Spin(8) Clifford modules, real forms, and D4 weights

On the ordered wedge basis $u_I=u_{i_1}\wedge \cdots \wedge u_{i_r}$, exterior multiplication and deletion satisfy ${ϵ}_j^2={\iota }_j^2=0$, ${ϵ}_j{ϵ}_k+{ϵ}_k{ϵ}_j=0$, ${\iota }_j{\iota }_k+{\iota }_k{\iota }_j=0$, and ${ϵ}_j{\iota }_k+{\iota }_k{ϵ}_j={\delta }_{jk}I$: if a deletion index is absent its term is zero, and when both orders are nonzero they differ by one transposition. Expanding $C_{2j-1}$ and $C_{2j}$ therefore gives $C_a{C}_b+C_b{C}_a=2{\delta }_{ab}I$ for all $a,b$, so [F4] defines a representation of the complexified Clifford algebra on $\mathrm{\Delta }$.

Put $\mathfrak{g}=\mathfrak{s}\mathfrak{o}(8,\mathbb{C})=\{X\in M_8(\mathbb{C}):X^T=-X\}$ with its commutator bracket. For $1\le a<b\le 8$ let $E_{ab}$ be the standard matrix unit and put $X_{ab}=E_{ab}-E_{ba}$; these form a complex basis of $\mathfrak{g}$, and write $X_{ba}=-X_{ab}$ when indices are reversed. Let $\mathcal{D}$ be the finite group of diagonal matrices $D=\mathrm{diag}({\epsilon }_1,\dots ,{\epsilon }_8)$ with ${\epsilon }_i\in \{\pm 1\}$ and ${\prod }_i{\epsilon }_i=1$. Each $D$ is a product of coordinate-plane rotations by $\pi$, so its adjoint action preserves every ideal $I⊴\mathfrak{g}$: each rotation is an exponential, and $\exp (t\mathrm{ad}X)$ preserves any subspace stable under $\mathrm{ad}X$. Since ${\mathrm{Ad}}_D(X_{ab})={\epsilon }_a{\epsilon }_b{X}_{ab}$, the characters $D\mapsto {\epsilon }_a{\epsilon }_b$ are distinct for distinct unordered pairs (their symmetric difference has size $2$ or $4$, so it is neither empty nor the full coordinate set). For $X={\sum }_{a<b}{c}_{ab}{X}_{ab}\in I$, the average $\mathrm{\mid }\mathcal{D}{\mathrm{\mid }}^{-1}{\sum }_{D\in \mathcal{D}}({\epsilon }_a{\epsilon }_b){\mathrm{Ad}}_D(X)$ equals $c_{ab}{X}_{ab}$: the coefficient of $X_{cd}$ is multiplied by the average of the character $D\mapsto ({\epsilon }_a{\epsilon }_b)({\epsilon }_c{\epsilon }_d)$, which is $1$ when the pairs agree and $0$ otherwise. For a nontrivial character the latter average is zero by pairing each $D$ with $D_{0}D$, where its value at $D_0$ is $-1$. If $I\mathrm{\ne }0$, one such coefficient is nonzero, hence $X_{ab}\in I$. For $k\mathrm{\notin }\{a,b\}$, brackets with $X_{bk}$ put each $X_{ak}$ in $I$, and brackets among these incident basis elements give every $X_{kl}$ with $k,l\mathrm{\ne }a$. Thus $I=\mathfrak{g}$. Since $[X_{12},X_{23}]=X_{13}\mathrm{\ne }0$, $\mathfrak{g}$ is nonabelian and simple. A solvable ideal is therefore $0$ or $\mathfrak{g}$; the latter is impossible because $[\mathfrak{g},\mathfrak{g}]$ is a nonzero ideal and hence equals $\mathfrak{g}$, so every term of the derived series is $\mathfrak{g}$. Thus $\mathfrak{g}$ is semisimple by [F7].

