An inversion formula for some Fock spaces

TL;DR

This paper establishes an inversion formula relating canonical and dual canonical bases in certain Fock spaces, providing an algorithm for computing these bases for arbitrary parameters.

Contribution

It introduces a bilinear form on a subspace of Fock space and derives an inversion formula linking canonical and dual canonical bases, extending computational methods.

Findings

Derived an inversion formula connecting basis coefficients.

Established duality between canonical and dual canonical bases.

Provided an algorithm for computing bases for arbitrary parameters.

Abstract

A symmetric bilinear form on a certain subspace $\widehat{\mathbb T}^{\bf b}$ of a completion of the Fock space $\mathbb T^{{\bf b}}$ is defined. The canonical and dual canonical bases of $\widehat{\mathbb T}^{\bf b}$ are dual with respect to the bilinear form. As a consequence, the inversion formula connecting the coefficients of the canonical basis and that of the dual canonical basis of $\widehat{\mathbb T}^{\bf b}$ expanded in terms of the standard monomial basis of $\mathbb T^{{\bf b}}$ is obtained. Combining with the Brundan's algorithm for computing the elements in the canonical basis of $\widehat{\mathbb{T}}^{{\bf b}_{\mathrm{st}}}$, we have an algorithm computing the elements in the canonical basis of $\widehat{\mathbb{T}}^{{\bf b}}$ for arbitrary ${\bf b}$.