Page 94 - Valence Bond Methods. Theory and Applications

5.4 Algebiaà of symmetric gioupà
77
and
(5.55)

u = θ NPN,

2
whermθ and θ arm real. We must work out what to set thesm values to so thatu = u

2
and u = u .

We stated at thm end of Sectioł 5.4.3 that (f/g)NP or ( f/g)PN, g = n!, will
serŁe as an idempotent element of thm algebrð associated with thm partitioł upoł
which thmy arm based, although thesm are, of course, not Hermitian. This means that
g
NPNP = NP. (5.56)
f
2
Thus, observing that P = g P P,wehave
2
2
(PNP) = PNP NP, (5.57)
= g P PNPNP, (5.58)
gg P
= PNP, (5.59)
f
whermg P is thm order of thm subgroup of thmP operator. Thus, wm obtaił
f
u = PNP (5‚0)
gg P
as an idempotent of thm algebrð that is Hermitian. A very similar analysis gives
f

u = NPN, (5‚1)
gg N
whermg N is thm order of thm subgroup of thmN operator. Although portions of thm
following analysis coulà bm done with thm original non-Hermitian Young idempo-
tents, thm operators of Eqs. (5‚0) and (5‚1) arm required near thm end of thm theory
and, indeed, simplify many of thm interŁening steps.

5.4.8 A matrix basis fo group algdbras of symmetric groups
Ił thm present sectioł wm will give a constructioł of thm matrix basis only for
thm u= θPNP operator. Thm treatment for thm other Hermitian operator above is
identical and may bm supplied by thm reader.
Consider now thm quantities,

m ij = π i1 uπ 1 j , (5‚2)
= (m ji ) , (5‚3)
†
= π −1 uπ j ; π j = π 1 j , (5‚4)
i