Fundamental Weight Research Articles

Quasi-constant fundamental weights in terms of Levi Weyl groups

In previous joint work with J.-S. Koskivirta, we introduced the notion of “quasi-constant” character (of a maximal torus of a connected reductive group over a field); we showed that over an algebraically closed field it naturally unifies the notions “minuscule” and “co-minuscule”. In this note we characterize quasi-constant fundamental weights in terms of the Weyl group of the corresponding maximal Levi subgroup. Equivalently, purely in the language of root systems, the result characterizes special and co-special vertices of Dynkin diagrams in terms of the Weyl group of the corresponding maximal sub-root system.

Representations of Brauer category and categorification

We study representations of the locally unital and locally finite dimensional algebra B associated to the Brauer category B(δ0) with defining parameter δ0 over an algebraically closed field K with characteristic p≠2. The Grothendieck group K0(B-modΔ) will be used to categorify the integrable highest weight slK-module V(ϖδ0−12) with the fundamental weight ϖδ0−12 as its highest weight, where B-modΔ is a subcategory of B-lfdmod in which each object has a finite Δ-flag, and slK is either sl∞ or slˆp depending on whether p=0 or 2∤p. As g-modules, C⊗ZK0(B-modΔ) is isomorphic to V(ϖδ0−12), where g is a Lie subalgebra of slK (see Definition 4.2). When p=0, standard B-modules and projective covers of simple B-modules correspond to monomial basis and so-called quasi-canonical basis of V(ϖδ0−12), respectively.

Real‐time implementation of active shunt compensator with adaptive SRLMMN control technique for power quality improvement in the distribution system

In this work, the development of sign regressor least mean mixed norm (SRLMMN) control technique for a distribution static compensator (D-STATCOM) is presented. This control technique extracts fundamental weight components from the non-sinusoidal load currents and generates reference grid currents. D-STATCOM is developed for harmonics eradication, reactive power injection and load balancing and its performance is investigated in several operating modes. The performance of SRLMMN control is compared with recursive least square (RLS) and variable step least mean square (VSLMS) control techniques in terms of convergence, steady-state error, harmonics elimination, sample time and computation complexity. The major advantages of SRLMMN control technique, are fast convergence, less steady-state error, low total harmonic distortion (THD) and offers less computation complexity when compared with RLS and VSLMS. A laboratory-scale prototype of compensator is realised using a voltage source converter with the controller implemented in the dSPACE-MicroLabBox. Both MATLAB simulation and experimental results are included to demonstrate the performance of shunt compensator under steady-state and dynamic loadings. The developed SRLMMN control technique mitigates power quality problems and effectively suppresses THD observed in the grid current with reference to IEEE Standard 519-2014.

N-Extended Lorentzian Kac\u2013Moody algebras

We investigate a class of Kac–Moody algebras previously not considered. We refer to them as n-extended Lorentzian Kac–Moody algebras defined by their Dynkin diagrams through the connection of an A_n Dynkin diagram to the node corresponding to the affine root. The cases n=1 and n=2 correspond to the well-studied over- and very-extended Kac–Moody algebras, respectively, of which the particular examples of E_{10} and E_{11} play a prominent role in string and M-theory. We construct closed generic expressions for their associated roots, fundamental weights and Weyl vectors. We use these quantities to calculate specific constants from which the nodes can be determined that when deleted decompose the n-extended Lorentzian Kac–Moody algebras into simple Lie algebras and Lorentzian Kac–Moody algebra. The signature of these constants also serves to establish whether the algebras possess SO(1, 2) and/or SO(3)-principal subalgebras.

BD‐VSS individual weighting factor sign based adaptive filter control for power quality improvement in weak distribution grid

This study presents the control of distribution static compensator (DSTATCOM) using band-dependent variable step size (BD-VSS) individual weighting factor with sign error based adaptive filter for power quality improvement in a weak distribution grid. Here, the proposed algorithm is used for estimation of fundamental active weight components from the distorted load currents in order to generate the reference grid currents to mitigate the grid currents power quality issues. This new control algorithm is proposed for fast and accurate estimation of active weight components with low steady-state error, without dynamic oscillation at fast convergence speed. Moreover, the frequency locked loop-double self-tuning second-order generalised integrator based voltage filter is used to extract the positive sequence voltages of the distorted grid voltages for estimating the harmonics, and noise-free voltage unit templates. Therefore, the DSTATCOM with the proposed control algorithm is capable of mitigating the harmonic currents, providing reactive power compensation and operating at unity power factor in a weak distribution grid. Test results demonstrate the viability and robustness of the proposed control algorithm under balanced and unbalanced non-linear load conditions.

