Dual spaces of mixed-norm martingale hardy spaces
Gökhan Mutlu
Gazi University, Faculty of Science, Department of Mathematics, Ankara-Turkey
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 [1,2,18]. In particular, it was shown that the quotient graph $ \Gamma/\mathbb{C}G $ is isospectral to $ \Gamma $ by using representation theory where $ \mathbb{C}G $ denotes the regular representation of $ G $ [18]. Further, it was conjectured that $ \Gamma $ can be obtained as a quotient $ \Gamma/\mathbb{C}G $ [18]. However, proving this by construction of the quotient quantum graphs has remained as an open problem. In this paper, we solve this problem by proving by construction that for a quantum graph $ \Gamma $ and a finite symmetry group $ G $ acting on it, the quotient quantum graph $ \Gamma / \mathbb{C}G $ is not only isospectral but rather identical to $ \Gamma $ for a particular choice of a basis for $ \mathbb{C}G $. Furthermore, we prove that, this result holds for an arbitrary permutation representation of $ G $ with degree $ |G| $, whereas it doesn't hold for a permutation representation of $ G $ with degree greater than $ |G|. $
Keywords: Isospectrality, quotient quantum graphs, regular representation, permutation representation, quantum graphs.
Mathematics Subject Classification: Primary: 58J53; Secondary: 20C30, 34L05, 35P05, 81Q50.
Citation: Gökhan Mutlu. On the quotient quantum graph with respect to the regular representation. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2020295
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Figure 1. A star graph with $ C_{2} $ symmetry
Figure 2. A star graph with 3 edges having the same length
Figure 3. An equilateral tetrahedron graph
Figure 4. We added dummy vertices and assigned directions arbitrarily
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