Dilatation operator is defined as
(1)
Let's see how it commutes with grading operator defined as
(2)
where sum is taken over Young diagrams and , are power sums and are just some numerical coefficients.
The first and simplest step is to compute
(3)
so
(4)
In other words
(5)
The next step is to evaluate
(6)
Therefore,
(7)
Taking all the pieces together
(8)
As a result
(9)
The commutator of with grading operator (2) by linearity is
(10)
That's why is also called grading operator.