Partial permutations and character evaluations

TL;DR

This work develops a comprehensive framework for evaluating irreducible characters of the symmetric group on partial permutations. It introduces path power sums and a path Murnaghan–Nakayama rule that, together with the classical rule, yields an explicit, finite combinatorial method to compute ${\\chi}^{\\lambda}([I,J])$ for partial permutations $(I,J)$, once $n \ge 2k$, by summing over monotonic ribbon tilings. The approach unifies local statistics via the Reynolds projection, giving a programmable expansion of class functions into irreducibles and providing polynomial dependence in $n$ for fixed local data. The results enable efficient character computations, stability analyses as $n$ grows, and concrete applications to permutation statistics such as exceedances and major index, with extensions to parabolic subgroups, unitary groups, and questions about coefficients of Schur functions in related bases. Overall, the paper creates a transparent, combinatorial path from partial permutations to character data and local statistics, with broad implications for asymptotic questions and symmetric-function techniques in combinatorics.

Abstract

Let $I = (i_1, \dots, i_k)$ and $J = (j_1, \dots, j_k)$ be two length $k$ sequences drawn from $\{1, \dots, n \}$. We have the group algebra element $[I,J] := \sum_{w(I) = J} w \in \mathbb{C}[\mathfrak{S}_n]$ where the sum is over permutations $w \in \mathfrak{S}_n$ which satisfy $w(i_p) = j_p$ for $p = 1, \dots, k$. We give an algorithm for evaluating irreducible characters $χ^λ: \mathbb{C}[\mathfrak{S}_n] \to \mathbb{C}$ of the symmetric group on the elements $[I,J]$. This algorithm is a hybrid of the classical Murnaghan--Nakayama rule and a new path Murnaghan--Nakayama rule which reflects the decomposition of a partial permutation into paths and cycles. These results first appeared in arXiv:2206.06567, which is no longer intended for publication. We originally used the character theoretic results in this paper to prove asymptotic results on moments of certain permutation statistics restricted to conjugacy classes. A referee generously shared a combinatorial argument which is strong enough to prove these results without recourse to character theory. These results now appear in our companion paper~\cite{HRMoment}. However, the approach in this paper is more explicit, as we demonstrate with several examples.

Figures (5)

• Figure 1: The graph $G(I,J)$ of $(I,J) \in {\mathfrak{S}}_{15,9}$ where $I = (11,10,2,7,13,9,3,5,15)$ and $J = (10,6,7,14,1,3,9,5,15)$. The cycle type of is $(2,1,1)$ and the path type is $(3,3,2,1,1,1)$.
• Figure 2: The standard ribbon tableaux computing $\chi^{431}_{3221}$ together with their signs. The numbers in the tails indicate the order in which ribbons are added.
• Figure 3: A monotonic ribbon tiling (left) and a tiling by ribbons which is not monotonic (right).
• Figure 4: The monotonic ribbon tilings used to calculate the Schur expansion of $\vec{p}_{321}$ using Theorem \ref{['thm:path-Murnaghan--Nakayama']}.
• Figure 5: The Schur expansion in Example \ref{['atomic-expansion-example']}.

Theorems & Definitions (69)

• Definition 1.1
• Theorem 1.2
• Theorem 2.2
• Corollary 2.3
• Theorem 2.4
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3