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Algebraic properties of codimension series of PI-algebras

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Abstract

Let c n (R), n = 0, 1, 2, …, be the codimension sequence of the PI-algebra R over a field of characteristic 0 with T-ideal T(R) and let c(R, t) = c 0(R) + c 1(R)t + c 2(R)t 2 + … be the codimension series of R (i.e., the generating function of the codimension sequence of R). Let R 1,R 2 and R be PI-algebras such that T(R) = T(R1)T(R 2). We show that if c(R 1, t) and c(R 2, t) are rational functions, then c(R, t) is also rational. If c(R 1, t) is rational and c(R 2, t) is algebraic, then c(R, t) is also algebraic. The proof is based on the fact that the product of two exponential generating functions behaves as the exponential generating function of the sequence of the degrees of the outer tensor products of two sequences of representations of the symmetric groups S n .

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Authors and Affiliations

  1. Higher School of Civil Engineering “Lyuben Karavelov”, 175 Suhodolska Str., 1373, Sofia, Bulgaria

    Silvia Boumova

  2. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113, Sofia, Bulgaria

    Silvia Boumova & Vesselin Drensky

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  1. Silvia Boumova
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  2. Vesselin Drensky
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Correspondence to Silvia Boumova.

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Boumova, S., Drensky, V. Algebraic properties of codimension series of PI-algebras. Isr. J. Math. 195, 593–611 (2013). https://doi.org/10.1007/s11856-012-0110-4

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