The monoid of semisimple multiclasses of the group $G=G_2(K)$

M. N. Kornienko

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 6, Tome 265 (1999), pp. 202-221

Résumé

Let $G$ be a group, and let $C_L,\ldots,C_K$ be a sequence of conjugacy classes of $G$. The product $C_1C_2\ldots C_K=\{c_1c_2\ldots c_k\mid c_i\in C_i\}$ is called a multiclass of $G$. Further, let $G$ be a simple algebraic group, and let $M_{cs}(G)$ be the set of closures (with respect to Zariski topology) of all multiclasses of $G$ which are generated by semisimple conjugacy classes of $G$. Then $M_{cs}(G)$ is a monoid with respect to the operation: $m_1\cdot m_2=\overline{m_1m_2}$, where $\overline m$ is the closure of $m$. In this paper we give a description of $M_{cs}(G)$ in the case $G=G_2(K)$, where $K$ is an algebraically closed field of the characteristic zero.