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

德利涅-辛普森问题

The Deligne-Simpson Problem

本文研究了德利涅-辛普森问题，通过根系关联猜想，证明了矩阵乘积为单位矩阵且无公共不变子空间的条件。

Quiver algebras, weighted projective lines, and the Deligne-Simpson problem

We describe recent work on preprojective algebras and moduli spaces of their representations. We give an analogue of Kac's Theorem, characterizing the dimension types of indecomposable coherent sheaves over weighted projective lines in terms of loop algebras of Kac-Moody Lie algebras, and explain how it is proved using Hall algebras. We discuss applications to the problem of describing the possible conjugacy classes of sums and products of matrices in known conjugacy classes.

The Deligne-Simpson problem -- a survey

The Deligne-Simpson problem (DSP) (resp. the weak DSP) is formulated like this: {\em give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\subset GL(n,{\bf C})$ or $c_j\subset gl(n,{\bf C})$ so that there exist irreducible (resp. with trivial centralizer) $(p+1)$-tuples of matrices $M_j\in C_j$ or $A_j\in c_j$ 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 surv…

Some Explicit Solutions of the Additive Deligne-Simpson Problem and Their Applications (Preprint)

In this paper we construct three infinite series and two extra triples of complex matrices B, C, and A=B+C of special spectral types associated to C. Simpson's classification in his paper ``Products of Matrices'' and a classification of multiple flag varieties with finitely many orbits of the diagonal action of the general linear group by P. Magyar, J. Weyman, and A. Zelevinsky. 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 tr…

On the weak Deligne-Simpson problem for index of rigidity 2

We consider the weak version of the Deligne-Simpson problem: give necessary and sufficient conditions upon the conjugacy classes $c_j\subset gl(n,{\bf C})$ (resp. $C_j\subset GL(n,{\bf C})$) so that there exist $(p+1)$-tuples of matrices $A_j\in c_j$, $A_1+... +A_{p+1}=0$ (resp. $M_1... M_{p+1}=I$) with trivial centralizers (i.e. reduced to scalars). The true Deligne-Simpson problem requires irreducibility instead of triviality of the centralizer. When the eigenvalues are generic, a Criterium on the Jordan normal forms defined by the conjugacy classes gives the necessary and sufficient conditi…

On some aspects of the Deligne-Simpson problem

The Deligne-Simpson problem in the multiplicative version is formulated like this: {\em give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\in SL(n,{\bf C})$ so that there exist irreducible $(p+1)$-tuples of matrices $M_j\in C_j$ satisfying the equality $M_1... M_{p+1}=I$}. We solve the problem for generic eigenvalues in the case when all the numbers $Σ_{j,m}(σ)$ of Jordan blocks of a given matrix $M_j$, with a given eigenvalue $σ$ and of a given size $m$ (taken over all $j$, $σ$, $m$) are divisible by $d>1$. Generic eigenvalues are defined by explicit algebra…