Im Bier, Beweis

(though wine may be needed for truth)

Worked out last night over a quiet pint:

Consequently: if , and lie in a commutative Banach algebra, and , , and all have norm , then the expression on the left hand side has norm at most .

[Title corrected 2011-12-08]

Latest from the Department of Unwanted Puzzles

(If you are called Matt and are folk of the Dales, mathematically speaking, then I guess I should disqualify both of you from leaving answers to this.)

In each of the following, identify the phrase being loosely paraphrased or indicated. What’s the connection, and which one is the odd one out?

• Smooth guy
• Entrance to place of rest
• Lux aeterna
• When does the present arrive?
• Honestly, Bill
• Erratically circle waterspout
• Petty thieves band together
• Hurry and heave to claim ground
• Transvestite clergyman
• Laterally pierced youth
• Violently vibrate
• Principal ceremony
• Couldn’t complete it
• Isn’t it all the same?
• Humour has worn off

Update 18/11/11: answered in comments.

wie es eigentlich gewesen

I’ve recently found myself wanting to cite a result, proved in an article of Leptin that formed part of some conference proceedings.

According to MathSciNet, the proceedings appeared as

Symposia Mathematica. Vol. XXII. (Italian)
Convegno sull’Analisi Armonica e Spazi di Funzioni su Gruppi Localmente Compatti. Tenuto al Istituto Nazionale di Alta Matematica (INDAM), Roma, 24–31 Marzo, 1976. Academic Press, London-New York, 1977. 464 pp.

On the other hand, the “inter-library loan” copy that I’ve just received today claims that Symposia Mathematica XXII was published in 1981, not 1977.

Any suggestions as to how the discrepancy might appear? Anyone able to confirm, by checking a hard copy, which is the correct year?

(For more on the title of this post, especially if you know enough German to wonder about the missing auxiliary verb, see this blogpost.)

Etiquette

Not-entirely-idle question for readers who have both published and refereed maths papers:

As I understand it, it is acceptable (though not always to be encouraged) if one makes public the comments received from referees, perhaps redacted to try and preserve anonymity.

What about feedback received directly from the handling editors?

Any other word would smell as sweet

So… in playing around with some calculations and ideas that might coalesce into a preprint, I find myself wanting to coin some new terminology.

For sake of argument, suppose we have a concept called slithiness, and that people have been studying slithy widgets for some time and some length. Moreover, people have been modifying the adjective slithy with various adjectives or adverbs, so that there are now a proliferation of flavours of slithiness.

One of the oldest, or at least one with claims to the best pedigree, is weak slithiness. Anything slithy is weakly slithy, but there are examples which are weakly slithy yet not slithy.

Now: for technical reasons, I’m toying with the idea of a slightly strengthened version of weak slithiness (i.e. a slightly more restrictive condition). But what do I call it? At present, the only reasonable candidates I can think of are

1) “strongly weak slithiness” – which is ungood, and sounds like an attempt to make Eric Blair spin in his grave, once he’s finished from what Endemol did to him;

2) “weak slithiness with WCB” – which is clunkier, and also commits the sin identified by G. K. Pedersen as ASHCEFLC.

Any suggestions?

Some additional remarks are in order. If we go with option (1), then I do know there are widgets which are slithy but not “strongly weakly slithy”. I am not 100% sure if slithy widgets are always “strongly weakly slithy”, but suspect they are.

A problem concerning character ratios for finite groups

While working on a paper with some colleagues (as part of the same project which motivated one of my questions on MathOverflow), I was led to a question that is not directly relevant, but seemed interesting, and which I don’t know how to go about answering. I thought it might be salutary to write it up as a blog post and see if people have suggestions.

The question seems slightly too open-ended to be asked on MathOverflow, and more importantly my hope is for co-operative discussion, which is not really what I think MO is suited for. Maybe it has already been answered in the literature, or is somehow known to be as hard as problems that are still open.

Alors.

The problem concerns the irreducible characters of finite groups (over the complex field, in case that makes a difference), and how big the values of such characters can be on elements of a given group. To avoid needless repetition and tiresome caveats: all my groups in this post are finite and have at least two elements.

If G is a group and is an irreducible character on G, then for every . (Writing for , just take an irrep whose trace is and note that it has to send to a scalar multiple of the identity matrix.) In general, though, can be much smaller than .

