scholarly journals Irreducible representations of E theory

2019 ◽  
Vol 34 (24) ◽  
pp. 1950133 ◽  
Author(s):  
Peter West
Keyword(s):  
Irreducible Representation ◽  
Direct Product ◽  
Simple Form ◽  
Light Cone ◽  
Degrees Of Freedom ◽  
Irreducible Representations ◽  
Vector Representation ◽  
Maximal Supergravity ◽  
Duality Relations ◽  
Cartan Involution

We construct the [Formula: see text] theory analogue of the particles that transform under the Poincaré group, that is, the irreducible representations of the semi-direct product of the Cartan involution subalgebra of [Formula: see text] with its vector representation. We show that one such irreducible representation has only the degrees of freedom of 11-dimensional supergravity. This representation is most easily discussed in the light cone formalism and we show that the duality relations found in [Formula: see text] theory take a particularly simple form in this formalism. We explain that the mysterious symmetries found recently in the light cone formulation of maximal supergravity theories are part of [Formula: see text]. We also argue that our familiar space–times have to be extended by additional coordinates when considering extended objects such as branes.

scholarly journals The massless irreducible representation in E theory and how bosons can appear as spinors

2021 ◽  
pp. 2150096
Author(s):  
Keith Glennon ◽  
Peter West
Keyword(s):  
Irreducible Representation ◽  
Infinite Number ◽  
Degrees Of Freedom ◽  
Spinor Representation ◽  
Covariant Theory ◽  
Massless Particles ◽  
Remarkable Similarity ◽  
Maximal Supergravity ◽  
Duality Relations

We study in detail the irreducible representation of [Formula: see text] theory that corresponds to massless particles. This has little algebra [Formula: see text] and contains 128 physical states that belong to the spinor representation of [Formula: see text]. These are the degrees of freedom of maximal supergravity in eleven dimensions. This smaller number of the degrees of freedom, compared to what might be expected, is due to an infinite number of duality relations which in turn can be traced to the existence of a subaglebra of [Formula: see text] which forms an ideal and annihilates the representation. We explain how these features are inherited into the covariant theory. We also comment on the remarkable similarity between how the bosons and fermions arise in [Formula: see text] theory.

scholarly journals An E11 invariant gauge fixing

2018 ◽  
Vol 33 (01) ◽  
pp. 1850009 ◽  
Author(s):  
Michaella Pettit ◽  
Peter West
Keyword(s):  
Direct Product ◽  
Tangent Space ◽  
Space Time ◽  
Vector Representation ◽  
Alternative Derivation ◽  
Four Dimensions ◽  
Gauge Fixing ◽  
Cartan Involution ◽  
Low Levels ◽  
Invariant Gauge

We consider the nonlinear realisation of the semi-direct product of [Formula: see text] and its vector representation which leads to a space-time with tangent group that is the Cartan involution invariant subalgebra of [Formula: see text]. We give an alternative derivation of the invariant tangent space metric that this space–time possesses and compute this metric at low levels in eleven, five and four dimensions. We show that one can gauge fix the nonlinear realisation in an [Formula: see text] invariant manner.

scholarly journals E11, brane dynamics and duality symmetries

2018 ◽  
Vol 33 (13) ◽  
pp. 1850080 ◽  
Author(s):  
Peter West
Keyword(s):  
Direct Product ◽  
Equations Of Motion ◽  
Space Time ◽  
Vector Representation ◽  
First Order ◽  
Duality Relations ◽  
The One

Following arXiv:hep-th/0412336 we use the nonlinear realisation of the semi-direct product of [Formula: see text] and its vector representation to construct brane dynamics. The brane moves through a space-time which arises in the nonlinear realisation from the vector representation and it contains the usual embedding coordinates as well as the worldvolume fields. The resulting equations of motion are first order in derivatives and can be thought of as duality relations. Each brane carries the full [Formula: see text] symmetry and so the Cremmer–Julia duality symmetries. We apply this theory to find the dynamics of the IIA and IIB strings, the M2 and M5 branes, the IIB D3 brane as well as the one and two branes in seven dimensions.

