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Character factorisations, $z$-asymmetric partitions and plethysm

Seamus Albion

TL;DR

The work unifies a broad family of character factorisations under the $t$-Verschiebung operator by introducing the universal symmetric function $\mathcal{X}_\lambda(z;q)$, which specializes to Schur, symplectic, and orthogonal cases. A new classification of $z$-asymmetric partitions under the Littlewood decomposition enables a uniform $(z,q)$-analogue of the factorisation, including explicit prefactors and signs that decompose multiplicatively over $t$-quotients. The paper further extends plethysm rules (the SXP framework) to universal characters, providing explicit expansion coefficients and stability phenomena, and links these results to the representation theory of symmetric and classical groups via restriction rules and branching data. Altogether, the results offer a coherent, combinatorial mechanism—via cores, quotients, and $z$-asymmetry—to understand how Verschiebung interacts with universal characters and plethysms, with implications for symmetric-group characters and related structures in algebraic combinatorics.

Abstract

The Verschiebung operators $\varphi_t $ are a family of endomorphisms on the ring of symmetric functions, one for each integer $t\geq2$. Their action on the Schur basis has its origins in work of Littlewood and Richardson, and is intimately related with the decomposition of a partition into its $t$-core and $t$-quotient. Namely, they showed that the action on $s_λ$ is zero if the $t$-core of the indexing partition is nonempty, and otherwise it factors as a product of Schur functions indexed by the $t$-quotient. Much more recently, Lecouvey and, independently, Ayyer and Kumari have provided similar formulae for the characters of the symplectic and orthogonal groups, where again the combinatorics of cores and quotients plays a fundamental role. We embed all of these character factorisations in an infinite family involving an integer $z$ and parameter $q$ using a very general symmetric function defined by Hamel and King. The proof hinges on a new characterisation of the $t$-cores and $t$-quotients of $z$-asymmetric partitions which generalise the well-known classifications for self-conjugate and doubled distinct partitions. We also explain the connection between these results, plethysms of symmetric functions and characters of the symmetric group.

Character factorisations, $z$-asymmetric partitions and plethysm

TL;DR

The work unifies a broad family of character factorisations under the -Verschiebung operator by introducing the universal symmetric function , which specializes to Schur, symplectic, and orthogonal cases. A new classification of -asymmetric partitions under the Littlewood decomposition enables a uniform -analogue of the factorisation, including explicit prefactors and signs that decompose multiplicatively over -quotients. The paper further extends plethysm rules (the SXP framework) to universal characters, providing explicit expansion coefficients and stability phenomena, and links these results to the representation theory of symmetric and classical groups via restriction rules and branching data. Altogether, the results offer a coherent, combinatorial mechanism—via cores, quotients, and -asymmetry—to understand how Verschiebung interacts with universal characters and plethysms, with implications for symmetric-group characters and related structures in algebraic combinatorics.

Abstract

The Verschiebung operators are a family of endomorphisms on the ring of symmetric functions, one for each integer . Their action on the Schur basis has its origins in work of Littlewood and Richardson, and is intimately related with the decomposition of a partition into its -core and -quotient. Namely, they showed that the action on is zero if the -core of the indexing partition is nonempty, and otherwise it factors as a product of Schur functions indexed by the -quotient. Much more recently, Lecouvey and, independently, Ayyer and Kumari have provided similar formulae for the characters of the symplectic and orthogonal groups, where again the combinatorics of cores and quotients plays a fundamental role. We embed all of these character factorisations in an infinite family involving an integer and parameter using a very general symmetric function defined by Hamel and King. The proof hinges on a new characterisation of the -cores and -quotients of -asymmetric partitions which generalise the well-known classifications for self-conjugate and doubled distinct partitions. We also explain the connection between these results, plethysms of symmetric functions and characters of the symmetric group.

Paper Structure

This paper contains 25 sections, 27 theorems, 112 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Historical background
  3. Generalisations to classical group characters
  4. Summary of the paper
  5. Partitions and the Littlewood decomposition
  6. Preliminaries
  7. Littlewood's decomposition
  8. Generalised universal characters
  9. Symmetric functions and plethysm
  10. Generalised universal characters
  11. Restriction rules
  12. Factorisations of universal characters
  13. Factorisations of classical characters
  14. A uniform $(z,q)$-analogue
  15. Proof of Theorem \ref{['Thm_chiz']}
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\lambda$ be a partition of $nt$. Then $\chi^\lambda(t\mu)=0$ unless the $t$-core of $\lambda$ is empty, in which case

Figures (6)

  • Figure 1: The partition $\lambda=(6,5,5,1)=(5,3,2~\vert~3,1,0)$ with its main diagonal shaded (left) and the same partition with hook length of each cell inscribed (right). We have $\lvert\lambda\rvert=17$, $l(\lambda) = 4$, $\mathrm{rk}(\lambda)=3$, $\mathrm{rk}_2(\lambda)=1$ and $\mathrm{rk}_{-3}(\lambda)=2$.
  • Figure 2: The pair of partitions $(4,4,2,1)\subseteq(6,5,5,1)$. The unshaded cells form a $6$-ribbon of height $2$ and the corresponding cell with hook length $6$ is marked.
  • Figure 3: The Maya diagram of $\lambda=(6,5,5,1)$ (top) and the $3$-Maya diagram of the same partition (bottom). We have that $3$-core$(\lambda)=(1,1)$, $\kappa_3((1,1))=(1,-1,0)$ and $(\lambda^{(0)},\lambda^{(1)},\lambda^{(2)})=((1),\varnothing,(2,2))$.
  • Figure 4: The $3$-Maya diagram of $(6,5,5,1)$ (top) and the $3$-Maya diagram of $(7,6,6,1)$ (bottom) corresponding to action the "cut and twist" map.
  • Figure 5: The Littlewood decomposition of $\lambda=(8,4,3,3,3,1,1)$ with $t=3$ and $\kappa_3(\lambda)=(0,1,-1)$. The marked cells explain the computation of the Frobenius rank: the left- and right-hand sides both contain three shaded cells since the first row of $\lambda^{(1)}$ and the first column of $\lambda^{(2)}$ are ignored.
  • ...and 1 more figures

Theorems & Definitions (38)

  • Theorem 1.1: Littlewood51
  • Theorem 1.2: Littlewood51
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1: Littlewood's decomposition
  • Proposition 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Corollary 2.5
  • ...and 28 more