Lie Superalgebras and Generalized Kazhdan-Lusztig Polynomials

Abstract: We present liesuperalg a SageMath package for representation-theoretic calculations involving Lie superalgebras in Type A. Our package introduces functionality to calculate invariants of weights and produce the associated cup diagrams. We expose functionality to calculate characters of irreducible representations, work with combinatorics of generalized Kazhdan-Lusztig polynomials, and determine composition factor multiplicities of indecomposable modules. Our package implements an algorithm to decompose arbitrary modules in terms of irreducible ones in the Grothendeick group of Lie superalgebra representations.

• The paper introduces a SageMath package that computes weight invariants, cup diagrams, and generalized Kazhdan-Lusztig polynomials for type-A Lie superalgebras.
• It employs recursive, combinatorial techniques to determine composition factors of Kac modules and provide explicit character formulas.
• Its algorithmic tools lower barriers in super representation theory and open avenues for research in categorification and quantum symmetry.

Lie Superalgebras and Generalized Kazhdan-Lusztig Polynomials: Algorithmic and Combinatorial Advances

Overview and Motivation

The problem of explicitly computing irreducible characters and composition factors for finite-dimensional representations of Lie superalgebras, particularly $\mathfrak{gl}(m|n)$, is central in modern representation theory, with implications in combinatorics, categorification, and mathematical physics. Classical highest weight theory and the Kazhdan-Lusztig polynomial machinery, which solved the analogous questions for complex semisimple Lie algebras, encounter substantial obstructions in the superalgebra context due to atypicality, non-semisimplicity, and the dependence of the representation category on the Borel subalgebra. Building on seminal work by Kac, Serganova, Brundan, and others, this paper presents the SageMath package liesuperalg, introducing systematic algorithmic tools for the computation of representation-theoretic invariants and combinatorial data for Type A Lie superalgebras, along with an exposition of the underlying mathematical structures and algorithms.

Mathematical Foundations

Structure of $\mathfrak{gl}(m|n)$ and Weight Invariants

$\mathfrak{gl}(m|n)$ is defined as the algebra of endomorphisms on a $\mathbb{Z}_2$-graded vector space $V = V_0 \oplus V_1$ with the dimension vector $(m,n)$. Weights are tuples $(\lambda^0\,|\,\lambda^1)$ and the Weyl group is $S_m \times S_n$ acting via the usual dot action with the super analog of the $\rho$ shift. However, unlike the setting for semisimple Lie algebras, representations are not governed by semisimple theory—the set of Kac modules (parabolically induced from the Levi $\mathfrak{gl}(m)\oplus\mathfrak{gl}(n)$) is not in bijection with irreducibles, and multiplicities appear at the level of composition factors.

Fundamental combinatorial invariants include:

• Degree of atypicality ($r$): the rank of the atypical root system for a given weight $\lambda$.
• Atypical/Tyical tuples: partitioning coordinates corresponding to roots orthogonal to $\lambda+\rho$ (atypical) and those not (typical), encoding block decomposition in the category $\mathcal{O}$.
• Height vector $h(\lambda)$: a collection of representation-theoretic invariants that govern the combinatorics of Kazhdan-Lusztig type polynomials.

The paper demonstrates how these invariants are computed via the Weight class in the package, supporting essential manipulations such as indexing, dominance checks, Weyl group actions, and conversion routines between data types.

Weight and Cup Diagrams

Cup diagrams provide a graphical combinatorial calculus for tracking the location and interaction of atypical roots along the integer lattice, capturing the intricate super-analog of weight diagrams. Their construction is essential for the recursive (Brundan-type) algorithms for Kazhdan-Lusztig polynomials and for the categorification context (diagrammatic Hecke/Chuang-Rouquier theory). The package provides automation for computing these diagrams in $\mathfrak{gl}(m|n)$, facilitating further calculation and visualization.

Generalized Kazhdan-Lusztig Polynomials and Characters

The central technical objects for constructing character formulas are the generalized Kazhdan-Lusztig polynomials $K_{\lambda,\mu}(q)$, which encode composition multiplicities and the structure of the category of representations. Utilizing the algorithmic paradigm developed by Brundan, the package implements:

• Explicit computation of $K_{\lambda,\mu}(q)$: leveraging the combinatorics of the cup/weight diagram, orderings on atypical roots, and permutations constrained by $c$-connectivity and strong $c$-connectivity relations.
• Multiplicity determination: composition multiplicities for Kac modules and irreducibles as the evaluation $K_{\lambda,\mu}(-1)$, reflecting the canonical character formula established by Serganova.

The approach is practically illustrated with explicit high-rank examples, emphasizing algorithmic transparency and reproducibility.

Decomposition Algorithms for Modules

Beyond isolated character calculations, the package supplies recursive algorithms for the decomposition of arbitrary modules (coded by highest weight data and multiplicities over the Levi) into the basis of irreducible characters via iterative subtraction of leading terms. This process relies upon invertibility at the generic level of the character ring and accurate computation of induced Kac module constituents and their images under the evaluated Kazhdan-Lusztig polynomials.

Composition Factors of Kac Modules

The structure theory for Kac modules is computationally nontrivial because their composition factors are governed by recursive actions of lowering operators on cup diagrams. The implementation closely follows the theoretical constructions developed in Su and Zhang, producing explicit lists of all composition factors for a given Kac module in terms of their shifted weights.

Implementation and Computational Aspects

The liesuperalg package is designed for integration with SageMath, exploiting its symbolic and algebraic data structures. Key features include:

• Class infrastructure for weights with full operator overloading and LaTeX compatibility,
• Core algorithms for diagrammatic, combinatorial, and character-theoretic quantities,
• Modular and extensible interface, assisting in experimental and theoretical investigations.

Implications and Future Research Directions

This work substantially lowers the barrier to concrete calculations in $\mathfrak{gl}(m|n)$ representation theory, with direct applications in categorification, combinatorial representation theory, and symmetry analysis in mathematical physics. The explicit, verifiable computation of Kazhdan-Lusztig polynomials for superalgebras further supports conjectures regarding character universality, supports investigations into super duality and equivalence of categories, and enables practical study of complicated blocks, linkage, and crystal structure.

Potential extensions include adaptation to other Lie superalgebras (e.g., orthosymplectic types), integration with higher categorical structures (2-representation theory), and the study of connections with knot invariants and quantum symmetry.

Conclusion

The presented work supplies both a rigorous combinatorial and algorithmic foundation for key aspects of $\mathfrak{gl}(m|n)$ representation theory and a highly practical computational toolset. The ability to compute weight invariants, cup diagrams, characters, composition multiplicities, and module decompositions in a controlled, reproducible manner opens new avenues for both theoretical research and applications in representation theory and related fields.