9
$\begingroup$

Let $p$ be a prime and let $f_p$ be the permutation on the set $\{1,2,\cdots,p-1\}$ which is given by taking inverses in $\mathbb{Z}/(p)$:

$$x \bmod(p) \mapsto \frac{1}{x} \bmod (p)$$

So for instance, written in cyclic notation:

$$f_2 = ()$$

$$f_3 = ()$$

$$f_5 = (2,3)$$

$$f_7 = (2,4)(3,5)$$

Looking at the permutation group generated by two primes $p,q$ not necessarily distinct, we observe that the generated group:

$$\langle f_p,f_q\rangle = \langle f_p,f_q| f_p^2 = 1, f_q^2 = 1, (f_p \circ f_q)^n = 1\rangle = D_n$$ is the Dihedral group of degree $n(p,q)=:n$ for some natural number $n$, which depends on $p,q$.

This is because $\frac{1}{\frac{1}{x}} = x \bmod (p)$, so $f_p^2 = 1$ and if we look at the product $f_p \circ f_q$ which has finite order $n:=n(p,q)$, we see that this is just the presentation of the dihedral group of degree $n$ and order $2n$: $$\langle f_p,f_q| f_p^2 = 1, f_q^2 = 1, (f_p \circ f_q)^n = 1\rangle = D_n$$

Let us look at the group matrix as defined by Dedekind, where we write $D(p,q):=D_{n(p,q)}$ for the dihedral group:

$$(\operatorname{ord}({gh^{-1}}))_{g,h \in D(p,q)}$$

where $\operatorname{ord}(g)$ is the order of the element $g$ in $D(p,q)$ and (so a divisor of $2n=2 n(p,q)$).

I have looked at Keith Conrad's "History of representation theory" which explains how one would compute the spectrum of this group matrix, but I am not sure how to apply this knowledge in this situation.

(Here is a link to the representation theory for dihedral groups, which might help in answering this question. )

The questions are: 1) Why does these group matrices seem to always have integer spectrum?

[ 2) Is there any interpretation of the spectrum of one such matrix $G_{p,q}$ in terms of number theoretic properties of the primes $p,q$? ]

Some data and SageMath code:

def perm(p):
    X = list(range(1,p))
    s = Permutation([(1/X[i])%p for i in range(len(X))])
    return s

def tmat(p,q):
    G = PermutationGroup([perm(r) for r in [p,q]])
    #print(G.order())
    M = matrix([[ (g*h.inverse()).order() for g in G] for h in G])
    return G,M

def printAttr(B):
    for d in dir(B):
        if d.startswith("__"):
            continue
        if d.startswith("is_"):
            try:
                if getattr(B,d)():
                    print(d,getattr(B,d)())
            except:
                pass
K=7
for M in range(1,K+1):
    for N in range(1,K+1):
        print(nth_prime(N),nth_prime(M))
        G,A=(tmat(nth_prime(N),nth_prime(M)))
        print((A))
        ev = sorted(A.eigenvalues())
        print(G.order(),ev)
        print([(g.order()) for g in G])
        plot(A).show()

Data:

p,q =  2 2
Matrix G_(p,q):
[1]
Order of dihedral group =  1 , Eigenvalues =  [1]
order of elements in group =  [1]
   
p,q =  3 2
Matrix G_(p,q):
[1]
Order of dihedral group =  1 , Eigenvalues =  [1]
order of elements in group =  [1]
   
p,q =  5 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  7 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  11 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  13 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  17 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  19 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  23 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  2 3
Matrix G_(p,q):
[1]
Order of dihedral group =  1 , Eigenvalues =  [1]
order of elements in group =  [1]
   
p,q =  3 3
Matrix G_(p,q):
[1]
Order of dihedral group =  1 , Eigenvalues =  [1]
order of elements in group =  [1]
   
