{T}he Gr\"obner Basis of a Catalan Path Ideal

TL;DR

This paper characterizes the reduced Gr"obner basis of a specific ideal in a polynomial ring using Catalan paths, revealing connections to Catalan numbers, Young tableaux, and symmetric group representations.

Contribution

It provides an explicit description of the Gr"obner basis for the Catalan path ideal and links the quotient's basis to Catalan paths and Young tableaux.

Findings

Reduced Gr"obner basis consists of path-based polynomials

Linear basis corresponds to Catalan paths

Dimension equals the number of certain Young tableaux

Abstract

For the ideal $I = \langle y_1 + \dots + y_n, y^2_1, \dots , y^2_n \rangle$ in $R = {\mathbb F}[y_1, \dots , y_n]$ with char($\mathbb F$) = 0, we show that the reduced Gr\"obner basis with lex-order consists of polynomials $g_\alpha$ that are represented in terms of paths, moving northeast in the Cartesian plane, that stay above the diagonal and cross the diagonal at the last step. This implies that a linear basis for the quotient ring $R/I$ is given by a set of Catalan paths. We show that the dimension is the number of standard Young tableaux of size $n$ and height at most two. The graded Frobenius characteristic of $R/I$ as a symmetric group module is given by $\sum_{k=0}^{\lfloor \frac{n}{2} \rfloor } s_{n-k,k}q^k$.