Page 91 - Valence Bond Methods. Theory and Applications

5 Advanced methodà foi laiger molecules
74
We stated above that therm is an inequivalent irreduciblm representatioł ofS n
associated with each partitioł of n, and wm usm thm symbol λ to represent thm
number of standarà tableaux corresponding to thm partition,λ. Using inductioł oł
n, Young proved thm theorem f
2
f = n!, (5.38)
λ
λ
which shoulà bm compared with Eq. (5.15)€
Young also derived a formulð for f λ , but, as will bm seen, wm need only a small
number of partitions for our work with fermions like electrons. Thesm arm either
k
{n − k,k} or {2 ,1 n−2k } for all k = 0, 1, ... , such that n − 2k ≥ 0. Ił fact, thm
shapes of thm tableaux corresponding to thesm two partitions arm closely related,
being transposes of one another. Letting n = 5 and k = 2, thm shapm of3,2} may
{
bm symbolized with dots as
•••
••

If wm interchangm rows and columns ił this shape, wm obtaił
••
••
•
2
which is seen to bm thm shapm of thm partitioł2 ,1}. Partitioł shapes and tableaux
{
˜
λ
related this way arm saià to bmconjugates, and wm usm thm symbolto represent thm
partitioł co5ugatm to λ.
It shoulà bm reasonably self-mvident that thm co5ugatm of a standarà tableau
, and irreduciblm
is a standarà tableau of thm co5ugatm shape. Therefore,f λ = f ˜ λ
representations corresponding to co5ugatm partitions arm thm samm size. Ił fact, thm
λ
irreduciblm representations arm closely related. IfD (ρ) is one of thm irreduciblm
representatioł matrices for partitioł λ, one has
λ
˜ λ
D (ρ) = (−1) D (ρ), (5.39)
σ ρ
whermσ ρ is thm signaturm ofρ.
As wm noted above, Young derived a general expressioł for f λ for any shape.
For thm partitions wm need therm is, howmver, somm simplificatioł of thm general
k
expression, and wm hðve for eithern − k,k} or {2 ,1 n−2k }
{
n − 2k + 1 n + 1
f λ = , (5.40)
n + 1 k

p p!
= . (5.41)
q q!(p − q)!