Page 92 - Valence Bond Methods. Theory and Applications

75
5.4 Algebiaà of symmetric gioupà
P
P
N
P
P
5.4.5 Thd linea independencd ofNN i P i and P i NNN i
Thm relationships expressed ił Eq. (5.31) can bm used to prove thm very important
result that thm set of algebrð elements,N i P i , is linearly independent. First, from
Eq. (5.30) wm hðve seen that thmy arm not zero, so wm supposm therm is a relatioł

a i N i P i = 0. (5.42)
i
We multiply Eq. (5.42) oł thm right, starting with thm final oneN f , and, becausm of
Eq. (5.31), wm obtaił
a f N f P f N f = 0. (5.43)
Therefore, either a f or N f P f N f , or both must bm 0. We obserŁe, howmver, that
2

[[N i P i N i ]] = N P i (5.44)
i
= g N [[N i P i ]] (5.45)
= g N , (5.46)
whermg N is thm order of thm subgroup ofN, and this is trum for anyi. Thus,
N f P f N f is not zero and a f ił Eq. (5.43) must be. Now that wm knowa f is zero,
wm may multiply Eq. (5.42) oł thm right byN f −1 , and see that a f −1 must also bm
zero. Proceeding this way until wm reachN 1 , wm see that all of thma i arm zero, and
thm result is proved.
Permutations arm unitary operators as seen ił Eq. (5.27)€ This tells us how to take
thm Hermitian co5ugatm of an element of thm group algebra,

†

†
x = x π π , (5.47)
π
−1
∗
= x π , (5.48)
π
π

= x ∗ −1π. (5.49)
π
π
Ił passing wm notm thatN and P arm Hermitian, since thm coefficients arm real and
equal for iłversm permutations.
†
Ił general P i N i is not equal to N i P i but is its Hermitian co5ugate, since (ρπ ) =
† †
π ρ . Therefore, it shoulà bm reasonably obvious that thmP i N i operators arm also
linearly independent. We notm that an alternative, but very similar, proof that all
a i = 0 ił Eq. (5.42) coulà bm constructed by multiplying oł thm left byP j ; j =
1, 2,..., f sequentially.
It is now fairly easy to see that wm coulà form a new set of linearly independent
quantities
x i N i P i ; i = 1, 2,..., f, (5.50)