Deformation of symmetric functions and the rational Steenrod algebra

TL;DR

This paper introduces a q-deformation of the ring of symmetric functions to explore the structure of the quotient by the rational Steenrod algebra, extending Wood's conjecture and providing partial results and conjectures.

Contribution

It proposes a non-commutative q-deformation of symmetric functions that generalizes the rational Steenrod algebra and extends Wood's conjecture to complex q values.

Findings

Partial results on the structure of the quotient

Extension of Wood's conjecture to q formal and complex values

Computer-based conjectures on the algebraic structure

Abstract

In 1999, Reg Wood conjectured that the quotient of Q[x_1,...,x_n] by the action of the rational Steenrod algebra is a graded regular representation of the symmetric group S_n. As pointed out by Reg Wood, the analog of this statement is a well known result when the rational Steenrod algebra is replaced by the ring of symmetric functions; actually, much more is known about the structure of the quotient in this case. We introduce a non-commutative q-deformation of the ring of symmetric functions, which specializes at q=1 to the rational Steenrod algebra. We use this formalism to obtain some partial results. Finally, we describe several conjectures based on an extensive computer exploration. In particular, we extend Reg Wood's conjecture to q formal and to any q complex not of the form -a/b, with a in {1,...,n} and b a positive natural number.