Symmetry, Representations, and Invariants

Abstract

Structure of Classical Groups 69 2.1 Semisimple Elements 69 2.1.1 Toral Groups 70 2.1.2 Maximal Torus in a Classical Group 72 2.1.3 Exercises 76 2.2 Unipotent Elements 77 2.2.1 Low-Rank Examples 77 2.2.2 Unipotent Generation of Classical Groups 2.2.3 Connected Groups 81 2.2.4 Exercises 2.3 Regular Representations of SL(2, C) 2.3.1 Irreducible Representations of .5((2, C) 2.3.2 Irreducible Regular Representations of SL(2, C) 2.3.3 Complete Reducibility of SL(2, (C) 2.3.4 Exercises 2.4 The Adjoint Representation 2.4.1 Roots with Respect to a Maximal Torus 2.4.2 Commutation Relations of Root Spaces 95 2.4.3 Structure of Classical Root Systems 99 2.4.4 Irreducibility of the Adjoint Representation 2.4.5 Exercises 2.5 Semisimple Lie Algebras 2.5.1 Solvable Lie Algebras 2.5.2 Root Space Decomposition 2.5.3 Geometry of Root Systems 2.5.4 Conjugacy of Cartan Subalgebras 2.5.5 Exercises x Contents

Lowest 𝔰𝔩(2)-types in 𝔰𝔩(𝔫)-representations

“…Put another way, for any irreducible representation, V , of K, the dimension of the M-invariant subspace, V M , is at most one-dimensional. These results are part of the Cartan-Helgason theorem (see [GW09]). …”

“…Furthermore, the functions on the cubic are a free module over the invariant subalgebra (see [GW09], Chapter 12). The fact that no lower F d 's appear other than the repeating pattern 0, 3, 2, 3 is an immediate consequence of the story for the cubic.…”

“…The full polynomial algebra is a free module over the invariants-a consequence of Kostant-Rallis theory (see [GW09], Chapter 12). Since we know we have a degree four invariant, f 4 (5), for the quintic we can predict that the final 1, 0, 1, 0 will continue to repeat by multiplying by powers.…”

Clifford Modules and Symmetries of Topological Insulators

“…We are now ready to describe the isotypic decomposition of W R . The following corollary generalizes the standard primary decomposition [12] to the case of representations over a non algebraically closed field such as R (the proof is given in the Appendix): Corollary 1. Let ǫ λ be the tautological evaluation map from the cartesian product R λ × E λ to W R :…”

Biological organisation as closure of constraints

“…In this paper, we base the theoretical and formal characterisation of closure on the concept of symmetry (see for example Weyl (1983); Goodman & Wallach (2009)). In very general terms, symmetries refer to transformations that do not change the relevant aspects of an object: symmetries and invariants (of energy, momentum, electrical charges, etc.)…”