Compound basis for the space of symmetric functions

TL;DR

This paper introduces a new compound basis for symmetric functions combining Schur and Q-functions, explores their connection through transition matrices, and reveals a combinatorial structure involving powers of 2.

Contribution

It presents a novel compound basis for symmetric functions derived from affine Lie algebra representations and investigates the transition matrix between this basis and Schur functions.

Findings

Transition matrix determinant is a power of 2.

Numerical results suggest a combinatorial pattern in the basis connection.

The basis is linked to twisted homogeneous realizations of affine Lie algebra representations.

Abstract

The aim of this note is to introduce a compound basis for the space of symmetric functions. Our basis consists of products of Schur functions and $Q$-functions. The basis elements are indexed by the partitions. It is well known that the Schur functions form an orthonormal basis for our space. A natural question arises. How are these two bases connected? In this note we present some numerical results of the transition matrix for these bases. In particular we will see that the determinant of the transition matrix is a power of 2. This is not a surprising fact. However the explicit formula involves an interesting combinatorial feature. Our compound basis comes from the twisted homogeneous realization of the basic representation of the affine Lie algebras. This note is not written in a standard style of mathematical articles. It is more like a draft of a talk. In particular proofs are not given here. Details and proofs will be published elsewhere.