arXiv subjects
Publications and source records attributed to Volodymyr Mazorchuk.
We provide a categorification for the hybrid Kazhdan-Lusztig bases of the Hecke algebra by constructing and investigating a series of new stratified structures on the principal block of BGG category $\mathcal{O}$.
In relation to Kostant's problem for simple highest weight modules over the general linear Lie algebra, we prove a persistence result for Kostant negative consecutive patterns. Inspired by it, we introduce the notion of a Kostant cuspidal permutation as a minimal Kostant negative consecutive pattern. It is shown that Kostant cuspidality is an invariant of a Kazhdan-Lusztig left cell. We describe four infinite families of Kostant cuspidal involutions, including a complete classification of Kostant cuspidal fully commutative involutions. In particular, we show that the number of new Kostant cuspidal elements can be arbitrarily large, when the rank grows. This provides some potential explanation why Kostant's problem is hard.
We provide a recurrence for computing the terms of the OEIS sequence A393920, introduced in \cite{KS}. We also describe a surprising connection between A393920 and the Fibonacci sequence A000045, obtain non-trivial lower and upper exponential bounds for its growth, and investigate connections with partial orders, Catalan numbers, and convex topologies on finite chains. For the representation-theoretic lattice underlying A393920, we describe its atoms, coatoms, join-irreducible and meet-irreducible elements.
We give an explicit classification as well as an explicit construction of simple $\mathfrak{sl}_2$-modules that are torsion-free of finite rank with respect to the action of the Cartan subalgebra and which admit a grading by a finite cyclic group corresponding to the rank of the module.
This paper discusses various aspects of the Hecke algebra combinatorics that are related to conditions appearing in Kåhrstr{ö}m's conjecture that addresses Kostant's problem for simple highest weight modules in the Bernstein-Gelfand-Gelfand category $\mathcal{O}$ for the complex Lie algebra $\mathfrak{sl}_n$. In particular, we study cyclic submodules of the regular Hecke module that are generated by the elements of the (dual) Kazhdan-Lusztig basis as well as the problem of left cell invariance for both categorical and combinatorial Kåhrstr{ö}m's conditions.
We provide an explicit classification of all simple $\mathfrak{sl}_2$-modules that are torsion free of rank $1$ over the Cartan subalgebra. We also establish a similar result for the first Weyl algebra and for the Lie superalgebra $\mathfrak{osp}(1|2)$.
For a permutation $z$ in the symmetric group $\mathrm{S}_{n}$, denote by $L_{z}$ the corresponding simple highest weight module in the principal block of the BGG category $\mathcal{O}$ for the Lie algebra $\mathfrak{sl}_{n}(\mathbb{C})$. In this paper, we provide a combinatorial answer to Kostant's problem for the modules $L_{z}$ when $z$ has shape (associated Young diagram/integer partition via Robinson-Schensted correspondence) equal to $(n-2,1,1)$ or $(n-3,2,1)$. Moreover, we verify that certain closely related conjectures hold for such permutations, including the Indecomposability Conjecture, which states that applying any indecomposable projective functor to the corresponding simple highest weight module outputs either an indecomposable module or zero.
For a permutation $w$ in the symmetric group $\mathfrak{S}_{n}$, let $L(w)$ denote the simple highest weight module in the principal block of the BGG category $\mathcal{O}$ for the Lie algebra $\mathfrak{sl}_{n}(\mathbb{C})$. We first prove that $L(w)$ is Kostant negative whenever $w$ consecutively contains certain patterns. We then provide a complete answer to Kostant's problem in type $A_{6}$ and show that the indecomposability conjecture also holds in type $A_{6}$, that is, applying an indecomposable projective functor to a simple module outputs either an indecomposable module or zero.
We determine the endomorphism categories of cell 2-representations of fiat 2-categories associated with strongly regular two-sided cells under some natural assumptions. Along the way, we completely describe J-simple fiat 2-categories which have only one two-sided cell J apart from the identities, under the same conditions as above. For positively graded 2-categories, we show that the additional restrictions are redundant.
