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

Jae-Hoon Kwon|ArXiv.org|Apr 29, 2006

Advanced Topics in Algebra12 references3 citations

TL;DR

This paper introduces $χ/ϒ$-semistandard tableaux as a combinatorial framework to unify the representation theory of the Lie superalgebra $χτγ_{\infty|\infty}$ and its dual $τγ_n$, establishing a duality between skew Littlewood-Richardson rules for these tableaux and rational semistandard tableaux. It provides a new character formula for quasi-finite highest weight modules via these tableaux, generalizing Kac-Radul and Cheng-Lam results and yielding a Jacobi-Trudi-type formula.

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 $\frak{gl}_n$.

Motivation & Objective

• To provide a unified combinatorial interpretation of the Howe dual pair $(\widehat{\mathfrak{gl}}_{\infty|\infty}, \mathfrak{gl}_n)$ acting on the Fock space $\mathfrak{F}^n$.
• To develop a new combinatorial framework—$\mathcal{A}/\mathcal{B}$-semistandard tableaux—for studying the character formulas of irreducible highest weight modules of $\widehat{\mathfrak{gl}}_{\infty|\infty}$.
• To establish a dual relationship between the combinatorics of $\mathcal{A}/\mathcal{B}$-semistandard tableaux and rational semistandard tableaux for $\mathfrak{gl}_n$, generalizing RSK and Littlewood-Richardson rules.
• To derive a Jacobi-Trudi-type character formula for $L_\lambda$, a new result not previously observed in the literature.
• To extend the duality to tensor product decompositions and branching rules, explaining the general structure of Howe dual pairs in terms of combinatorial reciprocity.

Proposed method

• Introduces $\mathcal{A}/\mathcal{B}$-semistandard tableaux as pairs of semistandard tableaux with entries in graded sets $\mathcal{A}$ and $\mathcal{B}$, constrained by generalized partitions $\lambda$.
• Develops an insertion scheme for $\mathcal{A}/\mathcal{B}$-tableaux, yielding analogues of the Robinson-Schensted-Knuth (RSK) correspondence and Littlewood-Richardson (LR) rule.
• Defines skew $\mathcal{A}/\mathcal{B}$-semistandard tableaux of shape $\lambda/\mu$ and establishes a skew LR rule, dually related to rational semistandard tableaux.
• Shows that the character of the set $SST_{\mathcal{A}/\mathcal{B}}(\lambda)$ of such tableaux reduces to the character of $L_\lambda$ under suitable choices of $\mathcal{A}$ and $\mathcal{B}$, via a branching-type formula.
• Uses the duality between $\mathcal{A}/\mathcal{B}$-tableaux and rational semistandard tableaux to explain the duality between tensor product decomposition and branching rules in the Howe duality framework.
• Establishes an isomorphism $\chi: \mathscr{R}^* \to \mathbf{K}(\mathfrak{g})$ between the Grothendieck group and the ring of symmetric functions, proving $\chi$ is a $\mathbb{Z}$-algebra isomorphism via the LR rule and duality.

Experimental results

Research questions

• RQ1 How can the character of a quasi-finite irreducible highest weight representation $L_\lambda$ of $\widehat{\mathfrak{gl}}_{\infty|\infty}$ be combinatorially realized in terms of tableaux?
• RQ2 What is the precise duality between the combinatorics of $\mathcal{A}/\mathcal{B}$-semistandard tableaux and rational semistandard tableaux for $\mathfrak{gl}_n$?
• RQ3 Can the Littlewood-Richardson rule for $\mathcal{A}/\mathcal{B}$-tableaux be derived from the skew LR rule of rational semistandard tableaux, and vice versa?
• RQ4 Does the character formula for $L_\lambda$ admit a Jacobi-Trudi-type expression, and if so, how is it constructed combinatorially?
• RQ5 How does the duality between tensor product decomposition and branching rules in the Howe dual pair $(\widehat{\mathfrak{gl}}_{\infty|\infty}, \mathfrak{gl}_n)$ emerge from the tableaux combinatorics?

Key findings

• The character of the irreducible highest weight module $L_\lambda$ for $\widehat{\mathfrak{gl}}_{\infty|\infty}$ is realized as the generating function of $\mathcal{A}/\mathcal{B}$-semistandard tableaux of shape $\lambda$, under appropriate choices of $\mathcal{A}$ and $\mathcal{B}$.
• A new Jacobi-Trudi-type character formula for $L_\lambda$ is derived, which had not been previously observed in the literature.
• The skew Littlewood-Richardson rule for $\mathcal{A}/\mathcal{B}$-semistandard tableaux is completely determined by the Littlewood-Richardson rule for rational semistandard tableaux, and vice versa, establishing a dual reciprocity.
• The RSK correspondence and LR rule for $\mathcal{A}/\mathcal{B}$-tableaux are constructed via a new insertion scheme, generalizing classical results to the superalgebra setting.
• The isomorphism $\chi: \mathscr{R}^* \to \mathbf{K}(\mathfrak{g})$ is established as a $\mathbb{Z}$-algebra isomorphism, linking the Grothendieck group of representations to symmetric functions via tableaux characters.
• The duality between tensor product decomposition and branching rules in the Howe dual pair is explained combinatorially through the reciprocal structure of the LR rules for $\mathcal{A}/\mathcal{B}$-tableaux and rational semistandard tableaux.