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Transition Matrices between Plethystic Bases of Polysymmetric Functions via Bijective Methods

Aditya Khanna

TL;DR

This work develops a comprehensive combinatorial framework for the transition matrices among the plethystic bases $H$, $E$, $E^+$, and $P$ in the polysymmetric function algebra $PSym$. By introducing bar tableaux and their poly-extensions (polyWBT/PSBT, polybrick tabloids, and their dyadic variants), the authors obtain explicit bijective proofs and sign-reversing involutions that realize all 12 base-pair expansions. The results unify recurrences and closed-form expansions, connect to rich combinatorial objects (polycompositions/types), and reveal multiple OEIS correspondences, enriching the algebraic theory with tangible combinatorial models. These insights extend classical symmetric-function theory to polysymmetric settings, offering robust tools for analyzing plethystic relationships and their enumerative aspects. The explicit appendix with concrete matrices for small $n$ demonstrates the practicality of the approach for computational exploration and potential applications in representation-theoretic contexts.

Abstract

Many identities involving symmetric functions can be proved through bijective manipulations of tableaux. In this paper, we prove identities involving polysymmetric functions through bijections and sign-reversing involutions. In their paper titled "Polysymmetric functions and motivic measures of configuration spaces", Asvin G and Andrew O'Desky introduced the algebra of polysymmetric functions (PSym) which can be defined as the tensor product of copies of the symmetric functions algebra (Sym) where the $i$th tensor factor is scaled by $i$. On one hand, we can obtain bases of this algebra by taking tensor products of the bases of Sym. On the other hand, the Asvin G and Andrew O'Desky paper introduces non-pure tensor bases families $H$, $E$, $E^+$, and $P$ that we call plethystic bases. In this paper, we present combinatorial interpretations of the entries of the transition matrices between all twelve pairs of distinct plethystic bases. We also provide new interpretations for six OEIS sequences that turn up in this context.

Transition Matrices between Plethystic Bases of Polysymmetric Functions via Bijective Methods

TL;DR

This work develops a comprehensive combinatorial framework for the transition matrices among the plethystic bases , , , and in the polysymmetric function algebra . By introducing bar tableaux and their poly-extensions (polyWBT/PSBT, polybrick tabloids, and their dyadic variants), the authors obtain explicit bijective proofs and sign-reversing involutions that realize all 12 base-pair expansions. The results unify recurrences and closed-form expansions, connect to rich combinatorial objects (polycompositions/types), and reveal multiple OEIS correspondences, enriching the algebraic theory with tangible combinatorial models. These insights extend classical symmetric-function theory to polysymmetric settings, offering robust tools for analyzing plethystic relationships and their enumerative aspects. The explicit appendix with concrete matrices for small demonstrates the practicality of the approach for computational exploration and potential applications in representation-theoretic contexts.

Abstract

Many identities involving symmetric functions can be proved through bijective manipulations of tableaux. In this paper, we prove identities involving polysymmetric functions through bijections and sign-reversing involutions. In their paper titled "Polysymmetric functions and motivic measures of configuration spaces", Asvin G and Andrew O'Desky introduced the algebra of polysymmetric functions (PSym) which can be defined as the tensor product of copies of the symmetric functions algebra (Sym) where the th tensor factor is scaled by . On one hand, we can obtain bases of this algebra by taking tensor products of the bases of Sym. On the other hand, the Asvin G and Andrew O'Desky paper introduces non-pure tensor bases families , , , and that we call plethystic bases. In this paper, we present combinatorial interpretations of the entries of the transition matrices between all twelve pairs of distinct plethystic bases. We also provide new interpretations for six OEIS sequences that turn up in this context.
Paper Structure (41 sections, 30 theorems, 134 equations, 2 tables)

This paper contains 41 sections, 30 theorems, 134 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Review of Symmetric Functions
  3. Polycompositions and types
  4. Polysymmetric functions
  5. Structure of this Paper
  6. Summary of results
  7. Combinatorial models for monomial expansions of the plethystic bases
  8. Bar tableaux
  9. Poly bar tableaux
  10. Marked poly bar tableaux
  11. Transition Matrices Between $H$, $E$, and $P$
  12. Recursions among $H$, $E$, and $P$
  13. $H$ and $E$
  14. $P$ in $H$ and $E$
  15. $H$ in $P$
  16. ...and 26 more sections

Key Result

Lemma 4

For $d\geq 0$,

Theorems & Definitions (94)

  • Example 1
  • Remark 2
  • Remark 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Proposition 7: polysymm, Remark 10
  • ...and 84 more