American Institute of Mathematical Sciences

February  2021, 20(2): 885-902. doi: 10.3934/cpaa.2020295

On the quotient quantum graph with respect to the regular representation

 Gazi University, Faculty of Science, Department of Mathematics, Ankara-Turkey

Received  June 2020 Revised  October 2020 Published  February 2021 Early access  December 2020

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 $|G|.$

Citation: Gökhan Mutlu. On the quotient quantum graph with respect to the regular representation. Communications on Pure and Applied Analysis, 2021, 20 (2) : 885-902. doi: 10.3934/cpaa.2020295
References:
 [1] R. Band, G. Berkolaiko, C. H. Joyner and W. Liu, Quotients of finite-dimensional operators by symmetry representations, arXiv: 1711.00918v3. [2] R. Band, O. Parzanchevski and G. Ben-Shach, The isospectral fruits of representation theory: quantum graphs and drums, J. Phys. A: Math. Theor., 42 (2009), 175202. doi: 10.1088/1751-8113/42/17/175202. [3] R. Band, T. Shapira and U. Smilansky, Nodal domains on isospectral quantum graphs: the resolution of isospectrality?, J. Phys. A: Math. Gen., 39 (2006), 13999-14014.  doi: 10.1088/0305-4470/39/45/009. [4] G. Berkolaiko, An elementary introduction to quantum graphs, arXiv: 1603.07356v2. [5] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, American Mathematical Society, Rhode Island, 2013. doi: 10.1090/surv/186. [6] R. Carlson, Inverse eigenvalue problems on directed graphs, T. Am. Math. Soc., 351 (1999), 4069-4088.  doi: 10.1090/S0002-9947-99-02175-3. [7] W. Fulton and J. Harris, Representation Theory, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0979-9. [8] S. Gnutzmann and U. Smilansky, Quantum graphs: applications to quantum chaos and universal spectral statistics, arXiv: nlin/0605028v2. [9] C. Gordon, P. Perry and D. Schüth, Isospectral and isoscattering manifolds: a survey of techniques and examples, in Geometry, Spectral Theory, Groups and Dynamics, Contemporary Mathematics vol. 387 (eds. M. Entov, Y. Pinchover and M. Sageev), American Mathematical Society, (2005), 157–179. doi: 10.1090/conm/387/07241. [10] C. Gordon, D. Webb and S. Wolpert, One cannot hear the shape of a drum, Bull. Am. Mat. Soc. (N.S.), 27 (1992), 134-138.  doi: 10.1090/S0273-0979-1992-00289-6. [11] C. Gordon, D. Webb and S. Wolpert, Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math., 110 (1992), 1-22.  doi: 10.1007/BF01231320. [12] B. Gutkin and U. Smilansky, Can one hear the shape of a graph?, J. Phys. A: Math. Gen., 31 (2001), 6061-6068.  doi: 10.1088/0305-4470/34/31/301. [13] M. Kac, Can one hear the shape of a drum?, Am. Math. Mon., 73 (1966), 1-23.  doi: 10.2307/2313748. [14] V. Kostrykin and R. Schrader, Kirchhoff's rule for quantum wires, J. Phys. A: Math. Gen., 32 (1999), 595-630.  doi: 10.1088/0305-4470/32/4/006. [15] T. Kottos and U. Smilansky, Quantum chaos on graphs, Phys. Rev. Lett., 79 (1997), 4794-4797.  doi: 10.1103/PhysRevLett.79.4794. [16] P. Kuchment, Quantum graphs: an introduction and a brief survey, arXiv: 0802.3442v1. [17] D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Springer, Cham, 2014. doi: 10.1007/978-3-319-04621-1. [18] O. Parzanchevski and R. Band, Linear representations and isospectrality with boundary conditions, J. Geom. Anal., 20 (2010), 439-471.  doi: 10.1007/s12220-009-9115-6. [19] T. Shapira and U. Smilansky, Quantum graphs which sound the same, in Non-Linear Dynamics and Fundamental Interactions. NATO Science Series Ⅱ: Mathematics, Physics and Chemistry, vol 213. (eds. F. Khanna and D. Matrasulov), Springer, (2005), 17–29. doi: 10.1007/1-4020-3949-2_2. [20] T. Sunada, Riemanninan coverings and isospectral manifolds, Ann. Math., 121 (1985), 169-186.  doi: 10.2307/1971195. [21] J. von Below, Can one hear the shape of a network?, in Partial Differential Equations on Multistructures (Luminy, 1999), Lecture Notes in Pure and Appl. Math., vol. 219 (eds. F. Mehmeti, J. von Below and S. Nicaise), Dekker, (2001), 19–36.