Disregarding the sage words of Pedersen, let’s say that the MCR of an irreducible character is the maximum value of as runs over all non-identity elements of the group in question. Here are some of the basic properties of the MCR:

• The MCR of any irreducible character is strictly positive and at most 1.
• Any character that comes from a 1-dimensional representation (a.k.a. a “linear character”) clearly has MCR equal to 1.
• By the previous remarks, if G has non-trivial centre, then every irreducible character on G has MCR equal to 1.
• If G has trivial centre, there exists at least one irreducible character on G whose MCR is strictly less than 1.

(The last of these facts is a paraphrase, for those who know of these things, of the result that the intersection of all centres of all irreducible characters is the centre of the group.)

Here is my question:

Does there exists a constant such that each group with trivial centre possesses at least one irreducible character whose MCR is ?

Or, for those who prefer formulas to wordy formulations:

Is ?

Some disjointed and simple-minded thoughts:

• My limited understanding of the literature is that people have mostly focused on getting upper bounds for the MCR that apply to all non-linear characters on particular classes of simple group: there is work of Gluck in this direction for groups of Lie type. However, this in some sense has different quantifiers from my question, and I have some vague hope that my question might be susceptible to global averaging/probabilistic arguments that don’t need as much detailed knowledge of the structure of finite groups…
• The simple-minded estimate , where denotes the size of the conjugacy class of in G, suggests that finding upper bounds on the minimal MCR we should focus on non-identity elements in the group with large centralizers, but I don't know how far one can get with that line of thought.
• In the absence of any particular proof strategy, one could start by taking one’s favourite family of groups with trivial centre and known character table, and then getting an upper bound on the minimal MCR by inspection of the formulas. For instance, if where F is a finite field with elements, the character tables tell us that G has an irreducible character^* of degree q which has absolute value at most 1 on all non-identity elements of G; hence the MCR of this character is .

[*] The character in question is the Steinberg character, obtainable as the non-trivial summand of the permutation representation given by the usual action of PSL(2,F) on the F-projective line; it is apparently Frightfully Important, for reasons which are too advanced for this Bear of Little Brain.

Allusion, or coincidence?

With various tasks that need to be done before term starts, it is perhaps no surprise that I find myself ignoring them in favour of various articles. One that recently caught my eye is a preprint of Borovik and Katz:

arXiv 1108.2285: Who Gave you the Cauchy-Weierstrass Tale? The Dual History of Rigorous Calculus

which on a first skim looks very interesting, and immediately scores points with me by taking apparent Whig history to task. (By mathematical upbringing, I am an analyst of sorts, schooled in orthodox fashion; so I don’t have immediate personal empathy with some of the misgivings about epsilontics. Nevertheless, I am interested in both the history — rather than the folklore accounts — of the subject’s development, and also in arguments for how the subject should be taught or viewed differently.)

None of that is really the reason for this post, though. Instead, I was just struck by the following acerbic aside (p. 21; the emphasis is mine):

It is sobering to realize that, forty years after A. Robinson, a logician named Walter Felscher still conceived of the history of analysis in terms of a triumphant march out of the dark age of the infinitesimal, and toward the yawning heights of Weierstrassian epsilontics.

While the phrase has no particular connotations for me, beyond the obvious irony, I wonder if it is making some deeper allusion, as used in the title of this book of Zinoviev?

Moses went back up the mountain for a bit

Nice to have a Prof. of Public Understanding of Science who still puts out research papers… (I vaguely remember the example on the first page being mentioned in the one and only talk by M du S which I’ve attended.)

Mais je n’aime pas bien le mot “liminaire”

Since efforts to write some mathematics on this blog keep stalling: here is the sort of thing which makes you feel better about digging through a textbook in search of a few key technical parts for citation. (I’d seen this a couple of years ago, but was reminded of it during my current foraging.)

The original name CCR means ‘completely continuous representations’ (completely continuous operators being another name for C(H)). The next layer in this hierarchy, ‘GCR’ indicates a generalization of the CCR condition. The modern names ‘liminary’ and ‘postliminary’ do not mean anything, which may be more aesthetic. In any case the Anglo-Saxon Habit of Condensing Every Formula into its Leading Characters (abbreviated ASHCEFLC) should not be tolerated in mathematics.

(From the notes to Section 6.2 of Pedersen’s book CAAG.)

WTF is “the post”?

Timothy Gowers on why people in Britain should vote to change their current voting system.

Hat-tip to Alistair Bird, for pointing out that somewhere in the post, Gowers all but asks (rhetorically) “WTF is the post?”. I look forward to the day when one of his research-oriented posts observes that “some m**herf***ers are always trying to ice-skate uphill”…

Also, I like to think that the author of The Complete Plain Words would approve of one of his descendants giving a bad and stale metaphor a good kicking.