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Projective Joint Spectra of operator tuples and representations of symmetry groups


Academic Report

Title: Projective Joint Spectra of operator tuples and represenntations of symmetry groups

Reportwe: Prof. Michael I. Stessin (State University of New York at Albany)

Time: May 19,2017(Friday) AM 9:00-10:30 

Location: A#1101 room, Innovation Park Building

Contact: Prof. LU Yufeng (tel: 84708351-8127) 

Abstract: Projective Joint Spectra of operator tuples were introduced by R.Yang in 2008 and have been actively investigated since then. They generalize determinantal manifolds whose scrutiny started by Dickson in 1920s, and this area of interplay of operator theory and algebraic geometry is still very active today. Recent paper of Stessin and Tchernev concentrated on properties of operator tuples reflected in the geometry of the Projective Joint Spectrum. One of the results proved there implies that if pairwise joint spectra of a tuple of $n\times n$ unitary matrices consists of lines and "complex ellipses", then the group generated by these matrices represents a Coxeter group whose Coxeter matrix is determined by those ellipses. This leads to a natural question whether the joint spectrum of images of generators of a Coxeter group determines the representation up to equivalence. In the talk we will show that the answer is affirmative for symmetry groups $A_n, \ B_n$ and $I(n)$.

 The brief introduction to the reporter: Prof.Stessin is the head of Department of Mathematics and Statistics, University of New York at Albany and he majors in functional analysis.