序集划分、广义余不变代数和Delta猜想
Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture
摘要：
对称群$\mathfrak{S}_n$通过变量置换作用在多项式环$\mathbb{Q}[\mathbf{x}_n] = \mathbb{Q}[x_1, \dots, x_n]$上。不变理想$I_n$是由所有具有消失常数项的$\mathfrak{S}_n$ -不变多项式生成的理想。商$R_n = \frac{\mathbb{Q}[\mathbf{x}_n]}{I_n}$称为共不变代数。上不变代数$R_n$在代数和几何组合学中得到了大量的研究。我们引入了由两个正整数$k \leq n$索引的理想$I_n$的推广$I_{n,k} \subseteq \mathbb{Q}[\mathbf{x}_n]$。相应的商$R_{n,k} := \frac{\mathbb{Q}[\mathbf{x}_n]}{I_{n,k}}$携带$\mathfrak{S}_n$的分级动作，并且当$k = n$时专用于$R_n$。我们将$R_n$的许多优良特性推广到$R_{n,k}$。特别地，我们描述了$R_{n,k}$的Hilbert级数，给出了$R_n$的Artin和Garsia-Stanton单项式基到$R_{n,k}$的推广，确定了$I_{n,k}$关于字典序单项式阶的约化Gröbner基，并描述了$R_{n,k}$的分次Frobenius级数.正如$R_n$的组合学是由$\mathfrak{S}_n$中的排列控制的一样，我们将证明$R_{n,k}$的组合学是由带有$k$块的$\{1, 2, \dots, n\}$的有序集划分控制的。Haglund，Remmel，and Wilson的{ \em Delta Conjecture}是Shuffle Conjecture在对角协不变理论中的推广。我们将表明，分级Frobenius系列的$R_{n,k}$是（到一个轻微的扭曲）$t = 0$专业化的组合侧的三角洲猜想。给出一个双阶化的$\mathfrak{S}_n$ -模$V_{n,k}$，它的Frobenius像甚至与Delta猜想中的任何一个表达式完全相等，这仍然是一个开放的问题;我们的模$R_{n,k}$在特殊化$t = 0$中解决了这个问题。
英文摘要：
The symmetric group $\mathfrak{S}_n$ acts on the polynomial ring $\mathbb{Q}[\mathbf{x}_n] = \mathbb{Q}[x_1, \dots, x_n]$ by variable permutation. The invariant ideal $I_n$ is the ideal generated by all $\mathfrak{S}_n$-invariant polynomials with vanishing constant term. The quotient $R_n = \frac{\mathbb{Q}[\mathbf{x}_n]}{I_n}$ is called the coinvariant algebra. The coinvariant algebra $R_n$ has received a great deal of study in algebraic and geometric combinatorics. We introduce a generalization $I_{n,k} \subseteq \mathbb{Q}[\mathbf{x}_n]$ of the ideal $I_n$ indexed by two positive integers $k \leq n$. The corresponding quotient $R_{n,k} := \frac{\mathbb{Q}[\mathbf{x}_n]}{I_{n,k}}$ carries a graded action of $\mathfrak{S}_n$ and specializes to $R_n$ when $k = n$. We generalize many of the nice properties of $R_n$ to $R_{n,k}$. In particular, we describe the Hilbert series of $R_{n,k}$, give extensions of the Artin and Garsia-Stanton monomial bases of $R_n$ to $R_{n,k}$, determine the reduced Gröbner basis for $I_{n,k}$ with respect to the lexicographic monomial order, and describe the graded Frobenius series of $R_{n,k}$. Just as the combinatorics of $R_n$ are controlled by permutations in $\mathfrak{S}_n$, we will show that the combinatorics of $R_{n,k}$ are controlled by ordered set partitions of $\{1, 2, \dots, n\}$ with $k$ blocks. The {\em Delta Conjecture} of Haglund, Remmel, and Wilson is a generalization of the Shuffle Conjecture in the theory of diagonal coinvariants. We will show that the graded Frobenius series of $R_{n,k}$ is (up to a minor twist) the $t = 0$ specialization of the combinatorial side of the Delta Conjecture. It remains an open problem to give a bigraded $\mathfrak{S}_n$-module $V_{n,k}$ whose Frobenius image is even conjecturally equal to any of the expressions in the Delta Conjecture; our module $R_{n,k}$ solves this problem in the specialization $t = 0$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
