Groups, Graphs & Isospectrality Rami Band Ori Parzanchevski Gilad Ben-Shach Uzy Smilansky.
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Transcript of Groups, Graphs & Isospectrality Rami Band Ori Parzanchevski Gilad Ben-Shach Uzy Smilansky.
Groups, Graphs Groups, Graphs & Isospectrality& Isospectrality
Rami BandRami Band
Ori ParzanchevskiOri Parzanchevski
Gilad Ben-ShachGilad Ben-Shach
Uzy SmilanskyUzy Smilansky
What is a graph ?What is a graph ?
The graph’s The graph’s spectrumspectrum is the is the sequence of basic sequence of basic frequenciesfrequencies at which the at which the graph can vibrate.graph can vibrate.
The spectrum depends on The spectrum depends on the shape of the graph.the shape of the graph.
‘‘Can one hear the shape of a drum ?’Can one hear the shape of a drum ?’was first asked by Marc Kac (1966). was first asked by Marc Kac (1966).
‘‘Can one hear the shape of a graph?’Can one hear the shape of a graph?’
Can one deduce the shape from the spectrum ?Can one deduce the shape from the spectrum ? Is it possible to have two different graphs with the Is it possible to have two different graphs with the
same spectrum (same spectrum (isospectral graphsisospectral graphs) ? ) ?
‘‘Can one hear the shape of a Can one hear the shape of a graph ?’graph ?’
Marc Kac (1914-1984)
Metric Graphs - IntroductionMetric Graphs - Introduction
A A graphgraph ΓΓ consists of a finite set consists of a finite set of vertices of vertices V={vV={vii}} and and a finite set of undirected edges E={eE={ejj}.}.
A metric graph has a finite length (Le>0) assigned to each edge.
Let EEvv be the set of all edges connected to a vertex v. The degree of v is
A function on the graph is a vector of functions on the edges:
2
1
3 4
5
6
3
4
5
1
2
L13
L23
L34
L45
L46
L15
L45
L34
L23
L12L25
L35
L14
),,(Eee ffF
1
C],[: jj ee Lf 0
vv Ed :
Quantum Graphs - Quantum Graphs - IntroductionIntroduction
A quantum graph is a metric graph equipped with an operator, such as the negative Laplacian:
For each vertex v, define:
We impose boundary conditions of the form:Where Av and Bv are complex dv x dv matrices.
To ensure the self-adjointness of the Laplacian, we require that the matrix (Av|Bv) has rank dv, and that the matrix AvBv
† is self-adjoint (Kostrykin and Schrader, 1999).
),...,(Eee ffF
1
0 'vvvv FBFA
,))(),...,(( teev vfvfFvd1
teev vfvfFdv
))(),...,(( '''
1
Quantum Graphs - Quantum Graphs - IntroductionIntroduction
A quantum graph is a metric graph equipped with an operator, such as the negative Laplacian:
For each vertex v, define:
We impose boundary conditions of the form:Where Av and Bv are complex dv x dv matrices.
A common boundary condition is the Kirchhoff condition, namely all functions agree at each vertex, and the sum of the derivatives vanishes. This corresponds to the matrices:
For dv =1 vertices, there are two specialcases of boundary conditions, denoted Dirichlet: Av= (1), Bv=(0) (so that )
Neumann: Av= (0), Bv=(1) (so that )
),...,(Eee ffF
1
0 'vvvv FBFA
00
101
1
101
0011
vA
111
000
00
vB
0ef v 0ef v
,))(),...,(( teev vfvfFvd1
teev vfvfFdv
))(),...,(( '''
1
Quantum Graphs - Quantum Graphs - IntroductionIntroduction
A quantum graph is a metric graph equipped with an operator, such as the negative Laplacian:
For each vertex v, define:
We impose boundary conditions of the form:Where Av and Bv are complex dv x dv matrices.
A common boundary condition is the Kirchhoff condition, namely all functions agree at each vertex, and the sum of the derivatives vanishes.
