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Shunsuke Hirota

Publications and source records attributed to Shunsuke Hirota.

6 recordsLinked to original sources
arXiv2026

Some closed formulas for super Kazhdan-Lusztig polynomials

Kazhdan--Lusztig theory for finite-dimensional representations of the general linear Lie superalgebra $\mathfrak{gl}(m|n)$ is well understood through the Khovanov arc-diagram combinatorics. On the other hand, the Kazhdan--Lusztig conjecture for the full super category $\mathcal O$ was established by Cheng--Lam--Wang and Brundan--Losev--Webster, and the corresponding Kazhdan--Lusztig polynomials can be computed by Brundan's algorithm. The aim of this paper is to deepen our understanding of super category $\mathcal O$ and its Kazhdan--Lusztig polynomials by bringing together three ingredients: auxiliary modules associated with changes of Borel subalgebras introduced by Cheng--Lam--Wang, the Koszul grading studied by Brundan--Losev--Webster, and Whittaker coinvariants functor investigated by Brundan--Goodwin. We obtain, in particular, closed formulas for certain Kazhdan--Lusztig polynomials associated with odd reflections, determine the socles of some auxiliary modules, and realize certain pullbacks of standard modules via parabolic Miura transforms. We also use Khovanov arc-diagram combinatorics to characterize certain regular dominant weights intrinsically in terms of the underlying abelian category. We discuss how this viewpoint can be used to obtain concrete examples relevant to the study of Morita equivalence classes of blocks of super category $\mathcal O$.

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arXiv2026

Categorification of generic Su-Zhang character formula

For semisimple Lie algebras, the BGG resolution is often viewed as a categorification of the Weyl character formula. For general linear Lie superalgebras, Brundan--Stroppel constructed an infinite resolution of the so-called Kostant simple modules by Kac modules, but their construction does not directly generalize the classical BGG resolution. In this paper we construct, for weights lying outside a neighborhood of the walls of the Weyl chambers, a resolution that categorifies a known Weyl-type finite-sum character formula in the same spirit as the Kac--Wakimoto formula. Our resolution is built from images of canonical homomorphisms between Verma modules attached to non-conjugate Borel subalgebras related by odd reflections. In particular, the construction developed here does generalize the classical BGG resolution.

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arXiv2025

$\mathfrak b_1$-Verma $\mathfrak b_2$-dual Verma supermodules

We show that if a module M over a basic classical Lie superalgebra of type type I is simultaneously a Verma module with respect to some Borel \(\mathfrak b_1\) and a dual Verma module with respect to Borel \(\mathfrak b_2\), then M is isomorphic to a Verma module with respect to either distinguished or an anti-distinguished Borel. Our method proceeds by analyzing edge contractions of the finite Young lattice that controls the combinatorics of odd reflections. In principle, the same strategy, for the most part, applies to all basic classical Lie superalgebras.

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arXiv2025

Odd Verma's Theorem

We formulate several basic properties of Verma supermodules over regular symmetrizable Kac--Moody Lie superalgebras, exhibiting $\mathfrak{gl}(1|1)$-nature as revealed through changing Borel subalgebras. We investigate variants of Verma modules obtained by changing Borel subalgebras, which enable us to realize the principal block of $\mathfrak{gl}(1|1)$ as an extension-closed abelian subcategory of category $\mathcal{O}$. This phenomenon is precisely formulated in terms of semibricks. On the other hand, by applying the exchange property of odd reflections, we describe compositions of homomorphisms between Verma modules associated with different Borel subalgebras that share the same character. As an application, we refine existing results on the associated varieties and projective dimensions of Verma modules.

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arXiv2025

Rainbow Boomerang Graphs

We generalize the well known exchange property of Coxeter groups to the setting of edge-colored graphs. This work aims to unify and extend the results of our companion article, "odd Verma's theorem", which were originally established for basic Lie superalgebras, to the broader setting of regular symmetrizable Kac-Moody Lie superalgebras and Nichols algebras of diagonal type, via the theory of Weyl groupoids in the sense of Heckenberger and Yamane. In particular, we show that the exchange property of odd reflections arises as a special case of the exchange property of Weyl groupoids. To study the exchange property itself, we analyze a class of edge-colored graphs introduced here, called rainbow boomerang graphs, which form an independently natural family of combinatorial objects. We also elaborate on odd Verma theorem in the specific setting of Nichols algebras of diagonal type.

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