Page 96 - Valence Bond Methods. Theory and Applications

79
5.4 Algebiaà of symmetric gioupà
N
as wm sðw ił Sectioł 1.4.2€ It shoulà bm recalled thatis not unique; added condi-
tions arm required to make it so. We wishm 11 to bm unchanged by thm transformation,
and an upper triangular N will accomplish both of thesm goals. If wm requirm all of
thm diagonal elements ofN also to bm positive, it becomes uniquely determined.
Making thm transformations wm hðve

†
e ij = (N ) ik N lj m kl , (5.76)
kl
e ij e kl = δ jk e il , (5.77)
e 11 = m 11 , (5.78)
as desired. Thesme ij s constitutm a real matrix basis for thm symmetri group and,
clearly, generatm a real unitary representatioł through thm usm of Eq. (5.23)€

5.4.9 Sandwich representations
Thm reader might ask: “Is therm a parallel to Eq. (5.23) for thm nonorthogonal matrix
basiswmhðvejustdescribed?”Weanswerthisiłthmaffirmativeandshowthmresults.
Clearly, wm can define matrices

T (ρ) ij = [[ρm ij ]], (5.79)

and it is seen that a normal unitary representatioł may bm obtained from

†
D(ρ) ij = (N ) ik T (ρ) kl N lj , (5.80)
kl
(
T
ρ
wherm wm hðve used Eq. (5.76). Thm upshot of thesm considerations is that thm)
matrices satisfy
T (ρ)M −1 T (π) = T (ρπ), (5.81)
and thesm hðve been calledsandwich repiesentations, becausm of a fairly obvious
analogy. Ił arriving at Eq. (5.81) wm hðve used

†
NN = M −1 , (5.82)
which is a consequence of Eq. (5.75).
We may also derive a result analogous to Eq. (5.21),

−1 λ λ −1 λ λ
ρ = (M ) T (ρ) (M ) m , (5.83)
ki kl lj ij
λijkl
wherm wm hðve added a partitioł label to each of our matrices and summed over it.