p,q =  5 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  7 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  11 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  13 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  17 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  19 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  23 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  2 5
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  3 5
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  5 5
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  7 5
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  11 5
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  13 5
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  17 5
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  19 5
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  23 5
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  2 7
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  3 7
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  5 7
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  7 7
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  11 7
Matrix G_(p,q):
[1 2 2 6 2 3 2 2 6 2 2 3]
[2 1 6 2 3 2 2 2 2 6 3 2]
[2 6 1 2 2 2 2 3 2 3 6 2]
[6 2 2 1 2 2 3 2 3 2 2 6]
[2 3 2 2 1 2 2 6 2 6 3 2]
[3 2 2 2 2 1 6 2 6 2 2 3]
[2 2 2 3 2 6 1 2 3 2 2 6]
[2 2 3 2 6 2 2 1 2 3 6 2]
[6 2 2 3 2 6 3 2 1 2 2 2]
[2 6 3 2 6 2 2 3 2 1 2 2]
[2 3 6 2 3 2 2 6 2 2 1 2]
[3 2 2 6 2 3 6 2 2 2 2 1]
Order of dihedral group =  12 , Eigenvalues =  [-7, -7, -6, -6, -6, -6, 2, 2, 2, 2, 9, 33]
order of elements in group =  [1, 3, 3, 2, 2, 2, 2, 2, 2, 6, 6, 2]
   
p,q =  13 7
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  17 7
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  19 7
Matrix G_(p,q):
[1 2 2 2 2 3 2 6 6 2 3 2]
[2 1 2 2 3 2 6 2 2 6 2 3]
[2 2 1 2 2 6 2 3 3 2 6 2]
[2 2 2 1 6 2 3 2 2 3 2 6]
[2 3 2 6 1 2 2 2 2 6 2 3]
[3 2 6 2 2 1 2 2 6 2 3 2]
[2 6 2 3 2 2 1 2 2 3 2 6]
[6 2 3 2 2 2 2 1 3 2 6 2]
[6 2 3 2 2 6 2 3 1 2 2 2]
[2 6 2 3 6 2 3 2 2 1 2 2]
[3 2 6 2 2 3 2 6 2 2 1 2]
[2 3 2 6 3 2 6 2 2 2 2 1]
Order of dihedral group =  12 , Eigenvalues =  [-7, -7, -6, -6, -6, -6, 2, 2, 2, 2, 9, 33]
order of elements in group =  [1, 3, 3, 2, 2, 2, 2, 2, 2, 6, 6, 2]
   
p,q =  23 7
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  2 11
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  3 11
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  5 11
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  7 11
Matrix G_(p,q):
[1 2 2 6 2 3 2 2 6 2 2 3]
[2 1 6 2 3 2 2 2 2 6 3 2]
[2 6 1 2 2 2 2 3 2 3 6 2]
[6 2 2 1 2 2 3 2 3 2 2 6]
[2 3 2 2 1 2 2 6 2 6 3 2]
[3 2 2 2 2 1 6 2 6 2 2 3]
[2 2 2 3 2 6 1 2 3 2 2 6]
[2 2 3 2 6 2 2 1 2 3 6 2]
[6 2 2 3 2 6 3 2 1 2 2 2]
[2 6 3 2 6 2 2 3 2 1 2 2]
[2 3 6 2 3 2 2 6 2 2 1 2]
[3 2 2 6 2 3 6 2 2 2 2 1]
Order of dihedral group =  12 , Eigenvalues =  [-7, -7, -6, -6, -6, -6, 2, 2, 2, 2, 9, 33]
order of elements in group =  [1, 3, 3, 2, 2, 2, 2, 2, 2, 6, 6, 2]
   
