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Extending the descent-to-peak map and its applications

Farid Aliniaeifard, Shu Xiao Li

Abstract

The descent-to-peak map serves as a bridge between algebra and combinatorics. We use it as a tool for proving the equidistribution of peak and valley sets of standard Young tableaux with a very short argument. We also introduce a new shuffle basis of quasisymmetric functions whose elements are eigenvectors of the descent-to-peak map. Using this basis, we then extend the notion of the peak algebra and of the descent-to-peak map to shuffle, tensor, and symmetric algebras.

Extending the descent-to-peak map and its applications

Abstract

The descent-to-peak map serves as a bridge between algebra and combinatorics. We use it as a tool for proving the equidistribution of peak and valley sets of standard Young tableaux with a very short argument. We also introduce a new shuffle basis of quasisymmetric functions whose elements are eigenvectors of the descent-to-peak map. Using this basis, we then extend the notion of the peak algebra and of the descent-to-peak map to shuffle, tensor, and symmetric algebras.

Paper Structure

This paper contains 17 sections, 19 theorems, 122 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Hopf algebras $\mathrm{QSym}, \mathrm{Sym}$ and $\mathrm{NSym}$
  4. Combinatorial Hopf algebras
  5. Peak algebras and the theta maps
  6. The peak and valley statistics are equidistributed
  7. The shuffle basis of $\mathrm{QSym}$
  8. The shuffle, tensor, and symmetric algebras
  9. The shuffle algebra
  10. The tensor algebra
  11. The symmetric algebra
  12. The Hopf subalgebras of odd and even elements
  13. The theta maps
  14. Examples of theta maps and peak algebras
  15. The classical cases
  16. ...and 2 more sections

Key Result

Theorem 2.2

A06 For any combinatorial Hopf algebra $(\mathsf{H},\zeta)$, there exists a unique morphism of combinatorial Hopf algebras Moreover, if $\mathsf{H}$ is cocommutative, then $\Phi(\mathsf{H})\subseteq\mathrm{Sym}$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Examples of standard Young tableaux.

Theorems & Definitions (37)

  • Example 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.4
  • Theorem 3.1
  • proof
  • Definition 4.1
  • Example 4.2
  • Definition 4.3
  • Proposition 4.4
  • ...and 27 more