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

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 $\frak{gl}_n$.