Inverses of Cartan matrices of Lie algebras and Lie superalgebras

The inverses of indecomposable Cartan matrices are computed for finite-dimensional Lie algebras and Lie superalgebras over fields of any characteristic, and for hyperbolic (almost affine) complex Lie (super)algebras. This enables one to express the fundamental weights in terms of simple roots corresponding to the Cartan matrix. We discovered three yet inexplicable new phenomena, of which (a) and (b) concern hyperbolic (almost affine) complex Lie (super)algebras: (a) several of the inverses of Cartan matrices have all their elements negative (not just non-positive, as they should be according to an a priori characterization due to Zhang Hechun); (b) the 0s only occur on the main diagonals of the inverses; (c) the determinants of inequivalent Cartan matrices of the simple Lie (super)algebra of any dimension may differ (in any characteristic).We interpret most of the results of Wei and Zou (2017) [31] as inverses of the Gram matrices of non-degenerate invariant symmetric bilinear forms on the (super)algebras considered, not of Cartan matrices, and give more adequate references. In particular, the inverses of Cartan matrices of simple Lie algebras were already published, starting with Dynkin's paper in 1952, see also Table 2 in Springer's book by Onishchik and Vinberg (1990).

Radiative decays and the SU(6) Lie algebra

We present research on radiative decays of vector [Formula: see text] to pseudoscalar [Formula: see text] particles ([Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] quark system) using broken symmetry techniques in the infinite-momentum frame and equal-time commutation relations and the [Formula: see text] Lie algebra, and conducted without ascribing any specific form to meson quark structure or intra-quark interactions. We utilize the physical electromagnetic current [Formula: see text] including its singlet [Formula: see text] term and focus on the [Formula: see text] 35-plet. We derive new relations involving the electromagnetic current (including its singlet — proportional to the [Formula: see text] singlet). Remarkably, we find that the electromagnetic current singlet plays an intrinsic role in understanding the physics of radiative decays and that the charged and neutral [Formula: see text] meson radiative decays into [Formula: see text] are due entirely to the singlet term in [Formula: see text]. Although there is insufficient radiative decay experimental data available at this time, parametrization of possible predicted values of [Formula: see text] is made. For conciseness and self-containment, we compute all [Formula: see text] Lie algebra simple roots, positive roots, weights and fundamental weights which allow the construction of all [Formula: see text] representations. We also derive all nonzero [Formula: see text] generator commutators and anticommutators — useful for further research on grand unified theories.

Root System Chip-Firing II: Central-Firing

Abstract Jim Propp recently proposed a labeled version of chip-firing on a line and conjectured that this process is confluent from some initial configurations. This was proved by Hopkins–McConville–Propp. We reinterpret Propp’s labeled chip-firing moves in terms of root systems; a “central-firing” move consists of replacing a weight $\lambda$ by $\lambda +\alpha$ for any positive root $\alpha$ that is orthogonal to $\lambda$. We show that central-firing is always confluent from any initial weight after modding out by the Weyl group, giving a generalization of unlabeled chip-firing on a line to other types. For simply-laced root systems we describe this unlabeled chip-firing as a number game on the Dynkin diagram. We also offer a conjectural classification of when central-firing is confluent from the origin or a fundamental weight.

Accelerometer-based physical activity levels, fundamental movement skills and weight status in British preschool children from a deprived area

Preschool children are recommended to participate in a minimum of 180-min physical activity (PA) per day to enhance their development and overall health. Low PA and increased obesity are thought to be linked to low mastery of fundamental movement skills (FMS) in preschool children. This study sought to investigate whether FMS influences PA levels and weight status in preschool children, in an area of low socioeconomic status. Secondary aims of this study were to determine whether gender or day of the week affected the primary outcomes. One hundred eighty-five preschool children aged 3–4 years old, participated in the study. FMS proficiency was determined using the Test of Gross Motor Development-2. PA was determined using triaxial accelerometry over a 4-day period. None of the samples met the recommended 180 min of PA. There were no significant differences in PA or weight status between preschool children with high, medium or low FMS mastery (P < 0.05). There were also no significant correlations between overall FMS and moderate to vigorous PA during the week or weekend days.Conclusion: Girls scored significantly greater at the hop, leap, and skip (locomotor skills) and the boys significantly higher at the kick (object control) (P < 0.05). There were no significant differences in PA or weight status between preschool children with high, medium, or low FMS mastery, possibly because FMS mastery had not developed to a high enough level to affect PA and FMS are considered independent of physical fitness and physical features, such as weight and height.What is Known:•FMS are commonly developed in early childhood, providing the building blocks for future motor skills, good health and lifelong PA.•No study to date has measured FMS, PA levels and weight status in preschool children, to determine whether FMS competency influences PA levels and weight status in preschool children, in an area of low SES.What is New:•FMS competency did not appear to influence the level of PA or weight status in this sample of UK preschool children from a low SES area.•PA and FMS may not be fully established and consequently not strongly linked at the ages of 3–4 years, therefore, the preschool years could be influential in providing a window to maximise input of good/optimal development of motor competence before the proficiency barrier sets in and we need remedial intervention.

