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A permutation based approach to the $q$-deformation of the Dynkin Operator

Darij Grinberg, Ekaterina A. Vassilieva

TL;DR

The paper studies a $q$-deformation of the Dynkin operator within Solomon’s descent algebra to understand its action on the $B$-basis. By expanding the $q$-deformed operator in the $D$-basis and applying Solomon’s Mackey formula, it derives a closed coefficient formula for $\\mathbf{V}_n^{(q)}\\mathbf{B}_I$ and analyzes resulting idempotence, eigenvalues, and image-dimension across roots of unity. Key contributions include a new combinatorial proof of $V_n B_I$ behavior, an explicit expression for action on the $B$-basis, and a complete description of the image’s dimension in terms of $q$ (with connections to peak algebras when $q=-1$). The results illuminate the representation-theoretic structure of deformed free Lie modules and reveal intricate dependence on $q$, including Fibonacci-type dimensions for roots of unity. Overall, the work connects permutation combinatorics, descent algebras, and deformations of Lie-theoretic objects, offering precise tools for further exploration of $q$-deformed Lie-type structures in symmetric groups.

Abstract

Introduced by Solomon, the descent algebra is a significant subalgebra of the group algebra of the symmetric group $\mathbf{k}S_n$ related to many important algebraic and combinatorial topics. It contains all the classical Lie idempotents of $\mathbf{k}S_n$, in particular the Dynkin operator, a fundamental tool for studying the free Lie algebra. We look at a $q$-deformation of the Dynkin operator and study its action over the descent algebra with classical combinatorial tools like Solomon's Mackey formula. This leads to elementary proofs that the operator is indeed an idempotent for $q=1$ as well as to interesting formulas and algebraic structures especially when $q$ is a root of unity.

A permutation based approach to the $q$-deformation of the Dynkin Operator

TL;DR

The paper studies a -deformation of the Dynkin operator within Solomon’s descent algebra to understand its action on the -basis. By expanding the -deformed operator in the -basis and applying Solomon’s Mackey formula, it derives a closed coefficient formula for and analyzes resulting idempotence, eigenvalues, and image-dimension across roots of unity. Key contributions include a new combinatorial proof of behavior, an explicit expression for action on the -basis, and a complete description of the image’s dimension in terms of (with connections to peak algebras when ). The results illuminate the representation-theoretic structure of deformed free Lie modules and reveal intricate dependence on , including Fibonacci-type dimensions for roots of unity. Overall, the work connects permutation combinatorics, descent algebras, and deformations of Lie-theoretic objects, offering precise tools for further exploration of -deformed Lie-type structures in symmetric groups.

Abstract

Introduced by Solomon, the descent algebra is a significant subalgebra of the group algebra of the symmetric group related to many important algebraic and combinatorial topics. It contains all the classical Lie idempotents of , in particular the Dynkin operator, a fundamental tool for studying the free Lie algebra. We look at a -deformation of the Dynkin operator and study its action over the descent algebra with classical combinatorial tools like Solomon's Mackey formula. This leads to elementary proofs that the operator is indeed an idempotent for as well as to interesting formulas and algebraic structures especially when is a root of unity.

Paper Structure

This paper contains 11 sections, 8 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Permutations, integer compositions and subsets
  3. The descent algebra of the symmetric group
  4. The Dynkin operator
  5. Action of $\mathbf{V}_n^{(q)}$ on the $\mathbf{B}$-basis of the descent algebra
  6. Decomposition according to descent classes
  7. Expansion of $\mathbf{V}_{n}^{(q)}\mathbf{B}_I$ in the $\mathbf{B}$-basis
  8. Consequences of Theorem \ref{['thm.vnq']}
  9. Idempotence
  10. Zero coefficients and eigenvalues
  11. Image space dimension

Key Result

Proposition 1

Let $I, J, K \subseteq [n-1]$. Let $\mathbb{N}^{I, J}_K$ be the set of nonnegative integer matrices with row sums vector $\operatorname{comp}(I)$, column sums vector $\operatorname{comp}(J)$ and reduced reading word $\operatorname{comp}(K)$. Then, $[\mathbf{B}_K]\mathbf{B}_J\mathbf{B}_I = |\mathbb{N

Theorems & Definitions (20)

  • Definition 1: composition refinement
  • Proposition 1: Solomon's Mackey formula
  • Example 1
  • Definition 2: Dynkin operator
  • Definition 3: $q$-deformation of the Dynkin operator, CalderbankHanlonSundaram94
  • Proposition 2
  • proof
  • Theorem 1
  • proof
  • Example 2
  • ...and 10 more