论文标题
lefschetz理论用于外部代数和费米子对角线共同变量
Lefschetz theory for exterior algebras and fermionic diagonal coinvariants
论文作者
论文摘要
让$ w $是一个不可约合的复杂反思小组，该小组在其反射表示$ v $上行动。我们考虑$ w $在外部代数$ \ wedge（v \ oplus v^*）$及其商$ dr_w：= \ wedge（v \ oplus v^*）/ \ langle \ wedge（v \ oplus v^*）agg y hom the $ fiustance the the Intuber the the Isport（ $ w $ invariants具有消失的持续学期。我们描述了$ dr_w $的大型同构类型；当$ W = \ Mathfrak {s} _n $是对称组时，答案是钩状$ \ mathfrak {s} _n $ -modules的Kronecker产品的区别。我们将Hilbert系列的$ DR_W $与（A型）加泰罗尼亚和Narayana编号联系起来，并使用Motzkin Paths的变体来描述$ DR_W $的标准单基础。我们的方法是类型均匀的，涉及一种类似lefschetz的理论，该理论适用于外部代数$ \ wedge（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 $\wedge (V \oplus V^*)$.
