Let \(G\) be a group of (finite) order \(n\). Show that any irreducible left module over the group algebra \(\mathbb CG\) has complex dimension at most \(\sqrt n\).
Give an example of a group \(G\) of order \(n\geq5\) and an irreducible left module over \(\mathbb CG\) of complex dimension \(\lfloor\sqrt n\rfloor\), the greatest integer to \(\sqrt n\).
Solution
1. Let\[ V_1,\dots,V_r \]be representatives of the irreducible complex representations of \(G\), and write\[ d_i=\dim_{\mathbb C}V_i. \]The left regular representation decomposes as\[ \mathbb C G\cong\bigoplus_{i=1}^r V_i^{\oplus d_i}. \]
By Maschke’s theorem, the regular representation is completely reducible. The multiplicity of an irreducible character \(\chi_i\) in the regular character \(\chi_{\mathrm{reg}}\) is \[ \langle\chi_{\mathrm{reg}},\chi_i\rangle =\frac1{|G|}\sum_{g\in G}\chi_{\mathrm{reg}}(g)\overline{\chi_i(g)}. \] The regular character satisfies \[ \chi_{\mathrm{reg}}(1)=|G|=n, \qquad \chi_{\mathrm{reg}}(g)=0\quad(g\ne1). \] Hence \[ \langle\chi_{\mathrm{reg}},\chi_i\rangle =\overline{\chi_i(1)}=d_i. \] Thus \(V_i\) occurs with multiplicity \(d_i\).
2. Therefore\[ n=\sum_{i=1}^r d_i^2. \]
1. The regular module \(\mathbb CG\) has dimension \(|G|=n\), while the summand \(V_i^{\oplus d_i}\) has dimension \(d_i^2\).
3. Every irreducible \(\mathbb CG\)-module \(V\) satisfies\[ \dim_{\mathbb C}V\le\sqrt n. \]
If \(V\cong V_j\), then 2 gives \[ (\dim V)^2=d_j^2\le\sum_{i=1}^r d_i^2=n. \] Taking nonnegative square roots yields the claim.
4. Let \(G=S_3\), so\[ n=|S_3|=6 \qquad\text{and}\qquad \lfloor\sqrt n\rfloor=2. \]Consider the subspace\[ V=\{(z_1,z_2,z_3)\in\mathbb C^3:z_1+z_2+z_3=0\} \]of the permutation representation of \(S_3\).Then\[ \dim_{\mathbb C}V=2. \]
The coordinate-sum map \[ \mathbb C^3\longrightarrow\mathbb C, \qquad (z_1,z_2,z_3)\longmapsto z_1+z_2+z_3 \] is nonzero and linear, and \(V\) is its kernel. Hence \(V\) has codimension \(1\) in \(\mathbb C^3\) and therefore dimension \(2\). Permuting coordinates preserves the coordinate sum, so \(V\) is \(S_3\)-stable.
5. The \(S_3\)-module \(V\) in 4 is irreducible.
Since \(\dim V=2\), a proper nonzero invariant subspace would be a one-dimensional invariant line. Suppose such a line existed and let \[ 0\ne v=(a,b,c)\in V \] span it. The transpositions \((1\,2)\) and \((2\,3)\) preserve this line. Since each has square equal to the identity, its scalar action on the line is \(+1\) or \(-1\).
For \((1\,2)\), either \[ a=b \] or \[ c=0,\qquad a=-b. \] For \((2\,3)\), either \[ b=c \] or \[ a=0,\qquad b=-c. \] Checking the four combinations, together with \[ a+b+c=0, \] forces \(a=b=c=0\) in every case, contradicting \(v\ne0\). Hence no invariant line exists, and \(V\) is irreducible.
6. Thus \(S_3\) provides the requested sharp example:\[ \dim_{\mathbb C}V=2=\lfloor\sqrt6\rfloor. \]