On Almost Nilpotent Varieties with Integer PI-Exponent
We study almost nilpotent varieties of algebras over a ﬁeld of zero characteristic. Earlier in th e class ofalgebras with identica l relation x(yz) ≡ 0 and in the class of all commutative metabelian algebras countablesets of varieties with integer PI-exponent were deﬁned. Only the existence of al most nilpotent subvarietyin each deﬁned variety was proved. In the paper by means of combinatorial methods and methods of therepresentation theory of sym metric groups we prove that earlie r d eﬁned varieties are almost nilpotent. Byanalogy with commutative case we deﬁne a countable set of al most nilpotent varieties with integer PI-exponentin the class of anticommutative metabelian algebras. For each variety in the corresponding relatively freealgebra we study multilinear part as a module of the symmetric group. More precisely we deﬁne restrictionson the shape of Young diagrams that correspond to nonzero irreducible submodules. For such diagrams wealso obtain the form of nonzero monomial s from the corresponding spaces of multihomogeneous elements.We give the detailed description of the results obtai ned for the algebras satisfying the identity x(yz) ≡ 0.Since similar results for commutative and anticommutative metabelian algebras were obt ained by analogywe present them without proofs but with remarks.