## 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).
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

Zbl 0661.20003