Deligne Simpson Problem Research Articles

On deformed preprojective algebras

Deformed preprojective algebras are generalizations of the usual preprojective algebras introduced by Crawley-Boevey and Holland, which have applications to Kleinian singularities, the Deligne-Simpson problem, integrable systems and noncommutative geometry . In this paper we offer three contributions to the study of such algebras: (1) the 2-Calabi-Yau property ; (2) the unification of the reflection functors of Crawley-Boevey and Holland with reflection functors for the usual preprojective algebras; and (3) the classification of tilting ideals in 2-Calabi-Yau algebras, and especially in deformed preprojective algebras for extended Dynkin quivers.

The Deligne–Simpson problem for connections on [formula omitted] with a maximally ramified singularity

The classical additive Deligne–Simpson problem is the existence problem for Fuchsian connections with residues at the singular points in specified adjoint orbits. Crawley-Boevey found the solution in 2003 by reinterpreting the problem in terms of quiver varieties. A more general version of this problem, solved by Hiroe, allows additional unramified irregular singularities. We apply the theory of fundamental and regular strata due to Bremer and Sage to formulate a version of the Deligne–Simpson problem in which certain ramified singularities are allowed. These allowed singular points are called toral singularities; they are singularities whose leading term with respect to a lattice chain filtration is regular semisimple. We solve this problem in the special case of connections on Gm with a maximally ramified singularity at 0 and possibly an additional regular singular point at infinity. Examples of such connections arise from Airy, Bessel, and Kloosterman differential equations. They play an important role in recent work in the geometric Langlands program. We also give a complete characterization of all such connections which are rigid, under the additional hypothesis of unipotent monodromy at infinity.

Motivic Donaldson-Thomas Invariants of Parabolic Higgs Bundles and Parabolic Connections on a Curve

Let $X$ be a smooth projective curve over a field of characteristic zero and let $D$ be a non-empty set of rational points of $X$. We calculate the motivic classes of moduli stacks of semistable parabolic bundles with connections on $(X,D)$ and motivic classes of moduli stacks of semistable parabolic Higgs bundles on $(X,D)$. As a by-product we give a criteria for non-emptiness of these moduli stacks, which can be viewed as a version of the Deligne-Simpson problem.

A New Approach to Simple Modules for Preprojective Algebras

The work of the first author on the moment map for representations of quivers included a classification of the possible dimension vectors of simple modules for deformed preprojective algebras. That classification was later used to solve an additive analogue of the Deligne-Simpson problem. The last step in the proof of the classification involved some general position arguments; here we give a new approach which avoids such arguments.

Linear differential equations on the Riemann sphere and representations of quivers

Our interest in this paper is a generalization of the additive Deligne-Simpson problem which is originally defined for Fuchsian differential equations on the Riemann sphere. We shall extend this problem to differential equations having an arbitrary number of unramified irregular singular points and determine the existence of solutions of the generalized additive Deligne-Simpson problems. Moreover we apply this result to the geometry of the moduli spaces of stable meromorphic connections of trivial bundles on the Riemann sphere. Namely, open embedding of the moduli spaces into quiver varieties is given and the non-emptiness condition of the moduli spaces is determined. Furthermore the connectedness of the moduli spaces is shown.

Moduli spaces of meromorphic connections and quiver varieties

We describe the moduli spaces of meromorphic connections on trivial holomorphic vector bundles over the Riemann sphere with at most one (unramified) irregular singularity and arbitrary number of simple poles as Nakajima's quiver varieties. This result enables us to solve partially the additive irregular Deligne–Simpson problem.

Additive Deligne-Simpson Problem for Non-Fuchsian Systems

We consider the problem for which tuples of conjugacy classes cj ⊂ gl (n, C) and for which tuples of non-resonant formal normal forms does there exist a non-Fuchsian linear system of ordinary differential equations on Riemann's sphere with residue matrices from the classes cj and with the given formal normal forms at its poles. It turns out that the answer is the same as in the case of Fuchsian systems and depends only on the conjugacy classes cj. We find also the dimension of the moduli space of such non-Fuchsian systems.

