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

On the quotient quantum graph with respect to the regular representation

Gökhan Mutlu

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|. $

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 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.

show all references

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.

Figure 1.  A star graph with $ C_{2} $ symmetry
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Figure 2.  A star graph with 3 edges having the same length
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Figure 3.  An equilateral tetrahedron graph
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Figure 4.  We added dummy vertices and assigned directions arbitrarily
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