Define $w_j^{+}=e_{2j-1}+i{e}_{2j}$ and $w_j^{-}=e_{2j-1}-i{e}_{2j}$. The commuting elements $H_j=-i{R}_j$ act on $w_j^{\pm }$ with eigenvalues $\pm 1$. Extend the Euclidean inner product on $V$ complex-bilinearly to $V_{\mathbb{C}}$. The map $u\wedge v\mapsto (x\mapsto ⟨u,x⟩v-⟨v,x⟩u)$ is an isomorphism ${\mathrm{\Lambda }}^2{V}_{\mathbb{C}}\cong \mathfrak{s}\mathfrak{o}(8,\mathbb{C})$: the basis wedges map to a basis of skew matrices. The identity $[H,X_{u,v}]=X_{Hu,v}+X_{u,Hv}$ shows its nonzero simultaneous $\mathfrak{h}$-weights are $\pm {\xi }_i\pm {\xi }_j$ for $i<j$, with one-dimensional eigenspaces; its zero-weight space is $\mathfrak{h}$. If $Y=Y_0+{\sum }_{\alpha }{Y}_{\alpha }$ normalizes $\mathfrak{h}$, then $[H,Y]\in \mathfrak{h}$ for every $H\in \mathfrak{h}$, so $\alpha (H)Y_{\alpha }=0$ for every $H$ and each nonzero root $\alpha$; hence all $Y_{\alpha }=0$ and $Y\in \mathfrak{h}$. Thus $\mathfrak{h}$ is self-normalizing; it is abelian and therefore nilpotent, so it is Cartan by [F5]. The displayed roots form a reduced crystallographic root system: they span ${\mathbb{R}}^4$; all have squared length $2$, so each root line contains only that root and its negative; reflections in ${\xi }_i-{\xi }_j$ swap coordinates $i,j$, while reflections in ${\xi }_i+{\xi }_j$ send $(x_i,x_j)$ to $(-x_j,-x_i)$, so both preserve the root set; and $2(\beta ,\alpha )\mathrm{/}(\alpha ,\alpha )=(\beta ,\alpha )\in \mathbb{Z}$ for all roots $\alpha ,\beta$. For the regular vector $(4,3,2,1)$, the positive roots are exactly ${\xi }_i-{\xi }_j$ and ${\xi }_i+{\xi }_j$ with $i<j$. The four ${\alpha }_i$ are linearly independent, and every positive difference is a nonnegative sum of adjacent differences; the positive sums decompose as ${\xi }_3+{\xi }_4={\alpha }_4$, ${\xi }_2+{\xi }_4={\alpha }_2+{\alpha }_4$, ${\xi }_2+{\xi }_3={\alpha }_2+{\alpha }_3+{\alpha }_4$, ${\xi }_1+{\xi }_4={\alpha }_1+{\alpha }_2+{\alpha }_4$, ${\xi }_1+{\xi }_3={\alpha }_1+{\alpha }_2+{\alpha }_3+{\alpha }_4$, and ${\xi }_1+{\xi }_2={\alpha }_1+2{\alpha }_2+{\alpha }_3+{\alpha }_4$. Thus these four simple roots are a basis of $E={\mathrm{span}}_{\mathbb{R}}\mathrm{\Phi }$. For $H_p,H_q$, the Killing form is $B(H_p,H_q)={\sum }_{\alpha \in \mathrm{\Phi }}\alpha (H_p)\alpha (H_q)=12{\delta }_{pq}$: each root space is one-dimensional with weight $\alpha$, the zero-weight space $\mathfrak{h}$ contributes $0$, and summing over the roots gives $12$ when $p=q$ and $0$ otherwise. For $\alpha ={\xi }_i\pm {\xi }_j$, its Killing-dual vector is therefore $H_{\alpha }=(H_i\pm H_j)\mathrm{/}12$, so $\alpha (H_{\alpha })=1\mathrm{/}6$ and its Lie-algebra coroot from [F8] is $h_{\alpha }=H_i\pm H_j$. In particular the simple coroots are $H_1-H_2,H_2-H_3,H_3-H_4,H_3+H_4$, and direct evaluation gives $({\lambda }_a(h_{{\alpha }_b}){)}_{a,b=1}^4=I_4$, so the displayed ${\lambda }_1,\dots ,{\lambda }_4$ are the fundamental weights.

The formulas ${ϵ}_j=(C_{2j-1}-i{C}_{2j})\mathrm{/}2$ and ${\iota }_j=(C_{2j-1}+i{C}_{2j})\mathrm{/}2$ put all wedge and deletion operators in the image of this representation. The projections $P_j={ϵ}_j{\iota }_j$ select precisely the basis wedges containing $u_j$, so ${\prod }_{j\in I}{P}_j{\prod }_{j\mathrm{\notin }I}(I-P_j)$ projects onto $\mathbb{C}{u}_I$. Composing this projection with deletions for the indices in $I$ and exterior multiplications for the indices in any $J$ gives, up to sign, the matrix unit sending $u_I$ to $u_J$ and all other basis vectors to zero. Thus the representation image is all of ${\mathrm{End}}_{\mathbb{C}}(\mathrm{\Delta })$; both algebras have dimension $256$ by [F3], so the representation is an isomorphism. The natural module for ${\mathrm{End}}_{\mathbb{C}}(\mathrm{\Delta })$ is irreducible: if $U\subseteq \mathrm{\Delta }$ is nonzero, then for any $u\mathrm{\ne }0$ in $U$ and any $z\in \mathrm{\Delta }$ there is an endomorphism sending $u$ to $z$, so invariance forces $z\in U$.

