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Ivan Dimitrov

Publications and source records attributed to Ivan Dimitrov.

18 recordsLinked to original sources
arXiv2025

On ${U}(\mathfrak{h})$-free modules over $\mathfrak{sl}(m|n)$

We study two categories of ${U}(\mathfrak h)$-free $\mathfrak{sl}(m|n)$-modules of total rank 2: $\mathcal{M}_{\mathfrak{sl}(m|n)}(2)$, whose objects are free of rank 2 over ${U}(\mathfrak h)$ which are not necessarily $\mathbb Z_2$-graded, and $\mathcal{M}_{\mathfrak{sl}(m|n)}(1|1)$, whose objects are supermodules with even and odd parts each isomorphic to ${U}(\mathfrak h)$. For $\mathfrak{sl}(m|1)$ we give a complete classification in both categories, and we prove that for $m,n\geq 2$ both categories are empty.

math.RT↗
arXiv2025

Generalized permutation matrices and non-weight modules over $\mathfrak{sl}(m|1)$

We study the category $\mathcal{M}_{\mathfrak{sl}(m|1)}(k|k)$ of $\mathcal U(\mathfrak h)\text{-free}$ $\mathcal U(\mathfrak{sl}(m|1))$-modules of rank $k$ in each parity (rank $(k|k)$), where $k\in\mathbb{Z}_{\geq1}$. We construct an explicit family of such modules, provide an isomorphism theorem, and establish an indecomposability criterion.

math.RT↗
arXiv2023

Inversion Sets and Quotient Root Systems

The main result of this paper is a recursive description of all decompositions \[ \Delta^+ = \Phi_1 \sqcup \Phi_2 \sqcup \dots \sqcup \Phi_k \] of the positive roots $\Delta^+$ of an arbitrary root system $\Delta$ into a disjoint union of inversion sets. Such decompositions play a central role in geometric invariant theory (GIT) in connection with studying the Littlewood-Richardson cone and related problems. This work can be considered as a continuation of the work of Dewji, Dimitrov, McCabe, Roth, Wehlau, and Wilson in which similar questions were studied for root systems of type $\mathbb{A}$. Their methods relied on properties of permutations and are not transferable to an arbitrary root system. In order to develop a type-independent approach, we go beyond root systems and consider quotient root systems (QRSs for short). We study subsets of positive roots in an arbitrary QRS $R$. We prove that every $\Phi \subseteq R^+$ can be represented in a canonical way as an inflation and develop methods to study recursively properties of such subsets. We extend the notion of an inversion to subsets of any QRS, i.e., beyond the case where a Weyl group is associated with $R$. If $\Phi \subseteq R^+$ is an inversion set, we introduce a graph $\text{G}(\Phi)$ and endow the set Comp$(\Phi)$ of connected components of $\text{G}(\Phi)$ with a partial addition. The resulting monoid-like structure (Comp$(\Phi),+)$ is a further generalization of root systems beyond QRSs. We study in detail the properties of (Comp$(\Phi),+)$ and their applications to studying the properties of $\Phi$. In particular, we investigate the relationship between $\Phi$ being primitive and $\Phi$ being irreducible. Apart from describing recursively all decompositions of $\Delta^+$ into the disjoint union of inversion sets, we provide applications to GIT and derive enumerative results which may be of independent interest.

math.CO↗
arXiv2023

Idempotents in the group algebra of the infinite dihedral group

We prove that over an algebraically closed field $\mathbb{K}$ of characteristic different from $2$, the group algebra $R=\mathbb{K} D_\infty$ of the infinite dihedral group $D_\infty$ has exactly six conjugacy classes of involutions (equivalently, of idempotents). This allows us to recover the fact that $R$ admits exactly four non-isomorphic indecomposable projective modules of the form $eR$ where $e$ is an idempotent, a result that was first established by Berman and Buz\'asi.

math.GR↗
arXiv2023

Generalized root systems

We generalize the notion of a root system by relaxing the conditions that ensure that it is invariant under reflections and study the resulting structures, which we call generalized root systems (GRSs for short). Since both Kostant root systems and root systems of Lie superalgebras are examples of GRSs, studying GRSs provides a uniform axiomatic approach to studying both of them. GRSs inherit many of the properties of root systems. In particular, every GRS defines a crystallographic hyperplane arrangement. We believe that GRSs provide an intrinsic counterpart to finite Weyl groupoids and crystallographic hyperplane arrangements, extending the relationship between finite Weyl groupoids and crystallographic hyperplane arrangements established by Cuntz. An important difference between GRSs and root systems is that GRSs may lack a (large enough) Weyl group. In order to compensate for this, we introduce the notion of a virtual reflection, building on a construction of Penkov and Serganova in the context of root systems of Lie superalgebras. The most significant new feature of GRSs is that, along with subsystems, one can define quotient GRSs. Both Kostant root systems and root systems of Lie superalgebras are equivalent to quotients of root systems and all root systems are isomorphic to quotients of simply-laced root systems. We classify all rank 2 GRSs and show that they are equivalent to quotients of root systems. Finally, we discuss in detail quotients of root systems. In particular we provide all isomorphisms and equivalences among them. Our results on quotient of root systems provide a different point of view on flag manifolds, reproving results of Alekseevsky and Graev.

