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**Two remarks on homogeneous varieties of representations of groups.**
*(Russian)*
Zbl 0705.20005

Let F be the free group with basis \(X=\{x_ i|\) \(i\in {\mathbb{N}}\}\) and kF be its group ring over a field k, char \(k\neq 2,3\). The Magnus embedding of kF into the algebra of formal power series \(k[[z_ i|\) \(i\in {\mathbb{N}}]]\) is given by \(x_ i\mapsto 1+z_ i\) (i\(\in {\mathbb{N}})\). To any \(u=u(x_ 1,...,x_ m)\in kF\) there corresponds the series \(\hat u\) which can be uniquely represented as the sum \(\sum^{\infty}_{n=0}u_{<n>}\) of its homogeneous components, the nth component \(u_{<n>}\) being a finite sum of monomials \(\lambda_{(i)}z_{i_ 1}...z_{i_ n}\) of degree n with coefficients \(\lambda_{(i)}\) in k.

A variety \({\mathfrak X}\) of group representations (over k) is said to be homogeneous iff from \(\hat u\) being an identity for \({\mathfrak X}\) it follows that all its homogeneous components are also identities for \({\mathfrak X}\). It appears that the variety \({\mathfrak S}_ 5\) of group representations, given by the identities \(z_ 1z_ 2z_ 3z_ 4z_ 5\) and \(2z_ 1z_ 2z_ 1-z_ 1z^ 2_ 2z_ 1\) is not homogeneous [the author, Sib. Mat. Zh. 29, 35-47 (1988; Zbl 0661.20003)]. Also, the variety of 3-unipotent representations of groups, given by \(z^ 3\) is not homogeneous [Th. 2 in the present paper].

To formulate the main concern in this paper, one more notion is needed. An element \(u\in kF\) is said to be a special word of type (n,m,k) iff the following holds: (1) \(\{z_ 1,...,z_ m\}\) serves as the support for every monomial in \(\hat z\) (2) t is minimal with \(u_{<t>}\) equals k and (3) \(u_{<i>}\not\in \hat Id(u,\Delta^ n)\) for some i, with \(Id(u,\Delta^ n)\) being the verbal ideal in kF containing u and \(\Delta^ n(k,F)\). The result about \({\mathfrak S}_ 5\) cited above follows from the fact that \(u(x_ 1,x_ 2)\) with \(\hat u=2z_ 1z_ 2z_ 1- z_ 1z^ 2_ 2z_ 1\) is a special word of type (5,2,3). The author proves (Th.1): there exists no special word of type \(<(5,2,3)\) with \(``<''\) being the lexicographical order (from the left).

A variety \({\mathfrak X}\) of group representations (over k) is said to be homogeneous iff from \(\hat u\) being an identity for \({\mathfrak X}\) it follows that all its homogeneous components are also identities for \({\mathfrak X}\). It appears that the variety \({\mathfrak S}_ 5\) of group representations, given by the identities \(z_ 1z_ 2z_ 3z_ 4z_ 5\) and \(2z_ 1z_ 2z_ 1-z_ 1z^ 2_ 2z_ 1\) is not homogeneous [the author, Sib. Mat. Zh. 29, 35-47 (1988; Zbl 0661.20003)]. Also, the variety of 3-unipotent representations of groups, given by \(z^ 3\) is not homogeneous [Th. 2 in the present paper].

To formulate the main concern in this paper, one more notion is needed. An element \(u\in kF\) is said to be a special word of type (n,m,k) iff the following holds: (1) \(\{z_ 1,...,z_ m\}\) serves as the support for every monomial in \(\hat z\) (2) t is minimal with \(u_{<t>}\) equals k and (3) \(u_{<i>}\not\in \hat Id(u,\Delta^ n)\) for some i, with \(Id(u,\Delta^ n)\) being the verbal ideal in kF containing u and \(\Delta^ n(k,F)\). The result about \({\mathfrak S}_ 5\) cited above follows from the fact that \(u(x_ 1,x_ 2)\) with \(\hat u=2z_ 1z_ 2z_ 1- z_ 1z^ 2_ 2z_ 1\) is a special word of type (5,2,3). The author proves (Th.1): there exists no special word of type \(<(5,2,3)\) with \(``<''\) being the lexicographical order (from the left).

Reviewer: U.Kaljulaid

### MSC:

20C15 | Ordinary representations and characters |

20C07 | Group rings of infinite groups and their modules (group-theoretic aspects) |

16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |

16R10 | \(T\)-ideals, identities, varieties of associative rings and algebras |

20E10 | Quasivarieties and varieties of groups |