A Remark by Philip Hall

Extract

The relationship between the representation theory of the full linear group GL(d) of all non-singular linear transformations of degree d over a field of characteristic zero and that of the symmetric group S_n goes back to Schur and has been expounded by Weyl in his classical groups, [4; cf also 2 and 3]. More and more, the significance of continuous groups for modern physics is being pressed on the attention of mathematicians, and it seems worth recording a remark made to the author by Philip Hall in Edmonton.

As is well known, the irreducible representations of S_n are obtainable from the Young diagrams [λ]=[λ₁, λ₂ ,..., λ_r] consisting of λ₁ nodes in the first row, λ₂ in the second row, etc., where λ₁≥λ₂≥ ... ≥λ_r and Σ λ_i = n. If we denote the j^th node in the i^th row of [λ] by (i,j) then those nodes to the right of and below (i,j), constitute, along with the (i,j) node itself, the (i,j)-hook of length h_ij.