Asymptotic behavior of \(S_n\) degrees associated with rational functions.
The paper under review is motivated by the result of \textit{A. Berele} and \textit{A. Regev}, [in Trans. Am. Math. Soc. 360, No. 10, 5155-5172 (2008; Zbl 1161.16016)] that over a field of characteristic 0 the codimension sequence \(c_n(A)\) of a finitely generated unitary PI-algebra \(A\) is asymptotically equal to \(\alpha\cdot n^g\cdot\ell^n\), where \(\alpha\) is a non-negative real, \(g\in\mathbb Z/2\), and \(\ell\) is a non-negative integer. (The general result for arbitrary unitary PI-algebras was established by the first author [in Adv. Appl. Math. 41, No. 1, 52-75 (2008; Zbl 1145.05052); corrigendum 43, No. 2, 199-200 (2009)].) Now the authors consider sequences of characters \(\chi_n=\sum_{\lambda\vdash n}m_\lambda\chi_\lambda\) of the symmetric groups \(S_n\) which share the main properties of the cocharacter sequences of finitely generated unitary PI-algebras: (i) All \(\lambda\)'s are partitions in not more than \(k\) parts; (ii) The associated symmetric function \(P(t_1,\ldots,t_k)=\sum m_\lambda S_\lambda(t_1,\ldots,t_k)\) (where \(S_\lambda(t_1,\ldots,t_k)\) is the Schur function corresponding to the partition \(\lambda\)) is a nice rational function, i.e., it can be written as a fraction of polynomials with denominator a product of terms of the form \((1-u)\), where \(u\) is a monic monomial; (iii) The symmetric function \(P(t_1,\ldots,t_k)\) is Young derived, i.e., both \(P(t_1,\ldots,t_k)\) and \(\prod_i(1-t_i)P(t_1,\ldots,t_k)\) can be expanded into formal series of Schur functions with non-negative integer coefficients. The main result of the paper states that in this case the sequence \(c_n=\sum_{\lambda\vdash n}m_\lambda\deg(\chi_\lambda)\) has asymptotic behaviour \(c_n\simeq\alpha\cdot n^g\cdot\ell^n\), where \(g\in\mathbb Z/2\), and \(\ell\) is algebraic over \(\mathbb Q\). If one replaces the condition to be Young derived with the weaker condition that each \(m_\lambda\) is a non-negative integer, then there exists a modulus \(d\) such that the sequence \(c_n\) splits into \(d\) subsequences \(c_{md+i}\), \(i=0,1,\ldots,d-1\), with \(c_{md+i}\simeq\alpha_i\cdot (md+i)^{g_i}\cdot\ell_i^{md+i}\) when \(m=0,1,2,\ldots\).
scientific article; zbMATH DE number 6509132