For dv =1 vertices, there are two special cases of boundary conditions, denoted Dirichlet: Av= (1), Bv=(0) (so that )
Neumann: Av= (0), Bv=(1) (so that )
),...,(Eee ffF
1
0 'vvvv FBFA
,))(),...,(( teev vfvfFvd1
teev vfvfFdv
))(),...,(( '''
1
0ef v 0ef v
The The spectrumspectrum is is
With the set of corresponding eigenfunctions:With the set of corresponding eigenfunctions:
Use the notation: Use the notation:
The Spectrum of Quantum The Spectrum of Quantum GraphsGraphs
Examples of several functions of the graph:
λλ88≈9.0≈9.0 λλ1313≈24≈24.0.0
λλ1616≈3≈37.27.2
2√2
D
D
N
N√3√3
√2
211 ; nn
nn FF n
FF|:)( F
One can hear the shape of a simple graph One can hear the shape of a simple graph if the lengths are incommensurate if the lengths are incommensurate (Gutkin, Smilansky 2001)(Gutkin, Smilansky 2001)
Otherwise, Otherwise, we do have isospectral graphs:we do have isospectral graphs: Von Below (2001)Von Below (2001) Band, Shapira, Smilansky (2006)Band, Shapira, Smilansky (2006)
There are several methods for There are several methods for construction of isospectralityconstruction of isospectrality – the main is due to Sunada (1985). – the main is due to Sunada (1985).
We present a method based on We present a method based on representation theory arguments.representation theory arguments.
a2b
a
2c
D N 2abb
cc
D
D
N
N
‘‘Can one hear the shape of a Can one hear the shape of a graph ?’graph ?’
rrxxee
Groups & GraphsGroups & Graphs Example: The Dihedral group – Example: The Dihedral group –
the symmetry group of the square the symmetry group of the squareDD44 = { e , a , a = { e , a , a22 , a , a33 , r , rxx , r , ryy , r , ruu , r , rvv } }
yy
xx
uuvv
aaHow does the Dihedral group act on a square ?How does the Dihedral group act on a square ?
A few subgroups of the Dihedral group: A few subgroups of the Dihedral group: HH11 = { e , a = { e , a2 2 , r, rxx , r , ryy}}HH22 = { e , a = { e , a2 2 , r, ruu , r , rvv } }HH33 = { e , a , a = { e , a , a22 , a , a33}}
RepresentationRepresentation – Given a group – Given a group GG, a representation , a representation RR is an assignment of a is an assignment of a matrixmatrix R(g)R(g) to each group element to each group element g g G G, such that: , such that: g g11,g,g2 2 G G R(gR(g11)R(g)R(g22)=R(g)=R(g11gg22))..
Example 1 - DExample 1 - D44 has the following 1-dimensional rep. S has the following 1-dimensional rep. S11::
Example 2 - DExample 2 - D44 has the following 2-dimensional rep. S has the following 2-dimensional rep. S22::
Restriction Restriction - is the following rep. of H- is the following rep. of H11::
InductionInduction - is the following rep. of G: - is the following rep. of G:
Groups - RepresentationsGroups - Representations
1e 1a 12 a 13 a 1xr 1yr 1ur 1vr
10
01e
01
10a
10
012a
01
103a
10
01xr
10
01yr
01
10ur
01
10vr
14
1SD
HResQ 1e 1a 12 a 13 a 1xr 1yr 1ur 1vr
QDH4
1Ind
10
01e
10
01a
10
012a
10
013a
10
01xr
10
01yr
10
01ur
10
01vr
F =F =F =F =
Knowing the matrix representation gives us Knowing the matrix representation gives us information on the functions.information on the functions.
The group can act on a graph and …The group can act on a graph and … the group can act on a function which is defined on the graph the group can act on a function which is defined on the graph and may give new functions: and may give new functions:
We have We have
So that, form a representation of the group.So that, form a representation of the group.
Groups & GraphsGroups & Graphs
...,)()(, aFaFFF
How does the group act on the function ?How does the group act on the function ?
aa
Example: The Dihedral group – Example: The Dihedral group – the symmetry group of the square the symmetry group of the squareDD44 = { e , a , a = { e , a , a22 , a , a33 , r , rxx , r , ryy , r , ruu , r , rvv } }
)xF(a(aF)(x) ,x -1
)(
Consider the following rep. Consider the following rep. RR11 of the subgroup of the subgroup HH11::
HH11 = { e , a = { e , a22, r, rxx , r , ryy}}
RR11::
This is a 1d rep. – what do we know This is a 1d rep. – what do we know about the function F ?about the function F ?
