I am working my way through Adams's "Lectures on Exceptional Lie Groups", and I am not satisfied with the proof for his proposition 4.2 (which states the even subalgebra for the Clifford Algebra \(Cl(V)_{0}\) has 1 irreducible representation when \(dim(V)=m=2n+1\) and 2 irreducible representations when \(dim(V)=m=2n\) with certain specific weights).

The argument seems to be to relate representations of an Abelian subgroup \(E=\{\prod^{m}_{j=1}e_{j}^{i_{j}}\mid i_{j}=0\mbox{ or }1\}\) and \(E_{0} = E\cap Cl(V)_{0}\) [where \(e_{j}\) form the canonical basis for \(V\)] to representations of \(\mathbb{R}[E]/(\nu + 1)\cong Cl(V)\), the quotient of the group algebra \(\mathbb{R}[E]\) modulo the identification of the square of the generators \(e_{j}^{2}=\nu\) with -1.

I'm with Adams until he picks a complex 1-dimensional representation \(W\) of \(F\), because he starts working with *COMPLEX* representations. But Adams triumphantly announces "We thus get a representation, \(\Delta\) of \(E_{0}\)..." then shows it is irreducible. I'm fine with it being irreducible from the character relations, that's fine.

Even supposing this is an irreducible representation for \(\mathbb{C}[E_{0}]\), I don't quite see how to obtain an irrep for \(Cl(V)_{0}\); I am guessing just extend it "in the obvious way"? Does this preserve irreducibility?

#RepresentationTheory #LieGroups #CliffordAlgebra #Mathematics #Proof

I learned from Jon Brett about physicist David #Bohm 's colleague Basil Hiley https://en.wikipedia.org/wiki/Basil_Hiley Their distinction between implicate and explicate order seems to appear in my study of #orthogonal #Sheffer #polynomial s. The Sheffer constraint of exponentiality yields an implicate order of #partition of a set https://www.math4wisdom.com/wiki/Exposition/20221122SpaceBuilders whereas orthogonality yields 5 possible explicate orders upon #measurement. Also curious how they use #CliffordAlgebra ,real and #symplectic. #BottPeriodicity ?