Trace cocharacters and the Kronecker products of Schur functions (Q1868770)

The Procesi-Razmyslov theory of trace polynomial identities for the algebra \(M_r(F)\) of \(r\times r\) matrices over a field \(F\) of characteristic 0 gives that the \(n\)-th trace cocharacter of \(M_r(F)\) has the presentation \[ TC_{r,n}=\sum_{\lambda\in\Lambda_r(n)}\chi^\lambda\otimes\chi^\lambda=\sum_{\mu\in\Lambda_{r^2}(n)}m_\mu(M_r(F))\chi^\mu, \] where \(\chi^\lambda\) is the complex irreducible character of the symmetric group \(S_n\) corresponding to the partition \(\lambda=(\lambda_1,\dots,\lambda_k)\), \(\lambda_1\leq\cdots\leq\lambda_k\), the summation is on the set \(\Lambda_r(n)\) of all partitions of \(n\) in not more than \(r\) parts and \(\chi^\lambda\otimes\chi^\lambda\) is the Kronecker square of characters of \(S_n\). For \(r>2\) it is a difficult open problem to give a satisfactory way of computing the multiplicity \(m_\mu(M_r(F))\) of \(\chi^\mu\). The goal of the paper under review is to study the behaviour of the coefficients \(m_\mu(M_r(F))\) for certain classes of partitions. The authors fix a partition \(\nu=(\nu_1,\dots,\nu_k)\) of \(m\) and for \(n\) large enough consider the partition \((\nu,n-|\nu|)\). Then they describe \(m_{(\nu,n-|\nu|)}(M_r(F))\) in the following way. For a fixed partition \(\nu\) and an explicitly constructed positive integer \(u_r\) there exist \(u_r\) rational polynomials \(P^\nu_{r,i}(n)\), \(i=0,1,\dots,u_r-1\), of degree \(r-1\) with the same leading term such that, for sufficiently large \(n\), \(m_{(\nu,n-|\nu|)}(M_r(F))=P^\nu_{r,i}(n)\), where \(i\equiv n\pmod{u_r}\). The authors provide also some tables of the coefficients \(m_{(\nu,n-|\nu|)}(M_r(F))\) and the polynomials \(P^\nu_{r,i}(n)\) for a few small values of \(r\) and partitions \(\nu\).