Consider the following rep. Consider the following rep. RR22 of the subgroup of the subgroup HH22::
HH22 = { e , a = { e , a22, r, ruu , r , rvv}}
RR22::
This is a 1d rep. – what do we know This is a 1d rep. – what do we know about the function F ?about the function F ?
D
N
N
D
Groups & GraphsGroups & Graphs
1R
Γ
1e 12 a 1xr 1yr
FF xr FFyr
1e 12 a 1ur 1vr
FFur FF vr
Examine the graphExamine the graph ΓΓ::
D N
2RΓ
DDDD
N
N
N
N
N
N
D
D
Theorem – Let Theorem – Let ΓΓ be a graph be a graphwhich obeys a symmetry group which obeys a symmetry group GG..Let Let HH11, , HH22 be two subgroups be two subgroupsof G with representations of G with representations RR11, , RR22 that obey that obey then the graphs , are isospectral.then the graphs , are isospectral.
In our example:In our example:
ΓΓ = =
HH11 = {e , a = {e , a2 2 , r, rx x , r, ryy} R} R11::
HH22 = {e , a = {e , a22, r, ruu , r , rvv} R} R22::
And it can be checked thatAnd it can be checked that
Groups, Graphs & Groups, Graphs & IsospectralityIsospectrality
1RΓ
2RΓ
D
N
N
D
D N
1e 12 a 1xr 1yr
1e 12 a 1ur 1vr
1RΓ
2RΓ
21 21IndInd RR G
HGH
214
2
4
1RR D
HDH IndInd
G = DG = D44
Extending our example:Extending our example:
ΓΓ = =
HH11 = { e , a = { e , a22, r, rxx , r , ryy} R} R11::
HH22 = { e , a = { e , a22, r, ruu , r , rvv} R} R22::
HH33 = { e , a, a = { e , a, a22 , a , a33} R} R33::
Extending the Isospectral pairExtending the Isospectral pair
D
N
N
D
D N
1RΓ
1e 12 a 1xr 1yr
1e 12 a 1ur 1vr
1e ia 12 a ia 3
FF ia
××ii××ii××ii
3RΓ
3RΓ
2RΓ
3214
3
4
2
4
1RRR D
HDH
DH IndIndInd
G = DG = D44
3RΓ
3RΓ
Extending our example:Extending our example:
ΓΓ = =
HH11 = { e , a = { e , a22, r, rxx , r , ryy} R} R11::
HH22 = { e , a = { e , a22, r, ruu , r , rvv} R} R22::
HH33 = { e , a , a = { e , a , a22 , a , a33} R} R33::
Extending the Isospectral pairExtending the Isospectral pair
D
N
N
D
D N
1RΓ
1e 12 a 1xr 1yr
1e 12 a 1ur 1vr
1e ia 12 a ia 3
××ii ××ii××ii
2RΓ
1f 2fv
(v)f
(v)fF
2
1v
(v)'f
(v)'f'F
2
1v
0
0'FBFA vvvv
00
1Av
i
i1
00Bv
G = DG = D44
3214
3
4
2
4
1RRR D
HDH
DH IndIndInd
TheoremTheorem – Let – Let ΓΓ be a graph which obeys a symmetry group be a graph which obeys a symmetry group GG..Let Let HH11, , HH22 be two subgroups of G with representations be two subgroups of G with representations RR11, , RR22 that obey then the graphs , are isospectral.that obey then the graphs , are isospectral.
ProofProof::LemmaLemma: There exists a quantum graph such that:: There exists a quantum graph such that:
Using the Lemma and Frobenius reciprocity theorem gives:Using the Lemma and Frobenius reciprocity theorem gives:
Hence , are isospectral.Hence , are isospectral.Applying the same for the rep. Applying the same for the rep. RR22 and using and using finishes the proof.finishes the proof.
Groups, Graphs & Groups, Graphs & IsospectralityIsospectrality
1RΓ
2RΓ
21 21IndInd RR G
HGH
1RΓ
)(,Hom)(1H 1
1
RC CR
)()(,IndHom
)(,Hom)(
G
H1
1
1
1
1
R
RC
GHC
CR
1RΓ
1RGH1
IndΓ
21 21IndInd RR G
HGH
TheoremTheorem – Let – Let ΓΓ be a graph which obeys a symmetry group be a graph which obeys a symmetry group GG..Let Let HH11, , HH22 be two subgroups of G with representations be two subgroups of G with representations RR11, , RR22 that obey then the graphs , are that obey then the graphs , are isospectral.isospectral.
