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Cocharacters for the weak polynomial identities of the Lie algebra of 3 × 3 skew-symmetric matrices
Advances in Mathematics ( IF 1.5 ) Pub Date : 2020-11-01 , DOI: 10.1016/j.aim.2020.107343
Mátyás Domokos , Vesselin Drensky

Let $so_3(K)$ be the Lie algebra of $3\times 3$ skew-symmetric matrices over a field $K$ of characteristic 0. The ideal $I(M_3(K),so_3(K))$ of the weak polynomial identities of the pair $(M_3(K),so_3(K))$ consists of the elements $f(x_1,\ldots,x_n)$ of the free associative algebra $K\langle X\rangle$ with the property that $f(a_1,\ldots,a_n)=0$ in the algebra $M_3(K)$ of all $3\times 3$ matrices for all $a_1,\ldots,a_n\in so_3(K)$. The generators of $I(M_3(K),so_3(K))$ were found by Razmyslov in the 1980's. In this paper the cocharacter sequence of $I(M_3(K),so_3(K))$ is computed. In other words, the ${\mathrm{GL}}_p(K)$-module structure of the algebra generated by $p$ generic skew-symmetric matrices is determined. Moreover, the same is done for the closely related algebra of $\mathrm{SO}_3(K)$-equivariant polynomial maps from the space of $p$-tuples of $3\times 3$ skew-symmetric matrices into $M_3(K)$ (endowed with the conjugation action). In the special case $p=3$ the latter algebra is a module over a $6$-variable polynomial subring in the algebra of $\mathrm{SO}_3(K)$-invariants of triples of $3\times 3$ skew-symmetric matrices, and a free resolution of this module is found. The proofs involve methods and results of classical invariant theory, representation theory of the general linear group and explicit computations with matrices.

中文翻译:

3 × 3 斜对称矩阵李代数的弱多项式恒等式的共特征

令 $so_3(K)$ 是特征为 0 的域 $K$ 上的 $3\times 3$ 斜对称矩阵的李代数。弱的理想 $I(M_3(K),so_3(K))$多项式恒等式 $(M_3(K),so_3(K))$ 由自由结合代数 $K\langle X\rangle$ 的元素 $f(x_1,\ldots,x_n)$ 组成,其性质为$f(a_1,\ldots,a_n)=0$ 在所有 $3\times 3$ 矩阵的代数 $M_3(K)$ 中,对于所有 $a_1,\ldots,a_n\in so_3(K)$。$I(M_3(K),so_3(K))$ 的生成器是由 Razmyslov 在 1980 年代发现的。本文计算了$I(M_3(K),so_3(K))$的共字符序列。换句话说,由$p$泛型斜对称矩阵生成的代数的${\mathrm{GL}}_p(K)$-模结构被确定。而且,对于 $\mathrm{SO}_3(K)$-等变多项式映射的密切相关代数,从 $3\times 3$ 偏斜对称矩阵的 $p$-元组空间到 $M_3(K) $(具有共轭作用)。在特殊情况 $p=3$ 中,后一个代数是 $3\times 3$ skew- 的三元组的 $\mathrm{SO}_3(K)$-invariants 的代数中的 $6$-变量多项式子环上的模对称矩阵,并找到该模块的自由分辨率。证明涉及经典不变论、一般线性群的表示论和矩阵显式计算的方法和结果。在特殊情况下 $p=3$ 后一个代数是 $3\times 3$ skew-对称矩阵,并找到该模块的自由分辨率。证明涉及经典不变论、一般线性群的表示论和矩阵显式计算的方法和结果。在特殊情况下 $p=3$ 后一个代数是 $3\times 3$ skew-对称矩阵,并找到该模块的自由分辨率。证明涉及经典不变论、一般线性群的表示论和矩阵显式计算的方法和结果。
更新日期:2020-11-01
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