Multiplicities of Schur functions in invariants of two \(3\times 3\) matrices (Q1399181)

With a symmetric power series \(f(x,y)\) the authors associate its multiplicity series\break \(\sum m(\lambda_1,\lambda_2)t^{\lambda_1}u^{\lambda_2}\), where \(f(x,y)=\sum m(\lambda_1,\lambda_2)S_{(\lambda_1,\lambda_2)}(x,y)\) expanded as a series of Schur functions. When \(f(x,y)\) is the Hilbert (or Poincaré) series of a polynomial module over \(\text{GL}_2\), the complex general linear group, the number \(m(\lambda_1,\lambda_2)\) is the multiplicity as a direct summand of the irreducible \(\text{GL}_2\)-module indexed by the partition \((\lambda_1,\lambda_2)\). The Hilbert series of the algebra of simultaneous conjugation invariants of pairs of \(3\times 3\) matrices is known from the work of \textit{Y. Teranishi} [Nagoya Math. J. 104, 149-161 (1986; Zbl 0615.16013)]. The main result of the present paper is an explicit rational expression for the multiplicity series of this Hilbert series. As a consequence, the asymptotics of the multiplicities in the trace cocharacter sequence of pairs of \(3\times 3\) matrices is determined. Along the way the authors express as a rational function the multiplicity series of the symmetric tensor algebra of the third symmetric tensor power of the vector representation of \(\text{GL}_2\). This is in the spirit of a classical result of \textit{R. M. Thrall} [Am. J. Math. 64, 371-388 (1942; Zbl 0061.04201)] on certain plethysms.