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Recently I found an arc-transitive graph (and then many more) for which some eigenspaces are reducible, something that I believed might not occure.

The example is the Shrikhande graph, a strongly regular graph with parameters $(16,6,2,2)$.

The spectrum consists of eigenvalues $6^1, -2^6, 2^9$ (multiplicities in the exponent), where only the eigenspace of $2$ is reducible. I cannot tell you how exactly the eigenspace decomposes since my knowledge about this stems from computing the characters and the Frobenius-Schur indicator as explained here.

There are other examples: e.g. $C_{10}\times C_{10}$ and some circulant graphs that have one suspiciously large eigenspace that decomposes. I have not investigated for what parameters these reducible eigenspaces occure, I just know that they are not always present.

What is interesting though, is, that the eigenspace to the second largest eigenvalue seems to be always irreducible. This is interesting since this eigenspace is related to the algebraic connectivity and is a central object of my research. I will have to investigate whether this is always true.

Update

Even the eigenspace of the second-largest eigenvalue does not have to be irreducible. I found some examples by computing Frobenius-Schur indicators for various arc-transitive graphs. However, counter-examples seem to be rare. E.g. there are none with $\le 30$ vertices.

Recently I found an arc-transitive graph (and then many more) for which some eigenspaces are reducible, something that I believed might not occure.

The example is the Shrikhande graph, a strongly regular graph with parameters $(16,6,2,2)$.

The spectrum consists of eigenvalues $6^1, -2^6, 2^9$ (multiplicities in the exponent), where only the eigenspace of $2$ is reducible. I cannot tell you how exactly the eigenspace decomposes since my knowledge about this stems from computing the characters and the Frobenius-Schur indicator as explained here.

There are other examples: e.g. $C_{10}\times C_{10}$ and some circulant graphs that have one suspiciously large eigenspace that decomposes. I have not investigated for what parameters these reducible eigenspaces occure, I just know that they are not always present.

What is interesting though, is, that the eigenspace to the second largest eigenvalue seems to be always irreducible. This is interesting since this eigenspace is related to the algebraic connectivity and is a central object of my research. I will have to investigate whether this is always true.

Update

Even the eigenspace of the second-largest eigenvalue does not have to be irreducible. I found some examples by computing Frobenius-Schur indicators for various arc-transitive graphs. However, counter-examples seem to be rare.

Recently I found an arc-transitive graph (and then many more) for which some eigenspaces are reducible, something that I believed might not occure.

The example is the Shrikhande graph, a strongly regular graph with parameters $(16,6,2,2)$.

The spectrum consists of eigenvalues $6^1, -2^6, 2^9$ (multiplicities in the exponent), where only the eigenspace of $2$ is reducible. I cannot tell you how exactly the eigenspace decomposes since my knowledge about this stems from computing the characters and the Frobenius-Schur indicator as explained here.

There are other examples: e.g. $C_{10}\times C_{10}$ and some circulant graphs that have one suspiciously large eigenspace that decomposes. I have not investigated for what parameters these reducible eigenspaces occure, I just know that they are not always present.

What is interesting though, is, that the eigenspace to the second largest eigenvalue seems to be always irreducible. This is interesting since this eigenspace is related to the algebraic connectivity and is a central object of my research. I will have to investigate whether this is always true.

Recently I found an arc-transitive graph (and then many more) for which some eigenspaces are reducible, something that I believed might not occure.

The example is the Shrikhande graph, a strongly regular graph with parameters $(16,6,2,2)$.

The spectrum consists of eigenvalues $6^1, -2^6, 2^9$ (multiplicities in the exponent), where only the eigenspace of $2$ is reducible. I cannot tell you how exactly the eigenspace decomposes since my knowledge about this stems from computing the characters and the Frobenius-Schur indicator as explained here.

There are other examples: e.g. $C_{10}\times C_{10}$ and some circulant graphs that have one suspiciously large eigenspace that decomposes. I have not investigated for what parameters these reducible eigenspaces occure, I just know that they are not always present.

What is interesting though, is, that the eigenspace to the second largest eigenvalue seems to be always irreducible. This is interesting since this eigenspace is related to the algebraic connectivity and is a central object of my research. I will have to investigate whether this is always true.

Update

Even the eigenspace of the second-largest eigenvalue does not have to be irreducible. I found some examples by computing Frobenius-Schur indicators for various arc-transitive graphs. However, counter-examples seem to be rare. E.g. there are none with $\le 30$ vertices.

Recently I found an arc-transitive graph (and then many more) for which some eigenspaces are reducible, something that I believed might not occure.

The example is the Shrikhande graph, a strongly regular graph with parameters $(16,6,2,2)$.

The spectrum consists of eigenvalues $6^1, -2^6, 2^9$ (multiplicities in the exponent), where only the eigenspace of $2$ is reducible. I cannot tell you how exactly the eigenspace decomposes since my knowledge about this stems from computing the characters and the Frobenius-Schur indicator as explained here.

There are other examples: e.g. $C_{10}\times C_{10}$ and some circulant graphs that have one suspicially large eigenspace that decomposes. I have not investigated for what parameters these reducible eigenspaces occure, I just know that they are not always present.

What is interesting though, is, that the eigenspace to the second largest eigenvalue seems to be always irreducible. This is interesting since this eigenspace is related to the algebraic connectivity and is a central object of my research. I will have to investigate whether this is always true.

Recently I found an arc-transitive graph (and then many more) for which some eigenspaces are reducible, something that I believed might not occure.

The example is the Shrikhande graph, a strongly regular graph with parameters $(16,6,2,2)$.

The spectrum consists of eigenvalues $6^1, -2^6, 2^9$ (multiplicities in the exponent), where only the eigenspace of $2$ is reducible. I cannot tell you how exactly the eigenspace decomposes since my knowledge about this stems from computing the characters and the Frobenius-Schur indicator as explained here.

There are other examples: e.g. $C_{10}\times C_{10}$ and some circulant graphs that have one suspiciously large eigenspace that decomposes. I have not investigated for what parameters these reducible eigenspaces occure, I just know that they are not always present.

What is interesting though, is, that the eigenspace to the second largest eigenvalue seems to be always irreducible. This is interesting since this eigenspace is related to the algebraic connectivity and is a central object of my research. I will have to investigate whether this is always true.