Multiline queues were introduced by Ferrari and Matrin as a tool for understanding the steady state of the Totally Asymmetric Simple Exclusion Process (TASEP) on a ring. Since then, they have attracted independent interest as combinatorial objects. A queue can be described as a transformation of words by a combinatorial rule (related to the Lascoux-Schützenberger action of the symmetric group). A multiline queue is merely a tuple of queues and can be applied to a word by successively applying each of the queues to it. Thus, for each given word, we can define a generating function (the “spectral weight” of the word) by summing over all multiline queues that give rise to this word. We prove a symmetry property as well as some Pieri-type and Jacobi-Trudi-type formulas for these generating functions, proving a conjecture by Aas and Linusson. As a consequence of this symmetry, we further prove a conjecture by Arita, Ayyer, Mallick and Prolhac about operators related to the TASEP.

Joint work with Erik Aas and Travis Scrimshaw.

The talk is based on a paper in progress:
http://www.cip.ifi.lmu.de/~grinberg/algebra/mlqs.pdf