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Einladung zum Vortrag (Zoom) von Herrn Dr. Gökhan Mutlu am 21.04.2021 - 14:30 Uhr im Rahmen des Forschungsseminars Analysis

[21.04.2021]

Im Rahmen des Forschungsseminars Analysis hält am Mittwoch, den 21. April 2021, um 14:30 Uhr,

Herr Dr. Gökhan Mutlu (Gazi University, Ankara, Turkey) einen Vortrag zum Thema

On the quotient quantum graph with respect to the regular representation.

Abstract: Given a quantum graph $ \Gamma $, a finite symmetry group $ G $ acting on it and a representation $R$ of $ G$, the quotient quantum graph $ \Gamma /R $ is described and constructed in the literature [1, 2, 3]. Different choices for the fundamental domain of the action of $G$ on $\Gamma $ and for the basis of $ R $ yield different quotient graphs $ \Gamma /R $ which are all isospectral to each other. In particular, it was proven that the quotient graph $ \Gamma / \mathbb{C} G $ is isospectral to $ \Gamma $ where $ \mathbb{C} G $ denotes the regular representation of $ G $ [3]. Choosing different fundamental domains for the action of $ G $ or different basis of $ \mathbb{C} G$ will yield new quantum graphs which are isospectral to $ \Gamma $. This provides an extremely useful way to create isospectral quantum graphs to a given quantum graph. It was conjectured that $ \Gamma $ is a $ \Gamma / \mathbb{C} G $ graph i.e. $ \Gamma $ can be obtained as a quotient $ \Gamma / \mathbb{C} G $ for a particular choice of a basis for $ \mathbb{C} G $ [3]. However, proving this by construction of the quotient quantum graphs has remained as an open problem. In this study, we construct the quotient quantum graph $ \Gamma / \mathbb{C} G $ by choosing G as a basis for $ \mathbb{C} G $ and show that the resulting graph is identical to $ \Gamma $. Moreover, we prove a more general result that $ \Gamma $ can be obtained as a quotient $ \Gamma / \rho $ where $ \rho $ is an arbitrary permutation representation of $ G $ with degree $ |G| $. We prove that if one constructs the quotient graph $ \Gamma / \rho $ by choosing the standard basis of $ \mathbb{C}^{|G|} $, one gets $ \Gamma $ where $ \rho $ is an arbitrary permutation representation of $ G $ with degree $|G|$. We also show by a counterexample that this does not hold for a permutation representation of $G$ with degree greater than $|G|$.

Der Vortrag findet im Rahmen eines Zoom-Meetings statt. Bei Interesse an den Zugangsdaten bitte bei Herrn Prof. Dr. Mugnolo (delio.mugnolo@fernuni-hagen.de) melden.

mathinf.webteam | 12.08.2021