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The (classical) Möbius function is the unique function that obeys the classical Möbius inversion formula:

Proposition 1 (Classical Möbius inversion)Let be functions from the natural numbers to an additive group . Then the following two claims are equivalent:

- (i) for all .
- (ii) for all .

There is a generalisation of this formula to (finite) posets, due to Hall, in which one sums over chains in the poset:

Proposition 2 (Poset Möbius inversion)Let be a finite poset, and let be functions from that poset to an additive group . Then the following two claims are equivalent:(Note from the finite nature of that the inner sum in (ii) is vacuous for all but finitely many .)

- (i) for all , where is understood to range in .
- (ii) for all , where in the inner sum are understood to range in with the indicated ordering.

Comparing Proposition 2 with Proposition 1, it is natural to refer to the function as the Möbius function of the poset; the condition (ii) can then be written as

*Proof:*If (i) holds, then we have for any . Iterating this we obtain (ii). Conversely, from (ii) and separating out the term, and grouping all the other terms based on the value of , we obtain (1), and hence (i).

In fact it is not completely necessary that the poset be finite; an inspection of the proof shows that it suffices that every element of the poset has only finitely many predecessors .

It is not difficult to see that Proposition 2 includes Proposition 1 as a special case, after verifying the combinatorial fact that the quantity

is equal to when divides , and vanishes otherwise.I recently discovered that Proposition 2 can also lead to a useful variant of the inclusion-exclusion principle. The classical version of this principle can be phrased in terms of indicator functions: if are subsets of some set , then

In particular, if there is a finite measure on for which are all measurable, we haveOne drawback of this formula is that there are exponentially many terms on the right-hand side: of them, in fact. However, in many cases of interest there are “collisions” between the intersections (for instance, perhaps many of the pairwise intersections agree), in which case there is an opportunity to collect terms and hopefully achieve some cancellation. It turns out that it is possible to use Proposition 2 to do this, in which one only needs to sum over chains in the resulting poset of intersections:

Proposition 3 (Hall-type inclusion-exclusion principle)Let be subsets of some set , and let be the finite poset formed by intersections of some of the (with the convention that is the empty intersection), ordered by set inclusion. Then for any , one has where are understood to range in . In particular (setting to be the empty intersection) if the are all proper subsets of then we have In particular, if there is a finite measure on for which are all measurable, we have

Using the Möbius function on the poset , one can write these formulae as

and
*Proof:* It suffices to establish (2) (to derive (3) from (2) observe that all the are contained in one of the , so the effect of may be absorbed into ). Applying Proposition 2, this is equivalent to the assertion that

Example 4If with , and are all distinct, then we have for any finite measure on that makes measurable that due to the four chains , , , of length one, and the three chains , , of length two. Note that this expansion just has six terms in it, as opposed to the given by the usual inclusion-exclusion formula, though of course one can reduce the number of terms by combining the factors. This may not seem particularly impressive, especially if one views the term as really being three terms instead of one, but if we add a fourth set with for all , the formula now becomes and we begin to see more cancellation as we now have just seven terms (or ten if we count as four terms) instead of terms.

Example 5 (Variant of Legendre sieve)If are natural numbers, and is some sequence of complex numbers with only finitely many terms non-zero, then by applying the above proposition to the sets and with equal to counting measure weighted by the we obtain a variant of the Legendre sieve where range over the set formed by taking least common multiples of the (with the understanding that the empty least common multiple is ), and denotes the assertion that divides but is strictly less than . I am curious to know of this version of the Legendre sieve already appears in the literature (and similarly for the other applications of Proposition 2 given here).

If the poset has bounded depth then the number of terms in Proposition 3 can end up being just polynomially large in rather than exponentially large. Indeed, if all chains in have length at most then the number of terms here is at most . (The examples (4), (5) are ones in which the depth is equal to two.) I hope to report in a later post on how this version of inclusion-exclusion with polynomially many terms can be useful in an application.

