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Polynomial Expressions for Symmetric Group Characters on Cycles

Tom Moshaiov, Shaul Zemel

TL;DR

This work extends Cohen–Zemel’s polynomial-dimension results for S_n to character values on cycles. It proves that χ^{(n−k,λ)}(σ) is a degree-k polynomial in n with coefficients expressible as signed counts of skew-tableaux, using r-primary partitions and Murnaghan–Nakayama/branching techniques. The special case of transpositions is made explicit via b^{(2)}_{λ,h} = b^{+}_{λ,h} − b^{-}_{λ,h}, including concrete base cases, while the general r-cycle case introduces Γ^{r}_{h} and ε^{r}_{ν} to describe the constant terms, with a stabilization result as r grows. A second derivation of the constants via vertical-strip recursions confirms the main coefficient structure and highlights limits for non-single-cycle permutations.

Abstract

In \cite{[CZ]}, Cohen and Zemel showed that for a partition $λ\vdash k$, the dimension of the irreducible representation of $S_{n}$ corresponding to the partition $(n-k,λ) \vdash n$ is a polynomial of degree $k$ in $n$, whose coefficients in the binomial basis count standard Young tableaux of shape $λ$ with special restrictions. In this paper, we generalize their results on the representation's dimension to character values on arbitrary cycles.

Polynomial Expressions for Symmetric Group Characters on Cycles

TL;DR

This work extends Cohen–Zemel’s polynomial-dimension results for S_n to character values on cycles. It proves that χ^{(n−k,λ)}(σ) is a degree-k polynomial in n with coefficients expressible as signed counts of skew-tableaux, using r-primary partitions and Murnaghan–Nakayama/branching techniques. The special case of transpositions is made explicit via b^{(2)}_{λ,h} = b^{+}_{λ,h} − b^{-}_{λ,h}, including concrete base cases, while the general r-cycle case introduces Γ^{r}_{h} and ε^{r}_{ν} to describe the constant terms, with a stabilization result as r grows. A second derivation of the constants via vertical-strip recursions confirms the main coefficient structure and highlights limits for non-single-cycle permutations.

Abstract

In \cite{[CZ]}, Cohen and Zemel showed that for a partition , the dimension of the irreducible representation of corresponding to the partition is a polynomial of degree in , whose coefficients in the binomial basis count standard Young tableaux of shape with special restrictions. In this paper, we generalize their results on the representation's dimension to character values on arbitrary cycles.
Paper Structure (4 sections, 19 theorems, 17 equations)

This paper contains 4 sections, 19 theorems, 17 equations.

Table of Contents

  1. Some Classical Results and their Consequences
  2. The Case of Transpositions
  3. The Result for Arbitrary Cycles
  4. A Second Determination of the Constant Coefficients

Key Result

Theorem 1

The $h$th coefficient $b_{\lambda,h}^{(r)}$ in the expansion of $\chi^{(n-k,\lambda)}(\sigma)$ equals $\sum_{\nu\in\Gamma^{r}_{h}}\varepsilon^{r}_{\nu}f^{\lambda\setminus\nu}$.

Theorems & Definitions (39)

  • Theorem
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • proof
  • Corollary 1.5
  • proof
  • Definition 1.6
  • Lemma 1.7
  • ...and 29 more