Let $W$ be an irreducible complex reflection group acting on its reflection representation $V$. We consider the doubly graded action of $W$ on the exterior algebra $\wedge (V \oplus V^*)$ as well as its quotient $DR_W := \wedge (V \oplus V^*)/ \langle \wedge (V \oplus V^*)^{W}_+ \rangle$ by the ideal generated by its homogeneous $W$-invariants with vanishing constant term. We describe the bigraded isomorphism type of $DR_W$; when $W = \mathfrak{S}_n$ is the symmetric group, the answer is a difference of Kronecker products of hook-shaped $\mathfrak{S}_n$-modules. We relate the Hilbert series of $DR_W$ to the (type A) Catalan and Narayana numbers and describe a standard monomial basis of $DR_W$ using a variant of Motzkin paths. Our methods are type-uniform and involve a Lefschetz-like theory which applies to the exterior algebra $\wedge (V \oplus V^*)$.

中文翻译：

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外代数和费米子对角共变体的 Lefschetz 理论

令 $W$ 是一个不可约复数反射群，作用于其反射表示 $V$。我们考虑 $W$ 对外部代数 $\wedge (V \oplus V^*)$ 及其商 $DR_W := \wedge (V \oplus V^*)/ \langle \wedge 的双重分级作用(V \oplus V^*)^{W}_+ \rangle$ 由具有消失常数项的齐次 $W$ 不变量生成的理想。我们描述了 $DR_W$ 的双级同构类型；当 $W = \mathfrak{S}_n$ 是对称群时，答案是钩形 $\mathfrak{S}_n$-modules 的 Kronecker 积的差。我们将 $DR_W$ 的希尔伯特级数与（A 型）Catalan 和 Narayana 数联系起来，并使用 Motzkin 路径的变体描述 $DR_W$ 的标准单项式基。