Thrall’s decomposition problem for higher Lie characters

Find a combinatorial interpretation of the multiplicities of irreducible symmetric-group characters in higher Lie characters, namely determine a combinatorial interpretation of c^{\lambda}_{\nu}=\langle\psi^{\lambda},\chi^{\nu}\rangle for partitions \lambda,\nu\vdash n.

Background

Higher Lie characters ψλ\psi^\lambda form a distinguished family of characters of the symmetric group. Their decomposition into irreducible characters is the central problem attributed to Thrall, and the paper studies this problem asymptotically rather than solving the decomposition problem in full.

The multiplicity cνλ=⟨ψλ,χν⟩c^{\lambda}_{\nu}=\langle\psi^{\lambda},\chi^{\nu}\rangle records how many copies of the irreducible character χν\chi^\nu occur in ψλ\psi^\lambda. Combinatorial interpretations are known for several special families, but no general interpretation is given.

References

Find a combinatorial interpretation of the inner products $c_\nu{\lambda}:=\langle \psi\lambda,\chi\nu\rangle$.

— Asymptotics of higher Lie characters  (2509.12904 - Adin et al., 16 Sep 2025) in Problem (Thrall), Section 1.1, “Higher Lie characters and Thrall’s problem”

This 83 years old open problem of Thrall has received significant attention over the years; see, e.g., .

— Asymptotics of higher Lie characters  (2509.12904 - Adin et al., 16 Sep 2025) in Section 1.1, “Higher Lie characters and Thrall’s problem”