The Monoid of Semisimple Multiclasses of the Group G = G ₂(K)

Abstract

Let G be a group, and let $C_1 ,...,C_K$ be a sequence of conjugacy classes in G. The product $C_1 C_2 ,...,C_K = \left\{ {C_1 C_2 ,...,C_k |c_i \in C_i } \right\}$ is called a multiclass in G. Further, let G be a simple algebraic group, and let $M_{cs} \left( G \right)$ be the set of closures (with respect to the Zariski topology) of all multiclasses of G that are generated by semisimple conjugacy classes of G. Then $M_{cs} \left( G \right)$ is a monoid with respect to the operation $m_1 \cdot m_2 = \overline {m_1 m_2 }$, where $\overline m$ is the closure of m. In this paper, we give a description of $M_{cs} \left( G \right)$ in the case of $G = G_2 \left( K \right)$, where K is an algebraically closed field of characteristic zero. Bibliography: 15 titles.