ProofProof::LemmaLemma: There exists a quantum graph such that:: There exists a quantum graph such that:
Interesting issues in the proof of the lemma:Interesting issues in the proof of the lemma: A group which does not act freely on the edges.A group which does not act freely on the edges. Representations which are not 1-d.Representations which are not 1-d. The dependence of in the choice of basis for the The dependence of in the choice of basis for the
representation.representation.
Groups, Graphs & Groups, Graphs & IsospectralityIsospectrality
1RΓ
2RΓ
21 21IndInd RR G
HGH
1RΓ
)(,Hom)(1H 1
1
RC CR
1RΓ
ΓΓ is the Cayley graph of is the Cayley graph of DD44
(with respect to the generators (with respect to the generators aa, , rrxx): ):
Take again Take again G=DG=D44 and the same subgroups: and the same subgroups:HH11 = { e , a = { e , a22, r, rxx , r , ryy} } with the rep.with the rep. R R11
HH22 = { e , a = { e , a22, r, ruu , r , rvv} } with the rep.with the rep. R R22
HH33 = { e , a , a = { e , a , a22 , a , a33} } with the rep.with the rep. R R33
Arsenal of isospectral Arsenal of isospectral examplesexamples
1RΓ
2RΓ
ea
a3 a
2
rx
ryrv
ruL1
L2
L2
L1
The resulting quotient graphs are:The resulting quotient graphs are:
3RΓ
L1
L1
L1
L1
L2
L2
L2
L2
L1
L1
L2
L2
G = DG = D66 = {e, a, a = {e, a, a22, a, a33, a, a44, a, a55, r, rxx, r, ryy, r, rzz, r, ruu, r, rvv, r, rww} } with the subgroups:with the subgroups:
HH11 = { e, a = { e, a22, a, a44, r, rxx, r, ryy, r, rz z } } with the rep.with the rep. R R11
HH22 = { e, a = { e, a22, a, a44, r, ruu, r, rvv, r, rw w } } with the rep.with the rep. R R22
HH33 = { e, a, a = { e, a, a22, a, a33, a, a44, a, a5 5 } } with the rep.with the rep. R R33
Arsenal of isospectral Arsenal of isospectral examplesexamples
1RΓ
2RΓ
The resulting quotient graphs are:The resulting quotient graphs are:
3RΓ
L2
L1 2L2
2L3
2L1L2
L3L3
2L2
2L3
2L1
2L3
2L3
2L2
2L2
2L1
L1
L1
2L2
2L2
2L3
2L3
2L1
2L1
G = SG = S44 acts on the tetrahedron.acts on the tetrahedron.
with the subgroups:with the subgroups:HH11 = S = S33 with the rep.with the rep. R R11
HH22 = S = S44 with the rep.with the rep. R R22
|S|S44|=24 , |S|=24 , |S33|=6|=6
Arsenal of isospectral Arsenal of isospectral examplesexamples
The resulting quotient graphs are:The resulting quotient graphs are:
L
3
4
1
2
1RΓ
D
L
L/2
L/2
2RΓ
D
N
L/2L/2
L/2
D
DN
D
L/2
(v)f
(v)fF
2
1v
(v)'f
(v)'f'F
2
1v
0
0'FBFA vvvv
00
11Av
21
00Bv
1f
2f
v
vvvv ABBA
ΓΓ is the a graph which obeys is the a graph which obeys the the OOh h (octahedral group (octahedral group with reflections).with reflections). |O |Ohh|=48. |=48.
Take Take G= OG= Ohh and the subgroups: and the subgroups:HH11 = O, = O, the octahedral group.the octahedral group.HH22 = T = Tdd, , the tetrahedral groupthe tetrahedral group with reflections.with reflections. |H|H11|= |H|= |H22|= 24|= 24
Arsenal of isospectral Arsenal of isospectral examplesexamples
Arsenal of isospectral Arsenal of isospectral examplesexamples
A puzzleA puzzle: construct an isospectral pair : construct an isospectral pair out of the following familiar graphout of the following familiar graph: :
G = SG = S33 (D (D33) ) acts on acts on ΓΓ with no fixed points. with no fixed points.
