Indecomposable representations in characteristic two of the simple groups of order not divisible by eight

The indecomposable representations in characteristic two of the groups PSL(2, q) where q is congruent to 3 or 5 modulo 8 are classified. For q = 3 or 5 the classification is obtained by explicit construction of modules, using the Green correspondence to prove completeness. For larger q, the classification is obtained using equivalences between appropriate categories of modules.

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The multistep homology of the simplex and representations of symmetric groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004119000124 ◽

2019 ◽

Vol 169 (2) ◽

pp. 231-253

Author(s):

MARK WILDON

Keyword(s):

Symmetric Group ◽

Symmetric Groups ◽

Explicit Construction ◽

Fixed Size ◽

Simplicial Homology ◽

Chain Complexes ◽

Basic Spin ◽

Characteristic Two

AbstractThe symmetric group on a set acts transitively on the set of its subsets of a fixed size. We define homomorphisms between the corresponding permutation modules, defined over a field of characteristic two, which generalize the boundary maps from simplicial homology. The main results determine when these chain complexes are exact and when they are split exact. As a corollary we obtain a new explicit construction of the basic spin modules for the symmetric group.

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Discrete Nonlinear Schrödinger Systems for Periodic Media with Nonlocal Nonlinearity: The Case of Nematic Liquid Crystals

Applied Sciences ◽

10.3390/app11104420 ◽

2021 ◽

Vol 11 (10) ◽

pp. 4420

Author(s):

Panayotis Panayotaros

Keyword(s):

Differential Equation ◽

Infinite System ◽

Linear Part ◽

Optical Power ◽

Explicit Construction ◽

Translation Invariance ◽

Nonlocal Nonlinearity ◽

Wannier Functions ◽

Nonlinear Schrödinger ◽

Nonlinear Schrödinger Systems

We study properties of an infinite system of discrete nonlinear Schrödinger equations that is equivalent to a coupled Schrödinger-elliptic differential equation with periodic coefficients. The differential equation was derived as a model for laser beam propagation in optical waveguide arrays in a nematic liquid crystal substrate and can be relevant to related systems with nonlocal nonlinearities. The infinite system is obtained by expanding the relevant physical quantities in a Wannier function basis associated to a periodic Schrödinger operator appearing in the problem. We show that the model can describe stable beams, and we estimate the optical power at different length scales. The main result of the paper is the Hamiltonian structure of the infinite system, assuming that the Wannier functions are real. We also give an explicit construction of real Wannier functions, and examine translation invariance properties of the linear part of the system in the Wannier basis.

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Instanton bundles on two Fano threefolds of index 1

Forum Mathematicum ◽

10.1515/forum-2019-0189 ◽

2020 ◽

Vol 32 (5) ◽

pp. 1315-1336

Author(s):

Gianfranco Casnati ◽

Ozhan Genc

Keyword(s):

Irreducible Component ◽

Moduli Space ◽

Blow Up ◽

Explicit Construction ◽

Fano Threefolds ◽

Instanton Bundles

AbstractWe deal with instanton bundles on the product {\mathbb{P}^{1}\times\mathbb{P}^{2}} and the blow up of {\mathbb{P}^{3}} along a line. We give an explicit construction leading to instanton bundles. Moreover, we also show that they correspond to smooth points of a unique irreducible component of their moduli space.

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Exceptional nonrelativistic effective field theories with enhanced symmetries

Journal of High Energy Physics ◽

10.1007/jhep02(2021)218 ◽

2021 ◽

Vol 2021 (2) ◽

Author(s):

Tomáš Brauner

Keyword(s):

Effective Field Theories ◽

Explicit Construction ◽

Effective Field ◽

Field Theories ◽

Relativistic Case ◽

Starting Point ◽

Higher Power ◽

Mere Presence ◽

Spatial Rotations

Abstract We initiate the classification of nonrelativistic effective field theories (EFTs) for Nambu-Goldstone (NG) bosons, possessing a set of redundant, coordinate-dependent symmetries. Similarly to the relativistic case, such EFTs are natural candidates for “exceptional” theories, whose scattering amplitudes feature an enhanced soft limit, that is, scale with a higher power of momentum at long wavelengths than expected based on the mere presence of Adler’s zero. The starting point of our framework is the assumption of invariance under spacetime translations and spatial rotations. The setup is nevertheless general enough to accommodate a variety of nontrivial kinematical algebras, including the Poincaré, Galilei (or Bargmann) and Carroll algebras. Our main result is an explicit construction of the nonrelativistic versions of two infinite classes of exceptional theories: the multi-Galileon and the multi-flavor Dirac-Born-Infeld (DBI) theories. In both cases, we uncover novel Wess-Zumino terms, not present in their relativistic counterparts, realizing nontrivially the shift symmetries acting on the NG fields. We demonstrate how the symmetries of the Galileon and DBI theories can be made compatible with a nonrelativistic, quadratic dispersion relation of (some of) the NG modes.

