玻色-费米对角共不变模猜想

三阶费米子反交换环的符号特征

The sign character of the triagonal fermionic coinvariant ring

研究三阶费米子反交换环R_n^(0,3)的符号特征的三重梯度乘重，并证明Bergeron(2020)的猜想，同时给出对角费米子反交换环R_n^(0,2)双钩特征的显式公式，并讨论计算R_n^(0,4)符号特征的方法。

$\hat{sl_2}$共不变量的玻色子公式

Bosonic formulas for $\hat{sl_2}$ coinvariants

我们推导出了$\hat{sl_2}$共不变量的特征标的玻色型公式。

A combinatorial model for the fermionic diagonal coinvariant ring

Let $Θ_n = (θ_1, \dots, θ_n)$ and $Ξ_n = (ξ_1, \dots, ξ_n)$ be two lists of $n$ variables and consider the diagonal action of $\mathfrak{S}_n$ on the exterior algebra $\wedge \{ Θ_n, Ξ_n \}$ generated by these variables. Jongwon Kim and Rhoades defined and studied the fermionic diagonal coinvariant ring $FDR_n$ obtained from $\wedge \{ Θ_n, Ξ_n \}$ by modding out by the $\mathfrak{S}_n$-invariants with vanishing constant term. In joint work with Rhoades we gave a basis for the maximal degree components of this ring where the action of $\mathfrak{S}_n$ could be interpreted combinatorially via n…

外代数的Lefschetz理论与费米子对角共不变量

Lefschetz theory for exterior algebras and fermionic diagonal coinvariants

令$W$是作用于其反射表示$V$的不可约复反射群。本文研究了外代数$W$在$\wedge (V \oplus V^*)$上的双分次作用以及它的商$DR_W := \wedge (V \oplus V^*)/ \langle \wedge (V \oplus V^*)^{W}_+ \rangle$与由它的齐次$W$ -不变量所生成的理想的关系，其中常数项为零.我们描述了$DR_W$的双阶同构类型，当$W = \mathfrak{S}_n$是对称群时，答案是钩形$\mathfrak{S}_n$ -模的Kronecker积的差.我们将$DR_W$的Hilbert级数与（A型）Catalan数和Narayana数联系起来，并使用Motzkin路径的变体描述了$DR_W$的标准单项式基。我们的方法是类型一致的，并涉及莱夫谢茨一样的理论，适用于外代数$\wedge (V \oplus V^*)$。

共不变III的$\hat{sl}_2$空间的组合学

Combinatorics of the $\hat{sl}_2$ Spaces of Coinvariants III

给出了由$\hat{\mathfrak sl}_2$的水平$k$可积表示得到的共不变量空间的费米子特征公式。我们建立了共不变空间的对偶空间的函数实现。我们利用操纵划分在对偶空间中参数化函数，并证明了操纵划分集的递归关系。

A proof of the fermionic Theta coinvariant conjecture

Let $(x_1, \dots, x_n, y_1, \dots, y_n)$ be a list of $2n$ commuting variables, $(θ_1, \dots, θ_n, ξ_1, \dots, ξ_n)$ be a list of $2n$ anticommuting variables, and $\mathbb{C}[X_n, Y_n] \otimes \wedge \{Θ_n, Ξ_n\}$ be the algebra generated by these variables. D'Adderio, Iraci, and Vanden Wyngaerd introduced the {\em Theta operators} on the ring of symmetric functions and used them to conjecture a formula for the quadruply-graded $\mathfrak{S}_n$-isomorphism type of $\mathbb{C}[X_n,Y_n] \otimes \wedge \{Θ_n, Ξ_n\}/I$ where $I$ is the ideal generated by $\mathfrak{S}_n$-invariants with vanishing…