[Paper Review] Dualities for Lie superalgebras

Shun‐Jen Cheng, Weiqiang Wang|arXiv (Cornell University)|Dec 31, 2009

Algebraic structures and combinatorial models51 references3 citations

TL;DR

This paper establishes a conceptual framework for Lie superalgebra representation theory through three dualities: Schur-Sergeev duality, Howe duality for gl and osp types, and super duality. Super duality provides a solution to the irreducible character problem for parabolic category O of Lie superalgebras by relating it to Kazhdan-Lusztig polynomials of classical Lie algebras, overcoming the absence of Weyl group control in super representation theory.

ABSTRACT

We explain how Lie superalgebras of types gl and osp provide a natural framework generalizing the classical Schur and Howe dualities. This exposition includes a discussion of super duality, which connects the parabolic categories O between classical Lie superalgebras and Lie algebras. Super duality provides a conceptual solution to the irreducible character problem for these Lie superalgebras in terms of the classical Kazhdan-Lusztig polynomials.

Motivation & Objective

• To generalize classical Schur and Howe dualities to Lie superalgebras of types gl and osp.
• To resolve the long-standing irreducible character problem in parabolic category O for Lie superalgebras.
• To establish super duality as an equivalence between parabolic categories O of Lie superalgebras and Lie algebras.
• To provide a conceptual framework using Kazhdan-Lusztig polynomials for computing characters despite the lack of Weyl group control in super representation theory.
• To unify the representation theory of Lie superalgebras and Lie algebras via a duality that treats them as two sides of the same coin.

Proposed method

• Employ Schur-Sergeev duality to relate the actions of gl(m|n) and the symmetric group S_d on the tensor space (C^{m|n})^{igotimes d} via a double centralizer theorem.
• Formulate Howe duality for (gl(m|n), gl(d)) and (Sp(d), osp(2m|2n)) dual pairs, showing multiplicity-free decompositions of tensor spaces.
• Use odd reflections and hook partitions to classify finite-dimensional irreducible modules for osp-type superalgebras.
• Construct super duality via an equivalence between parabolic categories O of Lie superalgebras and classical Lie algebras, using master diagrams and truncation functors.
• Realize irreducible characters of Lie superalgebras as images under truncation functors Tr_n from the category O of infinite-rank Lie algebras.
• Apply Kazhdan-Lusztig polynomials from classical Lie algebras to compute characters of irreducible modules in the super setting via the super duality equivalence.

Experimental results

Research questions

• RQ1 How can classical Schur duality be generalized to Lie superalgebras of type gl(m|n) and osp?
• RQ2 What is the structure of the multiplicity-free decomposition of tensor spaces under the actions of gl(m|n) and S_d?
• RQ3 How does Howe duality extend to Lie superalgebras beyond type A, particularly for osp(2m|2n) and Sp(d)?
• RQ4 Can the irreducible character problem for Lie superalgebras be solved conceptually using classical Kazhdan-Lusztig polynomials?
• RQ5 What is the role of super duality in relating the representation theories of Lie superalgebras and Lie algebras in parabolic category O?

Key findings

• Super duality establishes an equivalence between parabolic categories O of Lie superalgebras and classical Lie algebras, providing a conceptual solution to the irreducible character problem.
• The irreducible character of any finite-dimensional irreducible module over an ortho-symplectic Lie superalgebra appears in some finite truncation O_n, and its character is determined by the Lusztig solution of the Kazhdan-Lusztig polynomials.
• The character of a simple gl(m|n)-module with extremal weights admits a diagrammatic interpretation via Young diagrams and hook partitions.
• The classification of finite-dimensional irreducible modules for osp-type superalgebras is achieved using odd reflections and hook partitions, offering a more natural labeling than previous methods.
• The multiplicity-free decomposition of (C^{m|n})^{igotimes d} under gl(m|n) × S_d is explicitly described, with highest weight vectors identified for each isotypical component.
• The super duality framework allows the transfer of Kazhdan-Lusztig polynomials from classical Lie algebras to compute characters in the super setting, even in the absence of Weyl group control.