Andrius: I'm unpacking Exercise 20.38 in William Fulton, Joe Harris. Representation Theory: A First Course. I am also comparing that to the Wikipedia article on spin representations. I want to understand the eightfold Bott periodicity of spinor representations.
{$W\cong \mathbb{C}^n$} is a subspace of {$V\cong \mathbb{C}^m$}. Let {$a_1,\cdots,a_n$} be a basis of {$W$}.
We have a quadratic form {$Q$} on {$V$}. We define the Clifford algebra {$C=Cl(V,Q)$}.
{$S = \wedge^\bullet W$} is called the spinor space.
We have an action of {$V$} on {$S$}.
This action extends to an action of {$C$} on {$S$}. The spin group {$\textrm{Spin}(m)$} is in {$C$} and thus also acts on {$S$}.
Note that {$\wedge^n W$} is one-dimensional with generator {$a_1\wedge a_2\wedge\cdots\wedge a_n$}. Thus we have an isomorphism {$\wedge^n W\rightarrow\mathbb{C}$}.
The reversing map, also know as the main automorphism, is {$\tau(v_1 \cdot \cdots \cdot v_r)=v_r\cdot \cdots \cdot v_1$}. This is an anti-automorphism.
Construct a nondegenerate bilinear pairing {$\beta$} on {$S = \wedge^\bullet W$} as follows. Let {$\beta(s, t)$} be the image of {$\tau(s)\wedge t \in \wedge^\bullet W$} by the projection to {$\wedge^n W\cong\mathbb{C}$}.
- This means that {$\beta$} is a bilinear pairing {$\beta:S\times S\rightarrow \mathbb{C}$} where the elements {$s=a_{k_1}\wedge a_{k_2}\wedge\cdots\wedge a_{k_s}$}, all {$k_j$} distinct, and {$t=a_{l_1}\wedge a_{l_2}\wedge \cdots \wedge a_{l_t}$}, all {$l_j$} distinct, yield a nonzero element only if the sets {$K=\{k_1,\cdots ,k_s\}$} and {$L=\{l_1,\cdots ,l_t\}$} have intersection {$K\cap L=\varnothing$} and union {$K\cup L=\{1,\cdots ,n\}$}. Then the scalar value is either {$+1$} or {$-1$} and is given by the ordering of the generators.
Wikipedia notes that {$S$} is isomorphic to the dual representation {$S^*$} by some isomorphism {$B:S\rightarrow S^*$}. It then defines the nondegenerate bilinear form {$\beta$} on {$S$} as {$\beta(\phi,\psi)=B(\phi)(\psi)$}.
(a) When {$m = 2n$}, show that {$\beta$} can also be defined by the identity {$\beta(s, t)f = \tau(s\cdot f)\cdot t\cdot f$} for an appropriate generator {$f$} of {$\wedge^n W'$}.
- Choose {$f$} spanning {$\wedge^n W'$} so that, for the chosen generator of {$\wedge ^n W$}, {$\tau(f)\cdot e \cdot f = f$}.
Deduce that the action of {$\textrm{Spin}(Q)$} on {$S$} respects the bilinear form {$\beta$}.
- When {$m$} is even
- Note the following fact. The left {$C(Q)$}-module {$\wedge^\bullet W$} is isomorphic to a left ideal in {$C(Q)$}. Show that if {$f$} is a generator for {$\wedge^n W'$}, then {$C(Q)\cdot f = \wedge^\bullet W\cdot f$}, and the map {$\zeta\rightarrow\zeta\cdot f$} gives an isomorphism {$\wedge^\bullet W\rightarrow \wedge^\bullet W\cdot f = C(Q)\cdot f$} of left {$C(Q)$}-modules.
- Then {$x(s)f = x\cdot s\cdot f$}
- So {$\beta(x(s), x(t))f = \tau(x\cdot s\cdot f)\cdot (x\cdot t\cdot f) = t(s\cdot f)\tau(x)\cdot x \cdot (t\cdot f) = \tau(s\cdot f)\cdot (t· f) = \beta(s, t)$}.
- The odd case can be reduced to the even case as follows. Let {$C_m$} denote the Clifford algebra of the vector space {$\mathbb{C}^m$} with our standard quadratic form {$Q_m$}.
- (a) The embedding of {$\mathbb{C}^{2n} = W\oplus W'$} in {$\mathbb{C}^{2n+1} = W\oplus W'\oplus U$} as indicated induces an embedding of {$\mathbb{C}^{2n}$} in {$\mathbb{C}^{2n+1}$} and corresponding embedding of {$\textrm{Spin}_{2n}\mathbb{C}$} in {$\textrm{Spin}_{2n+1}\mathbb{C}$} and of {$\textrm{SO}_{2n}\mathbb{C}$} in {$\textrm{SO}_{2n+1}\mathbb{C}$}. Show that the spin representation {$S$} of {$\textrm{Spin}_{2n+1}\mathbb{C}$} restricts to the spin representation {$S^\oplus S^-$} of {$\textrm{Spin}_{2n}\mathbb{C}$}.
- (b) Similarly there is an embedding of {$\textrm{Spin}_{2n+1}\mathbb{C}$} in {$\textrm{Spin}_{2n+2}\mathbb{C}$} coming from an embedding of {$\mathbb{C}^{2n+1} = W\oplus W'\oplus U$} in {$\mathbb{C}^{2n+1} = W\oplus W'\oplus U_1\oplus U_2$}; here {$U_1\oplus U_2=\mathbb{C}\oplus\mathbb{C}$} with the quadratic form {$\begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}$}, and {$U = \mathbb{C}$} is embedded in {$U_1\oplus U_2$} by sending {$1$} to {$(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$}. Show that each of the half-spin representations of {$\textrm{Spin}_{2n+2}\mathbb{C}$} restricts to the spin representation of {$\textrm{Spin}_{2n+1}\mathbb{C}$}.
- In this way, embed {$C(Q)$} into a larger Clifford algebra.
(b) Show that {$\beta$} is symmetric if {$n$} is congruent to {$0$} or {$3$} modulo {$4$}, and skew-symmetric otherwise. So the spin representation is a homomorphism
{$\textrm{Spin}_{2n+l}\mathbb{C}\rightarrow \textrm{SO}_{2^n}\mathbb{C}$} if {$n\equiv 0,3 \mod 4$},
{$\textrm{Spin}_{2n+l}\mathbb{C}\rightarrow \textrm{Sp}_{2^n}\mathbb{C}$} if {$n\equiv 1,2 \mod 4$}
(c) If {$m = 2n$}, the restrictions of {$\beta$} to {$S^+$} and {$S^-$} are zero if {$n$} is odd. For {$n$} even, deduce that the half-spin representations are homomorphisms
{$\textrm{Spin}_{2n}\mathbb{C}\rightarrow \textrm{SO}_{2^{n-1}}\mathbb{C}$} if {$n\equiv 0 \mod 4$},
{$\textrm{Spin}_{2n}\mathbb{C}\rightarrow \textrm{Sp}_{2^{n-1}}\mathbb{C}$} if {$n\equiv 2 \mod 4$}