Lefschetz Theory for Exterior Algebras and Fermionic Diagonal Coinvariants
Lefschetz Theory for Exterior Algebras and Fermionic Diagonal Coinvariants
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DOI：
10.1093/imrn/rnaa203
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发表时间：
2020-03
影响因子：
1
通讯作者：
Jongwon Kim;B. Rhoades
中科院分区：
文献类型：
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作者：
Jongwon Kim;B. Rhoades
设$W$是作用于其反射表示$V$上的不可约复反射群。我们考虑了$W$在外代数$wedge(V+V^*)$上的双重分次作用，以及它的商$dr_W：=\wedge(V\o+V^*)/\lange\wedge(V\o+V^*)^{W}+\Range$是由它的齐次$W$不变量与零常数项生成的理想所生成的。我们刻画了$DR_W$的二重同构类型；当$W={{\mathfrak{S}}_n$是对称群时，答案是钩形${{\mathfrak{S}_n$-模的Kronecker乘积之差。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，并且涉及一个类似Lefschetz的理论，该理论适用于外代数$\wdge(V\oplus V^*)$。
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 $\wedge (V \oplus V^*)$.