The Deligne–Simpson Problem and the Connectedness of some Varieties

The Deligne---Simpson problem is formulated as follows: give necessary and sufficient conditions for the choice of the conjugacy classes $C_{j} \subset {\text{GL}}{\left( {n,\mathbb{C}} \right)}$ or $c_{j} \subset {\text{gl}}{\left( {n,\mathbb{C}} \right)}$ so that there exist irreducible (p+1)-tuples of matrices M j ? C j or A j ? c j satisfying the equality M 1 ? M p+1 = I or A 1 + ? + A p+1 = 0. The matrices M j and A j are interpreted as monodromy operators of regular linear systems and as matrices-residues of Fuchsian ones on the Riemann sphere. We prove that in the so-called simple case the subset $\mathcal{W}^{\prime }$ or $\mathcal{V}^{\prime }$ of the variety $\mathcal{W}: = {\left\{ {{\left( {M_{1} , \ldots ,M_{{p + 1}} } \right)}\left| {M_{j} \in C_{j} ,M_{1} \cdots M_{{p + 1}} = I} \right.} \right\}}$ or $\mathcal{V}: = {\left\{ {{\left( {A_{1} , \ldots ,A_{{p + 1}} } \right)}\left| {A_{j} \in c_{j} ,A_{1} + } \right. \cdots + A_{{p + 1}} = 0} \right\}}$ consisting of all irreducible (p+1)-tuples (if nonempty) is connected. "Simple" means that the greatest common divisor of all quantities of Jordan blocks of a given size, of a given matrix M j or A j , and with a given eigenvalue is 1.

Multiplicative preprojective algebras, middle convolution and the Deligne–Simpson problem

We introduce a family of algebras which are multiplicative analogues of preprojective algebras, and their deformations, as introduced by M.P. Holland and the first author. We show that these algebras provide a natural setting for the ‘middle convolution’ operation introduced by N.M. Katz in his book ‘Rigid local systems’, and put in an algebraic setting by M. Dettweiler and S. Reiter, and H. Völklein. We prove a homological formula relating the dimensions of Hom and Ext spaces, study varieties of representations of multiplicative preprojective algebras, and use these results to study simple representations. We apply this work to the Deligne–Simpson problem, obtaining a sufficient (and conjecturally necessary) condition for the existence of an irreducible solution to the equation A 1 A 2 … A k = 1 with the A i in prescribed conjugacy classes in GL n ( C ) .

The Connectedness of Some Varieties and the Deligne—Simpson Problem

The Deligne--Simpson problem (DSP) (respectively, the weak DSP) is formulated as follows: give necessary and sufficient conditions for the choice of the conjugacy classes Cj � GL(n, �) or cj � gl(n, �) so that there exist irreducible (respectively, with trivial centralizer) (p + 1)-tuples of matrices Mj � Cj or Aj � cj satisfying the equality M1 ... Mp+1 = I or A1 + ... + Ap+1 = 0. The matrices Mj and Aj are interpreted as monodromy operators of regular linear systems and as matrices-residua of Fuchsian ones on the Riemann sphere. For ((p + 1))-tuples of conjugacy classes one of which is with distinct eigenvalues we prove that the variety {(M1, ..., Mp+1) | Mj � Cj, M1 ... Mp+1 = I} or {(A1, ..., Ap+1) | Aj � cj, A1 + ... + Ap+1 = 0| is connected if the DSP is positively solved for the given conjugacy classes and give necessary and sufficient conditions for the positive solvability of the weak DSP.

New deformations of group algebras of Coxeter groups

In our previous paper math.QA/0409261, we defined a deformation of the group algebra of the group of even elements of a Coxeter group W, and showed that it is flat for all values of parameters if and only if all the rank 3 parabolic subgroups of W are infinite. In this paper, we study what happens in the general case. Then the deformation is flat only for some values of parameters, and the set of all such values is called the flatness locus. The main result of the paper is an explicit description of the this flatness locus as a scheme over Z. More specifically, we show that this scheme is the intersection of the flatness loci for the subalgebras corresponding to parabolic subgroups of rank 3. The latter are determined by solving the rigid multiplicative Deligne-Simpson problem. We also define additive analogs of our algebras and study their properties.

