Deformation of Symmetric Functions and the Rational Steenrod Algebra

In 1999, Reg Wood conjectured that the quotient of Q[x1, . . . , xn] by the action of the rational Steenrod algebra is a graded regular representation of the symmetric group Sn. 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 ∈ C not of the form −a/b, with a ∈ {1, . . . , n} and b ∈ N.