Meeting Notes 2025-03-25

This Week’s Finds

No finds this week.

 

Focus Session

Speaker: Joshua
Title: Representation Theory and Spinor Construction

 

1. Overview

• This talk explores how representation theory underpins spinor construction via Clifford algebras.
• Goals:
  1. Introduce basic representation theory notions relevant to Clifford algebras.
  2. Show how spin representations emerge as irreducible modules over the algebra.

 

2. Main Content

1. Representation Theory Basics
   • Definition of group representation: homomorphism \(\rho: G \to \mathrm{GL}(V)\).
   • Module over an algebra: \(V\) as a left \(\mathrm{Cl}(p,q)\)-module.
   • Irreducibility and Schur’s Lemma: criteria for simple modules.
2. Spin Representations from Clifford Algebras
   • Spin group \(\mathrm{Spin}(p,q)\) as a subgroup of invertible elements in \(\mathrm{Cl}(p,q)\).
   • Construction of spinor module: minimal left ideal \(\mathrm{Cl}(p,q)\,P\) yields irreducible representation.
   • Explicit matrix realizations: Dirac and Weyl spinors as representation spaces.
   • Relationship between algebra action and Lorentz transformations \(\Lambda \in \mathrm{SO}(p,q)\) via the double cover.

 

3. Discussion Points

• Differences between real and complex representations in physics applications.
• Role of primitive idempotents in selecting irreducible modules.
• Physical interpretation of module structure under symmetry transformations.

 

Up Next

Next Talk:

• Alice on Characteristic Classes and Their Applications

 

References

1. Fulton & Harris, Representation Theory: A First Course.
2. Lounesto, Clifford Algebras and Spinors.
3. Representation Theory (Wikipedia)
4. Clifford Algebra (Wikipedia)