The Great Orthogonality

We define a new term which helps greatly in stating and in understanding the Great Orthogonality. Consider the irreducible representations of some group. In the figure below, we take the octahedral group O, with reps named A ₁ , A ₂ , E, T ₁ , and T ₂ . Each representation is a List of h matrices, where h is the group order. In each rep, think of the h matrices as stacked up like a deck of cards. (For the octahedral group, h is 24, but we show only the first 6 below.) For the two T reps the matrices are 3×3; for E, 2×2, for the two A reps, 1×1. Now thrust a skewer through the deck at the i,j position, making a shishkebab of all the i,j elements in the stack of matrices. This list we call the “skewer of the matrix list at the i,j position”.