Consider tensor fields on the flat -dimensional space with coordinates:

They are functions with indices: , where is the rank of the tensor. There are some differential equations which only have finite-dimensional spaces of solutions. For example:

The solutions of this equation are parameterized by constant antisymmetric tensors :

More generally, consider the equation:

We want to classify the solutions of this equation. Consider the Taylor expansion of near . Since Eq. (62) is homogeneous in , we can consider each order of the Taylor expansion separately. In other words, it is enough to consider a homogeneous polynomial of . Let us introduce auxiliary variable and consider the generating function:

Homogeneous polynomials of of the order form a finite-dimensional representation of , with the generators defined as follows:

Eq. (62) implies that is a highest weigh vector:

On the other hand, being a polynomial of the order in implies:

Therefore, the space of polynomial solutions of Eq. (62) decomposes into the direct sum of representations of dimension . They correspond to polynomials of degree in . We conclude that all solutions of Eq. (62) are polynomials of order in (not necessarily homogeneous).

Let us now consider a weaker equation. Instead of requiring be zero, we require the existence of such that:

(We can think of Eq. (64) as having a gauge symmetry , , and Eq. (65) as fixing the gauge.) The solutions of Eq. (64) are higher spin conformal Killing tensors. They correspond to traceless Killing tensors in AdS Mikhailov:2002bp. Given a traceless Killing tensor in AdS, we can consider the leading Taylor coefficient of its expansion around a point in AdS. It will satisfy Eq. (62) (with an additional condition ) implying that the space of solutions is finite-dimensional.