Spin representations, Spin representation periodicity
Bott Periodicity Models the Assembly of Mental Contexts
Abstract
The Folk Dance
The mechanism of Bott periodicity.
Given {$N$} dancers (or perspectives or dimensions or agents)
- Pair up the dancers into couples. There may be one dancer left over.
- Line up the couples from left to right: {$c_1, \dots, c_m$}
- Have adjacent couples swap positions {$c_jc_k\rightarrow c_kc_j$}. where {$j<k$}, until the line is inverted: {$c_m, \dots, c_1$}.
If the number of swaps is even, then we say the dance matches the music, and the dance is a success, but if the number of swaps is odd, then it does not, and the dance is a failure.
We can think of swaps as taking place one after the other. Alternatively, we can think of pairs of swaps occurring together. For this to happen, we need four couples.
Understand
- Why is the bilinear form {$\beta(s,t)$} important in the case when the irreducible representation is even {$S^+$} or odd {$S^-$} and the overall {$S$} is generated from an odd-dimensional {$W$}?
- What does it mean when {$\beta(s,t)=0$} for all {$s,t$}?
- Why are spin representations irreducible (when {$n$} is odd) or reducible (when {$n$} is even)?
- When matrices satisfy a symmetric bilinear form, why are they orthogonal?
- When matrices satisfy a skew-symmetric bilinear form, why are they symplectic?
- Why are these matrices all special linear?
Metaphysical interpretation
Insights from various manifestations of Bott periodicity
Spinor representations
Clifford algebra representations
Mutually anticommuting linear complex structures
Lie group embeddings
Dimensions of Lie groups
Quantum symmetries
Hamiltonians
Super division algebras
Symmetric spaces
Homotopy groups of compact Lie groups
Octonions
{$(\phi, \chi)$}-representations of {$CT$}-groups
Random matrices
Homotopy groups of spheres
References
- Pierre Deligne. Notes on Spinors. Proposition 2.2(i) describes the Clifford action on the exterior algebra.
- William Fulton, Joe Harris. Representation Theory: A First Course
- Wikipedia: Spin representation