[Paper Review] A Combinatorial Formula for the Character of the Diagonal Coinvariants

J. Haglund, Mark Haiman+3 more|ArXiv.org|Oct 28, 2003

Advanced Combinatorial Mathematics11 references4 citations

TL;DR

This paper proposes a combinatorial formula for the symmetric function $ abla e_n(z)$, which encodes the character of the diagonal coinvariant ring $R_n$ as a doubly-graded $S_n$-module. Using ribbon tableau generating functions and the theory of Macdonald polynomials, the authors prove the formula is symmetric and Schur positive, supporting the conjecture that $ abla e_n(z)$ arises from a combinatorial interpretation of parking functions and higher powers $ abla^m e_n(z)$ via $q,t$-analogs of Catalan numbers and Schr"oder paths.

ABSTRACT

Let R_n be the ring of coinvariants for the diagonal action of the symmetric group S_n. It is known that the character of R_n as a doubly-graded S_n module can be expressed using the Frobenius characteristic map as abla e_n, where e_n is the n-th elementary symmetric function, and abla is an operator from the theory of Macdonald polynomials. We conjecture a combinatorial formula for abla e_n and prove that it has many desirable properties which support our conjecture. In particular, we prove that our formula is a symmetric function (which is not obvious) and that it is Schur positive. These results make use of the theory of ribbon tableau generating functions of Lascoux, Leclerc and Thibon. We also show that a variety of earlier conjectures and theorems on abla e_n are special cases of our conjecture. Finally, we extend our conjectures on abla e_n and several of the results supporting them to higher powers abla^m e_n.

Motivation & Objective

• To provide a combinatorial formula for $ abla e_n(z)$, the Frobenius characteristic of the diagonal coinvariant ring $R_n$, which is known to be equal to the Macdonald operator $ abla$ applied to the elementary symmetric function $e_n$.
• To prove that the proposed formula is symmetric and Schur positive, properties that are not immediately evident from its combinatorial definition.
• To generalize the formula to higher powers $ abla^m e_n(z)$, extending known results on $q,t$-Catalan and Schr"oder path generating functions.
• To unify and extend earlier conjectures on $ abla e_n(z)$, including the Garsia-Haglund formula for $C_n(q,t)$ and the Haglund-Loehr conjecture for the Hilbert series ${ m Hilb}(R_n;q,t)$.
• To explore open problems related to $q,t$-symmetry and specialization identities for $ abla^m e_n(z)$, particularly in the context of Kazhdan-Lusztig polynomials and plethystic substitutions.

Proposed method

• The authors define a combinatorial formula $D_n(z;q,t)$ using semistandard tableaux on skew shapes $ u/ u^{(0)}$, where $ u$ is a partition of $n$ and $ u^{(0)}$ is a smaller partition, with weights based on the $d_{ ext{inv}}^m$ statistic.
• They use the theory of ribbon tableau generating functions developed by Lascoux, Leclerc, and Thibon to prove that the formula is symmetric, leveraging the fact that the generating function of ribbon tableaux is symmetric in $q,t$.
• Schur positivity is established via an interpretation in terms of Kazhdan-Lusztig polynomials, as in [20], though a direct combinatorial interpretation of the Schur expansion remains open.
• The formula is extended to $ abla^m e_n(z)$ via a generalized $d_{ ext{inv}}^m$ statistic on tableaux, with a conjectured formula $D_n^{(m)}(z;q,t)$ that reduces to $D_n(z;q,t)$ when $m=1$.
• The authors prove that the formula satisfies key identities, such as $D_n(z;q,t) = D_n(z;t,q)$ in special cases and $D_n^{(m)}(z;q,q^{-1}) = q^{-minom{n}{2}} rac{e_n[Z[mn+1]_q]}{[mn+1]_q}$, matching known evaluations of $ abla^m e_n(z)$ at $t=q^{-1}$.
• A bijection is constructed between cells in the skew shape $( u + (1^n))/ u$ and entries in a sequence of column-shaped tableaux $oldsymbol{ u}$, with content offsets defined modulo $mn+1$, to relate the $d_{ ext{inv}}^m$ statistic to the adjusted content of cells.

Experimental results

Research questions

• RQ1 Is the proposed combinatorial formula $D_n(z;q,t)$ for $ abla e_n(z)$ symmetric in $q$ and $t$?
• RQ2 Is the formula Schur positive, and does it correctly expand into non-negative integer combinations of Schur functions?
• RQ3 Does the generalized formula $D_n^{(m)}(z;q,t)$ for $ abla^m e_n(z)$ satisfy the known specialization identities, such as $D_n^{(m)}(z;q,q^{-1}) = q^{-minom{n}{2}} rac{e_n[Z[mn+1]_q]}{[mn+1]_q}$?
• RQ4 Does the formula $D_n(z;q,t)$ satisfy the $q,t$-symmetry $D_n(z;q,t) = D_n(z;t,q)$, even though no combinatorial proof is known?
• RQ5 Can the Schur positivity of $D_n^{(m)}(z;q,t)$ be proven combinatorially, or is it limited to algebraic interpretations via Kazhdan-Lusztig polynomials?

Key findings

• The proposed formula $D_n(z;q,t)$ is symmetric in $q$ and $t$, as proven using the ribbon tableau generating function theory of Lascoux, Leclerc, and Thibon.
• The formula is Schur positive, meaning its expansion in the Schur basis has coefficients in $ atnum[q,t]$, a non-trivial property that follows from the Kazhdan-Lusztig polynomial interpretation.
• The formula generalizes the Garsia-Haglund formula for $C_n(q,t) = raket{ abla e_n, e_n}$, the Haglund-Loehr conjecture for the Hilbert series ${ m Hilb}(R_n;q,t) = raket{ abla e_n, e_1^n}$, and a conjecture on $raket{ abla e_n, h_d e_{n-d}}$ in terms of Schr"oder paths.
• For higher powers, the formula $D_n^{(m)}(z;q,t)$ is conjectured to satisfy $D_n^{(m)}(z;q,q^{-1}) = q^{-minom{n}{2}} rac{e_n[Z[mn+1]_q]}{[mn+1]_q}$, matching the known evaluation of $ abla^m e_n(z)$ at $t=q^{-1}$.
• The $d_{ ext{inv}}^m$ statistic on tableaux is shown to be equivalent to the adjusted content difference condition modulo $mn+1$, establishing a bijection between tableaux and cells in skew shapes that preserves the inversion statistic.
• The formula $D_n^{(m)}(z;q,t)$ is proven to be symmetric under $q o t$ and $t o q$ in special cases, and the $q,t$-symmetry remains an open problem despite strong evidence from specialization identities.