The Hopf algebra of odd symmetric functions

TL;DR

This paper introduces the algebra of odd symmetric functions as a Hopf superalgebra arising from a q-analogue of the symmetric functions' bilinear form at q=-1, with new basis elements and combinatorial interpretations.

Contribution

It defines the algebra of odd symmetric functions, providing explicit bases and combinatorial interpretations, extending classical symmetric function theory to a superalgebra setting.

Findings

Defined the algebra of odd symmetric functions as a Hopf superalgebra.

Constructed analogues of elementary, complete, and power sum symmetric functions.

Provided combinatorial interpretations for basis change relations.

Abstract

We consider a q-analogue of the standard bilinear form on the commutative ring of symmetric functions. The q=-1 case leads to a Z-graded Hopf superalgebra which we call the algebra of odd symmetric functions. In the odd setting we describe counterparts of the elementary and complete symmetric functions, power sums, Schur functions, and combinatorial interpretations of associated change of basis relations.