Let $\mathcal{P}_k(δ)$, where $k$ is a positive integer and $δ$ some complex parameter, be the classical partition algebra over the complex numbers. In the case when $δ=n$, it is well-known that the algebra $\mathcal{P}_k(δ)$ is the centralizer of the symmetric group $S_n$ acting on the $k$-fold tensor space of the natural representation of $S_n$, for $n\geq 2k$. The algebra $\mathcal{P}_k(δ)$ is semi-simple for generic values of $δ$. In this paper, we show that semi-simple partition algebras appear as the centralizer algebras for certain representations of the rook monoids given by an iterative restriction-induction of the trivial representation. Along the way, we also give a decomposition of this iterative representation of the rook monoid into various tensor spaces and show that the corresponding dimensions are given by generalized Bell numbers.
This paper is, essentially, a survey related to the problem of understanding the combinatorics of the action of the monoidal category of finite dimensional modules over a simple finite dimensional Lie algebra on various categories of Lie algebra modules. A special attention is payed to the Lie algebras $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$. A few new general results are collected at the end.
We study various categories of Whittaker modules over a type I Lie superalgebra realized as cokernel categories that fit into the framework of properly stratified categories. These categories are the target of the Backelin functor $Γ_ζ$. We show that these categories can be described, up to equivalence, as Serre quotients of the BGG category $\mathcal O$ and of certain singular categories of Harish-Chandra $(\mathfrak g,\mathfrak g_{\bar 0})$-bimodules. We also show that $Γ_ζ$ is a realization of the Serre quotient functor. We further investigate a $q$-symmetrized Fock space over a quantum group of type A and prove that, for general linear Lie superalgebras our Whittaker categories, the functor $Γ_ζ$ and various realizations of Serre quotients and Serre quotient functors categorify this $q$-symmetrized Fock space and its $q$-symmetrizer. In this picture, the canonical and dual canonical bases in this $q$-symmetrized Fock space correspond to tilting and simple objects in these Whittaker categories, respectively.
We show that the action of the Serre functor on the subcategory of projective-injective modules in a parabolic BGG category $\mathcal O$ of a quasi-reductive finite dimensional Lie superalgebra is given by tensoring with the top component of the symmetric power of the odd part of our superalgebra. As an application, we determine, for all strange Lie suepralgebras, when the subcategory of projective injective modules in the parabolic category $\mathcal O$ is symmetric.
We prove that, when $n$ goes to infinity, the expression, with respect to the dual Kazhdan-Lusztig basis, of the product $\hat{\underline{H}}_x\underline{H}_y$ of elements of the dual and the usual Kazhdan-Lusztig bases in the Hecke algebra of the symmetric group $S_n$ stabilizes. As an application, we define the action of projective functors on the principal block of category $\mathcal{O}$ for $\mathfrak{sl}_\infty$ and show that the subcategory of finite length objects is stable under this action. As a bonus, we also prove that this latter block is Koszul, answering, for this block, a question from \cite{CP}.
We describe Calabi-Yau objects in the regular block of the (parabolic) BGG category $\mathcal{O}$ associated to a semi-simple finite dimensional complex Lie algebra. Each such object comes with a natural transformation from the Serre functor to a shifted identity whose evaluation at that object is an isomorphism.
We determine the combinatorics of transitive module categories over the monoidal category of finite dimensional $\mathfrak{sl}_3$-modules which arise when acting by the latter monoidal category on arbitrary simple $\mathfrak{sl}_3$-modules. This gives us a family of eight graphs which can be viewed as $\mathfrak{sl}_3$-generalizations of the classical infinite Dynkin diagrams.
We prove that, when $n$ goes to infinity, Kostant's problem has negative answer for almost all simple highest weight modules in the principal block of the BGG category $\mathcal{O}$ for the Lie algebra $\mathfrak{sl}_n(\mathbb{C})$.
We study locally finitary realizations of simple transitive module categories of infinite rank over the monoidal category $\mathscr{C}$ of finite dimensional modules for the complex Lie algebra $\mathfrak{sl}_2$. Combinatorics of such realizations is governed by six infinite Coxeter diagrams. We show that five of these are realizable in our setup, while one (type $B_\infty$) is not. We also describe the $\mathscr{C}$-module subcategories of $\mathfrak{sl}_2$-mod generated by simple modules as well as the $\mathscr{C}$-module categories coming from the natural action of $\mathscr{C}$ on the categories of finite dimensional modules over Lie subalgebras of $\mathfrak{sl}_2$.