Past exam of the groups and representations course of the University of Oxford / 2025 / 1

A group representation on a vector space $V$ is a group homomorphism $R:G\to \mathrm{GL}(V)$, so $R(gh)=R(g)R(h)$ and $R(e)=I$. A nonzero group representation is an irreducible representation if its only invariant subspaces are $0$ and $V$. Here the representations and characters are finite-dimensional over the complex numbers.

The character of a representation is ${\chi }_R(g)=\mathrm{tr}R(g)$. Three useful properties are:

In particular, ${\chi }_R(e)=\dim V$ and ${\chi }_R(g^{-1})={\chi }_R(g{)}^{\ast }$, the latter following from unitarization of a finite-group representation. The character inner product $⟨{\chi }_r,{\chi }_R⟩$ counts the multiplicity of an irreducible representation $r$ in $R$.

For any $h\in G$, left multiplication permutes the elements of the finite group. Consequently averaging over a finite group givesThus $Pv$ belongs to the fixed-point subspace of a group action $V^G$; any nonzero such vector spans a trivial representation, or singlet. The zero vector is included in the invariance statement.

Conversely $Pw=w$ for every $w\in V^G$, soThis is a projection operator onto all singlets, rather than necessarily onto a one-dimensional vector subspace. Its trace is the singlet multiplicity $|G{|}^{-1}{\sum }_g{\chi }_R(g)$.

Put $r=g_1$ and $s=g_2$. Direct matrix multiplication givesMoving every $s$ to the right shows that every word in these generators of a group has the form $r^a{s}^b$, with $a\in \{0,1,2\}$ and $b\in \{0,1\}$. The three $r^a$ are distinct diagonal matrices, and the three $r^{a}s$ are distinct off-diagonal matrices. HenceSince $srs=r^2\ne r$, this is a non-abelian group. Its group presentation identifies it with the dihedral group of an equilateral triangle, equivalently the symmetric group $S_3$.

The conjugacy classes areIndeed $s$ interchanges $r$ and $r^2$ by conjugation, while $r^{a}s{r}^{-a}=r^{2a}s$ runs through all three reflections. No reflection is conjugate to a rotation, since conjugation preserves order of a group element.

There are three inequivalent complex irreducible representations, one for each conjugacy class. The sum of squares of irreducible representation dimensions gives $d_1^2+d_2^2+d_3^2=6$. One representation is trivial, and the only possible positive integer dimensions are

Write $1$ for the trivial representation, $\epsilon$ for the sign representation, and $2$ for the given two-dimensional group representation. Their matrix representations on the generators of a group arewith $\alpha =e^{2\pi i/3}$. The two eigenvalues of $r$ in $2$ are distinct, and $s$ exchanges its two eigenlines; no line is invariant under both. Thus $2$ is an irreducible representation. The dimensions already exhaust the sum of squares of irreducible representation dimensions, so this is the complete list.

Since $\alpha +{\alpha }^{-1}=-1$, the character table isMultiplying characters and using character orthogonality gives every tensor product of group representations:For example ${\chi }_2^2=(4,1,0)={\chi }_1+{\chi }_{\epsilon }+{\chi }_2$. Explicit Clebsch-Gordan coefficients can be read from two doublets $x=(x_1,x_2{)}^T$, $y=(y_1,y_2{)}^T$:The first two combinations are invariant under $r$ and respectively even and odd under $s$; the last transforms by the displayed doublet matrices.

Choose the doublet bases so that the column of barred fields $B=({\bar{\psi }}_L^1,{\bar{\psi }}_L^2{)}^T$ and $R=({\psi }_R^1,{\psi }_R^2{)}^T$ both transform by the doublet matrices in d. This choice follows the source's assignment of the barred doublet itself. Since $H$ transforms in $\epsilon$, its Yukawa coupling must multiply a fermion bilinear in $\epsilon$.

There is exactly one such bilinear in $2\otimes 2$: the Clebsch-Gordan coefficients in d give $B_1{R}_2-B_2{R}_1$. Equivalently, writing the upper block of the Yukawa matrix as $A$, invariance requiresThe first equation sets $A_{11}=A_{22}=0$, because ${\alpha }^2\ne 1$; the second sets $A_{21}=-A_{12}$.

A mixed doublet-singlet bilinear remains a doublet after multiplication by $H$, so all four mixed entries vanish. Finally ${\bar{\psi }}_L^3{\psi }_R^3$ transforms in $\epsilon \otimes 1=\epsilon$, and multiplication by $H$ makes it invariant. ThereforeThe allowed Yukawa interaction is $H\{a({\bar{\psi }}_L^1{\psi }_R^2-{\bar{\psi }}_L^2{\psi }_R^1)+b{\bar{\psi }}_L^3{\psi }_R^3\}$, together with its Hermitian conjugate when forming a real Lagrangian density.

The displayed texture depends on basis. If instead the unbarred left doublet is assigned $D$ in the same ordering as the right doublet, its barred doublet has the dual representation $D^{\ast }$ and the equivalent upper block is $a\mathrm{diag}(1,-1)$. Exchanging the two barred components converts between these descriptions. In either convention there are exactly two independent complex Yukawa couplings.