A new extremal property of the variety of the Lie algebras \({\mathcal A}N_2\) (Q1596164)

Let \(\chi_\lambda\) be a character of irreducible \(S_n\)-representation corresponding to \(\lambda\vdash n\). The expansion \[ \chi_n(V)= \chi_n (P_n(V))= \sum_{\lambda\vdash n} m_\lambda \chi_\lambda \tag{1} \] is considered for an arbitrary variety \({\mathcal V}\). The authors study the multiplicity of \(m_\lambda\) in the expansion (1) for both the variety \({\mathcal A}N_2\) and for its proper subvarieties. They prove that the multiplicities of irreducible \(S_n\)-modules of the multilinear component cannot be bounded by a common constant that does not depend on \(n\). At the same time, such a constant exists for any proper subvariety in \({\mathcal A}N_2\).