Since $C_{2j-1}{C}_{2j}=i(I-2P_j)$ and these four even products commute, $\omega =i^4{\prod }_{j=1}^4(I-2P_j)$ acts on $u_I$ by $(-1{)}^{\mathrm{\mid }I\mathrm{\mid }}$. There are eight even and eight odd subsets of $\{1,2,3,4\}$, hence ${\dim }_{\mathbb{C}}{\mathrm{\Delta }}^{+}={\dim }_{\mathbb{C}}{\mathrm{\Delta }}^{-}=8$. Each $C_a$ anticommutes with the product of the other seven Clifford generators in $\omega$, so $c(v)\omega =-\omega c(v)$ for every $v\in V$; thus $c(v)$ interchanges ${\mathrm{\Delta }}^{+}$ and ${\mathrm{\Delta }}^{-}$.

Each $C_{2j-1}$ has real coefficients and each $C_{2j}$ has purely imaginary coefficients. The four factors of $A=C_2{C}_4{C}_6{C}_8$ anticommute and square to $I$, so $A^2=(-1{)}^{6}I=I$ and coefficient conjugation gives $\bar{A}=A$. Conjugation by $A$ fixes the four odd-indexed $C_a$ and negates the four even-indexed ones; coefficient conjugation $K$ has the same respective signs on the $C_a$. Therefore $J=AK$ is conjugate-linear, $J^2=A\bar{A}=I$, and $J{C}_a=C_{a}J$ for each $a$. The volume operator is real and commutes with $A$, so $J$ preserves ${\mathrm{\Delta }}^{+}$ and ${\mathrm{\Delta }}^{-}$.

For each $i<j$, $\exp (t{e}_i{e}_j\mathrm{/}2)=\cos (t\mathrm{/}2)+\sin (t\mathrm{/}2)e_i{e}_j=e_i(\cos (t\mathrm{/}2)e_i+\sin (t\mathrm{/}2)e_j)$ is a product of two unit vectors in $G$. Since $[e_i{e}_j,e_i]=-2e_j$ and $[e_i{e}_j,e_j]=2e_i$, its vector action is the rotation in the $ij$-plane; its action on $\mathrm{\Delta }$ is $\exp (t{C}_i{C}_j\mathrm{/}2)$. Thus the infinitesimal rotation generators act on $\mathrm{\Delta }$ by $C_i{C}_j\mathrm{/}2$; their commutators, computed from step 1.1, are the standard $\mathfrak{s}\mathfrak{o}(8)$ brackets. Every element of $SO(8)$ is a product of coordinate-plane rotations: successively rotating its first column to $e_1$ and repeating on the orthogonal complement reduces it to a one-dimensional determinant-$+1$ block, which is $1$. By [F2], any element of Spin(8) differs from a product of the corresponding lifted plane-rotation curves by an element of $\{1,-1\}$. Hence invariance under the plane-rotation Lie algebra and this scalar kernel is equivalent to invariance under Spin(8).

The inner product making the ordered wedge basis orthonormal has ${ϵ}_j^{\ast }={\iota }_j$, so every $C_a$ is self-adjoint and satisfies $C_a^2=I$. More generally, for a unit vector $u={\sum }_a{u}_a{e}_a$ with real coefficients, $c(u)={\sum }_a{u}_a{C}_a$ is self-adjoint and $c(u{)}^2=Q(u)I=I$ by step 1.1, so every unit-vector generator is unitary and Spin(8) acts unitarily on $\mathrm{\Delta }$. The volume eigenspaces are preserved by Spin(8), and $J$ commutes with every $c(u)$ because its coefficients are real and $J$ commutes with each $C_a$; hence $J$ commutes with the group action. For any $z\in {\mathrm{\Delta }}^{\pm }$, the vectors $(z+Jz)\mathrm{/}2$ and $(z-Jz)\mathrm{/}(2i)$ are $J$-fixed and $z=(z+Jz)\mathrm{/}2+i(z-Jz)\mathrm{/}(2i)$; hence the fixed real subspace complexifies to ${\mathrm{\Delta }}^{\pm }$ and has real dimension $8$.

Differentiating the spin action of the plane-rotation lifts in step 2.4, the complexified Cartan generator $H_j=-i{R}_j$ acts on $\mathrm{\Delta }$ by $-i{C}_{2j-1}{C}_{2j}\mathrm{/}2$. On $u_I$, this operator has eigenvalue $+1\mathrm{/}2$ when $j\mathrm{\notin }I$ and $-1\mathrm{/}2$ when $j\in I$, so the corresponding weight has coordinate $+1\mathrm{/}2$ or $-1\mathrm{/}2$, respectively. Thus ${\mathrm{\Delta }}^{+}$ has the eight half-integral weights with an even number of minus signs, ${\mathrm{\Delta }}^{-}$ has the eight with an odd number of minus signs, and $V_{\mathbb{C}}$ has weights $\{\pm {\xi }_j\}$. Permuting the four coordinate two-planes and reversing an even number of their orientations gives elements of $SO(V)$ inducing the even signed permutations of the ${\xi }_j$; these act transitively on each half-spin parity set and on the vector weights.

