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Some results concerning the multiplicities of cocharacters of superalgebras with graded involution. (English) Zbl 1439.16028

Let \(A\) be a finitely generated superalgebra with graded involution \(\ast\) over a field \(F\) of characteristic zero and let \(\chi_{n_1,\cdots,n_4} (A)\), \(n_1+\cdots+n_4 = n\), be the \((n_1, \cdots , n_4)\)-cocharacter of \(A\). Recall that the \((n_1, \cdots, n_4)\)-cocharacter is the character corresponding to the action of the group \(S_{n_1}\times \cdots \times S_{n_4}\) on \(P_{n_1,\cdots,n_4} (A)\), the space of multilinear \(\ast\)-polynomials in \(n_1\) even symmetric variables, \(n_2\) even skew variables, \(n_3\) odd symmetric variables and \(n_4\) odd skew variables, modulo the \(\ast\)-identities of \(A\), by permutation of the variables of the same homogeneous degree which are all symmetric or all skew at the same time with respect to the graded involution \(\ast\).
In this paper, the author discusses certain results concerning the multiplicities of cocharacters of \(\ast\)-algebras. Firstly, he discusses \(\ast\)-algebra to be excluded from the variety generated by \(A\) in order to have the multiplicities of the \((n_1, \cdots,n_4)\)-cocharacter of \(A\) bounded by a constant. Next, he gives a condition which ensuring that such multiplicities are zero. Lastly, he presents a result which is related to the growth of the \(\ast\)-codimension sequence of \(A\) with its \(\ast\)-colengths.

MSC:

16R50 Other kinds of identities (generalized polynomial, rational, involution)
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
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