A mathematics preprint shows that forgotten symmetric functions arise naturally when class functions are evaluated on partial permutations. The calculation concerns a group-algebra element: a formal sum of all permutations that extend a partial permutation, meaning a mapping specified only in part.
The work is symbolic rather than empirical. It studies partial permutations, symmetric functions, class functions, irreducible characters of symmetric groups, set partitions and group-algebra elements, developing identities among those mathematical objects rather than analyzing an empirical sample.
A translation between two mathematical languages
The central evaluation identity gives a way to move between the language of permutations and the language of symmetric functions. For a class function and a partial permutation, the value on the corresponding extending-permutation group-algebra element is expressed as a signed multiplicity factor times a Hall inner product involving a forgotten symmetric function and a power-sum function.
The authors make this translation with the Frobenius characteristic and the Hall inner product. In the resulting framework, class functions are represented through symmetric functions, allowing the evaluation to be written as the specified inner-product calculation.
That structure places forgotten symmetric functions at the point where the partial-permutation evaluation is computed. In the paper’s framework, they are the functions that naturally encode the relevant class-function calculation.
How the identities are built
The proof relies on a result from P. Doubilet’s Ph.D. thesis and uses the Möbius function on the lattice of set partitions to obtain coefficients for the expansions under study.
The paper then expresses a forgotten symmetric function in the Schur basis, a standard named basis of symmetric functions used in the paper’s expansion. The coefficients are not left abstract: they are described as signed sums over monotonic ribbon tilings.
Together, these steps take the argument from an extending-permutation sum to an expression in symmetric functions and then to combinatorial objects. The paper’s evidence is therefore made up of exact algebraic identities and combinatorial constructions within the mathematical setting it defines.
Tilings make the character formulas visible
The Schur-basis expansion assigns coefficients through monotonic ribbon tilings. These tilings are indexed by compositions that sort to the target partition, giving a combinatorial way to describe the expansion of the forgotten symmetric function.
The work also identifies the path power sum as a sign-and-multiplicity rescaling of the forgotten symmetric function. This lets the authors combine the expansion with the Murnaghan–Nakayama rule to derive a combinatorial expression for irreducible-character evaluations.
The character calculation can consequently be organized through ribbon tilings and rim-hook tableaux, rather than only through formal manipulation of symmetric-function symbols. The paper presents those constructions as rules for the evaluations attached to its partial-permutation framework.
A sharper outcome for a set-mapping element
The preprint applies the framework to the group-algebra elements written as {𝐼, 𝐽}, referred to in the paper as set-mapping elements. Their expanded path-power-sum expression contains terms that cancel, and the cancellation is organized by a sign-reversing involution that pairs terms with opposite signs.
After the cancellation, the relevant expansion involves only path types whose parts are no larger than 2. This is a structural reduction of the formula within the algebraic calculation.
For the same set-mapping element, irreducible-character evaluations vanish unless the indexing partition has at most 2 parts. The calculation therefore has support only on the one- and two-part cases described by the paper.
When the indexing partition is written as 𝜆 = (𝑛 − 𝑖, 𝑖) and has at most 2 parts, the paper supplies an explicit summation formula for the evaluation. That formula is the concrete endpoint of the cancellation and character-expansion argument.
What the result does—and does not—claim
The authors interpret the identities as showing that the forgotten basis is naturally suited to class-function evaluations on partial permutations. They also interpret the associated character calculations as admitting combinatorial rules based on the tiling and tableau constructions.
The conclusions remain within the mathematical objects and assumptions defined in the paper. The work does not establish computational runtime, practical superiority over another method or validation of the identities outside that mathematical setting.
No empirical validation or numerical uncertainty analysis is reported. The paper does not provide confidence intervals, p-values, risk estimates or causal effects because it does not study an empirical population.
The supplied document is arXiv version 1, dated 20 August 2026. It is identified as a preprint rather than a reported journal article in the supplied publication information.
Questions left open
The framework leaves open whether the identities extend to related algebraic structures beyond the symmetric groups and partial permutations considered in the paper. The supplied analysis does not answer that question.
It also remains unclear whether the formulas provide substantial computational advantages for larger instances. The symbolic derivations establish the identities, but they do not establish computational performance.
The paper requires background in symmetric functions, Frobenius characteristics, Hall inner products, symmetric-group characters, partition lattices and Möbius functions. Its contribution is a mathematical framework and a set of formulas, not a result with an immediate human-facing outcome.
The supplied funding statement says that Brendon Rhoades was partially supported by NSF grant DMS-2246846.
Paper data and sources
Original title: Forgotten characters
Authors: Kyle Celano, Brendon Rhoades
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text