p,q =  11 11
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  13 11
Matrix G_(p,q):
[ 1  2  2  2  2 10  2  5  2  5 10  2  2 10  5  2  5  2  2 10]
[ 2  1  2  2 10  2  5  2  5  2  2 10 10  2  2  5  2  5 10  2]
[ 2  2  1  2  2  5  2 10  2 10  5  2  2  5 10  2 10  2  2  5]
[ 2  2  2  1  5  2 10  2 10  2  2  5  5  2  2 10  2 10  5  2]
[ 2 10  2  5  1  2 10  2  2  2  2  5  5  2  2 10  2 10  5  2]
[10  2  5  2  2  1  2 10  2  2  5  2  2  5 10  2 10  2  2  5]
[ 2  5  2 10 10  2  1  2  5  2  2 10  2  2  2  5  2  5 10  2]
[ 5  2 10  2  2 10  2  1  2  5 10  2  2  2  5  2  5  2  2 10]
[ 2  5  2 10  2  2  5  2  1  2  2 10 10  2  2  5  2  5 10  2]
[ 5  2 10  2  2  2  2  5  2  1 10  2  2 10  5  2  5  2  2 10]
[10  2  5  2  2  5  2 10  2 10  1  2  2  5  2  2 10  2  2  5]
[ 2 10  2  5  5  2 10  2 10  2  2  1  5  2  2  2  2 10  5  2]
[ 2 10  2  5  5  2  2  2 10  2  2  5  1  2  2 10  2 10  5  2]
[10  2  5  2  2  5  2  2  2 10  5  2  2  1 10  2 10  2  2  5]
[ 5  2 10  2  2 10  2  5  2  5  2  2  2 10  1  2  5  2  2 10]
[ 2  5  2 10 10  2  5  2  5  2  2  2 10  2  2  1  2  5 10  2]
[ 5  2 10  2  2 10  2  5  2  5 10  2  2 10  5  2  1  2  2  2]
[ 2  5  2 10 10  2  5  2  5  2  2 10 10  2  2  5  2  1  2  2]
[ 2 10  2  5  5  2 10  2 10  2  2  5  5  2  2 10  2  2  1  2]
[10  2  5  2  2  5  2 10  2 10  5  2  2  5 10  2  2  2  2  1]
Order of dihedral group =  20 , Eigenvalues =  [-21, -21, -12, -12, -12, -12, -12, -12, -12, -12, 4, 4, 4, 4, 4, 4, 4, 4, 43, 83]
order of elements in group =  [1, 5, 5, 5, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 10, 10, 2, 10, 10]
   
p,q =  17 11
Matrix G_(p,q):
[ 1  2  2 10 10  2 10  2  2  5  2  5  5  2  2  2  2  5  2 10]
[ 2  1 10  2  2 10  2 10  5  2  5  2  2  5  2  2  5  2 10  2]
[ 2 10  1  2  2  5  2  5 10  2  2  2  2 10  5  2 10  2  5  2]
[10  2  2  1  5  2  5  2  2 10  2  2 10  2  2  5  2 10  2  5]
[10  2  2  5  1  2  5  2  2 10  2 10  2  2  2  5  2 10  2  5]
[ 2 10  5  2  2  1  2  5 10  2 10  2  2  2  5  2 10  2  5  2]
[10  2  2  5  5  2  1  2  2  2  2 10 10  2  2  5  2 10  2  5]
[ 2 10  5  2  2  5  2  1  2  2 10  2  2 10  5  2 10  2  5  2]
[ 2  5 10  2  2 10  2  2  1  2  5  2  2  5 10  2  5  2 10  2]
[ 5  2  2 10 10  2  2  2  2  1  2  5  5  2  2 10  2  5  2 10]
[ 2  5  2  2  2 10  2 10  5  2  1  2  2  5 10  2  5  2 10  2]
[ 5  2  2  2 10  2 10  2  2  5  2  1  5  2  2 10  2  5  2 10]
[ 5  2  2 10  2  2 10  2  2  5  2  5  1  2  2 10  2  5  2 10]
[ 2  5 10  2  2  2  2 10  5  2  5  2  2  1 10  2  5  2 10  2]
[ 2  2  5  2  2  5  2  5 10  2 10  2  2 10  1  2 10  2  5  2]
[ 2  2  2  5  5  2  5  2  2 10  2 10 10  2  2  1  2 10  2  5]
[ 2  5 10  2  2 10  2 10  5  2  5  2  2  5 10  2  1  2  2  2]
[ 5  2  2 10 10  2 10  2  2  5  2  5  5  2  2 10  2  1  2  2]
[ 2 10  5  2  2  5  2  5 10  2 10  2  2 10  5  2  2  2  1  2]
[10  2  2  5  5  2  5  2  2 10  2 10 10  2  2  5  2  2  2  1]
Order of dihedral group =  20 , Eigenvalues =  [-21, -21, -12, -12, -12, -12, -12, -12, -12, -12, 4, 4, 4, 4, 4, 4, 4, 4, 43, 83]
order of elements in group =  [1, 5, 5, 5, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 10, 10, 10, 10]
   
