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The ring of $ω$-invariant symmetric functions in characteristic 2

Sebastian Ørsted

Abstract

We provide a simple presentation by generators and relations of the ring of $ω$-invariant symmetric functions over the field $\mathbb{F}_{2}$. Here, $ω$ denotes the standard involution on the ring of symmetric functions, interchanging the elementary symmetric functions with the complete homogeneous symmetric functions. Along the way, we prove several important properties of this involution in the specific setting of characteristic 2.

The ring of $ω$-invariant symmetric functions in characteristic 2

Abstract

We provide a simple presentation by generators and relations of the ring of -invariant symmetric functions over the field . Here, denotes the standard involution on the ring of symmetric functions, interchanging the elementary symmetric functions with the complete homogeneous symmetric functions. Along the way, we prove several important properties of this involution in the specific setting of characteristic 2.

Paper Structure

This paper contains 6 sections, 33 theorems, 38 equations.

Table of Contents

  1. Thick differentials, transversality, and normality
  2. Basic properties of the ring R
  3. The subring of power sums
  4. The quotient ring S/I
  5. Mixed coordinates, the Transversality Theorems, and standard forms
  6. Generators and relations for S

Key Result

Proposition 1.4

For a transverse involution $\omega$, we have a short exact sequence of vector spaces \begin{tikzcd}[sep=small] 0 \ar[r] & S/I \ar[r] & R/RI \ar[r, "\diff{d}"] & I/I^{2} \ar[r] & 0 \text{.} \end{tikzcd}

Theorems & Definitions (77)

  • Definition 1.3
  • Proposition 1.4
  • Proof 1
  • Lemma 1.5
  • Proof 2
  • Definition 1.6
  • Proposition 1.7
  • Proof 3
  • Definition 1.8
  • Proposition 1.9
  • ...and 67 more