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Lower bounds on the dimensions of irreducible representations of symmetric groups and on the exponents of varieties of Lie algebras. (English. Russian original) Zbl 0873.17007
Sb. Math. 187, No. 1, 81-92 (1996); translation from Mat. Sb. 187, No. 1, 83-94 (1996).
The first main result of the paper under review estimates the dimensions of the irreducible representations of the symmetric group over a field of characteristic 0. For a fixed positive integer \(k\) the author considers representations corresponding to partitions \(\lambda=(\lambda_1,\ldots,\lambda_p)\) such that \(\lambda_1\leq n/k\), \(p\leq n/k\), i.e. the related Young diagram lies in an \(n/k\times n/k\) square. He proves that for any real \(b<k\) and for sufficiently large \(n\) the dimension of the representation is bigger than \(b^n\).
Then the author applies this result to varieties of Lie algebras. He studies the codimension sequence \(c_n(V)\), \(n=1,2,\ldots\), of a T-ideal \(V\) of the free Lie algebra and shows that the inequality \(\text{lim\;inf}_{n\to\infty}(c_n(V))^{1/n}<2\) implies that the sequence \(c_n(V)\) is of polynomial growth. The author also includes several open problems from the theory of varieties of Lie algebras. Although most of the problems are well known, the paper contains interesting remarks and suggestions to attack them.
Reviewer: V.Drensky (Sofia)

MSC:
17B01 Identities, free Lie (super)algebras
20C30 Representations of finite symmetric groups
17B30 Solvable, nilpotent (super)algebras
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