p,q =  19 11
Matrix G_(p,q):
[ 1  2  2  3  3  2  2  2  2  6  6  2  2 12  4  2  2 12  2 12  4  2  2 12]
[ 2  1  3  2  2  3  2  2  6  2  2  6 12  2  2  4 12  2 12  2  2  4 12  2]
[ 2  3  1  2  2  3  2  6  2  2  2  6  4  2  2 12 12  2  4  2  2 12 12  2]
[ 3  2  2  1  3  2  6  2  2  2  6  2  2  4 12  2  2 12  2  4 12  2  2 12]
[ 3  2  2  3  1  2  6  2  2  6  2  2  2 12 12  2  2  4  2 12 12  2  2  4]
[ 2  3  3  2  2  1  2  6  6  2  2  2 12  2  2 12  4  2 12  2  2 12  4  2]
[ 2  2  2  6  6  2  1  2  2  3  3  2  2 12  4  2  2 12  2 12  4  2  2 12]
[ 2  2  6  2  2  6  2  1  3  2  2  3 12  2  2  4 12  2 12  2  2  4 12  2]
[ 2  6  2  2  2  6  2  3  1  2  2  3  4  2  2 12 12  2  4  2  2 12 12  2]
[ 6  2  2  2  6  2  3  2  2  1  3  2  2  4 12  2  2 12  2  4 12  2  2 12]
[ 6  2  2  6  2  2  3  2  2  3  1  2  2 12 12  2  2  4  2 12 12  2  2  4]
[ 2  6  6  2  2  2  2  3  3  2  2  1 12  2  2 12  4  2 12  2  2 12  4  2]
[ 2 12  4  2  2 12  2 12  4  2  2 12  1  2  2  3  3  2  2  2  2  6  6  2]
[12  2  2  4 12  2 12  2  2  4 12  2  2  1  3  2  2  3  2  2  6  2  2  6]
[ 4  2  2 12 12  2  4  2  2 12 12  2  2  3  1  2  2  3  2  6  2  2  2  6]
[ 2  4 12  2  2 12  2  4 12  2  2 12  3  2  2  1  3  2  6  2  2  2  6  2]
[ 2 12 12  2  2  4  2 12 12  2  2  4  3  2  2  3  1  2  6  2  2  6  2  2]
[12  2  2 12  4  2 12  2  2 12  4  2  2  3  3  2  2  1  2  6  6  2  2  2]
[ 2 12  4  2  2 12  2 12  4  2  2 12  2  2  2  6  6  2  1  2  2  3  3  2]
[12  2  2  4 12  2 12  2  2  4 12  2  2  2  6  2  2  6  2  1  3  2  2  3]
[ 4  2  2 12 12  2  4  2  2 12 12  2  2  6  2  2  2  6  2  3  1  2  2  3]
[ 2  4 12  2  2 12  2  4 12  2  2 12  6  2  2  2  6  2  3  2  2  1  3  2]
[ 2 12 12  2  2  4  2 12 12  2  2  4  6  2  2  6  2  2  3  2  2  3  1  2]
[12  2  2 12  4  2 12  2  2 12  4  2  2  6  6  2  2  2  2  3  3  2  2  1]
Order of dihedral group =  24 , Eigenvalues =  [-35, -35, -22, -22, -22, -22, -7, -7, -7, -7, 2, 2, 2, 2, 2, 2, 2, 2, 10, 10, 10, 10, 53, 101]
order of elements in group =  [1, 3, 3, 2, 2, 2, 2, 2, 2, 12, 12, 4, 2, 2, 2, 12, 12, 4, 2, 6, 6, 2, 2, 2]
   
