Group Theory query based on Green Schwarz Witten volume 2

[maverick280857]

Hi,

In chapter 12 of GSW volume 2, the authors remark, "spinors form a representation of SO(n) that does not arise from a representation of GL(2,R)."

What do they mean by this?

More generally, since SO(n) is a subgroup of GL(2,R) won't every representation of GL(2,R) be a representation of SO(n) as well?

I know the Dirac matrices of the spinor representation of SO(n) will have different matrix dimension depending on whether n is even or odd. Is that related?

Thanks!

 

[Eredir]

Hi,

In chapter 12 of GSW volume 2, the authors remark, "spinors form a representation of SO(n) that does not arise from a representation of GL(2,R)."

What do they mean by this?

More generally, since SO(n) is a subgroup of GL(2,R) won't every representation of GL(2,R) be a representation of SO(n) as well?

I know the Dirac matrices of the spinor representation of SO(n) will have different matrix dimension depending on whether n is even or odd. Is that related?

Thanks!

Check out this answer from MathOverflow:
http://mathoverflow.net/questions/121620/why-does-gln-have-no-spinor-representations

 

[lpetrich]

Hi,

In chapter 12 of GSW volume 2, the authors remark, "spinors form a representation of SO(n) that does not arise from a representation of GL(2,R)."

What do they mean by this?

Here's what I think is going on.

GL(2,R) ~ GL(1,R+) * SL(2,R)
SL(2,R) ~ SO(2,1)
SO(2,1) is an analytic continuation of SO(3)
For SO(3),
U(2) ~ U(1) * SU(2)
SU(2) ~ SO(3)

More generally, since SO(n) is a subgroup of GL(2,R) won't every representation of GL(2,R) be a representation of SO(n) as well?

Only SO(2) is. SO(n) is a subgroup of GL(n,R) ~ GL(1,R+) * SL(n,R), and of SL(n,R) also. Reps of GL(n,R) can indeed be decomposed into reps of SO(n), but an irreducible rep of GL(n,R) is not in general irreducible in SO(n).

I know the Dirac matrices of the spinor representation of SO(n) will have different matrix dimension depending on whether n is even or odd. Is that related?

That's a separate issue. The Dirac matrices are matrices in a "Clifford algebra", and one indeed uses Clifford algebras to get spinor-rep generators. For algebra elements X_i:
X_i.X_j + X_j.X_i = 2 g_ij

for algebra metric g. One finds algebra generator L_ij from a multiple of the commutator of X_i and X_j.

For SO(2n) and SO(2n+1), one constructs 2n+1 Clifford-algebra matrices using outer products of n Pauli matrices, giving their spinor reps dimension 2ⁿ. One needs all 2n+1 for the generators of the SO(2n+1) spinor rep, and it is thus irreducible. But one needs only 2n of them for the generators of the SO(2n) spinor rep, with the remaining one splitting that rep into two equal irreducible halves, both with dimension 2^n-1.