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


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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