The Deligne–Simpson problem—a survey

The Deligne–Simpson problem (DSP) (respectively the weak DSP) is formulated like this: give necessary and sufficient conditions for the choice of the conjugacy classes C j ⊂ GL ( n , C ) or c j ⊂ gl ( n , C ) so that there exist irreducible ( respectively with trivial centralizer) ( p + 1 ) -tuples of matrices M j ∈ C j or A j ∈ c j satisfying the equality M 1 ⋯ M p + 1 = I or A 1 + ⋯ + A p + 1 = 0 . The matrices M j and A j are interpreted as monodromy operators of regular linear systems and as matrices-residua of Fuchsian ones on Riemann's sphere. The present paper offers a survey of the results known up to now concerning the DSP.

On the Deligne–Simpson problem and its weak version

We consider the Deligne–Simpson problem (DSP) (respectively the weak DSP): Give necessary and sufficient conditions upon the choice of the p+1 conjugacy classes c j⊂gl(n, C ) or C j⊂GL(n, C ) so that there exist irreducible ( p+1)-tuples (respectively ( p+1)-tuples with trivial centralizers) of matrices A j ∈ c j with zero sum or of matrices M j ∈ C j whose product is I. The matrices A j (respectively M j ) are interpreted as matrices-residua of Fuchsian linear systems (respectively as monodromy matrices of regular linear systems) of differential equations with complex time. In the paper we give sufficient conditions for solvability of the DSP in the case when one of the matrices is with distinct eigenvalues.

On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero

We determine those $k$-tuples of conjugacy classes of matrices from which it is possible to choose matrices that have no common invariant subspace and have sum zero. This is an additive version of the Deligne-Simpson problem. We deduce the result from earlier work of ours on preprojective algebras and the moment map for representations of quivers. Our answer depends on the root system for a Kac-Moody Lie algebra.

Some explicit solutions of the additive Deligne–Simpson problem and their applications

In this paper we construct three infinite series and two extra triples ( E 8 and E ̂ 8 ) of complex matrices B, C, and A= B+ C of special spectral types associated to Simpson's classification in Amer. Math. Soc. Proc. 1 (1992) 157 and Magyar et al. classification in Adv. Math. 141 (1999) 97. This enables us to construct Fuchsian systems of differential equations which generalize the hypergeometric equation of Gauss–Riemann. In a sense, they are the closest relatives of the famous equation, because their triples of spectral flags have finitely many orbits for the diagonal action of the general linear group in the space of solutions. In all the cases except for E 8, we also explicitly construct scalar products such that A, B, and C are self-adjoint with respect to them. In the context of Fuchsian systems, these scalar products become monodromy invariant complex symmetric bilinear forms in the spaces of solutions. When the eigenvalues of A, B, and C are real, the matrices and the scalar products become real as well. We find inequalities on the eigenvalues of A, B, and C which make the scalar products positive-definite. As proved by Klyachko, spectra of three hermitian (or real symmetric) matrices B, C, and A= B+ C form a polyhedral convex cone in the space of triple spectra. He also gave a recursive algorithm to generate inequalities describing the cone. The inequalities we obtain describe non-recursively some faces of the Klyachko cone.

On Some Aspects of the Deligne–Simpson Problem

The Deligne-Simpson problem in the multiplicative version is formulated as follows: give necessary and sufficient conditions for the choice of the conjugacy classes Cj e GL(n, \Bbb C), so that there exist irreducible (p + 1)-tuples of matrices Mj e Cj satisfying the equality M1 . . . Mp+1 e I. We solve the problem for generic eigenvalues in the case where all the numbers ∑j,m(σ) of Jordan blocks of a given matrix Mj with a given eigenvalue σ and of a given size m (taken over all j, σ, and m) are divisible by d > 1. Generic eigenvalues are defined by explicit algebraic inequalities of the form a ≠ 0. For such eigenvalues there exist no reducible (p + 1)-tuples. The matrices Mj are interpreted as monodromy operators of regular linear systems on the Riemann sphere.

On the Deligne-Simpson problem

Consider the Deligne-Simpson problem: give necessary and sufficient conditions for the choice of the conjugacy classes C j ∃( n, C) (or C j λ( n, C) so that there exist irreducible (p + 1) -tuples of matrices M j ε C j (or A j ε c j) satisfying the equality m 1 … M p+1 = I (or A 1 + … + A p+1 = 0) . We solve the problem for generic eigenvalues; their set is defined by explicit algebraic inequalities. There exist no reducible ( p + l)-tuples of matrices for generic eigenvalues. The matrices M 1 and A j are interpreted as monodromy operators and as matrices-residua of fuchsian systems on Riemann's sphere.