2024/11/05 by Alexander Kleshchev, Kleshchev, Alexander · 1 citation
Mathematics · #18N25 #20C20 #20C25 #20C30 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2411.02735
openalex publication_date 2024/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The irreducible modules over quiver Hecke superalgebras Rθ can be classified in terms of cuspidal modules. To an indivisible positive root α and a non-negative integer d, one associates a quotient Rdα of Rdα called the cuspidal algebra. If the root α is real, the cuspidal algebra is well-understood. But if α=δ, the imaginary null-root, the \em imaginary cuspidal algebra Rdδ is rather mysterious. It has been known that the number of the isomorphism classes of the irreducible Rdδ-modules equals the number of the ℓ-multipartitions of d, but there has been no way to canonically associate an irreducible Rdδ-module to such a multipartiton. The imaginary cuspidal algebra is especially important because of its connections to the RoCK blocks of the double covers of symmetric and alternating groups. We undertake a detailed study of the imaginary cuspidal algebra and its representation theory. We use the so-called Gelfand-Graev idempotents and subtle degree and parity shifts to construct a (graded) Morita (super)equivalent algebra C(n,d) (for any n≥ d). The advantage of the algebra C(n,d) is that, unlike Rdδ, it is non-negatively graded. Moreover, the degree zero component C(n,d)0 is shown to be isomorphic to the direct sum of tensor products of ℓ copies of the classical Schur algebras. This gives the classification (and the description of dimensions/characters, etc.) of the irreducible C(n,d)-modules, and hence of the irreducible Rdδ-modules, in terms of the classical Schur algebras. In particular, this allows us to canonically label these by the ℓ-multipartitions of d. The results of this paper will be used in our future work on RoCK blocks of the double covers of symmetric groups.