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Trace identities and central polynomials in matrix superalgebras $$M_{n,k}$$. (Russian) Zbl 0601.16016
This paper extends the one reviewed above and contains its results. The definitions of $$M_{n,k}$$ and central polynomials are there. Let $$M_{n,k}$$ be a matrix superalgebra over a field $$K$$. Let $$str(A)=tr(a_{11})-tr(a_{22})$$. It is the so called supertrace of $$A\in M_{n,k}$$. The usual identities satisfied by the trace in matrix algebras are satisfied by the supertrace in $$M_{n,k}$$. Therefore it is clearly what a trace identity in $$M_{n,k}$$ means. See the exact definitions also in the author’s paper [Izv. Akad. Nauk SSSR, Ser. Mat. 38, 723-756 (1974; Zbl 0311.16016)].
Theorem 1. If the characteristic of $$K$$ is zero then all trace identities in $$M_{n,k}$$ follow from a subset of the polylinear trace identities of $$M_{n,k}$$ with degree $$nk+n+k$$. This result follows from some stronger one (Theorem 4) which gives a full description (by means of Young tableaux) of the ideal of trace identities of $$M_{n,k}$$ in the language of the representation theory of symmetric groups.
Reviewer: G.I.Zhitomirskij

##### MSC:
 16R10 $$T$$-ideals, identities, varieties of associative rings and algebras 15A75 Exterior algebra, Grassmann algebras 17A70 Superalgebras 20C30 Representations of finite symmetric groups 16S50 Endomorphism rings; matrix rings
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