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A basis of the alternating diagonal coinvariants

Yuhan Jiang

TL;DR

The paper constructs an explicit basis for the alternating component of the diagonal coinvariant ring $DR_n$ using antisymmetrized bivariate Vandermonde determinants $\Delta_\pi = \det(x_i^{d_j(\pi)} y_i^{a_j(\pi)})$ indexed by Dyck paths, recovering the $q,t$-Catalan numbers through the degree statistics $\deg_y$ and $\deg_x$. It situates this within the Carlsson–Oblomkov monomial basis for $DR_n$, and proves a unitriangular change of basis to establish the basis rigorously, thereby providing a combinatorial realization of $\mathcal{A}$ and its bigraded Hilbert series. The work extends to $q,t$-Fuß-Catalan combinatorics by proposing a conjectural basis built from an $m$-Dyck path decomposition into an $m$-tuple of Dyck paths, connected via the $\zeta$-map (sweep map) and bounce/area statistics. If true, the conjecture would yield a concrete basis for the alternating component of generalized diagonal coinvariants and illuminate the structure of rational Dyck-path statistics in the Fuss-Catalan regime. Overall, the approach provides a tangible algebraic/combinatorial framework that ties Vandermonde determinants to Dyck-path statistics and suggests a path toward generalized bases beyond the classical Catalan case.

Abstract

We construct an explicit vector space basis in terms of bivariate Vandermonde determinants for the alternating component of the diagonal coinvariant ring $DR_n$, answering a question of Stump. As a Corollary, we recover the combinatorial formula of the $q,t$-Catalan numbers. Moreover, we construct a decomposition of an $m$-Dyck path into an $m$-tuple of Dyck paths such that the area sequence and bounce sequence of the $m$-Dyck path is entrywise the sum of the area sequences and bounce sequences of the Dyck paths in the tuple.

A basis of the alternating diagonal coinvariants

TL;DR

The paper constructs an explicit basis for the alternating component of the diagonal coinvariant ring using antisymmetrized bivariate Vandermonde determinants indexed by Dyck paths, recovering the -Catalan numbers through the degree statistics and . It situates this within the Carlsson–Oblomkov monomial basis for , and proves a unitriangular change of basis to establish the basis rigorously, thereby providing a combinatorial realization of and its bigraded Hilbert series. The work extends to -Fuß-Catalan combinatorics by proposing a conjectural basis built from an -Dyck path decomposition into an -tuple of Dyck paths, connected via the -map (sweep map) and bounce/area statistics. If true, the conjecture would yield a concrete basis for the alternating component of generalized diagonal coinvariants and illuminate the structure of rational Dyck-path statistics in the Fuss-Catalan regime. Overall, the approach provides a tangible algebraic/combinatorial framework that ties Vandermonde determinants to Dyck-path statistics and suggests a path toward generalized bases beyond the classical Catalan case.

Abstract

We construct an explicit vector space basis in terms of bivariate Vandermonde determinants for the alternating component of the diagonal coinvariant ring , answering a question of Stump. As a Corollary, we recover the combinatorial formula of the -Catalan numbers. Moreover, we construct a decomposition of an -Dyck path into an -tuple of Dyck paths such that the area sequence and bounce sequence of the -Dyck path is entrywise the sum of the area sequences and bounce sequences of the Dyck paths in the tuple.

Paper Structure

This paper contains 10 sections, 12 theorems, 27 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. $q,t$-Catalan combinatorics
  3. Main theorem
  4. Carlsson--Oblomkov basis
  5. Proof of the main theorem
  6. $q,t$-Fuß-Catalan combinatorics
  7. Bounce
  8. The decomposition algorithm
  9. The sweep map
  10. A conjectural basis of alternating generalized diagonal coinvariants

Key Result

Theorem 1

For any Dyck path $\pi$ from $(0,0)$ to $(n,n)$, let $(a_1(\pi),\dots,a_n(\pi))$ be the area sequence of $\pi$ and let $(d_1(\pi),\dots,d_n(\pi))$ be the dinv sequence (see def:dinvpi) of $\pi$. Let The set of bivariate Vandermonde determinants $\{\Delta_\pi\}$ over all Dyck paths $\pi$ of semilength $n$ form a vector space basis for $\mathcal{A}/\langle \mathbf x,\mathbf y \rangle \mathcal{A}$ i

Figures (8)

  • Figure 1: The area sequence and dinv sequence of all Dyck paths in $L_{3,3}^+$.
  • Figure 2: We draw a Dyck path and its image under $\phi$.
  • Figure 3: A Dyck path on the left with bounce path on the right, with bounce sequence $(0,0,1,2)$.
  • Figure 4: A rational Dyck path in $L_{mn,m}^+$ for $m=2, n=3$, with bounce sequence (0,1,2).
  • Figure 5: A rational Dyck path (left) with bounce path shown in \ref{['fig:111']}, bounce sequence $(0,1,2)$, and area sequence $(0,1,2)$, and its decomposition into two Dyck paths (right).
  • ...and 3 more figures

Theorems & Definitions (42)

  • Theorem 1
  • Corollary 1
  • Definition 1
  • Example 1
  • Definition 2
  • Definition 3: lex-labelling
  • Proposition 1
  • proof
  • Definition 4: MR2371044
  • Theorem 2: MR2371044carlsson2025affineschubertcalculusdouble
  • ...and 32 more