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Michael Stessin

Publications and source records attributed to Michael Stessin.

9 recordsLinked to original sources
arXiv2026

Reducibility of linear representations, free ideals, and Kippenhahn's conjecture

This paper presents a comprehensive study of the characteristic polynomial of matrix tuples and finite dimensional group representations. Among other things, several key concepts are introduced, including the minimal polynomial, spectral index, spectral stability, and characteristic graph. Notably, this framework provides a complete resolution to Kippenhahn's conjecture, settling a long-standing and influential problem in the theory of matrix tuples.

math.RT↗
arXiv2026

Kippenhahn's Conjecture Revisited

In 1951 paper \cite{Ki} Kippenhahn conjectured that if the characteristic polynomial \ $P_A(x_1,x_2,x_3)=\mbox{det}(x_1A_1+x_2A_2-x_3I)$, \ where $A_1$ and $A_2$ are $n\times n$ Hermitian matrices, has a repeated factor in the polynomial ring $\C[x_1,x_2,x_3]$, then the pair $(A_1,A_2)$ is unitary equivalent to a direct sum $(C_1\oplus C_2, \ D_1\oplus D_2)$ where $C_i, D_i\in M_{n_i}(\C) $ for some $1\leq n_i<n, \ n_1+n_2=n, i=1,2$. Kippenhahn verified the conjecture whenever the degree of the minimal polynomial of $x_1A_1 + x_2A_2$ is 1 or 2. In subsequent works \cite{Sh1,Sh2} Shapiro obtained a number of results which supported the conjecture. In particular, she showed that it held if $n \leq 5$. In 1983 Laffey \cite{La} showed that, in general, Kippenhahn's conjecture was not true by constructing a counterexample for $n=8$. Since then additional counterexamples were worked out (see \cite{Wa} for example). Some positive results in this direction including the quantum version of the conjecture can be found in \cite{F1, F2, KVo1, Law}. In this paper we use methods of recently developed local spectral analysis to give some necessary and sufficient conditions for the affirmative answer to Kippenhahn's conjecture in terms of the characteristic polynomials of certain elements of the algebra generated by the matrices in the tuple.

math.FA↗
arXiv2025

Spectral test of reducibility for Matrix tuples

If a tuple of matrices has a common invariant subspace, its projective joint spectrum has an algebraic component. In general, the converse is not true, and there might be algebraic components in the projective joint spectrum without corresponding common invariant subspaces. In this paper we give necessary and sufficient conditions for the occurrence of such correspondence.

math.FA↗
arXiv2023

Spectral reconstruction and representations of finitely generated groups

It is well-known that characters classify linear representations of finite groups, that is if characters of two representations of a finite group are the same, these representations are equivalent. It is also well-known that, in general, this is not true for representations of infinite groups, even if they are finitely generated. The goal of this paper is to establish a characterization of representations of finitely generated groups in terms of projective joint spectra. This approach has a clear advantage compared to character classification as it is valid for a much wider family of groups and for both finite and infinite dimensional representations. The main tool in establishing our spectral characterization is a reconstruction of an operator acting on a separable Hilbert space from the proper projective joint spectrum of the quadruple containing this operator along with a certain triple of bounded operators acting on the same space.

math.GR↗
arXiv2022

Spectral analysis and representations]{Spectral analysis near regular point of reducibility and representations of Coxeter groups

For a tuple of square matrices $A_1,...,A_n$ the determinantal hypersurface is defined as \begin{eqnarray*} &σ(A_1,...,A_n)= \\ &\Big\{[x_1:\cdots :x_n]\in \C{\mathbb P}^{n-1}: det(x_1A_1+\cdots +x_nA_n)=0\Big \}. \end{eqnarray*} In this paper we develop a local spectral analysis near a regular point of reducibility of a determinantal hypersurface. We prove a rigidity type theorem for representations of Coxeter groups as an application

math.SP↗
arXiv2018

Determinantal hypersurfaces and representations of Coxeter groups

Given a finite generating set $T=\{g_0,\dots, g_n\}$ of a group $G$, and a representation $ρ$ of $G$ on a Hilbert space $V$, we investigate how the geometry of the set $D(T,ρ)=\{ [x_0 : \dots : x_n] \in\mathbb C\mathbb P^n \mid \sum x_iρ(g_i) \text{ not invertible} \}$ reflects the properties of $ρ$. When $V$ is finite-dimensional this is an algebraic hypersurface in $\mathbb C\mathbb P^n$. In the special case $T=G$ and $ρ=$ the left regular representation of $G$, this hypersurface is defined by the \emph{group determinant}, an object studied extensively in the founding work of Frobenius that lead to the creation of representation theory. We focus on the classic case when $G$ is a finite Coxeter group, and make $T$ by adding the identity element $1_G$ to a Coxeter generating set for $G$. Under these assumptions we show in our first main result that if $ρ$ is the left regular representation, then $D(T,ρ)$ determines the isomorphism class of $G$. Our second main result is that if $G$ is not of exceptional type, and $ρ$ is any finite dimensional representation, then $D(T,ρ)$ determines $ρ$.

math.RT↗
arXiv2015

Spectral algebraic curves and decomposable operator tuples

Joint spectra of tuples of operators are subsets in complex projective space. The corresponding tuple of operators can be viewed as an infinite dimensional analog of a determinantal representation of the joint spectrum. We investigate the relationship between the geometry of the spectrum and the properties of the operators in the tuple when these operators are self-adjoint. In the case when the spectrum contains an algebraic curve passing through an isolated spectral point of one of the operators we give necessary and sufficient geometric conditions for the operators in the tuple to have a common reducing subspace. We also address spectral continuity and obtain a norm estimate for the commutant of a pair of self-adjoint matrices in terms of the Hausdorff distance of their joint spectrum to a family of lines.

math.SP↗
arXiv2013

Geometric spectral theory for compact operators

We introduce a notion of joint spectrum for a tuple of compact operators on a separable Hilbert space and show that in many situations these operators commute if and only if the joint spectrum consists of countably many, locally finite, complex hyperplanes. In particular, we show that normal matrices (of the same size) $A_1,\cdots,A_n$ commute if and only if the polynomial $\det(z_1A_1+\cdots+z_nA_n+I)$ is completely reducible, that is, it can be factored into a product of linear polynomials.

math.FA↗