Schur-Weyl dualities for symmetric inverse semigroups
We obtain Schur-Weyl dualities in which the algebras, acting on both sides, are semigroup algebras of various symmetric inverse semigroups and their deformations.
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arXiv subjects
Volodymyr Mazorchuk
Publications and source records attributed to Volodymyr Mazorchuk.
Categorification of (induced) cell modules and the rough structure of generalized Verma modules
This paper presents categorifications of (right) cell modules and induced cell modules for Hecke algebras of finite Weyl groups. In type $A$ we show that these categorifications depend only on the isomorphism class of the cell module, not on the cell itself. Our main application is multiplicity formulas for parabolically induced modules over a reductive Lie algebra of type $A$, which finally determines the so-called rough structure of generalized Verma modules. On the way we present several categorification results and give the positive answer to Kostant's problem from \cite{Jo} in many cases. We also give a general setup of decategorification, precategorification and categorification.
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On the semigroup of square matrices
We study the structure of nilpotent subsemigroups in the semigroup $M(n,\mathbb{F})$ of all $n\times n$ matrices over a field, $\mathbb{F}$, with respect to the operation of the usual matrix multiplication. We describe the maximal subsemigroups among the nilpotent subsemigroups of a fixed nilpotency degree and classify them up to isomorphism. We also describe isolated and completely isolated subsemigroups and conjugated elements in $M(n,\mathbb{F})$.
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Koszul duality for extension algebras of standard modules
We define and investigate a class of Koszul quasi-hereditary algebras for which there is a natural equivalence between the bounded derived category of graded modules and the bounded derived category of graded modules over (a proper version of) the extension algebra of standard modules. Examples of such algebras include, in particular, the multiplicity free blocks of the BGG category $\mathcal{O}$, and some quasi-hereditary algebras with Cartan decomposition in the sense of K{ö}nig.
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Partialization of categories and inverse braid-permutation monoid
We show how the categorial approach to inverse monoids can be described as a certain endofunctor (which we call the partialization functor) of some category. In this paper we show that this functor can be used to obtain several recently defined inverse monoids, and use it to define a new object, which we call the inverse braid-permutation monoid. A presentation for this monoid is obtained. Finally, we study some abstract properties of the partialization functor and its iterations. This leads to a categorification of a monoid of all order preserving maps, and series of orthodox generalizations of the symmetric inverse semigroup.
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A categorification of integral Specht modules
We suggest a simple definition for categorification of modules over rings and illustrate it by categorifying integral Specht modules over the symmetric group and its Hecke algebra via the action of translation functors on some subcategories of category $\mathcal{O}$ for the Lie algebra $\mathfrak{sl}_n(\mathbb{C})$.
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Some homological properties of the category $\mathcal{O}$
In the first part of this paper the projective dimension of the structural modules in the BGG category $\mathcal{O}$ is studied. This dimension is computed for simple, standard and costandard modules. For tilting and injective modules an explicit conjecture relating the result to Lusztig's $\mathbf{a}$-function is formulated (and proved for type $A$). The second part deals with the extension algebra of Verma modules. It is shown that this algebra is in a natural way $\mathbb{Z}^2$-graded and that it has two $\mathbb{Z}$-graded Koszul subalgebras. The dimension of the space $\mathrm{Ext}^1$ into the projective Verma module is determined. In the last part several new classes of Koszul modules and modules, represented by linear complexes of tilting modules, are constructed.
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Presentation of the singular part of the Brauer monoid
We obtain a presentation for the singular part of the Brauer monoid with respect to an irreducible system of generators, consisting of idempotents. As an application of this result we get a new construction of the symmetric group via connected sequences of subsets. Another application describes the lengths of elements in the singular part of the Brauer monoid with respect to the system of generators, mentioned above.
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On finitistic dimension of stratified algebras
In this survey we discuss the results on the finitistic dimension of various stratified algebras. We describe what is already known, present some recent estimates, and list some open problems.