A Spin-invariant complex subspace of any of $V_{\mathbb{C}},{\mathrm{\Delta }}^{+},{\mathrm{\Delta }}^{-}$ is invariant under the Lie algebra action by differentiation. The commuting Cartan generators act diagonally in the explicit weight bases of step 3.2; their spectral projections are polynomials in these generators, so the subspace is the sum of its simultaneous weight spaces and a nonzero subspace contains a weight line. The matrices realizing the even signed permutations from step 3.2 lie in $SO(V)$ and by [F2] have Spin(8) lifts. Conjugation by these matrices permutes and changes signs of the $R_j$, so the lifted group elements carry each Cartan weight space to the space with the corresponding signed-permuted weight. These lifts act transitively on the weight sets in step 3.2. Spin invariance therefore forces the subspace to contain every weight line, so it is the whole module; hence all three complex modules are irreducible. If a real invariant subspace $W$ of $8_s$ or $8_c$ is nonzero, then its complexification is all of ${\mathrm{\Delta }}^{+}$ or ${\mathrm{\Delta }}^{-}$. Since ${\dim }_{\mathbb{C}}{W}_{\mathbb{C}}={\dim }_{\mathbb{R}}W$ and ${\dim }_{\mathbb{C}}{\mathrm{\Delta }}^{\pm }=8$, this forces ${\dim }_{\mathbb{R}}W=8$; the full fixed real space has dimension $8$ by step 3.1, so $W$ is that full space. For a nonzero real invariant subspace $W\subseteq V$, the same argument gives $W_{\mathbb{C}}=V_{\mathbb{C}}$ and ${\dim }_{\mathbb{R}}W={\dim }_{\mathbb{C}}{V}_{\mathbb{C}}=8={\dim }_{\mathbb{R}}V$, hence $W=V$.

Relative to ${\lambda }_4$, each weight of ${\mathrm{\Delta }}^{+}$ with two minus signs differs by the positive root ${\xi }_i+{\xi }_j$, and the all-minus weight differs by $({\xi }_1+{\xi }_2)+({\xi }_3+{\xi }_4)$; hence every weight is below ${\lambda }_4$ in the root order. In ${\mathrm{\Delta }}^{-}$ the reference sign pattern $+++-$ has weight ${\lambda }_3$. Every other odd-minus pattern differs from it in an even number of signs: two flips among the first three give ${\xi }_i+{\xi }_j$, a flip at $i<4$ and at $4$ gives ${\xi }_i-{\xi }_4$, and four flips give $({\xi }_1+{\xi }_2)+({\xi }_3-{\xi }_4)$, all positive roots or sums of positive roots. Thus every weight is below ${\lambda }_3$. The vector weights are $\{\pm {\xi }_j\}$; each lies below ${\lambda }_1={\xi }_1$, since the differences are positive roots ${\xi }_1-{\xi }_j$ or ${\xi }_1+{\xi }_j$ for $j>1$, or $2{\xi }_1=({\xi }_1-{\xi }_2)+({\xi }_1+{\xi }_2)$ for $-{\xi }_1$. For $M=V_{\mathbb{C}},{\mathrm{\Delta }}^{+},{\mathrm{\Delta }}^{-}$, take $\mu ={\lambda }_1,{\lambda }_4,{\lambda }_3$, respectively, and $0\mathrm{\ne }v\in M_{\mu }$. For $x\in {\mathfrak{g}}_{\alpha }$, the relation $[H,x]=\alpha (H)x$ gives $xv\in M_{\mu +\alpha }$; if $xv$ were nonzero for a positive root $\alpha$, then $\mu +\alpha$ would be a weight strictly above the displayed maximal weight $\mu$, by the root-order definition in [F6]; hence every positive-root space annihilates its top weight line. By step 4.1 each module is irreducible, so those vectors generate the corresponding modules and their highest weights are ${\lambda }_1,{\lambda }_4,{\lambda }_3$ for $V_{\mathbb{C}},{\mathrm{\Delta }}^{+},{\mathrm{\Delta }}^{-}$, respectively.

The volume operator is the product of the oriented orthonormal Clifford generators; replacing the orientation reverses its sign and exchanges its $+1$ and $-1$ eigenspaces. Therefore the labels $8_s,8_c$ are exchanged when the chirality convention is reversed, while each fixed real space has dimension eight and is irreducible by steps 3.1 and 4.1.