Quasi-symmetric functions as polynomial functions on Young diagrams

TL;DR

This paper characterizes the algebra of smooth functions on Young diagrams that depend only on shape, showing it is isomorphic to quasi-symmetric functions and providing a noncommutative analog.

Contribution

It establishes a precise isomorphism between shape-dependent smooth functions on Young diagrams and quasi-symmetric functions, including a noncommutative extension.

Findings

The algebra of shape-dependent smooth functions on Young diagrams is isomorphic to quasi-symmetric functions.

A noncommutative analog of this isomorphism is also established.

The results provide a new algebraic perspective on functions defined by Young diagram shapes.

Abstract

We determine the most general form of a smooth function on Young diagrams, that is, a polynomial in the interlacing or multirectangular coordinates whose value depends only on the shape of the diagram. We prove that the algebra of such functions is isomorphic to quasi-symmetric functions, and give a noncommutative analog of this result.