Given a finite permutation group *G* on a set *X*, the *permutation character* π of *G* is the function on *G* mapping an element *g* to its number of fixed points in *X*. This is a *character* of *G*, the function giving the trace of a matrix representation of *G* (in this case, the representation by permutation matrices). So it can be decomposed into irreducible characters of *G*; say,

π = ∑*m*_{i}χ_{i}.

Now the number of orbits of *G* is equal to the multiplicy *m*_{0} of the trivial character χ_{0} in this expression (by the *Orbit-Counting Lemma*), while the rank of *G* (the number of orbits on *X*^{2}) is equal to the sum of squares of the multiplicities *m*_{i}.

Thus, the permutation character is the sum of the trivial character and one non-trivial irreducible if and only if *G* is doubly transitive. An old result asserts that the permutation character cannot have the form χ_{0}+*m*χ for a non-trivial irreducible χ if *m* > 1.

The proof goes like this. By Jordan’s Theorem, since *G* is transitive, it contains an element *g* with no fixed points. Now the expression for π shows that χ(*g*) = −1/*m*. But any character value must be an algebraic integer, and so an integer if it is also rational, as this value is.

Now I have described before the notion of *coherent configuration*, a combinatorial gadget which describes (among other things) the orbits of a permutation group *G* on the set of ordered pairs. The relation matrices for the group orbits span an algebra (over the complex numbers) which is a direct sum of complete matrix algebras of dimensions equal to the multiplicities *m*_{i}, this matrix algebra occurring with multiplicity in the regular representation equal to the degree of the character χ_{i}. For a general coherent configuration, we have a similar algebraic theory, but we do not have the group to give us these numbers.

**Problem** Is there a coherent configuration which “looks like” one coming from a group whose permutation character has this forbidden form, that is, the sum of a 1-dimensional algebra and a complete matrix algebra of degree greater than i?

We no longer have Jordan’s theorem to help us. I suspect that there is a simple algebraic argument which substitutes, but I cannot at the moment see how it would go. Any ideas?

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About Peter Cameron

I count all the things that need to be counted.

Hi Peter,

I think the answer is “no”. If the adjacency algebra A of a CC (X,S) is isomorphic to C\oplus M_d(C), d > 1, then (X,S) is homogeneous. A has two irreducible characters \chi_0 and \chi_1 with \chi_0(A_0)=1,\chi_1(A_0)=d (A_0 is the identity matrix). The character \chi of the standard module CX has a decomposition: \chi = chi_0 + m\chi_1 for some positive integer m. Then $\chi(A_0)=|X|$ implies $|X|=1+md$. If A_i, i >0 is a basic matrix, then $\chi(A_i)=0$ implying $\chi_1(A_i)m+k_i=0$ (k_i is the valency of A_i). Since $\chi_1(A_i)$ is an algebraic integer, the valency $k_i$ is divisible by $m$. Therefore, $k_i \geq m$ for each $i > 0$. Thus $|X|\geq 1 + m(|S|-1) = 1 + md^2$. Together with $|X|=1+md$ we obtain $d=1$. I hope this argument is mistake-free.

More general statements are proven in [1](Theorem 1) and [2] (Theorem 6).

If you assume that A \cong M_a(C)\oplus M_d(C) with a > 1 than it becomes possible. Inhomogeneous coherent algebra could be a direct sum of only two full matrix algebras: M_a(C)\oplus M_a(C). It happens if the CC comes from a system of linked block designs with the same parameters.

All the best, Misha

[1] H. Blau, Association schemes, fusion rings, C-algebras, and reality-based algebras where all nontrivial multiplicities are equal”, J Algebr Comb (2010) 31: 491–499.

[2] Herman, A.; Muzychuk, M.; Xu, B. The recognition problem for table algebras and reality-based algebras. J. Algebra 479 (2017), 173–191.

Dear Misha,

Thank you. I was hoping you could come up with something like this.

Peter.