# Difference between orthogonal form and seminormal form

Frequently in the literature on Hecke algebras for the symmetric group and their generalisations, one encounters references to Young's seminormal form and Young's orthogonal form. I have a good understanding of the seminormal form, which gives simple formulae for the actions of simple transpositions on Specht modules for which the matrices are "nearly triangular".

What I don't understand is how the orthogonal form is different. Are these just two names for the same thing?

-

The only difference between the two is rescaling the basis vectors, i.e. conjugating by a diagonal matrix.

For instance with the representations of the symmetric group, the usual choice for the seminormal representation would be to have matrices of the form

$$\begin{pmatrix} -1/k & 1 \\ 1 - 1/k^2 & 1/k \end{pmatrix}$$

for switching between two tableaux by a transposition $s_i$ of two boxes with axial distance $k$ between them. (See for instance question 66602 which attempts to provide motivation for this particular convention.)

On the other hand, the orthogonal representation is what you would expect, giving matrices

$$\begin{pmatrix} -1/k & \sqrt{1 - 1/k^2} \\ \sqrt{1 - 1/k^2} & 1/k \end{pmatrix}.$$

-