ON TENSORING WITH THE STEINBERG REPRESENTATION

Let G be a simple, simply connected algebraic group over an algebraically closed field of prime characteristic p > 0. Recent work of Kildetoft and Nakano and of Sobaje has shown close connections between two long-standing conjectures of Donkin: one on tilting modules and the lifting of projective modules for Frobenius kernels of G and another on the existence of certain filtrations of G-modules. A key question related to these conjectures is whether the tensor product of the rth Steinberg module with a simple module with prth restricted highest weight admits a good filtration. In this paper we verify this statement (i) when p ≥ 2h − 4 (h is the Coxeter number), (ii) for all rank two groups, (iii) for p ≥ 3 when the simple module corresponds to a fundamental weight and (iv) for a number of cases when the rank is less than or equal to five.

On truncated Weyl modules

We study structural properties of truncated Weyl modules. A truncated Weyl module is a local Weyl module for , where is a finite-dimensional simple Lie algebra. It has been conjectured that, if N is sufficiently small with respect to λ, the truncated Weyl module is isomorphic to a fusion product of certain irreducible modules. Our main result proves this conjecture when λ is a multiple of certain fundamental weights, including all minuscule ones for simply laced . We also take a further step towards proving the conjecture for all multiples of fundamental weights by proving that the corresponding truncated Weyl module is isomorphic to a natural quotient of a fusion product of Kirillov–Reshetikhin modules. One important part of the proof of the main result shows that any truncated Weyl module is isomorphic to a Chari–Venkatesh module and explicitly describes the corresponding family of partitions. This leads to further results in the case that related to Demazure flags and chains of inclusions of truncated Weyl modules.

Explicit construction of the Voronoi and Delaunay cells of W(An) and W(Dn) lattices and their facets.

Voronoi and Delaunay (Delone) cells of the root and weight lattices of the Coxeter-Weyl groups W(An) and W(Dn) are constructed. The face-centred cubic (f.c.c.) and body-centred cubic (b.c.c.) lattices are obtained in this context. Basic definitions are introduced such as parallelotope, fundamental simplex, contact polytope, root polytope, Voronoi cell, Delone cell, n-simplex, n-octahedron (cross polytope), n-cube and n-hemicube and their volumes are calculated. The Voronoi cell of the root lattice is constructed as the dual of the root polytope which turns out to be the union of Delone cells. It is shown that the Delone cells centred at the origin of the root lattice An are the polytopes of the fundamental weights ω1, ω2,…, ωn and the Delone cells of the root lattice Dn are the polytopes obtained from the weights ω1, ωn-1 and ωn. A simple mechanism explains the tessellation of the root lattice by Delone cells. It is proved that the (n-1)-facet of the Voronoi cell of the root lattice An is an (n-1)-dimensional rhombohedron and similarly the (n-1)-facet of the Voronoi cell of the root lattice Dn is a dipyramid with a base of an (n-2)-cube. The volume of the Voronoi cell is calculated via its (n-1)-facet which in turn can be obtained from the fundamental simplex. Tessellations of the root lattice with the Voronoi and Delone cells are explained by giving examples from lower dimensions. Similar considerations are also worked out for the weight lattices An* and Dn*. It is pointed out that the projection of the higher-dimensional root and weight lattices on the Coxeter plane leads to the h-fold aperiodic tiling, where h is the Coxeter number of the Coxeter-Weyl group. Tiles of the Coxeter plane can be obtained by projection of the two-dimensional faces of the Voronoi or Delone cells. Examples are given such as the Penrose-like fivefold symmetric tessellation by the A4 root lattice and the eightfold symmetric tessellation by the D5 root lattice.

Weight multiplicity formulas for bivariate representations of classical Lie algebras

A bivariate representation of a complex simple Lie algebra is an irreducible representation having highest weight a combination of the first two fundamental weights. For a complex classical Lie algebra, we establish an expression for the weight multiplicities of bivariate representations.

Action of Weyl group on zero-weight space

For any simple complex Lie group, we classify irreducible finite-dimensional representations ρ for which the longest element w0 of the Weyl group acts non-trivially on the zero-weight space. Among irreducible representations that have zero among their weights, w0 acts by ±Id if and only if the highest weight of ρ is a multiple of a fundamental weight, with a coefficient less than a bound that depends on the group and on the fundamental weight.