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Explicit Construction of ($k+r, k$) MDS Code with Small Sub-packetization level and Optimal Access Property for All Nodes

2021 International Conference on Communications, Information System and Computer Engineering (CISCE) ◽

10.1109/cisce52179.2021.9445943 ◽

2021 ◽

Author(s):

Yi Liu ◽

Xiaohu Tang ◽

Xianfu Lei

Keyword(s):

Explicit Construction ◽

Mds Code

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A diameter bound for finite simple groups of large rank

Journal of the London Mathematical Society ◽

10.1112/jlms.12018 ◽

2017 ◽

Vol 95 (2) ◽

pp. 455-474 ◽

Cited By ~ 2

Author(s):

Arindam Biswas ◽

Yilong Yang

Keyword(s):

Simple Groups ◽

Finite Simple Groups ◽

Large Rank

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NONRELATIVISTIC HOLOGRAPHY — A GROUP-THEORETICAL PERSPECTIVE

International Journal of Modern Physics A ◽

10.1142/s0217751x14300014 ◽

2014 ◽

Vol 29 (03n04) ◽

pp. 1430001 ◽

Cited By ~ 10

Author(s):

V. K. DOBREV

Keyword(s):

Representation Theory ◽

Theoretical Perspective ◽

Explicit Construction ◽

Intertwining Operators ◽

Difference Operators ◽

Semisimple Lie Groups ◽

Boundary Operators ◽

Boundary Fields ◽

Theoretical Results ◽

Schrödinger Algebra

We give a review of some group-theoretical results related to nonrelativistic holography. Our main playgrounds are the Schrödinger equation and the Schrödinger algebra. We first recall the interpretation of nonrelativistic holography as equivalence between representations of the Schrödinger algebra describing bulk fields and boundary fields. One important result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory, and that these operators and the bulk-to-boundary operators are intertwining operators. Further, we recall the fact that there is a hierarchy of equations on the boundary, invariant with respect to Schrödinger algebra. We also review the explicit construction of an analogous hierarchy of invariant equations in the bulk, and that the two hierarchies are equivalent via the bulk-to-boundary intertwining operators. The derivation of these hierarchies uses a mechanism introduced first for semisimple Lie groups and adapted to the nonsemisimple Schrödinger algebra. These require development of the representation theory of the Schrödinger algebra which is reviewed in some detail. We also recall the q-deformation of the Schrödinger algebra. Finally, the realization of the Schrödinger algebra via difference operators is reviewed.

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An explicit construction of the dimension-9 operator basis in the standard model effective field theory

Journal of High Energy Physics ◽

10.1007/jhep11(2020)152 ◽

2020 ◽

Vol 2020 (11) ◽

Author(s):

Yi Liao ◽

Xiao-Dong Ma

Keyword(s):

Field Theory ◽

Standard Model ◽

Effective Field Theory ◽

Equations Of Motion ◽

Baryon Number ◽

Explicit Construction ◽

Effective Field ◽

Starting Point ◽

The Standard Model ◽

Symmetry Relations

Abstract We investigate systematically dimension-9 operators in the standard model effective field theory which contains only standard model fields and respects its gauge symmetry. With the help of the Hilbert series approach to classifying operators according to their lepton and baryon numbers and their field contents, we construct the basis of operators explicitly. We remove redundant operators by employing various kinematic and algebraic relations including integration by parts, equations of motion, Schouten identities, Dirac matrix and Fierz identities, and Bianchi identities. We confirm counting of independent operators by analyzing their flavor symmetry relations. All operators violate lepton or baryon number or both, and are thus non-Hermitian. Including Hermitian conjugated operators there are $$ {\left.384\right|}_{\Delta B=0}^{\Delta L=\pm 2}+{\left.10\right|}_{\Delta B=\pm 2}^{\Delta L=0}+{\left.4\right|}_{\Delta B=\pm 1}^{\Delta L=\pm 3}+{\left.236\right|}_{\Delta B=\pm 1}^{\Delta L=\mp 1} $$ 384 Δ B = 0 Δ L = ± 2 + 10 Δ B = ± 2 Δ L = 0 + 4 Δ B = ± 1 Δ L = ± 3 + 236 Δ B = ± 1 Δ L = ∓ 1 operators without referring to fermion generations, and $$ {\left.44874\right|}_{\Delta B=0}^{\Delta L=\pm 2}+{\left.2862\right|}_{\Delta B=\pm 2}^{\Delta L=0}+{\left.486\right|}_{\Delta B=\pm 1}^{\Delta L=\pm 3}+{\left.42234\right|}_{\Delta B=\mp 1}^{\Delta L=\pm 1} $$ 44874 Δ B = 0 Δ L = ± 2 + 2862 Δ B = ± 2 Δ L = 0 + 486 Δ B = ± 1 Δ L = ± 3 + 42234 Δ B = ∓ 1 Δ L = ± 1 operators when three generations of fermions are referred to, where ∆L, ∆B denote the net lepton and baryon numbers of the operators. Our result provides a starting point for consistent phenomenological studies associated with dimension-9 operators.

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