Row orthogonality vs Column orthogonality

Question

Let $G$ be a finite group.

If $K_1,K_2,\ldots, K_m$ are conjugacy classes of $G$ and if $\chi_1,\ldots,\chi_m$ are irreducible characters of $G$ over complex field, then we have row and column orthogonality relations:

• (row) $\frac{1}{|G|} \sum_{x\in G} \chi_i(x) \overline{\chi_j(x)}=\delta_{i,j}$.

• (column) $\sum_{i=1}^m \chi_i (K_r) \overline{\chi_i(K_s)} = \frac{|G|}{\sqrt{|K_r|\cdot |K_s|}}\delta_{r,s}$.

Q. Is there application of column-orthogonality, which we do not get easily by row-orthogonality?

Since one of the above relations can be derived from other, I do not know how much my question makes sense. The application of row-orthogonality, or more precisely, orthogonality of different irreducible character comes in proving that all the irreducible characters of $G$ form an orthonormal basis for the space of class functions on $G$; but what about use of column orthogonality?

One may say that if we know all the irreducible characters except one character $\psi$, then we can get it from column-orthogonality. But, it is much easier to cover $\psi$ from the remaining and the regular character than the column orthogonality. This raised question of practical use of column-orthogonality.

Answer 1

$\newcommand{\Size}[1]{\left\lvert #1 \right\rvert}\newcommand{\Irr}[0]{\mathrm Irr}\newcommand{\norm}{\trianglelefteq}$Let me rewrite the column orthogonality relations as \begin{equation*} \sum_{\chi \in \Irr(G)} \chi(x) \chi(y^{-1}) = \begin{cases} \#{C_{G}(x)} & \text{if $x$ and $y$ are conjugate,}\\ 0 & \text{otherwise.} \end{cases} \end{equation*} In particular, one gets \begin{equation*} \sum_{\chi \in \Irr(G)} \Size{\chi(x)}^{2} = \#C_{G}(x) \end{equation*} From this it follow immediately that if $N \norm G$, and $x \in G$, then

\begin{equation*} \#C_{G}(x) \ge \#C_{G/N}(xN). \end{equation*}

This can also be proved handily without characters, but the point is that it is an immediate consequence of the column orthogonality relations.