to a charge of $N-1$ electrons. The optimal effective charge density (electron repulsion density, $\rho_{\rm rep}$) and the corresponding optimal effective potential (electron repulsion potential $v_{\rm rep}$) are obtained by minimizing the electronic total energy in any density functional approximation. The two constraints are sufficient to remove the self-interaction errors from $v_{\rm rep}$, correcting its asymptotic behavior at large distances from the system. In the present work, we describe, in complete detail, the constrained minimization method, including recent refinements. We also assess its performance in removing the self-interaction errors for three popular density functional approximations, namely LDA, PBE, and B3LYP, by comparing the obtained ionization energies to their experimental values for an extended set of molecules. We show that the results of the constrained minimizations are almost independent of the specific approximation with average percentage errors 15%, 14%, 13% for the above DFAs respectively. These errors are substantially smaller than the corresponding errors of the plain (unconstrained) Kohn-Sham calculations at 38%, 39%, and 27% respectively. Finally, we showed that this method correctly predicts negative values for the HOMO energies of several anions.

Difference-frequency combs in cold atom physics

Optical frequency combs provide the clockwork to relate optical frequencies to radio frequencies. Hence, combs allow to measure optical frequencies with respect to a radio frequency where the accuracy is limited only by the reference signal. In order to provide a stable link between the radio and optical frequencies, the two parameters of the frequency comb must be fixed: the carrier envelope offset frequency $f_{\rm ceo}$ and the pulse repetition-rate $f_{\rm rep}$. We have developed the first optical frequency comb based on difference frequency generation (DFG) that eliminates $f_{\rm ceo}$ by design - specifically tailored for applications in cold atom physics. An $f_{\rm ceo}$-free spectrum at 1550 nm is generated from a super continuum spanning more than an optical octave. Established amplification and frequency conversion techniques based on reliable telecom fiber technology allow generation of multiple wavelength outputs. In this paper we discuss the frequency comb design, characterization, and optical frequency measurement of Sr Rydberg states. The DFG technique allows for a compact and robust, passively $f_{\rm ceo}$ stable frequency comb significantly improving reliability in practical applications.

Twisting the $q$-deformations of compact semisimple Lie groups

Given a compact semisimple Lie group $G$ of rank $r$, and a parameter $q$ > 0, we can define new associativity morphisms in ${\rm Rep}(G_q)$ using a $3$-cocycle $\Phi$ on the dual of the center of $G$, thus getting a new tensor category ${\rm Rep}(G_q)^\Phi$. For a class of cocycles $\Phi$ we construct compact quantum groups $G^\tau_q$ with representation categories ${\rm Rep}(G_q)^\Phi$. The construction depends on the choice of an $r$-tuple $\tau$ of elements in the center of $G$. In the simplest case of $G=SU(2)$ and $\tau=-1$, our construction produces Woronowicz's quantum group $SU_{-q}(2)$ out of $SU_q(2)$. More generally, for $G=SU(n)$, we get quantum group realizations of the Kazhdan–Wenzl categories.

Reflection equation algebras, coideal subalgebras, and their centres

Reflection equation algebras and related $${U{_q}(\mathfrak g)}$$ -comodule algebras appear in various constructions of quantum homogeneous spaces and can be obtained via transmutation or equivalently via twisting by a cocycle. In this paper we investigate algebraic and representation theoretic properties of such so called ‘covariantized’ algebras, in particular concerning their centres, invariants, and characters. The locally finite part $${F_l(U{_q} (\mathfrak g))}$$ of $${U{_q}(\mathfrak g)}$$ with respect to the left adjoint action is a special example of a covariantized algebra. Generalising Noumi’s construction of quantum symmetric pairs we define a coideal subalgebra B f of $${U{_q}(\mathfrak g)}$$ for each character f of a covariantized algebra. We show that for any character f of $${F_l(U{_q}(\mathfrak g))}$$ the centre Z(B f ) canonically contains the representation ring $${{\rm Rep}(\mathfrak g)}$$ of the semisimple Lie algebra $${\mathfrak g}$$ . We show moreover that for $${\mathfrak g = {\mathfrak sl}_n(\mathbb C)}$$ such characters can be constructed from any invertible solution of the reflection equation and hence we obtain many new explicit realisations of $${{\rm Rep}({\mathfrak sl}_n(\mathbb C))}$$ inside $${U_q({\mathfrak sl}_n(\mathbb C))}$$ . As an example we discuss the solutions of the reflection equation corresponding to the Grassmannian manifold Gr(m,2m) of m-dimensional subspaces in $${{\mathbb C}^{2m}}$$ .

The Action of the Mapping Class Group on Maximal Representations

Let Γ g be the fundamental group of a closed oriented Riemann surface Σ g , g ≥ 2, and let G be a simple Lie group of Hermitian type. The Toledo invariant defines the subset of maximal representations Repmax(Γ g , G) in the representation variety Rep(Γ g , G). Repmax(Γ g , G) is a union of connected components with similar properties as Teichmuller space $$\mathcal{T}(\Sigma_g) = {\rm Rep}_{\max}(\Gamma_g, {\rm PSL}(2,\mathbb{R}))$$ . We prove that the mapping class group $$Mod_{\Sigma_g}$$ acts properly on Repmax(Γ g , G) when $$G= {\rm Sp}(2n,\mathbb{R})$$ , SU(n,n), SO*(4n), Spin(2,n).

On the variation of the Poisson structures of certain moduli spaces

Let \(\pi\) be the fundamental group of a compact connected orientable topological surface with finitely many boundary circles. Our aim is to construct Poisson algebras of continuous functions on certain open subspaces \({\cal R}(\pi,G)_{\vec w}\) of the space of representations of \(\pi\) in a Lie group G whose Lie algebra is supposed to be endowed with a nondegenerate invariant symmetric bilinear form. When G is compact and connected, the spaces \({\cal R}(\pi,G)_{\vec w}\) may be taken to be dense in the space of all representations. The spaces \({\cal R}(\pi,G)_{\vec w}\) contain spaces of representations \({\rm Rep}(\pi,G)_{\vec C}\) where the values of those generators of the fundamental group which correspond to the boundary circles are constrained to lie in fixed conjugacy classes and, the representation spaces \({\rm Rep}(\pi,G)_{\vec C}\) being endowed with universal Poisson algebras constructed elsewhere earlier, the restrictions from the spaces \({\cal R}(\pi,G)_{\vec w}\) to any of the spaces \({\rm Rep}(\pi,G)_{\vec C}\) are Poisson maps. Thus the Poisson algebra on each \({\cal R}(\pi,G)_{\vec w}\) gives a description of the variation of the universal Poisson structures on the spaces \({\rm Rep}(\pi,G)_{\vec C}\) as the chosen conjugacy classes move across G. We also give twisted versions of these results.