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The representation of the symmetric group on m-Tamari intervals

Mireille Bousquet-Mélou, Guillaume Chapuy, Louis-François Préville-Ratelle

TL;DR

The paper determines the character of the symmetric group action on labelled $m$-Tamari intervals by introducing a refined Frobenius series with catalytic variables and solving a novel functional equation derived from a recursive interval construction. The authors obtain an explicit formula for $\chi_m(\sigma)$ depending only on the cycle type of $\sigma$, and hence the dimension of the representation, via a deep generating-function analysis that blends divided differences and Lagrange inversion. The work reveals a tight connection between $m$-Tamari intervals, parking functions, and diagonal coinvariant-space structures, while providing a powerful, though non-bijective, framework for refined enumeration and character calculations. The methodology yields exact enumerative formulas and suggests rich algebraic structures underlying Tamari-related representations, with potential extensions to $q$-analogues and unlabelled interval counts.

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.

The representation of the symmetric group on m-Tamari intervals

TL;DR

The paper determines the character of the symmetric group action on labelled -Tamari intervals by introducing a refined Frobenius series with catalytic variables and solving a novel functional equation derived from a recursive interval construction. The authors obtain an explicit formula for depending only on the cycle type of , and hence the dimension of the representation, via a deep generating-function analysis that blends divided differences and Lagrange inversion. The work reveals a tight connection between -Tamari intervals, parking functions, and diagonal coinvariant-space structures, while providing a powerful, though non-bijective, framework for refined enumeration and character calculations. The methodology yields exact enumerative formulas and suggests rich algebraic structures underlying Tamari-related representations, with potential extensions to -analogues and unlabelled interval counts.

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.

Paper Structure

This paper contains 24 sections, 22 theorems, 168 equations, 4 figures.

Table of Contents

  1. Introduction and main result
  2. The refined Frobenius series
  3. Definitions and notation
  4. A refined theorem
  5. A functional equation
  6. Recursive construction of Tamari intervals
  7. From the construction to the functional equation
  8. Principle of the proof, and the case $m=1$
  9. Principle of the proof
  10. The case $m=1$
  11. A homogeneous differential equation and its solution
  12. Reconstruction of $\tilde{G}(u,y)$
  13. The case $y=1$
  14. Solution of the functional equation: the general case
  15. A homogeneous differential equation and its solution
  16. ...and 9 more sections

Key Result

Theorem 2

Let $\lambda=(\lambda_1, \ldots, \lambda_\ell)$ be a partition of $n$ and $\sigma$ a permutation of ${\mathfrak S}_n$ having cycle type $\lambda$. Then for the $m$-Tamari representation of ${\mathfrak S}_n$, Since ${\mathfrak S}_n$ acts by permuting labelled intervals, this is also the number of labelled $m$-Tamari intervals left unchanged under the action of $\sigma$. The value of the character

Figures (4)

  • Figure 1: The covering relation between $m$-ballot paths ($m=2$).
  • Figure 2: The $m$-Tamari lattice $\mathcal{T}_{n}^{(m)}$ for $m=1$ and $n=4$ (left) and for $m=2$ and $n=3$ (right). The three walks surrounded by a line in $\mathcal{T}_{4}^{(1)}$ form a lattice that is isomorphic to $\mathcal{T}_{2}^{(2)}$ (see Proposition \ref{['prop:sublattice']}).
  • Figure 3: A labelled $2$-Tamari interval, and its image under the action of $\sigma= 2\, 3\, 5\, 6\, 1\, 4$.
  • Figure 4: The recursive construction of Tamari intervals.

Theorems & Definitions (38)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • Proposition 6: bousquet-fusy-preville
  • Proposition 7
  • Lemma 8
  • proof
  • Lemma 9
  • ...and 28 more