Show that if = {1,, } is a partition of the indices j1, ,jn and

Show that if ϒ = {υ1,…, υν} is a partition of the indices j1, …,jn and ω = e2πi/n is a primitive nth root of unity, then

∑

j

ω

j1+…+jn δ (υ1)…δ (υν) = {

where the sum extends over all positive integer vectors having components in the range

(l,n).

∂aij/∂a rs = −aisarj

∂a ij/∂rs = −a isa rj

.

2κ2 = E(X1 − X2)

2 3κ3 = E(X1 + ωX2 + ω

2X3)

3

(ω = e 2πi/3)

4κ4 = E(X1 + ωX2 + ω

2X3 + ω

3X4)

4

(ω = e 2πi/4)

0 if ϒ < 1 n if ϒ = 1