Orbits of groups generated by transvections over $\FF_2$

Simply-laced Coxeter groups and groups generated by symplectic transvections

Let W be an arbitrary Coxeter group of simply-laced type (possibly infinite but of finite rank), u,v be any two elements in W, and i be a reduced word (of length m) for the pair (u,v) in the Coxeter group W\times W. We associate to i a subgroup Gamma_i in GL_m(Z) generated by symplectic transvections. We prove among other things that the subgroups corresponding to different reduced words for the same pair (u,v) are conjugate to each other inside GL_m(Z). We also generalize the enumeration result of the first three authors (see AG/9802093) by showing that, under certain assumptions on u and v, …

有限群的三维上三角表示的不变式环

Rings of invariants of three-dimensional upper-triangular representations of finite groups

本文证明了奇特征域上有限三维上三角群的不变式环为多项式环的充要条件，即极大幂幺子群由伪反射生成或不含平延。

p-Groups and the Polynomial Ring of Invariants Question

Let G be a finite p-subgroup of GL(V), where p = char(F), and V is finite-dimensional over the field F. Let S(V) be the symmetric algebra of V, S(V)^G the subring of G-invariants, and V* the F-dual space of V. The following presents our solution to the above question. Theorem A. Suppose dim(V)=3. Then S(V)^G is a polynomial ring if and only if G is generated by transvections. Theorem B. Suppose dim(V) > 3. Then S(V)^G is a polynomial ring if and only if: (1) S(V)^{G_U} is a polynomial ring for each subspace U of V* with dim(U)=2, where G_U = {g in G | g(u) = u, for all u in U}, and (2) S(V)^G …

计算仿射平面中轨道 Zariski 闭包的算法

Algorithm to Compute Orbit Zariski Closure in Affine Plane

本文提出一种算法，通过判断有限生成群是否包含于代数群，并计算单个三角自同构的不变子簇，进而计算仿射平面上有限生成群轨道的 Zariski 闭包。

Regular orbit closures in module varieties

Let A be a finitely generated associative algebra over an algebraically closed field. We characterize the finite dimensional modules over A whose orbit closures are regular varieties.

轨道闭包与矩阵群的判定问题

Determination Problems for Orbit Closures and Matrix Groups

本文针对矩阵群作用下的轨道闭包判定问题，提出多项式空间流程，可判定给定代数簇是否为s生成交换代数矩阵群作用下单点的轨道闭包，非交换场景留作开放问题。