Rational semistandard tableaux and character formula for the Lie superalgebra $\hat{\frak{gl}}_{\infty|\infty}$

TL;DR

This paper introduces a combinatorial framework using rational semistandard tableaux to compute characters of certain representations of the infinite-dimensional Lie superalgebra e1a1e9a9e9a7e1a1e9a7e1a1e9a7e1a1e9a7, connecting it with classical combinatorics and Howe duality.

Contribution

The paper develops a new combinatorial interpretation of the character formula for e1a1e9a9e9a7e1a1e9a7e1a1e9a7e1a1e9a7 representations, linking rational semistandard tableaux with Howe dual pairs.

Findings

Character formulas expressed via bitableaux of skew shapes.

Established combinatorial rules including RSK correspondence and Littlewood-Richardson rule.

Unified framework explaining Howe dual pairs involving e1a1e9a9e9a7e1a1e9a7 and e1a1e9a9e9a7.

Abstract

A new combinatorial interpretation of the Howe dual pair $(\hat{\frak{gl}}_{\infty|\infty},\frak{gl}_n)$ acting on an infinite dimensional Fock space $\frak{F}^n$ of level $n$ is presented. The character of a quasi-finite irreducible highest weight representation of $\hat{\frak{gl}}_{\infty|\infty}$ occurring in $\frak{F}^n$ is realized in terms of certain bitableaux of skew shapes. We study a general combinatorics of these bitableaux, including Robinson-Schensted-Knuth correspondence and Littlewood-Richardson rule, and then its dual relation with the rational semistandard tableaux for $\frak{gl}_n$. This result also explains other Howe dual pairs including $\frak{gl}_n$.