p,q =  23 11
Matrix G_(p,q):
[1 2 2 8 8 2 2 4 2 2 2 4 8 2 8 2]
[2 1 8 2 2 8 4 2 2 2 4 2 2 8 2 8]
[2 8 1 2 2 2 8 2 2 8 8 2 2 4 2 4]
[8 2 2 1 2 2 2 8 8 2 2 8 4 2 4 2]
[8 2 2 2 1 2 2 8 8 2 2 8 4 2 4 2]
[2 8 2 2 2 1 8 2 2 8 8 2 2 4 2 4]
[2 4 8 2 2 8 1 2 2 4 2 2 2 8 2 8]
[4 2 2 8 8 2 2 1 4 2 2 2 8 2 8 2]
[2 2 2 8 8 2 2 4 1 2 2 4 8 2 8 2]
[2 2 8 2 2 8 4 2 2 1 4 2 2 8 2 8]
[2 4 8 2 2 8 2 2 2 4 1 2 2 8 2 8]
[4 2 2 8 8 2 2 2 4 2 2 1 8 2 8 2]
[8 2 2 4 4 2 2 8 8 2 2 8 1 2 2 2]
[2 8 4 2 2 4 8 2 2 8 8 2 2 1 2 2]
[8 2 2 4 4 2 2 8 8 2 2 8 2 2 1 2]
[2 8 4 2 2 4 8 2 2 8 8 2 2 2 2 1]
Order of dihedral group =  16 , Eigenvalues =  [-21, -21, -5, -5, -5, -5, -1, -1, -1, -1, -1, -1, -1, -1, 27, 59]
order of elements in group =  [1, 2, 2, 2, 2, 2, 8, 8, 2, 2, 4, 4, 8, 8, 2, 2]
   
