Past exam of the groups and representations course of the University of Oxford / 2024 / 3

For a real linear change of coordinates $x^{\prime }=Ax$, one has $d{x}^{\prime }=Adx$. The Euclidean metric therefore transforms toIt agrees with $d{x}^{T}dx$ for every displacement precisely when $A^{T}A=I$. For example, equality of the associated quadratic forms determines their symmetric coefficient matrices by polarization identity. The orthogonal group is defined by this condition, soTaking determinants gives $(\det A{)}^2=1$. Thus these matrices are invertible automatically; both signs of the determinant are allowed, since the metric alone does not specify an orientation.

Write $x=(x_1,x_2,x_3{)}^T$, $v=(x_4,x_5,x_6{)}^T$ and $z=x+iv$. Here products of differentials in a metric tensor are symmetric products, unlike the wedge products used later. ThenIndeed $d{z}_{k}d{\bar{z}}_k=d{x}_k^2+d{x}_{k+3}^2$. A transformation $z^{\prime }=Uz$ preserves this metric whenever $U^{†}U=I$, becauseThus the entire unitary group $U(3)$ preserves it, including its special unitary group $SU(3)$.

To obtain a real matrix embedding, write $U=A+iB$ with $A,B$ real. ThenThe unitarity equations imply $A^{T}A+B^{T}B=I$ and $A^{T}B=B^{T}A$, hence $\rho (U{)}^T\rho (U)=I$. Multiplication of the block matrices gives $\rho (UV)=\rho (U)\rho (V)$, and the blocks recover $U$, proving injectivity. FinallyFor this last identity, over the complex numbers the change of basis from $(x,v)$ to $(z,\bar{z})$ conjugates $\rho (U)$ to $\mathrm{diag}(U,\bar{U})$. This proves that the image lies in the special orthogonal group.

The change of variables in b diagonalizes the complexified action into $z\mapsto Uz$ and $\bar{z}\mapsto \bar{U}\bar{z}$. Consequently the conventional complex branching rule isPrecisely, this is a statement about the complexification of a real vector space carrying the real defining representation. Over the real numbers, the six-dimensional space is the realification of a complex representation $3$; the conjugate summands above are not separate three-dimensional real invariant subspaces.

The block embedding $SO(6)\hookrightarrow SO(7)$ fixes the seventh coordinate. It adds a trivial representation, givingThe dimensions are complex dimensions after complexification of a real vector space, or equivalently $6+1$ as real dimensions before it.

Let ${\rho }_3(P)=P\otimes P\otimes P$ act on the alternating coefficient tensor $\phi$, and setThe coefficient convention here is the one in the source's hint; using the usual covariant pullback gives the same stabilizer subgroup, since an invertible transformation preserves a differential form if and only if its inverse does. The identity matrix fixes $\phi$. If $P,Q$ fix it, then ${\rho }_3(PQ)\phi ={\rho }_3(P){\rho }_3(Q)\phi =\phi$. Multiplying ${\rho }_3(P)\phi =\phi$ by ${\rho }_3(P^{-1})$ shows that $P^{-1}$ fixes it as well. Thus $H$ is a subgroup of $SO(7)$. Its defining equations are polynomial, so it is closed and hence a Matrix Lie group.

For $P(t)=I+tT+O(t^2)\in H$, differentiating $P^{T}P=I$ gives $T^T=-T$. Differentiating the three factors in ${\rho }_3(P(t))\phi =\phi$ givesThe skew-symmetric matrix condition records the ambient special orthogonal group; it must be included when differentiating that group's definition.

These linear conditions are sufficient as well as necessary. On the tensor product, the infinitesimal action isIts three terms commute, so ${\rho }_3(e^{tT})=e^{tD(T)}$. If $D(T)\phi =0$, every positive power of $D(T)$ annihilates $\phi$, giving $e^{tD(T)}\phi =\phi$. Also $T\in \mathfrak{so}(7)$ implies $e^{tT}\in SO(7)$. Therefore $e^{tT}\in H$, proving the asserted Lie algebra description. For the given positive three-form this is the compact G2 Lie algebra ${\mathfrak{g}}_2$.

For each complex coordinate, exchanging the factors of a wedge product of differential forms reverses its sign, givingConsequently the Kähler form is $\omega =d{x}_{14}+d{x}_{25}+d{x}_{36}$, and $\omega \wedge d{x}_7=d{x}_{147}+d{x}_{257}+d{x}_{367}$. Expanding the holomorphic volume form givesThe sign of $d{x}_{426}$ changes on exchanging $4,2$, while reordering $4,5,3$ to $3,4,5$ requires two exchanges. This verifies every sign inUnder $z\mapsto Uz$, unitarity preserves $\omega$:The determinant is the factor by which a transformation acts on a top exterior power, so ${\Omega }^{\prime }=(\det U)\Omega =\Omega$ for $U\in SU(3)$. Extend the real embedding by fixing $y$. It preserves $\omega$, $dy$ and $\Omega$, and therefore $\varphi$. ThusThe induced seven-dimensional representation is exactly the restriction already computed in c. In the conventional complex notation its branching rule isOver the real numbers, the six-dimensional summand is the underlying real space of the complex fundamental representation, and the last real line is the fixed $y$ axis.