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arXiv2022

Left-symmetric Superalgebras on Special Linear Lie Superalgebras

In this paper, we study the existence and classification problems of left-symmetric superalgebras on special linear Lie superalgebras ${\mathfrak{sl}}(m|n)$ with $m\neq n$. The main three results of this paper are: (i) a complete classification of the left-symmetric superalgebras on ${\mathfrak{sl}}(2|1)$, (ii) ${\mathfrak{sl}}(m|1)$ does not admit left-symmetric superalgebras for $m \geq 3$, and (iii) ${\mathfrak{sl}}(m+1|m)$ admits a left-symmetric superalgebra for every $m \geq 1$. To prove these results we combine existing results on the existence and classification of left-symmetric algebras on the Lie algebras ${\mathfrak{gl}}_m$ with a detailed analysis of small representations of the Lie superalgebras ${\mathfrak{sl}}(m|1)$. We also conjecture that ${\mathfrak{sl}}(m|n)$ admits left-symmetric superalgebras if and only if $m = n+1$.

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arXiv2022

Representations of free products of semisimple algebras via quivers

Let $\mathbb{K}$ denote an algebraically closed field and $A$ a free product of finitely many semisimple associative $\mathbb{K}$-algebras. We associate to $A$ a finite acyclic quiver $\Gamma$ and show that the category of finite dimensional $A$-modules is equivalent to a full subcategory of the category ${\rm rep}(\Gamma)$ of finite dimensional representations of $\Gamma$. Under this equivalence, the simple $A$-modules correspond exactly to the $\theta$-stable representations of $\Gamma$ for some stability parameter $\theta$. This gives us necessary conditions for an $A$-module to be simple, conditions which are also sufficient if the module is in general position. Even though there are indecomposable modules that are not simple, we prove that a module in general position is always semisimple. We also discuss the construction of arbitrary finite dimensional modules using nilpotent representations of quivers. Finally, we apply our results to the case of a free product of finite groups when $\mathbb{K}$ has characteristic zero.

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arXiv2021

Subregular $J$-rings of Coxeter systems via quiver path algebras

We study the subregular $J$-ring $J_C$ of a Coxeter system $(W,S)$, a subring of Lusztig's $J$-ring. We prove that $J_C$ is isomorphic to a quotient of the path algebra of the double quiver of $(W,S)$ by a suitable ideal that we associate to a family of Chebyshev polynomials. As applications, we use quiver representations to study the category mod-$A_K$ of finite dimensional right modules of the algebra $A_K=K\otimes_\Z J_C$ over an algebraically closed field $K$ of characteristic zero. Our results include classifications of Coxeter systems for which mod-$A_K$ is semisimple, has finitely many simple modules up to isomorphism, or has a bound on the dimensions of simple modules. Incidentally, we show that every group algebra of a free product of finite cyclic groups is Morita equivalent to the algebra $A_K$ for a suitable Coxeter system; this allows us to specialize the classifications to the module categories of such group algebras.

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arXiv2017

Intersection multiplicity one for classical groups

In this paper we show that when $\mathrm{G}$ is a classical semi-simple algebraic group, $\mathrm{B}\subset\mathrm{G}$ a Borel subgroup, and $\mathrm{X} = \mathrm{G}/\mathrm{B}$, then the structure coefficients of the Belkale-Kumar product $\odot_{0}$ on $\mathrm{H}^{*}(\mathrm{X}, \mathbf{Z})$ are all either $0$ or $1$.

