论文标题
{t}他是加泰罗尼亚路径理想的基础
{T}he Gröbner Basis of a Catalan Path Ideal
论文作者
论文摘要
对于理想的$ 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 with Char（$ \ Mathbb f $）= 0 $g_α$以路径表示，在笛卡尔飞机上向东北移动，在最后一步保持在对角线上并越过对角线。这意味着商环$ r/i $的线性基础由一组加泰罗尼亚路径给出。我们表明，尺寸是$ n $和高度的标准Young Tableaux数量。 $ r/i $作为对称组模块的分级Frobenius特性由 $ \ sum_ {k = 0}^{\ lfloor \ frac {n} {2} \ rfloor} s_ {n-k，k} q^k $。
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 Gröbner basis with lex-order consists of polynomials $g_α$ 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 $\sum_{k=0}^{\lfloor \frac{n}{2} \rfloor } s_{n-k,k}q^k$.
