Modular representations in type \(A\) with a two-row nilpotent central character. (English) Zbl 07799759
Summary: We study the category of representations of \(\mathfrak{sl}_{m + 2n}\) over a field of characteristic \(p\) with \(p \gg 0\), whose \(p\)-character is a nilpotent whose Jordan type is the two-row partition \((m + n, n)\). In a previous paper with R. Anno [Zbl 1524.14037], we used Bezrukavnikov-Mirković-Rumynin’s theory [Zbl 1220.17009] of positive characteristic localization and exotic t-structures to give a geometric parametrization of the simples using annular crossingless matchings. Building on this, here we give combinatorial dimension formulae for the simple objects, and compute the Jordan-Hölder multiplicities of the simples inside the baby Vermas. We use Cautis-Kamnitzer’s geometric categorification [Zbl 1145.14016] of the tangle calculus to study the images of the simple objects under the BMR equivalence. Our results generalize Jantzen’s formulae in the subregular nilpotent case (i.e. when \(n = 1\)), and may be viewed as a positive characteristic analogue of the combinatorial description for Kazhdan-Lusztig polynomials of Grassmannian permutations due to Lascoux and Schutzenberger.
MSC:
| 17B50 | Modular Lie (super)algebras |
| 20C20 | Modular representations and characters |
| 20G05 | Representation theory for linear algebraic groups |
| 22E47 | Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) |
| 14L30 | Group actions on varieties or schemes (quotients) |
Keywords:
modular representation theoryReferences:
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