p,q =  2 13
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  3 13
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  5 13
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  7 13
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  11 13
Matrix G_(p,q):
[ 1  2  2  2  2 10  2  5  2  5 10  2  2 10  5  2  5  2  2 10]
[ 2  1  2  2 10  2  5  2  5  2  2 10 10  2  2  5  2  5 10  2]
[ 2  2  1  2  2  5  2 10  2 10  5  2  2  5 10  2 10  2  2  5]
[ 2  2  2  1  5  2 10  2 10  2  2  5  5  2  2 10  2 10  5  2]
[ 2 10  2  5  1  2 10  2  2  2  2  5  5  2  2 10  2 10  5  2]
[10  2  5  2  2  1  2 10  2  2  5  2  2  5 10  2 10  2  2  5]
[ 2  5  2 10 10  2  1  2  5  2  2 10  2  2  2  5  2  5 10  2]
[ 5  2 10  2  2 10  2  1  2  5 10  2  2  2  5  2  5  2  2 10]
[ 2  5  2 10  2  2  5  2  1  2  2 10 10  2  2  5  2  5 10  2]
[ 5  2 10  2  2  2  2  5  2  1 10  2  2 10  5  2  5  2  2 10]
[10  2  5  2  2  5  2 10  2 10  1  2  2  5  2  2 10  2  2  5]
[ 2 10  2  5  5  2 10  2 10  2  2  1  5  2  2  2  2 10  5  2]
[ 2 10  2  5  5  2  2  2 10  2  2  5  1  2  2 10  2 10  5  2]
[10  2  5  2  2  5  2  2  2 10  5  2  2  1 10  2 10  2  2  5]
[ 5  2 10  2  2 10  2  5  2  5  2  2  2 10  1  2  5  2  2 10]
[ 2  5  2 10 10  2  5  2  5  2  2  2 10  2  2  1  2  5 10  2]
[ 5  2 10  2  2 10  2  5  2  5 10  2  2 10  5  2  1  2  2  2]
[ 2  5  2 10 10  2  5  2  5  2  2 10 10  2  2  5  2  1  2  2]
[ 2 10  2  5  5  2 10  2 10  2  2  5  5  2  2 10  2  2  1  2]
[10  2  5  2  2  5  2 10  2 10  5  2  2  5 10  2  2  2  2  1]
Order of dihedral group =  20 , Eigenvalues =  [-21, -21, -12, -12, -12, -12, -12, -12, -12, -12, 4, 4, 4, 4, 4, 4, 4, 4, 43, 83]
order of elements in group =  [1, 5, 5, 5, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 10, 10, 2, 10, 10]
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17
  • 1
    $\begingroup$ I suspect that your integer eigenvalues are due to perhaps not checking big enough dihedral groups. The general formula for the eigenvalues for any finite group $G$ is there is one eigenvalue of each irreducible character $\chi$ of $G$. The corresponding eigenvalue is $\sum_{g\in G}\frac{ord(g)\chi(g)}{\chi(1)}$. If your group has Q as a splitting field, then these will take on integer values. Very small order dihedral groups have Q as a splitting field. $\endgroup$
    – Benjamin Steinberg
    Commented Aug 17 at 1:46
  • 4
    $\begingroup$ I just checked that all groups of order less than 128 have integral spectrum, so this seems to hold not only for dihedral groups! $\endgroup$
    – Kasper Andersen
    Commented Aug 17 at 9:34
  • 4
    $\begingroup$ @StevenStadnicki he is looking at the matrix in the regular representation of the element of the center if the group algebra that adds all the elements of the group weighted by their order. This particular combination maybe always has rational coefficients even when the splitting field is bigger but the formula is correct $\endgroup$
    – Benjamin Steinberg
    Commented Aug 17 at 13:37
  • 1
    $\begingroup$ Everyone else seems to understand, so I guess I'm just being dense, but what does $f_p \circ f_q$ mean when $f_p$ is a permutation of $1, \dotsc, p - 1$ and $f_q$ is a permutation of $1, \dotsc, q - 1$? $\endgroup$
    – LSpice
    Commented Aug 17 at 15:08
  • 2
    $\begingroup$ @LSpice Take $p < q$ and extend $f_p$ by the identity. $\endgroup$
    – mme
    Commented Aug 17 at 15:25

2 Answers 2

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12
$\begingroup$

To add to my comment and @SeanEberhard's answer, for any finite group $G$ the eigenvalues of the matrix are integers. You are looking at the matrix of $\sum_{g\in G} \operatorname{ord}(g)g$ under the regular representation, which belongs to the center of $\mathbb ZG$. Therefore its image under the irreducible representation corresponding to a character $\chi$ is $\lambda_\chi I$ where $\lambda_\chi=\sum_{g\in G}\operatorname{ord}(g)\frac{\chi(g)}{\chi(1)}$ and this value is an algebraic integer. So the eigenvalues are the $\lambda_\chi$ with $\chi$ an irreducible character and $\lambda_{\chi}$ has multiplicity $\chi(1)^2$.

But if $G$ has exponent $n$, the Galois group of $\mathbb Q(\zeta_n)/\mathbb Q$ acts on the characters of $G$ and if $\sigma$ is in this Galois group, then it satisfies $\sigma(\zeta_n) = \zeta_n^m$ for some $m$ with $\gcd(m,n)=1$. Then $\sigma(\chi)(g) =\chi(g^m)$ since the eigenvalues of $g$ under the representation affording $\chi$ are $n^{th}$-roots of unity. Since $g$ and $g^m$ generate the same cyclic group (because $\gcd(m,n)=1$), it follows that $g^m$ and $g$ have the same order. Thus the Galois action fixes $\lambda_\chi$ and so $\lambda_\chi$ is a rational algebraic integer hence an integer.