scholarly journals On the different formulations of the E11 equations of motion

2017 ◽  
Vol 32 (18) ◽  
pp. 1750096 ◽  
Author(s):  
Peter West
Keyword(s):  
Direct Product ◽  
Equations Of Motion ◽  
Degree Of Freedom ◽  
Equivalence Relations ◽  
Vector Representation ◽  
Nonlinear Realization ◽  
Duality Relations ◽  
Alternative Approach

The nonlinear realization of the semi-direct product of [Formula: see text] with its vector representation leads to equation of motions for the field graviton, three-form, six-form, dual graviton and the level four fields which correctly describe the degree of freedom of 11-dimensional supergravity at the linearized level. The equations with one derivative generically hold as equivalence relations and are often duality relations. From these equations, by taking derivatives, one can derive equations that are equations of motion of the familiar kind. The entire hierarchy of equations is [Formula: see text] invariant and the construction does not require any steps beyond [Formula: see text]. We review these past developments with an emphasis on the features that have been overlooked in hep-th:1703.01305, whose alternative approach we also comment on.

scholarly journals A decomposition formula for representations

1987 ◽  
Vol 107 ◽  
pp. 63-68 ◽  
Author(s):  
George Kempf
Keyword(s):  
Irreducible Representation ◽  
General Formula ◽  
Parabolic Subgroup ◽  
Characteristic Zero ◽  
Maximal Torus ◽  
Reductive Group ◽  
Irreducible Representations ◽  
Levi Subgroup ◽  
Know How ◽  
Decomposition Formula

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.

scholarly journals Description of unitary representations of the group of infinite 𝑝-adic integer matrices

2021 ◽  
Vol 25 (21) ◽  
pp. 606-643
Author(s):  
Yury Neretin
Keyword(s):  
Irreducible Representation ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Infinite Matrices ◽  
Natural Topology ◽  
Content Type ◽  
Residue Ring ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations

We classify irreducible unitary representations of the group of all infinite matrices over a p p -adic field ( p ≠ 2 p\ne 2 ) with integer elements equipped with a natural topology. Any irreducible representation passes through a group G L GL of infinite matrices over a residue ring modulo p k p^k . Irreducible representations of the latter group are induced from finite-dimensional representations of certain open subgroups.

scholarly journals Dimensions of the Irreducible Representations of the Symmetric and Alternating Group

10.37236/4909 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Korneel Debaene
Keyword(s):  
Irreducible Representation ◽  
Irreducible Representations ◽  
Alternating Group ◽  
Main Ingredient ◽  
Short Intervals ◽  
Prime Factors

We establish the existence of an irreducible representation of $A_n$ whose dimension does not occur as the dimension of an irreducible representation of $S_n$, and vice versa. This proves a conjecture by Tong-Viet. The main ingredient in the proof is a result on large prime factors in short intervals. 

Representations Subduced on an Ideal of a Lie Algebra

1962 ◽  
Vol 14 ◽  
pp. 293-303 ◽  
Author(s):  
B. Noonan
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Kronecker Product ◽  
Point Of View ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Representations Of Lie Algebras ◽  
The Ideal

This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).

scholarly journals Dirac equation based on the vector representation of the Lorentz group

2020 ◽  
Vol 135 (10) ◽  
Author(s):  
Eckart Marsch ◽  
Yasuhito Narita
Keyword(s):  
Gauge Theory ◽  
Dirac Equation ◽  
Lorentz Group ◽  
Degrees Of Freedom ◽  
Lagrangian Density ◽  
Degree Of Freedom ◽  
Vector Representation ◽  
Massive Fermion

AbstractIn this paper, we derive an expanded Dirac equation for a massive fermion doublet, which has in addition to the particle/antiparticle and spin-up/spin-down degrees of freedom explicity an isospin-type degree of freedom. We begin with revisiting the four-vector Lorentz group generators, define the corresponding gamma matrices and then write a Dirac equation for the fermion doublet with eight spinor components. The appropriate Lagrangian density is established, and the related chiral and SU(2) symmetry is discussed in detail, as well as applications to an electroweak-style gauge theory. In “Appendix,” we present some of the relevant matrices.

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