To construct the quotient graph, To construct the quotient graph, we take the same rep. of we take the same rep. of GG, , but use two different bases but use two different bases for the matrix representation. for the matrix representation.
Arsenal of isospectral Arsenal of isospectral examplesexamples
The resulting quotient graphs are:The resulting quotient graphs are:
L1
L2
L3
L3 L2
L1L3
L3
L3
L2L2
L2
L1L3
L3
L3
L2L2
L2
L1
L2
L2
L3
L3
L1
L2 L3
L1
L2 L3
L1 L2
L3
L3
L2
Arsenal of isospectral Arsenal of isospectral examplesexamples
‘‘One One cannotcannot hear the shape of a drum’ hear the shape of a drum’ Gordon, Webb and Wolpert (1992)
Isospectral drumsIsospectral drums
D
N
D
N
GG00=*444=*444 (using Conway’s orbifold notation) acts on the hyperbolic plane.
Considering a homomorphism of GG00 onto G=PSL(3,2)G=PSL(3,2)and taking two subgroups HH11, HH22 such that: and RR11, RR22 are the sign representations, we obtain the known isospectral drums of Gordon et al.but with new boundary conditions:
4241 SHSH ;
Arsenal of isospectral Arsenal of isospectral examplesexamples
‘Spectral problems with mixed Dirichlet-Neumann boundary conditions: isospectrality and beyond’ D. Jacobson, M. Levitin, N. Nadirashvili, I. Polterovich (2004)
‘Isospectral domains with mixed boundary conditions’ M. Levitin, L. Parnovski, I. Polterovich (2005)
Isospectral drumsIsospectral drums
This isospectral quartet can be obtained when This isospectral quartet can be obtained when acting with the group Dacting with the group D44xDxD44 on the following on the following torus and considering the subgroups torus and considering the subgroups HH11X HX H11, H, H11X HX H22, H, H22X HX H11, H, H22X HX H22 with the reps. with the reps. RR11X RX R11, R, R11X RX R22, R, R22X RX R11, R, R22X RX R22 (using the notation presented before for the (using the notation presented before for the main dihedral example).main dihedral example).
The relation to Sunada’s The relation to Sunada’s constructionconstruction
21 HH 11 GH
GH IndInd
21
Let G be a group. Let HH11, , HH22 be two subgroups of G be two subgroups of G.
Then the triple (G, H1, H2) satisfies Sunada’s condition if:
where [g] is the cunjugacy class of g in G.For such a triple (G, H1, H2) we get that , are isospectral.
Pesce (94) proved Sunada’s theorem using the observation that Sunada’s condition is equivalent to the following:
The relation to the construction method presented so far is via the identification:
21 HgHgGg ][][
1H
21, iH i iH1
2H
Further on …Further on … What is the strength of this method ?What is the strength of this method ?
Having two isospectral graphs – how to construct the ‘parent’ graph Having two isospectral graphs – how to construct the ‘parent’ graph from which they were born ?from which they were born ?
Having such a ‘parent’ graph, can it be shown that it obeys a Having such a ‘parent’ graph, can it be shown that it obeys a symmetry group such that the conditions of the theorem are symmetry group such that the conditions of the theorem are fulfilled ?fulfilled ?
What are the conditions which guarantee that the quotient What are the conditions which guarantee that the quotient graphs are not isometric ?graphs are not isometric ?
A graph with a self-adjoint operator might be isospectral to a A graph with a self-adjoint operator might be isospectral to a graph with a non self-adjoint one.graph with a non self-adjoint one.
What other properties of the functions can be used to What other properties of the functions can be used to resolve isospectrality ?resolve isospectrality ?
Groups, Graphs Groups, Graphs & Isospectrality& Isospectrality
Rami BandRami Band
Ori ParzanchevskiOri Parzanchevski
Gilad Ben-ShachGilad Ben-Shach
Uzy SmilanskyUzy Smilansky
Acknowlegments: M. Sieber, I. Yaakov.Acknowlegments: M. Sieber, I. Yaakov.