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.