9
$\begingroup$

If you're in a hurry scroll down until the questions:

First the known part:

A sequence $\left(b_1,b_2,b_3,...\right)$ of integers will be called a ghost-Witt sequence if there exists a sequence $\left(x_1,x_2,x_3,...\right)$ of integers such that $\left(b_n=\sum\limits_{d\mid n}dx_d^{n/d}\text{ for every positive integer }n\right)$. (The reason for this naming lies in the big/universal Witt polynomials, of course.)

There exist several equivalent criteria for when a sequence $\left(b_1,b_2,b_3,...\right)$ of integers is a ghost-Witt sequence. One of them is that $b_{n/p}\equiv b_n\mod p^{v_p\left(n\right)}$ for any positive integer $n$ and any prime divisor $p$ of $n$, where $v_p\left(n\right)$ denotes the $p$-adic valuation of $n$. Another is that $n\mid \sum\limits_{d\mid n}\phi\left(d\right)b_{n/d}$ for every positive integer $n$, where $\phi$ is the Euler totient function. There are more (I listed some of them in Witt#5 Theorem 15), but there are yet more - I plan to write them up yet.

This can be generalized several times; besides, the above criteria (well, some of them) easily show that the sum and the product of two ghost-Witt sequences are ghost-Witt sequences again (which leads to the integrality of big/universal Witt polynomials, but only if one generalizes this properly to polynomials instead of integers; otherwise you just get that the big Witt polynomials are integral-valued, but this doesn't mean that their coefficients are integers).

Enough ramblings - here is what I am wondering about:

Questions:

A sequence $\left(b_1,b_2,b_3,...\right)$ of integers will be called a ghost-Burnside sequence if $\left(n\mid \sum\limits_{d\mid n}\phi\left(d\right)b_d^{n/d}\text{ for every positive integer }n\right)$.

  • Is the sum / the product of two ghost-Burnside sequences another ghost-Burnside sequence?
  • Are there any nice equivalent criteria? In particular, is there a criterion similar to the definition of a ghost-Witt sequence?

Why do I care? Because the polynomial $\frac{1}{n}\sum\limits_{d\mid n}\phi\left(d\right)X_d^{n/d}$ is the cycle index of the cyclic group $\mathbb Z / n\mathbb Z$. It is not an integer polynomial, and not even an integer-valued polynomial, but one of the results of Polya enumeration theory states that $\frac{1}{n}\sum\limits_{d\mid n}\phi\left(d\right)\left(Y_1^d+Y_2^d+...+Y_k^d\right)^{n/d}\in\mathbb Z\left[Y_1,Y_2,...,Y_k\right]$ for any $k\geq 0$ (actually, this can be shown using Theorem 13 in Witt#5 as well). This makes every sequence of the form

$\left(u_1^d+u_2^d+...+u_k^d\right)_{d\text{ positive integer}}$

for integers $u_1$, $u_2$, ..., $u_k$ a ghost-Burnside sequence. I hope for a kind of converse; of course, it will require considering finite sequences only and possibly only up to congruence modulo $n$. EDIT: Sorry, this approach is doomed to fail. All the sequences constructed this way are not only ghost-Burnside, but also ghost-Witt, and we can't get the remaining ghost-Burnside sequences. (Yes, being ghost-Witt is stronger than being ghost-Burnside; but I don't have a nice proof.)

Results so far: If we let $m$ be a positive integer and require $n\mid \sum\limits_{d\mid n}\phi\left(d\right)b_d^{n/d}$ to hold for $n\mid m$ only (and not necessarily for all positive integers $n$), then I can prove that the sum and the product of two ghost-Burnside sequences are ghost-Burnside sequences again, if $m$ is a prime power or a squarefree integer. I think I also did it for the case $m=p^2q$, which already required some casebash. The other cases are scaring me off...

I am recording the results for myself:

(1) If $m=p^k$ is a power of a prime $k$, then a sequence $\left(b_1,b_2,b_3,...\right)$ of integers (of course, only the $b_{p^i}$ members matter) is ghost-Burnside if and only if $b_{p^u}\equiv b_{p^{u+1}}\mod p$ for every $0\leq u\leq k-1$. No typo here: it's $\mod p$, not $\mod p^u$ as in the ghost-Witt case!

(2) If $m=pq$ is the product of two distinct primes $p$ and $q$, then a sequence $\left(b_1,b_2,b_3,...\right)$ of integers (of course, it's all in the terms $b_1$, $b_p$, $b_q$, $b_{pq}$) is ghost-Burnside if and only if

$b_1\equiv b_p\mod p$; $b_1\equiv b_q\mod q$; ($b_q\equiv b_{pq}\mod p$ OR $p\mid q-1$); ($b_p\equiv b_{pq}\mod q$ OR $q\mid p-1$).

(3) If $m=p^2q$ for two distinct primes $p$ and $q$, then a sequence $\left(b_1,b_2,b_3,...\right)$ of integers (this times six terms matter) is ghost-Burnside if and only if

$b_1\equiv b_p\equiv b_{p^2}\mod p$; $b_1\equiv b_q\mod q$; ($b_q\equiv b_{pq}\mod p$ OR $p\mid q-1$); ($b_p\equiv b_{pq}\mod q$ OR $q\mid p-1$); ($b_{p^2}\equiv b_{p^2q}\mod q$ OR $q\mid p-1$); [($b_{pq}\equiv b_{p^2q}\mod p$ AND $p\not\mid q-1$) OR ($b_q\equiv b_{pq}\mod p$ AND $p\mid q-1$) OR ($p^2\mid q-1$)].

(4) Generalizing (2): If $m$ is the product of finitely many distinct primes (i. e. if $m$ is squarefree), then a sequence $\left(b_1,b_2,b_3,...\right)$ of integers is ghost-Burnside if and only if

($\phi\left(d\right)b_d\equiv \phi\left(d\right)b_{pd}\mod p$ for every $d\mid m$ and every prime $p$ such that $pd\mid m$).

Dreams:

If this works, we could move on to groups more interesting than cyclic groups.

$\endgroup$
5
  • $\begingroup$ Have you tried writing down the Dirichlet generating function of the cycle index polynomials of Z/nZ? $\endgroup$ Mar 3, 2010 at 18:00
  • $\begingroup$ It's the logarithm of $\prod_{d=1}^{\infty} \left(1-a_dT^d\right)^{-\phi\left(d\right)/d}$. Nice idea, but I don't have a clue how to proceed from here. Reminds me of the Artin-Hasse exponential, though. $\endgroup$ Mar 3, 2010 at 18:29
  • $\begingroup$ Read $a_d$ as $X_d$ in my comment, sorry. $\endgroup$ Mar 3, 2010 at 18:30
  • 2
    $\begingroup$ Concerning your dream: there already is a Witt vector construction beyond the case of cyclic groups. In 1988, Dress and Siebeneicher introduced a functor W_G for any profinite group G. For any ring A, the new ring W_G(A) has operations defined in terms of certain universal "G-Witt polynomials", and the key point is that the sum and product of two such polynomials (over independent sets of variables) is again such a polynomial. See A. Dress, C. Siebeneicher, "The Burnside ring of profinite groups and the Witt vector construction", Adv. Math 70 (1988), 87--132. $\endgroup$
    – KConrad
    Mar 4, 2010 at 3:18
  • 1
    $\begingroup$ Thanks for the reference, but I know the construction (Witt-Burnside functors). I should have mentioned it. However, ghost-Burnside is not the same as ghost-Witt, and their respective generalizations should be even less equivalent. $\endgroup$ Mar 4, 2010 at 13:15

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.