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On presentations of Brauer-type monoids
We obtain presentations for the Brauer monoid, the partial analogue of the Brauer monoid, and for the greatest factorizable inverse submonoid of the dual symmetric inverse monoid. In all three cases we apply the same approach, based on the realization of all these monoids as Brauer-type monoids.
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Representation type of ${}^{\infty}_λ\mathcal{H}_μ^1$
For a semi-simple finite-dimensional complex Lie algebra $\mathfrak{g}$ we classify the representation type of the category ${}^{\infty}_λ\mathcal{H}_μ^1$ of Harish-Chandra bimodules for $\mathfrak{g}$.
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Combinatorics and distributions of partial injections
We obtain several combinatorial results about chains, cycles and orbits of the elements of the symmetric inverse semigroup $\IS_n$ and the set $T_n$ of nilpotent elements in $\IS_n$. We also get some estimates for the growth of $|\IS_n|$ and $|T_n|$, and study random products of elements from $\IS_n$.
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On Kiselman's semigroup
We study the algebraic properties of the series $\mathrm{K}_n$ of semigroups, which is inspired by \cite{Ki} and has origins in convexity theory. In particular, we describe Green's relations on $\mathrm{K}_n$, prove that there exists a faithful representation of $\mathrm{K}_n$ by $n\times n$ matrices with non-negative integer coefficients (and even explicitly construct such a representation), and prove that $\mathrm{K}_n$ does not admit a faithful representation by matrices of smaller size. We also describe the maximal nilpotent subsemigroups in $\mathrm{K}_n$, all isolated and completely isolated subsemigroups, all automorphisms and anti-automorphisms of $\mathrm{K}_n$. Finally, we explicitly construct all irreducible representations of $\mathrm{K}_n$ over any field and describe primitive idempotents in the semigroup algebra (which we prove is basic).
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On nilpotent subsemigroups in some matrix semigroups
We describe maximal nilpotent subsemigroups of a given nilpotency class in the semigroup $Ω_n$ of all $n\times n$ real matrices with non-negative coefficients and the semigroup $\mathbf{D}_n$ of all doubly stochastic real matrices.
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On classification of maximal nilpotent subsemigroups
We develop a general approach to the study of maximal nilpotent subsemigroups of finite semigroups. This approach can be used to recover many known classifications of maximal nilpotent subsemigroups, in particular, for the symmetric inverse semigroup, the symmetric semigroup, and the factor power of the symmetric group. We also apply this approach to obtain a classification of maximal nilpotent subsemigroups in the semigroup of binary relations and in certain 0-simple finite semigroups.
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Projective-injective modules, Serre functors and symmetric algebras
We describe Serre functors for (generalisations of) the category O associated with a semi-simple complex Lie algebra. In our approach, projective-injective modules play an important role. They control the Serre functor in the case of a quasi-hereditary algebra having a double centraliser property with respect to a symmetric algebra. As an application of the double centraliser property and our description of Serre functors, we prove three conjectures of Khovanov about the projective-injective modules in the parabolic category O for sl_n.
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$\mathcal{L}$-- and $\mathcal{R}$--cross-sections in the Brauer semigoup
We classify all cross-sections of Green's relations $\mathcal{L}$ and $\mathcal{R}$ in the Brauer semigroup. The regular behavior of such cross-sections starts from $n=7$. We show that in the regular case there are essentially two different cross-sections and all others are $\mathcal{S}_n$-conjugated to one of these two. We also classify all cross-sections up to isomorphism.
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Classification of simple weight Virasoro modules with a finite-dimensional weight space
We show that the support of a simple weight module over the Virasoro algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all non-trivial weight spaces of such module are infinite dimensional. As a corollary we obtain that every simple weight module over the Virasoro algebra, having a non-trivial finite-dimensional weight space, is a Harish-Chandra module (and hence is either a simple highest or lowest weight module or a simple module from the intermediate series). This implies positive answers to two conjectures about simple pointed and simple mixed modules over the Virasoro algebra.
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