# Row orthogonality vs Column orthogonality

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.

$$\newcommand{\Size}{\left\lvert #1 \right\rvert}\newcommand{\Irr}{\mathrm Irr}\newcommand{\norm}{\trianglelefteq}$$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
$$\begin{equation*} \#C_{G}(x) \ge \#C_{G/N}(xN). \end{equation*}$$
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