Actually in our application we need an abstraction of the above formula, in which the indicator functions are replaced by more abstract idempotents:

Proposition 6 (Hall-type inclusion-exclusion principle for idempotents)Let be pairwise commuting elements of some ring with identity, which are all idempotent (thus for ). Let be the finite poset formed by products of the (with the convention that is the empty product), ordered by declaring when (note that all the elements of are idempotent so this is a partial ordering). Then for any , one has where are understood to range in . In particular (setting ) if all the are not equal to then we have

Morally speaking this proposition is equivalent to the previous one after applying a “spectral theorem” to simultaneously diagonalise all of the , but it is quicker to just adapt the previous proof to establish this proposition directly. Using the Möbius function for , we can rewrite these formulae as

and
*Proof:* Again it suffices to verify (6). Using Proposition 2 as before, it suffices to show that

A finite group is said to be a Frobenius group if there is a non-trivial subgroup of (known as the *Frobenius complement* of ) such that the conjugates of are “disjoint as possible” in the sense that whenever . This gives a decomposition

where the *Frobenius kernel* of is defined as the identity element together with all the non-identity elements that are not conjugate to any element of . Taking cardinalities, we conclude that

A remarkable theorem of Frobenius gives an unexpected amount of structure on and hence on :

Theorem 1 (Frobenius’ theorem)Let be a Frobenius group with Frobenius complement and Frobenius kernel . Then is a normal subgroup of , and hence (by (2) and the disjointness of and outside the identity) is the semidirect product of and .

I discussed Frobenius’ theorem and its proof in this recent blog post. This proof uses the theory of characters on a finite group , in particular relying on the fact that a character on a subgroup can induce a character on , which can then be decomposed into irreducible characters with *natural number* coefficients. Remarkably, even though a century has passed since Frobenius’ original argument, there is no proof known of this theorem which avoids character theory entirely; there are elementary proofs known when the complement has even order or when is solvable (we review both of these cases below the fold), which by the Feit-Thompson theorem does cover all the cases, but the proof of the Feit-Thompson theorem involves plenty of character theory (and also relies on Theorem 1). (The answers to this MathOverflow question give a good overview of the current state of affairs.)

I have been playing around recently with the problem of finding a character-free proof of Frobenius’ theorem. I didn’t succeed in obtaining a completely elementary proof, but I did find an argument which replaces character theory (which can be viewed as coming from the representation theory of the non-commutative group algebra ) with the Fourier analysis of class functions (i.e. the representation theory of the centre of the group algebra), thus replacing non-commutative representation theory by commutative representation theory. This is not a particularly radical depature from the existing proofs of Frobenius’ theorem, but it did seem to be a new proof which was technically “character-free” (even if it was not all that far from character-based in spirit), so I thought I would record it here.

The main ideas are as follows. The space of class functions can be viewed as a commutative algebra with respect to the convolution operation ; as the regular representation is unitary and faithful, this algebra contains no nilpotent elements. As such, (Gelfand-style) Fourier analysis suggests that one can analyse this algebra through the idempotents: class functions such that . In terms of characters, idempotents are nothing more than sums of the form for various collections of characters, but we can perform a fair amount of analysis on idempotents directly without recourse to characters. In particular, it turns out that idempotents enjoy some important integrality properties that can be established without invoking characters: for instance, by taking traces one can check that is a natural number, and more generally we will show that is a natural number whenever is a subgroup of (see Corollary 4 below). For instance, the quantity

is a natural number which we will call the *rank* of (as it is also the linear rank of the transformation on ).

In the case that is a Frobenius group with kernel , the above integrality properties can be used after some elementary manipulations to establish that for any idempotent , the quantity

is an integer. On the other hand, one can also show by elementary means that this quantity lies between and . These two facts are not strong enough on their own to impose much further structure on , unless one restricts attention to *minimal* idempotents . In this case spectral theory (or Gelfand theory, or the fundamental theorem of algebra) tells us that has rank one, and then the *integrality gap* comes into play and forces the quantity (3) to always be either zero or one. This can be used to imply that the convolution action of every minimal idempotent either preserves or annihilates it, which makes itself an idempotent, which makes normal.

In this lecture, we use topological dynamics methods to prove some other Ramsey-type theorems, and more specifically the polynomial van der Waerden theorem, the hypergraph Ramsey theorem, Hindman’s theorem, and the Hales-Jewett theorem. In proving these statements, I have decided to focus on the ultrafilter-based proofs, rather than the combinatorial or topological proofs, though of course these styles of proof are also available for each of the above theorems.

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