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

Abstract

A new combinatorial interpretation of the Howe dual pair $({\widehat{\mathfrak{gl}}}_{\infty |\infty },{\mathfrak{gl}}_n)$ acting on an infinite dimensional Fock space ${\mathfrak{F}}^n$ of level $n$ is presented. The character of a quasi-finite irreducible highest weight representation of ${\widehat{\mathfrak{gl}}}_{\infty |\infty }$ occurring in ${\mathfrak{F}}^n$ 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 ${\mathfrak{gl}}_n$. This result also explains other Howe dual pairs including ${\mathfrak{gl}}_n$.

39pages

References in corpus (1)

• Crystal graphs for Lie superalgebras and Cauchy decomposition

Cited by in corpus (1)

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