Bialgebras for Stanley symmetric functions

TL;DR

This paper constructs a new algebraic structure called a bialgebra that encodes permutations and their associated Stanley symmetric functions, providing new formulas, identities, and extensions to classical types.

Contribution

It introduces a novel non-commutative, non-cocommutative bialgebra related to permutations and Stanley symmetric functions, extending to classical types with module coalgebras.

Findings

Defined a new bialgebra with basis indexed by permutations

Established a morphism to quasi-symmetric functions mapping permutations to Stanley symmetric functions

Derived new enumerative identities and extended constructions to classical types

Abstract

We construct a non-commutative, non-cocommutative, graded bialgebra $\mathbf{\Pi}$ with a basis indexed by the permutations in all finite symmetric groups. Unlike the formally similar Malvenuto-Poirier-Reutenauer Hopf algebra, this bialgebra does not have finite graded dimension. After giving formulas for the product and coproduct, we show that there is a natural morphism from $\mathbf{\Pi}$ to the algebra of quasi-symmetric functions, under which the image of a permutation is its associated Stanley symmetric function. As an application, we use this morphism to derive some new enumerative identities. We also describe analogues of $\mathbf{\Pi}$ for the other classical types. In these cases, the relevant objects are module coalgebras rather than bialgebras, but there are again natural morphisms to the quasi-symmetric functions, under which the image of a signed permutation is the corresponding Stanley symmetric function of type B, C, or D.