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9
  • $\begingroup$ Thanks for your insight in this question. Would your arguments also work if we change $\operatorname{ord}$ with any function $f: G \rightarrow \mathbb{N}$? Thanks for your help. $\endgroup$
    – mathoverflowUser
    Commented Aug 17 at 16:11
  • 3
    $\begingroup$ No, you would need the property that if $g,g'$ generate the same subgroup of $G$ then $f(g)=f(g')$. $\endgroup$
    – Benjamin Steinberg
    Commented Aug 17 at 16:12
  • 2
    $\begingroup$ Also you need f constant on conjugacy classes $\endgroup$
    – Benjamin Steinberg
    Commented Aug 17 at 16:37
  • $\begingroup$ Thanks for your explanation and nice answer. $\endgroup$
    – mathoverflowUser
    Commented Aug 17 at 16:38
  • 1
    $\begingroup$ The multiplicities are character degrees squared although some characters might give the same eigenvalue. I don't have an interpretation of the eigenvalues $\endgroup$
    – Benjamin Steinberg
    Commented Aug 18 at 16:40
8
$\begingroup$

You can just compute the spectrum exactly using the formula mentioned by Benjamin Steinberg: $$\sum_g \operatorname{ord}(g) \frac{\chi(g)}{\chi(1)}.$$ There are two or four characters of degree $1$ according to the parity of $n$, and these obviously give an integer value since $\chi(g) = \pm1$ and $\chi(1) = 1$. The characters of degree $2$ have the form $r^i \mapsto \zeta^i + \zeta^{-i}$ where $\langle r\rangle$ is the rotation subgroup and $\zeta \ne \pm1$ is an $n$th root of unity. The value of the character on $G \setminus \langle r \rangle$ is zero. Therefore the sum above is $$\sum_{i=0}^{n-1} \frac{n}{\gcd(i, n)} \frac{\zeta^i + \zeta^{-i}}2 = \sum_{i=0}^{n-1} \frac{n}{\gcd(i, n)} \zeta^i.$$ This sum is invariant under the action of the Galois group and therefore a rational algebraic integer, a.k.a., an integer. You can compute its value exactly if you want using the fact that the sum of the primitive $d$th roots of unity is $\mu(d)$.

The observation that the sum is an integer for other finite groups sounds more interesting. I wonder if it is related to the result of Frobenius that the number of solutions to $g^n = 1$ is a multiple of $n$.

Your second question is about how $n = n(p, q)$ depends on $p$ and $q$. This is completely unrelated and a bit unnatural. One may be able to model $f_p f_q$ as the product of random involutions supported on $\{1, \dots, p-1\}$ and $\{1, \dots, q-1\}$.

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3
  • $\begingroup$ Isn’t the formula I gave invariant under the Galois action for any finite group? $\endgroup$
    – Benjamin Steinberg
    Commented Aug 17 at 15:57
  • $\begingroup$ Thanks for your answer. I will need some time to understand what your wrote. $\endgroup$
    – mathoverflowUser
    Commented Aug 17 at 16:00
  • 1
    $\begingroup$ It is explained in the answer of @BenjaminSteinberg why you obtain that (for any finite group $G$) the eigenvalues of any element of the form $\sum_{g \in G} f(g)g$ are rational integers if $f$ is a class function which is also constant on generators of any given cyclic subgroup. I don't think this is related in any obvious way to Frobenius's theorem about the number of solutions in $G$ of $x^{n}=1$, though it's difficult to entirely exclude that possibility. $\endgroup$
    – Geoff Robinson
    Commented Aug 17 at 20:21

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