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arXiv · 2201.04006

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

Nantel Bergeron·Xavier Mootoo·Vedarth Vyas
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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 $\sum_{k=0}^{\lfloor \frac{n}{2} \rfloor } s_{n-k,k}q^k$.

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Nantel Bergeron, Xavier Mootoo, Vedarth Vyas. 2022-01-11. {T}he Gr\"obner Basis of a Catalan Path Ideal. https://arxiv.org/abs/2201.04006

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