To formalize the last step of your proof, let’s carefully work through the problem of showing that any linear permutation-equivariant function ( F: \mathbb{R}^n \to \mathbb{R}^n ) can be written as ( F(X) = aI X + b 11^T X ), where ( I ) is the identity matrix, ( 11^T ) is the matrix corresponding to the average function, and ( a, b \in \mathbb{R} ). The key property is that ( F ) is linear and permutation-equivariant, meaning ( FPX = PFX ) for any permutation matrix ( P ). Your insight about setting ( X = 11^T ) is a good starting point, and we’ll use it to derive the result.
Since ( F ) is a linear function from ( \mathbb{R}^n \to \mathbb{R}^n ), it can be represented by an ( n \times n ) matrix, say ( A ), such that ( F(X) = AX ). The permutation-equivariance condition ( FPX = PFX ) translates to:
[ A(PX) = P(AX) ]