  Using the description of multiline queues as functions on words, we introduce
the notion of a spectral weight of a word by defining a new weighting on
multiline queues. We show that the spectral weight of a word is invariant under
a natural action of the symmetric group, giving a proof of the commutativity
conjecture of Arita, Ayyer, Mallick, and Prolhac. We give a determinant formula
for the spectral weight of a word, which gives a proof of a conjecture of the
first author and Linusson.
