On the \(*\)-cocharacter sequence of \(3\times 3\) matrices
The authors consider the \(*\)-polynomial identities of the \(k\times k\) matrix algebra \(M_k(F)\) with transpose involution over a field \(F\) of characteristic 0. By analogy with the case of ordinary polynomial identities, they study the cocharacter sequence \(\chi_n(M_k(F))\), \(n=1,2,\ldots\), of the \(*\)-multilinear polynomials of \(M_k(F)\), where instead of characters of the symmetric group \(S_n\) one considers the characters of the hyperoctahedral group \(B_n=Z_2\wr S_n\) (the wreath product of the cyclic group of order 2 and the symmetric group of degree \(n\)). It is well known that \(\chi_n(M_k(F))=\sum m_{\lambda,\mu}\chi_{\lambda,\mu}\), where \(\chi_{\lambda,\mu}\) is the character of \(B_n\) associated with the pair of partitions \(\lambda\vdash r\), \(\mu\vdash n-r\). The exact values of the multiplicities \(m_{\lambda,\mu}\) are known for \(k=2\) only [\textit{V. Drensky} and \textit{A. Giambruno}, Can. J. Math. 46, No. 4, 718-733 (1994; Zbl 0814.16021)]. In the present paper the authors study the problem for which pairs of partitions the multiplicities \(m_{\lambda,\mu}\) in the cocharacter sequence for \(3\times 3\) matrices are different from 0. The approach involves the technique of gluing of Young diagrams developed by \textit{A. Regev} [Commun. Algebra 8, 1417-1467 (1980; Zbl 0437.16012)] which has already been used by one of the authors for a similar investigation of the ordinary polynomial identities of \(3\times 3\) matrices [\textit{F. Benanti}, Commun. Algebra 24, No. 13, 4263-4279 (1996; Zbl 0888.16012)]. The main result gives that for all possible pairs of partitions \(\lambda=(\lambda_1,\ldots,\lambda_6),\;\mu=(\mu_1,\mu_2,\mu_3)\) one has \(m_{\lambda,\mu}\geq 1\). The only exception is the case \(\lambda=(1^6),\;\mu=\emptyset\) when \(m_{(1^6),\emptyset}=0\) and the corresponding multilinear polynomials are consequences of the standard identity \(s_6=0\). The authors also give some information concerning the multiplicities of the cocharacter sequence for the \(*\)-trace identities of \(M_3(F)\). (Also submitted to MR).
scientific article; zbMATH DE number 1485559
scientific article