Multiline Queues with Spectral Parameters

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.

Connections with Other Constructions

We describe how our definition of q(u) (when q is a queue and u a word) is connected to the construction of [30], which was coined bully paths in [2], and the tableaux combinatorics of [8].

To clarify the connection, we recall our visualization of a queue q in (2.3) by a row of circles and squares. We modify this visualization slightly: Namely, we still represent each $i \in q$ by a circle as in Example 2.1, but we no longer represent the $i \notin q$ by squares. We refer to the circles as “boxes”.

1.1 Connection with Ferrari–Martin and Aas–Linusson MLQs

We first connect our MLQs with those from [30]. We will use the language of [2], where we are only considering “discrete MLQs” as we consider our ring to have a finite number of sites.

Consider a MLQ ${\mathbf {q}}= (q_1, q_2, \dotsc , q_{\ell })$. A labeling of ${\mathbf {q}}$ is a sequence of maps ${\mathbf {f}}= (f_1, \dotsc , f_{\ell })$, where $f_i :q_i \rightarrow \left[ i \right]$. We represent this by placing an $f_i(j)$, which we call the label, inside of the circle corresponding to $j \in q_i$. The canonical labeling${\mathbf {f}}_{{\mathbf {q}}}$ of ${\mathbf {q}}$ is the labeling $(f_1, \dotsc , f_{\ell })$ defined by

$$\begin{aligned} f_k(j) = \bigl ( q_k( \cdots q_1(1^n) \cdots ) \bigr )_j. \end{aligned}$$

(A.1)

For example, the labeling of each MLQ in Example 7.1 is precisely the circled values. When ${\mathbf {q}}$ is an ordinary MLQ, we can also construct this canonical labeling recursively as follows:

• 1. Set $q_0 = \emptyset$, and let $f_0 :q_0 \rightarrow \left[ 0 \right]$ be the trivial map.

• 2. For each $k = 0, 1, \ldots , \ell -1$:

  • Suppose $f_k :q_k \rightarrow \left[ k \right]$ is already defined. Let $\left( j_1, j_2, \ldots , j_r \right)$ be a list of all elements of $q_k$ in the order of increasing label in $f_k$; that is, $f_k(j_1) \le f_k(j_2) \le \cdots \le f_k(j_r)$. (The relative order between elements with equal label is immaterial.)

  • For $i = i_1, i_2, \ldots , i_r$, do the following: Find the first site j weakly to the right (cyclically) of i such that $j \in q_{k+1}$ and $f_{k+1}(j)$ is not set; then set $f_{k+1}(j) = f_k(i)$.

  • For all sites $j \in q_{k+1}$ for which $f_{k+1}(j)$ is not set yet, set $f_{k+1}(j) = k+1$.

  This defines $f_{k+1} :q_{k+1} \rightarrow \left[ k+1 \right]$.

Note that the canonical labeling ${\mathbf {f}}_{{\mathbf {q}}}$ are the elements in the circles in the graveyard diagram of ${\mathbf {q}}$ (as in Example 5.5).

Now consider the labeling procedure in [2, §2.2] given by k-bully paths. Note that a k-bully path from one queue to the next precisely corresponds to the path of a letter k under Phase II of our definition of a queue as a function on words. For example, the bully paths would correspond to the blue paths in Example 2.1. In addition, this is exactly the recursive labeling procedure given above. Thus, the labeling ${\mathbf {f}}_{{\mathbf {q}}}$ is equivalently constructed following the labeling procedure of [2] using bully paths.

1.2 Connection with Kohnert diagrams and Assaf–Searles theory

Next we relate the action of queues on words with the Kohnert labelings in [8, Def. 2.5] and the thread decomposition in [8, Def. 3.5]. We remark that the thread decomposition is the same as the Kohnert labeling when the shape is an antipartition (i.e. a weakly increasing sequence of positive integers). Roughly speaking, Kohnert diagrams are MLQs built of queues that live on a half-line (instead of a circle), and the construction of the Kohnert labeling (and the thread decomposition) is a standardization of the bully path construction.

In more detail (and using the notations of [8]): If $\alpha$ is a weak composition, and if $D \in {{\,\mathrm{KM}\,}}(\alpha )$ is a Kohnert diagram, then we view the columns of D as queues. This time, our queues are subsets of ${\mathbb {N}}$ (or ${\mathbb {Z}}$) instead of ${\mathbb {Z}}/n{\mathbb {Z}}$; thus, there is no “wrapping around”. We consider the reflection of D across the line $x = y$ as an MLQ ${\mathbf {q}}_D = (q_1, \dotsc , q_{\ell })$: namely, a cell in row i and column j in the reflected diagram corresponds to a $j \in q_i$, and $\ell$ is the number of columns in D. We then apply the bully path construction to the boxes of this reflected Kohnert diagram. To obtain the thread decomposition we need to distinguish paths with a fixed label such that these paths are also constructed by the bully path algorithm, where we consider the labels to be decreasing from left to right. Hence, this can be considered as a standardization of our construction or, equivalently, as fixing specific permutations for how the queues act on words.

Example A.1

Consider the thread decomposition of the Kohnert diagram

given in [8, Fig. 11]. Thus, the corresponding MLQ is $\left( \left\{ 2 \right\} , \left\{ 1,2,3 \right\} , \left\{ 2,3,4 \right\} \right)$; we can draw it using bully paths as follows:

where each bully path matches with a thread in the decomposition. Note that for each fixed k, the k-bully paths must be constructed from left to right.⁶ If not, the diamond and square in the bottom row would be interchanged.

Note that the distinction between ${\mathbb {N}}$ and ${\mathbb {Z}}/n{\mathbb {Z}}$ never arises since, for ${\mathbf {q}}_D$ and all i, we have

$$\begin{aligned} \left| \left\{ j \in q_i \mid j \ge k \right\} \right| \le \left| \left\{ j \in q_{i+1} \mid j \ge k \right\} \right| , \end{aligned}$$

(by [8, Lemma 2.2]), which means that there is no “wrapping” around the cylinder.

To obtain a Kohnert labeling from a Kohnert diagram of height K (for this, we require $n \gg 1$), we can construct an MLQ

$$\begin{aligned} \widetilde{{\mathbf {q}}}_D = (\widetilde{q}_1, \widetilde{q}_2, \dotsc , \widetilde{q}_{\widetilde{\ell }}, q_1, q_2, \dotsc , q_{\ell }) \end{aligned}$$

from the MLQ ${\mathbf {q}}_D = (q_1, q_2, \dotsc , q_{\ell })$ to obtain the correct labelings, where $\ell + \ell '$ is the largest label appearing in the Kohnert labeling. Indeed, a label added in column i comes from a $k \in q$ with $k > K$ for sufficiently many queues q before $q_i$. In particular, when a smaller label appears, it must be in the bottom row of the Kohnert diagram, which would correspond to the bully path wrapping around the cylinder. We then only consider the labels in the regime $k \in q_i$ for all $1 \le k \le K$ and all $1 \le i \le \ell$. We leave the precise details for the interested reader.

Example A.2

We consider the Kohnert labeling from Example A.1. The following MLQ, given as a graveyard diagram, is an MLQ that gives the corresponding Kohnert labeling:

where we have suppressed the $(i+1)$’s that appear in row i. Note that all circles that appear to the left of the dashed line correspond to the Kohnert labeling and that the labels match.