论文标题
在$ \ mathrm {gl}（n）$ - 谎言的模块结构nilpotent关联相对自由代数
On the $\mathrm{GL}(n)$-module structure of Lie nilpotent associative relatively free algebras
论文作者
论文摘要
令$ k \ left \ langle x \ right \ rangle $表示由特征$ 0 $的字段$ k $上的field $ k $上的一组$ x = \ {x_1，\ dots，x_n \} $生成的免费关联代数。让$ i_p $，对于$ p \ geq 2 $，表示$ k \ weft \ langle x \ langle x \ ryct \ rangle $ in Form $ $ $ [u_1，\ dots，u_p] $生成的双面理想，其中$ _1，$ u_1，u_1，u_p \ dots，u_p \ in u__p \ in k \ weft \ weft \ weft \ langle x \ rangle x \ rangle x \。我们讨论$ \ mathrm {gl}（n，k）$ - 商的模块结构$ k \ left \ langle x \ right x \ right \ rangle / i_ {p+1} $在标准对角线下，$ p \ geq 1 $。我们对分区$λ$的值进行了限制，以使不可修复的$ \ mathrm {gl}（n，k）$ - 模块$v_λ$出现在$ k \ left \ weft \ langle x \ rangle x \ ryn \ rangle \ rangle / i_ _ {p+1} $ as a $ \ mathrm {gl mathrm {gl} $ {作为一个应用程序，我们采用$ k = \ mathbb {c} $，我们考虑了不变的代数$（\ mathbb {c} \ left \ langle x \ rangle x \ rangle \ rangle / i_ {p+1}） $ \ mathrm {o}（n，\ mathbb {c}）$，$ \ mathrm {so}（n，\ mathbb {c}）$，或$ \ mathrm {sp}（2s，\ m mathbb {c}）$（for $ n = 2s $ n = 2s $）。由Domokos和Drensky定理，$（\ Mathbb {C} \ left \ langle x \ right \ rangle / i_ {p+1}）^g $有限地生成。在最小生成集中，我们在$（\ mathbb {c} \ langle x \ rangle / i_ {p+1}）^g $的$（\ mathbb {c} \ left \ langle x \ rangle x \ rangle x \ rangle x \ rangle x \ rangle x \ rangle x^g $中，我们给出了上限。 In a similar way, we consider also the algebra of invariants $(\mathbb{C}\left\langle X \right\rangle / I_{p+1})^{G}$, where $G=\mathrm{UT}(n, \mathbb{C})$, and give an upper bound on the degree of generators in a minimal generating set.这些结果从经典不变理论的角度从$ \ mathbb {c} \ left \ langle x \ rangle^g $中提供了有关不变性的有用信息。特别是，对于上面的所有$ g $，我们给出了一个标准，当$ \ mathbb {c} \ left \ langle x \ right \ rangle $属于$ i_p $时。
Let $K\left\langle X \right\rangle$ denote the free associative algebra generated by a set $X = \{x_1, \dots, x_n\}$ over a field $K$ of characteristic $0$. Let $I_p$, for $p \geq 2$, denote the two-sided ideal in $K\left\langle X \right\rangle$ generated by all commutators of the form $[u_1, \dots, u_p]$, where $u_1, \dots, u_p \in K\left\langle X \right\rangle$. We discuss the $\mathrm{GL}(n, K)$-module structure of the quotient $K\left\langle X \right\rangle / I_{p+1}$ for all $p \geq 1$ under the standard diagonal action. We give a bound on the values of partitions $λ$ such that the irreducible $\mathrm{GL}(n, K)$-module $V_λ$ appears in the decomposition of $K\left\langle X \right\rangle / I_{p+1}$ as a $\mathrm{GL}(n, K)$-module. As an application, we take $K = \mathbb{C}$ and we consider the algebra of invariants $(\mathbb{C}\left\langle X \right\rangle / I_{p+1})^G$ for $G = \mathrm{SL}(n, \mathbb{C})$, $\mathrm{O}(n, \mathbb{C})$, $\mathrm{SO}(n, \mathbb{C})$, or $\mathrm{Sp}(2s, \mathbb{C})$ (for $n=2s$). By a theorem of Domokos and Drensky, $(\mathbb{C}\left\langle X \right\rangle / I_{p+1})^G$ is finitely generated. We give an upper bound on the degree of generators of $(\mathbb{C}\left\langle X \right\rangle / I_{p+1})^G$ in a minimal generating set. In a similar way, we consider also the algebra of invariants $(\mathbb{C}\left\langle X \right\rangle / I_{p+1})^{G}$, where $G=\mathrm{UT}(n, \mathbb{C})$, and give an upper bound on the degree of generators in a minimal generating set. These results provide useful information about the invariants in $\mathbb{C}\left\langle X \right\rangle^G$ from the point of view of Classical Invariant Theory. In particular, for all $G$ as above we give a criterion when a $G$-invariant of $\mathbb{C}\left\langle X \right\rangle$ belongs to $I_p$.
