arXiv subjects
Publications and source records attributed to Vladimir Petrov Kostov.
We consider the family of polynomials $P(x,a)=x^n+a_1x^{n-1}+... +a_n$, $x,a_i\in {\bf R}$, and the stratification of ${\bf R}^n\cong \{(a_1,... ,a_n)|a_i\in {\bf R}\}$ defined by the multiplicity vector of the real roots of $P$. We prove smoothness of the strata and a transversality property of their tangent spaces.
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 survey of the results known up to now concerning the DSP.
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 conditions for solvability of the true problem. For index of rigidity 2 (i.e. when the sum of the dimensions of the conjugacy classes equals $2n^2-2$) we show that for a sufficiently large class of $(p+1)$-tuples of conjugacy classes the answer to the weak problem is negative. These conjugacy classes define Jordan normal forms that satisfy the Criterium.
We consider the variety of $(p+1)$-tuples of matrices $A_j$ (resp. $M_j$) from given conjugacy classes $c_j\subset gl(n,{\bf C})$ (resp. $C_j\subset GL(n,{\bf C})$) such that $A_1+... +A_{p+1}=0$ (resp. $M_1... M_{p+1}=I$). This variety is connected with the weak {\em Deligne-Simpson problem: give necessary and sufficient conditions on the choice of 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 with trivial centralizers of matrices $A_j\in c_j$ (resp. $M_j\in C_j$) whose sum equals 0 (resp. whose product equals $I$).} The matrices $A_j$ (resp. $M_j$) are interpreted as matrices-residua of Fuchsian linear systems (resp. as monodromy operators of regular linear systems) on Riemann's sphere. We consider examples of such varieties of dimension higher than the expected one due to the presence of $(p+1)$-tuples with non-trivial centralizers; in one of the examples the difference between the two dimensions is O(n).
We consider the {\em Deligne-Simpson problem}: {\em Give necessary and sufficient conditions for the choice of the conjugacy classes $c_j\subset gl(n,{\bf C})$ or $C_j\subset GL(n,{\bf C})$, $j=1,..., p+1$, so that there exist irreducible $(p+1)$-tuples of matrices $A_j\in c_j$ whose sum is 0 or of matrices $M_j\in C_j$ whose product is $I$.} The matrices $A_j$ (resp. $M_j$) are interepreted as matrices-residua of Fuchsian linear systems (resp. as monodromy operators of regular systems) on Riemann's sphere. We consider the case when the sum of the dimensions of the conjugacy classes $c_j$ or $C_j$ is $2n^2$ and we prove a theorem of non-existence of such irreducible $(p+1)$-tuples.
The Deligne-Simpson problem is formulated like this: give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\subset SL(n,{\bf C})$ or $c_j\subset sl(n,{\bf C})$ so that there exist irreducible $(p+1)$-tuples of matrices $M_j\in C_j$ or $A_j\in c_j$ satisfying the equality $M_1... M_{p+1}=I$ or $A_1+... +A_{p+1}=0$. We solve the problem for generic eigenvalues with the exception of the case of matrices $M_j$ when the greatest common divisor of the numbers $Σ_{j,l}(σ)$ of Jordan blocks of a given matrix $M_j$, with a given eigenvalue $σ$ and of a given size $l$ (taken over all $j$, $σ$, $l$) is $>1$. Generic eigenvalues are defined by explicit algebraic inequalities. For such eigenvalues there exist no reducible $(p+1)$-tuples. 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.
We consider the variety of $(p+1)$-tuples of matrices $M_j$ from given conjugacy classes from $GL(n,{\bf C})$ such that $M_1... M_{p+1}=I$. This variety is connected with the Deligne-Simpson problem and the matrices $M_j$ are interpreted as monodromy operators of regular systems on Riemann's sphere. We consider among others cases when the dimension of the variety is higher than the expected one due to the presence of $(p+1)$-tuples with non-trivial centralizers.
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 algebraic inequalities of the form $a\neq 0$. For such eigenvalues there exist no reducible $(p+1)$-tuples. The matrices $M_j$ are interpreted as monodromy operators of regular linear systems on Riemann's sphere.