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References:
 [1] R. Band, G. Berkolaiko, C. H. Joyner and W. Liu, Quotients of finite-dimensional operators by symmetry representations, arXiv: 1711.00918v3. [2] R. Band, O. Parzanchevski and G. Ben-Shach, The isospectral fruits of representation theory: quantum graphs and drums, J. Phys. A: Math. Theor., 42 (2009), 175202. doi: 10.1088/1751-8113/42/17/175202. [3] R. Band, T. Shapira and U. Smilansky, Nodal domains on isospectral quantum graphs: the resolution of isospectrality?, J. Phys. A: Math. Gen., 39 (2006), 13999-14014.  doi: 10.1088/0305-4470/39/45/009. [4] G. Berkolaiko, An elementary introduction to quantum graphs, arXiv: 1603.07356v2. [5] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, American Mathematical Society, Rhode Island, 2013. doi: 10.1090/surv/186. [6] R. Carlson, Inverse eigenvalue problems on directed graphs, T. Am. Math. Soc., 351 (1999), 4069-4088.  doi: 10.1090/S0002-9947-99-02175-3. [7] W. Fulton and J. Harris, Representation Theory, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0979-9. [8] S. Gnutzmann and U. Smilansky, Quantum graphs: applications to quantum chaos and universal spectral statistics, arXiv: nlin/0605028v2. [9] C. Gordon, P. Perry and D. Schüth, Isospectral and isoscattering manifolds: a survey of techniques and examples, in Geometry, Spectral Theory, Groups and Dynamics, Contemporary Mathematics vol. 387 (eds. M. Entov, Y. Pinchover and M. Sageev), American Mathematical Society, (2005), 157–179. doi: 10.1090/conm/387/07241. [10] C. Gordon, D. Webb and S. Wolpert, One cannot hear the shape of a drum, Bull. Am. Mat. Soc. (N.S.), 27 (1992), 134-138.  doi: 10.1090/S0273-0979-1992-00289-6. [11] C. Gordon, D. Webb and S. Wolpert, Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math., 110 (1992), 1-22.  doi: 10.1007/BF01231320. [12] B. Gutkin and U. Smilansky, Can one hear the shape of a graph?, J. Phys. A: Math. Gen., 31 (2001), 6061-6068.  doi: 10.1088/0305-4470/34/31/301. [13] M. Kac, Can one hear the shape of a drum?, Am. Math. Mon., 73 (1966), 1-23.  doi: 10.2307/2313748. [14] V. Kostrykin and R. Schrader, Kirchhoff's rule for quantum wires, J. Phys. A: Math. Gen., 32 (1999), 595-630.  doi: 10.1088/0305-4470/32/4/006. [15] T. Kottos and U. Smilansky, Quantum chaos on graphs, Phys. Rev. Lett., 79 (1997), 4794-4797.  doi: 10.1103/PhysRevLett.79.4794. [16] P. Kuchment, Quantum graphs: an introduction and a brief survey, arXiv: 0802.3442v1. [17] D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Springer, Cham, 2014. doi: 10.1007/978-3-319-04621-1. [18] O. Parzanchevski and R. Band, Linear representations and isospectrality with boundary conditions, J. Geom. Anal., 20 (2010), 439-471.  doi: 10.1007/s12220-009-9115-6. [19] T. Shapira and U. Smilansky, Quantum graphs which sound the same, in Non-Linear Dynamics and Fundamental Interactions. NATO Science Series Ⅱ: Mathematics, Physics and Chemistry, vol 213. (eds. F. Khanna and D. Matrasulov), Springer, (2005), 17–29. doi: 10.1007/1-4020-3949-2_2. [20] T. Sunada, Riemanninan coverings and isospectral manifolds, Ann. Math., 121 (1985), 169-186.  doi: 10.2307/1971195. [21] J. von Below, Can one hear the shape of a network?, in Partial Differential Equations on Multistructures (Luminy, 1999), Lecture Notes in Pure and Appl. Math., vol. 219 (eds. F. Mehmeti, J. von Below and S. Nicaise), Dekker, (2001), 19–36.
A star graph with $C_{2}$ symmetry
A star graph with 3 edges having the same length
An equilateral tetrahedron graph
We added dummy vertices and assigned directions arbitrarily
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