math.AG↗
arXiv2017

Positive Systems of Kostant Roots

Let $\mathfrak{g}$ be a simple complex Lie algebra and let $\mathfrak{t} \subset \mathfrak{g}$ be a toral subalgebra of $\mathfrak{g}$. As a $\mathfrak{t}$-module $\mathfrak{g}$ decomposes as \[\mathfrak{g} = \mathfrak{s} \oplus \big(\oplus_{ν\in \mathcal{R}} \mathfrak{g}^ν\big)\] where $\mathfrak{s} \subset \mathfrak{g}$ is the reductive part of a parabolic subalgebra of $\mathfrak{g}$ and $\mathcal{R}$ is the Kostant root system associated to $\mathfrak{t}$. When $\mathfrak{t}$ is a Cartan subalgebra of $\mathfrak{g}$ the decomposition above is nothing but the root decomposition of $\mathfrak{g}$ with respect to $\mathfrak{t}$; in general the properties of $\mathcal{R}$ resemble the properties of usual root systems. In this note we study the following problem: "Given a subset $\mathcal{S} \subset \mathcal{R}$, is there a parabolic subalgebra $\mathfrak{p}$ of $\mathfrak{g}$ containing $\mathcal{M} = \oplus_{ν\in \mathcal{S}} \mathfrak{g}^ν$ and whose reductive part equals $\mathfrak{s}$?". Our main results is that, for a classical simple Lie algebra $\mathfrak{g}$ and a saturated $\mathcal{S} \subset \mathcal{R}$, the condition $(\operatorname{Sym}^\cdot(\mathcal{M}))^{\mathfrak{s}} = \mathbf{C}$ is necessary and sufficient for the existence of such a $\mathfrak{p}$. In contrast, we show that this statement is no longer true for the exceptional Lie algebras $\mathrm{F}_4, \mathrm{E}_6, \mathrm{E}_7$, and $\mathrm{E}_8$. Finally, we discuss the problem in the case when $\mathcal{S}$ is not saturated.

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arXiv2009

A Bott-Borel-Weil theorem for diagonal ind-groups

We establish a theorem computing the cohomology groups of line bundles on homogeneous ind-varieties $G/B$ for diagonal ind-groups $G$. The main difficulty in proving this analog of the classical Bott-Borel-Weil theorem is in defining an appropriate analog $W_B$ of the Weyl group so that the action of $W_B$ on weights of $G$ is compatible with the analog of the Demazure "action" of the Weyl group on the cohomology of line bundles.

math.AG↗
arXiv2009

Cup products of line bundles on homogeneous varieties and generalized PRV components of multiplicity one

Let X=G/B be a complete flag variety, and L' and L" two line bundles on X. Consider the cup product map H^{d'}(X,L') x H^{d"}(X, L") --> H^{d}(X,L), where L=L' x L" and d=d'+d". We answer two natural questions about the map above: When is it a nonzero map of irreducible G-representations? Conversely, given generic irreducible representations V' and V" of G, which irreducible components of V' x V" may appear in the right hand side of the map above? We also give bounds on the multiplicities appearing in a tensor product, and relate these considerations to the boundary of the Littlewood-Richardson cone.

math.AG↗
arXiv2009

Parabolic sets of roots

We compare two combinatorial definitions of parabolic sets of roots. We show that these definitions are equivalent for simple finite dimensional Lie algebras, affine Lie algebras, and toroidal Lie algebras. In contrast, these definitions are not always equivalent for simple finite dimensional Lie superalgebras.

math.RT↗
arXiv2004

Ind--varieties of generalized flags as homogeneous spaces for classical ind--groups

The purpose of the present paper is twofold: to introduce the notion of a generalized flag in an infinite dimensional vector space $V$ (extending the notion of a flag of subspaces in a vector space), and to give a geometric realization of homogeneous spaces of the ind--groups $SL(\infty)$, $SO(\infty)$ and $Sp(\infty)$ in terms of generalized flags. Generalized flags in $V$ are chains of subspaces which in general cannot be enumerated by integers. Given a basis $E$ of $V$, we define a notion of $E$--commensurability for generalized flags, and prove that the set $\cFl (\cF, E)$ of generalized flags E$--commensurable with a fixed generalized flag $\cF$ in $V$ has a natural structure of an ind--variety. In the case when $V$ is the standard representation of $G = SL(\infty)$, all homogeneous ind--spaces $G/P$ for parabolic subgroups $P$ containing a fixed splitting Cartan subgroup of $G$, are of the form $\cFl (\cF, E)$. We also consider isotropic generalized flags. The corresponding ind--spaces are homogeneous spaces for $SO(\infty)$ and $Sp(\infty)$. As an application of the construction, we compute the Picard group of $\cFl (\cF, E)$ (and of its isotropic analogs) and show that $\cFl (\cF, E)$ is a projective ind--variety if and only if $\cF$ is a usual, possibly infinite, flag of subspaces in $V$.

math.AG↗
arXiv1998

Weight modules of direct limit Lie algebras

In this article we initiate a systematic study of irreducible weight modules over direct limits of reductive Lie algebras, and in particular over the simple Lie algebras $A(\infty)$, $B(\infty)$, $C(\infty)$ and $D(\infty)$. Our main tool is the shadow method introduced recently in \cite{DMP}. The integrable irreducible modules are an important particular class and we give an explicit parametrization of the finite integrable modules which are analogues of finite-dimensional irreducible modules over reductive Lie algebras. We then introduce the more general class of pseudo highest weight modules. Our most general result is the description of the support of any irreducible weight module.

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