Power quality improvement using DSTATCOM with affine projection algorithm

A distribution static compensator (DSTATCOM) is implemented at distribution level for overcoming several power quality problems commonly encountered today. Some of these problems considered here deal with issues of harmonics, unbalancing of loads, meeting reactive power demand of the load for improving power factor to unity and regulating the voltage. A new technique is proposed based on affine projection control algorithm and its design and implementation on a shunt compensator are discussed. The proposed control approach estimates the fundamental weight values of load currents. These weights are utilised to compute the magnitude and phase of reference grid currents and consequently gating pulses for three-phase voltage source converter (VSC). The control approach is based on the convergence of the weighted values and is independent of the property of the input signal. This control approach has characteristic of updating weights on the basis of multiple, delayed input signal vectors. Fast convergence of active and reactive weighted values is added advantage of the proposed control approach with the above-mentioned characteristics. A working prototype of the DSTATCOM is implemented using three-phase VSC and digital signal processor (DSP) (dSPACE 1104). The affine projection control approach is developed in MATLAB/SIMULINK and verified on a prototype using DSP.

A bilateral relationship between stable profiles of pinned–pinned bistable shallow arches

Arch-profiles in the two force-free stable equilibrium states of shallow bistable arches are related to each other. We derive a two-way, i.e., bilateral, relationship between stress-free initial profile and stressed toggled profile so that pinned–pinned bistable arches of arbitrary profiles can be efficiently analyzed and designed. The derivation relies on representing the initial and toggled profiles with two sets of mode weights corresponding to the buckling mode shapes of a pinned–pinned column. Furthermore, we prove that the fundamental mode weights should be non-zero for an arch to be bistable. The following corollaries arise from the aforementioned relation: (1) symmetry in initial and toggled profiles remains unchanged; (2) all the mode weights other than the fundamental mode weight have the same sign in both stable states; (3) magnitudes of corrugations in stable force-free arch-profiles are approximately equal. Derivations and proofs of the principal relationship and its corollaries as well as examples of analysis and design of bistable arches of arbitrary arch-profiles are presented in the paper.

Annotazioni sul testo del Salmo 116

This article deals with many philological and textual issues of fundamental weight for an accurate analysis of Psalm 116. The study isolates linguistic evidence that substantiate a post-exilic date of the text. Furthermore, Hebrew manuscripts attest to its unitary composition.

Generating functions for characters and weight multiplicities of irreducible sl(4)-modules

Generating functions for the characters of the irreducible representations of simple Lie algebras are rational functions where both the numerator and denominator can be expressed as polynomials in the characters corresponding to the fundamental weights. They encode much information on the representation theory of the algebra, but their explicit expressions are in general very complicated. In fact, it seems that rank three is the highest rank tractable. In this paper, we use a method based on the quantum Calogero-Sutherland model to compute the full generating function for the characters of irreducible modules over the complex Lie algebra sl(4), and exploit this result to obtain also generating functions giving the multiplicities of some low order weights in all representations. We have applied the same method to compute the generating function for the characters of the modules the other rank three simple Lie algebras, but in these cases the full expressions are very long and appear only in the arXiv version of the paper (arXiv:1705.03711 [math-ph]). Nevertheless, when the generating functions are limited to some particular subsets of characters, the results are quite simple and we present them here.

Lakshmibai–Seshadri paths for hyperbolic Kac–Moody algebras of rank 2

ABSTRACTLet 𝔤 be a hyperbolic Kac–Moody algebra of rank 2, and set , where are the fundamental weights for 𝔤; note that λ is neither dominant nor antidominant. Let 𝔹(λ) be the crystal of all Lakshmibai–Seshadri paths of shape λ. We prove that (the crystal graph of) 𝔹(λ) is connected. Furthermore, we give an explicit description of Lakshmibai–Seshadri paths of shape λ.

Pictorial Scale of Physical Self-Concept for Younger Children (P-PSC-C): A Feasibility Study

Children’s self-perception of motor skills and physical fitness is said to be an important mediator between skills and physical fitness on the one hand and physical activity on the other hand. An age-appropriate self-perception scale is needed to understand the development and the differentiation of the physical self-concept of children and its components. Therefore, the objectives of this study were (1) to develop a pictorial scale of physical fitness for pre-school children (3–6 years old), and (2) to describe the face validity and feasibility of the scale. The study sample included 27 kindergarten children. In order to determine the psychometric properties, validity was assessed by administrating the Pictorial Scale for Physical Self-Concept in Kindergarten Children (P-PSC-C) compared with children’s fundamental movement skill competency (Test of Gross Motor Development [TGMD]-3; six locomotor and seven object-control skills), height, weight, and demographics. The face validity was favorable. Expectable negatively skewed response distributions were found in all items. Medium correlations with related constructs and with sport enjoyment were found. The results indicate that the new scale is usable for kindergarten children. Future validation studies are needed so that the new scale can contribute to the research about physical self-concept development in kindergarten children.