  As part of the proof of the Bethe ansatz conjecture for the Gaudin model for
$\mathfrak{gl}_n$, Mukhin, Tarasov, and Varchenko described a correspondence
between inverse Wronskians of polynomials and eigenspaces of the Gaudin
Hamiltonians. Notably, this correspondence afforded the first proof of the
Shapiro-Shapiro conjecture. In the present paper, we give an identity in the
group algebra of the symmetric group, which allows one to establish the
correspondence directly, without using the Bethe ansatz.
