We say that a multilinear polynomial $P(x_1,\ldots,x_n)$ in $n$ commuting variables over $\mathbb{R}$ has zero trace if $$ \frac{d}{dt} P(t,\ldots,t) = 0. $$ Equivalently, $$ \left(\sum_{i=1}^n \frac{\partial}{\partial x_i}\right) P = 0. $$ (Suggestions for a better name are welcome.)
Such functions come up, for example, when considering the Johnson association scheme: every function on $\binom{[n]}{k} = \{(x_1,\ldots,x_n\}) \in \{0,1\}^n : \sum_i x_i = k\}$ can be represented as a zero trace multilinear polynomial of degree at most $k$ (assuming $k \leq n/2$); see Bannai and Ito, Association schemes, Proposition III.2.7.
For $d \leq n/2$, the dimension of the linear space of all zero trace multilinear polynomials of degree at most $d$ is $\binom{n}{d}$.
Are there known orthogonal bases for the linear span of all zero-trace multilinear polynomials in $n$ variables of degree at most $n/2$?
Orthogonality is with respect to the uniform measure on $\binom{[n]}{k}$, though the orthogonal basis proposed below is orthogonal with respect to any measure which is permutation-invariant.
Is the following orthogonal basis known?
Here is a conjectured orthogonal basis which can be gleaned from Young's orthogonal representation of $S_n$. Given two sequences $A = a_1,\ldots,a_d$ and $B = b_1,\ldots,b_d$ of distinct numbers in $[n]$, we say that $A < B$ if $A$ and $B$ are disjoint and $a_i < b_i$ for all $i$. We say that a sequence $B$ is a top set if $B$ is increasing and there exists a sequence $A$ smaller than $B$. It turns out that there are $\binom{n}{d} - \binom{n}{d-1}$ top sets of size $d$, a fact mentioned by Frankl and Graham (this can be proved using Bertrand's ballot problem). For each top set $B$ of size $d$ there is a basis vector $$ \sum_{A < B} \prod_{i=1}^d (x_{a_i} - x_{b_i}). $$ For example, for $d=1$ the $n-1$ basis vectors are $$ \sum_{i=1}^{b-1} (x_i - x_b), \quad 2 \leq b \leq n. $$
