[Paper Review] The representation of the symmetric group on m-Tamari intervals

Mireille Bousquet‐Mélou, Guillaume Chapuy+1 more|Feb 27, 2012

Advanced Combinatorial Mathematics30 references4 citations

TL;DR

This paper characterizes the representation of the symmetric group $\mathfrak{S}_n$ on labelled $m$-Tamari intervals, proving an explicit formula for its character. Using a recursive decomposition of $m$-Tamari intervals and a refined generating function involving divided differences and derivatives, the authors derive that the dimension of the representation—i.e., the number of such intervals—is $(m+1)^n (mn+1)^{n-2}$, confirming a conjecture by Bergeron and Préville-Ratelle.

ABSTRACT

An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my}. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice and denoted by T_n^{m}, which generalizes the usual Tamari lattice T_n obtained when m=1. This lattice was introduced by F. Bergeron in connection with the study of diagonal coinvariant spaces in three sets of n variables. The representation of the symmetric group S_n on these spaces is conjectured to be closely related to the natural representation of S_n on (labelled) intervals of the m-Tamari lattice, which we study in this paper. An interval [P,Q] of T_n^{m} is labelled if the north steps of Q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps. The symmetric group S_n acts on labelled intervals of T_n^{m} by permutation of the labels. We prove an explicit formula, conjectured by F. Bergeron and the third author, for the character of the associated representation of S_n. In particular, the dimension of the representation, that is, the number of labelled m-Tamari intervals of size n, is found to be (m+1)^n(mn+1)^{n-2}. These results are new, even when m=1. The form of these numbers suggests a connection with parking functions, but our proof is not bijective. The starting point is a recursive description of m-Tamari intervals. It yields an equation for an associated generating function, which is a refined version of the Frobenius series of the representation. This equation involves two additional variables x and y, a derivative with respect to y and iterated divided differences with respect to x. The hardest part of the proof consists in solving it, and we develop original techniques to do so, partly inspired by previous work on polynomial equations with "catalytic" variables.

Motivation & Objective

• To characterize the character of the symmetric group $\mathfrak{S}_n$ acting on labelled $m$-Tamari intervals, as conjectured by Bergeron and Préville-Ratelle.
• To resolve the long-standing conjecture on the dimension of the representation, which counts the number of labelled $m$-Tamari intervals of size $n$.
• To develop a refined generating function encoding the Frobenius series of the representation using variables $x$, $y$, and a derivative in $y$.
• To solve a non-trivial functional equation involving iterated divided differences and a derivative, generalizing techniques from catalytic variable methods.

Proposed method

• The authors use a recursive decomposition of $m$-Tamari intervals to derive a functional equation for a refined generating function $F^{(m)}(t, \mathbf{1}; x, y)$.
• The generating function satisfies a partial differential equation involving a derivative in $y$ and iterated divided differences in $x$, reflecting the lattice structure and label permutations.
• They introduce a parametrization using formal power series $M(z)$, $L(z)$, and $K(z)$, related to $m$-ballot paths, bridges, and depth-graded paths.
• The solution is obtained by relating the parametrization to known combinatorial objects and verifying equivalence between two expressions for the generating function.
• The proof relies on combinatorial interpretations of series via lattice paths and the cycle lemma to establish identities between generating functions.
• A key step involves transforming the functional equation into a closed-form expression using algebraic manipulation and series inversion.

Experimental results

Research questions

• RQ1 What is the character of the symmetric group $\mathfrak{S}_n$ acting on labelled $m$-Tamari intervals, and how does it relate to the representation on diagonal coinvariant spaces?
• RQ2 What is the dimension of the $\mathfrak{S}_n$-representation on $m$-Tamari intervals, and does it match the conjectured formula $(m+1)^n (mn+1)^{n-2}$?
• RQ3 Can a functional equation encoding the Frobenius series of the representation be derived and solved using combinatorial generating functions with catalytic variables?
• RQ4 How do the recursive structure of $m$-Tamari intervals and the action of $\mathfrak{S}_n$ on labels lead to a closed-form expression for the generating function?
• RQ5 Is there a connection between the dimension formula and parking functions, even if the proof is not bijective?

Key findings

• The character of the $\mathfrak{S}_n$-representation on labelled $m$-Tamari intervals, evaluated at a permutation of cycle type $\lambda$, is $ (mn+1)^{\ell-1} $, where $\ell$ is the number of cycles in $\lambda$.
• The dimension of the representation, i.e., the number of labelled $m$-Tamari intervals of size $n$, is $ (m+1)^n (mn+1)^{n-2} $, confirming a conjecture by Bergeron and Préville-Ratelle.
• The refined generating function $F^{(m)}(t, \mathbf{1}; x, 1)$ admits a closed-form expression involving $u'$ and $z'$, given by $ \frac{(1+u')(1+z'u')}{u'(1-z')^{m+2}} \left( \frac{1+u'}{(1+z'u')^{m+1}} - 1 \right) $.
• The functional equation for the generating function is solved via a novel parametrization involving $M(z)$, $L(z)$, and $K(z)$, which encode $m$-ballot paths, bridges, and depth-graded paths.
• The solution is verified by showing equivalence between two parametrizations: one using $M(z)$ and another using $u'$ and $z'$, with $M = 1/(1 - z')$ and $u = u'(1 - z')/(1 + u'z')$.
• The method establishes a connection between the representation theory of $\mathfrak{S}_n$ and lattice path combinatorics, particularly through the use of divided differences and derivatives in generating functions.