An orthogonal basis for functions over a slice of the Boolean cube

Yuval Filmus
Electronic Journal of Combinatorics

The Johnson and Kneser are graphs defined on the $k$-sets of $[n]$, a vertex set known as a slice of the Boolean cube. Two sets are connected in the Johnson graph if they have Hamming distance two, and in the Kneser graph if they are disjoint. Both graphs belong to the Bose–Mesner algebra of the Johnson association scheme; this just means that whether an edge connects two sets $S,T$ depends only on $|S \cap T|$.

All graphs belonging to the Bose–Mesner algebra have the same eigenspaces, and these are well-known, arising from certain representations of the symmetric group. The multiplicity of the $k$th eigenspace is rather large, ${n \choose k} – {n \choose k-1}$. As far as we can tell, prior to our work no explicit orthogonal basis for these eigenspace has been exhibited.

We present a simple orthogonal basis for the eigenspaces of the Bose–Mesner algebra of the Johnson association scheme, arising from Young’s orthogonal basis for the symmetric group. Our presentation is completely elementary and makes no mention of the symmetric group.

As an application, we restate Wimmer’s proof of Friedgut’s theorem for the slice. The original proof makes heavy use of computations over the symmetric group. We are able to do these computations directly in the slice using our basis.

Update: Qing Xiang and his student Rafael Plaza pointed out that the same basis has been constructed by Murali K. Srinivasan in his paper Symmetric chains, Gelfand–Tsetlin chains, and the Terwilliger algebra of the binary Hamming scheme.

Another update: The proof of Theorem 3.1 was clarified according to suggestions of Bruno Loff. See also this writeup for another approach, relying on the work of Murali K. Srinivasan mentioned above.


 author = {Yuval Filmus},
 title = {Orthogonal basis for functions over a slice of the
 {B}oolean hypercube},
 journal = {Electronic Journal of Combinatorics},
 volume = {23},
 number = {1},
 year = {2016},
 pages = {P1.23}
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