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Generators for the Algebra of Symmetric Functions

Velmurugan S

Abstract

The algebra of symmetric functions contains several interesting families of symmetric functions indexed by integer partitions or skew partitions. Given a sequence $\{u_n\}$ of symmetric functions taken from one of these families such that $u_n$ is homogeneous of degree $n$, we provide necessary and sufficient conditions for the sequence to form a system of algebraically independent generators for the algebra of symmetric functions.

Generators for the Algebra of Symmetric Functions

Abstract

The algebra of symmetric functions contains several interesting families of symmetric functions indexed by integer partitions or skew partitions. Given a sequence of symmetric functions taken from one of these families such that is homogeneous of degree , we provide necessary and sufficient conditions for the sequence to form a system of algebraically independent generators for the algebra of symmetric functions.
Paper Structure (16 sections, 26 theorems, 32 equations, 1 table)

This paper contains 16 sections, 26 theorems, 32 equations, 1 table.

Table of Contents

  1. Introduction
  2. The Lemmas
  3. Generators for the familiar bases over $\mathbb{F}$
  4. Skew monomial symmetric functions
  5. Skew complete and skew elementary symmetric functions
  6. Schur functions
  7. Skew Schur functions $s_{\lambda/\mu}$
  8. Generators for the familiar bases over $\mathbb{F}(t)$
  9. Hall-Littlewood symmetric functions and their specializations.
  10. Big Schur functions $S_{\lambda}(x;t)$ and the functions $q_{\lambda}(x;t)$
  11. Skew Hall-Littlewood symmetric functions
  12. Schur's $P$-functions and $Q$-functions
  13. Generators for the familiar bases over $\mathbb{F}(q,t)$
  14. The Macdonald symmetric function $P_{\lambda}(x;q,t)$
  15. $Q$-Whittaker symmetric functions $W_\lambda(x;q)$
  16. ...and 1 more sections

Key Result

Theorem 1.2

For any graded partition sequence $\{\lambda^n\}$, $\{ m_{\lambda^n}\}_{n\geq0}$ generates $\Lambda_\mathbb{F}$.

Theorems & Definitions (52)

  • Definition 1.1
  • Theorem 1.2: Mead
  • Theorem 1.3: Timofte
  • Definition 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 3.1: Domino tabloid, MR3443860
  • Definition 3.2: Refinement
  • ...and 42 more