Jacobians of Reflection Groups

Module Structure on Invariant Jacobians

In this paper we show that a conjecture of Stephen Yau on highest weights of invariant Jacobians is true for arbitrary connected semisimple algebraic groups.

不变雅可比与齐次形式的中心

Invariant Jacobian and the Center of a Homogeneous Form

本文研究群作用下具有不变雅可比（invariant Jacobian）的齐次形式，通过中心定义的典范分解确定雅可比的模结构，并应用于对称群分类自然置换模上的此类形式。

Realizations of Unisingular Representations by Hyperelliptic Jacobians

A representation of a finite group $G$ on a finite dimensional vector space $V$ is called \textbf{unisingular} if every $g\in G$ has 1 as an eigenvalue in its action on $V$. In this paper we show that certain unisingular representations can be realized as mod 2 representations of hyperelliptic Jacobians over $\Q$. We additionally identify new unisingular representations of the symmetric and alternating groups.

Skew group algebras of Jacobian algebras

For a quiver with potential $(Q,W)$ with an action of a finite cyclic group $G$, we study the skew group algebra $ΛG$ of the Jacobian algebra $Λ= \mathcal P(Q, W)$. By a result of Reiten and Riedtmann, the quiver $Q_G$ of a basic algebra $η( ΛG) η$ Morita equivalent to $ΛG$ is known. Under some assumptions on the action of $G$, we explicitly construct a potential $W_G$ on $Q_G$ such that $η(ΛG) η\cong \mathcal P(Q_G , W_G)$. The original quiver with potential can then be recovered by the skew group algebra construction with a natural action of the dual group of $G$. If $Λ$ is self-injective, t…

A combinatorial identity for the Jacobian of $t$-shifted invariants

Let $\mathfrak g$ be a simple Lie algebra. There are classical formulas for the Jacobians of the generating invariants of the Weyl group of $\mathfrak g$ and of the images under the Harich-Chandra projection of the generators of $ZU(\mathfrak g)$. We present a modification of these formulas related to Takiff Lie algebras.

Two-dimensional families of hyperelliptic jacobians with big monodromy

Let $K$ be a global field of characteristic different from 2 and $u(x)\in K[x]$ be an irreducible polynomial of even degree $2g\ge 6$, whose Galois group over $K$ is either the full symmetric group $S_{2g}$ or the alternating group $A_{2g}$. We describe explicitly how to choose (infinitely many) pairs of distinct elements $t_1, t_2$ of $K$ such that the $g$-dimensional jacobian of a hyperelliptic curve $y^2=(x-t_1)(x-t_2))u(x)$ has no nontrivial endomorphisms over an algebraic closure of $K$ and has big $\ell$-adic monodromy.