Multiline queues with spectral parameters

TL;DR

This paper introduces a spectral weight concept for multiline queues, proves its invariance under symmetric group actions, and provides determinant formulas, confirming several conjectures in the field.

Contribution

It defines spectral weights for multiline queues, proves their invariance under symmetric group actions, and derives determinant formulas, advancing understanding of related algebraic structures.

Findings

Spectral weight is invariant under symmetric group actions.

Determinant formulas for spectral weights are established.

The work confirms key conjectures in multiline queue theory.

Abstract

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.