mm-4029
===
Subject: Proving a number transcendental
http://ioannis.virtualcomposer2000.com/math/Naturals.html
I would like some assistance in proving that c(n) is transcendental. c(n)
doesn't appear to be Liouville, so I have no idea how to proceed.
Note that c(n)=0.a(n) with a(n) given by the recursion:
a(n)= 0, iff n=0,
102, iff n=1,
[a(n-1)/10]*10^{D(a(n-1))+1}+10*a(n-1)+2, otherwise.
where [] denotes the floor function and D(n)=log[10](n)+1 is the length of
the
number n.
I posted the question to sci.math, but only got some speculative answers.
If
anyone has suggestions on methods of attack, references, etc, they are all
appreciated.
>
--
I.N. Galidakis --- http://ioannis.virtualcomposer2000.com/
----------------------------------------------------------
There's ALWAYS a mistake somewhere
===
Subject: Re: Proving a number transcendental
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
> suggestions on ... references
Baker, Transcendental Number Theory
--
G. A. Edgar
http://www.math.ohio-state.edu/~edgar/
===
Subject: Re: generalized catalan numbers
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
> I am trying to count something that appears to be related to a
Have you looked at Dyck paths? Your characterization of the catalan
number is isomorphic to counting Dyck paths as follows. Given one of
your strings, let each 1 be a move diagonally up to the right and let
each 0 be a move diagonally down to the right. Then if you start at
(0,0) in the xy plane, since the string is of length n there will be n
moves and since any prefix of the string has more 1's then 0's where
exactly half of them are 1's the path will end at (n,0) and always be
weakly above the x-axis. which is exactly a dyck path.
For your generalization, your first requirement is the same as
requiring a dyck path to end not at (n,0) but rather at (n,nk-(n-
nk))=(n,n(2k-1). Since the furthest up or down you could get in n
moves is between -n and n, this forces k to between 0 and 1. Your
second requirement is the same as requiring the path to below the line
y = kx.
As for your first requirement check out
http://geomblog.blogspot.com/2007/04/generalized-dyck-paths.html. As
for your second it should be just a matter of taking all the paths and
then checking how many paths it over counts by.
Also there have been a few new entries on dyck paths over at the arxiv
which answer some other interesting questions.
Best,
-m
> generalization of catalan numbers:
Catalan(n/2) =
> number of binary (0/1) strings containing n/2 ones and n/2 zeroes
such
> that any prefix has no more 1's than 0's =
> number of binary strings of length n containing exactly n/2 ones
such
> that in any prefix of length i there are no more than i/2 ones.
What about this generalization where 0<k<1 :
> X(k,n) = number of binary (0/1) strings of length n containing
exactly
> kn ones such that in any prefix of length i there are no more than ki
ones.
This seems general enough that someone in combinatorics probably already
> solved it. Anyone seen this before or know where i could look?
===
Subject: Re: a classical diophantine quadratic equation
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
On May 21, 7:56 pm, Giovanni Resta
> I'm looking for a link or a reference
> for the problem of determining for which
> integer positive values x and y the
> expression
a x^2 + b x y + c y^2
is a square (a,b,c are integer constants).
Presently I'm interested in a case in which a = c,
> such as 9x^2+42xy+9y^2 = w^2,
> but of course the general solution will be
> more interesting.
I'm sure I've seen something about this problem,
> but I do not remember where...
g.
The following is a quick response to your question:
The formula f(x,y,z)=a x^2 + b x y + c y^2 can be written as
f= [x,y]A[x,y]^T
where
$A=[ a b/2;
b/2 c ]
$
So f is a square iff the matrix $A$ can be factorized as
A=BB^T
where $B$ is a $2xm$ nonnegative integral matrix. Surely this matrix A
shall be positive semidefinite, so $4ac >=b^2$. But generally, that is
not enough. You can find this kind of facorization from some early
work of Mordell and Deza, which may go back to 1930's.
Oh, for this 2 by 2 case, things will be much more easier, especially
in your case, i.e., $a=c$.
===
Subject: Doubly-Affine Extensions of GL(n)
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
Is there a standard name and body of results for the semidirect
product of the Lie algebra for GL(n) with the Heisenberg algebra H(n)
on n pairs of generators?
If the generators are denoted E(a,b) for (a,b = 0,...,n), this is the
algebra given by the Lie bracket
[E(a,b), E(c,d)] = g(b,c) E(a,d) - g(a,d) E(b,c)
where the matrix g = (g(a,b): a,b = 0,...,n) is just g =
diag(0,1,1,...,1).
The generator h = E(0,0) is in the center, the generators q(a) =
E(0,a) and p(a) = E(a,0) (for a = 1,...,n) play the role of the
complementary sets of translation operators, with (h, q(1), ...,
q(n), p(1), ..., p(n)) forming the Heisenberg algebra H(n). The
generators (E(a,b): a,b = 1,...,n) form the Lie algebra for GL(n).
===
Subject: Re: Intersecting Circles: a question
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
> I have the following problem.
Let C(x,r) be the notation for a circle centered in x
> (vector in R^2)
> and having radius r.
Let us define the following four circles:
C(0,gamma)
> C(a,alpha-gamma)
> C(b,beta-gamma)
> C(a+b,2+gamma-alpha-beta)
Now, suppose we are given a quadruple of the form
> <a,b,alpha,beta>.
Is there a necessary and sufficient condition in
> a,b,alpha,beta such
> that there exists gamma for which the four circles
> have a nonzero
> overlapping area?
A necessary condition is that ||a+b||+||a-b|| is
> smaller or equal than
> 2. However, this is not sufficient.
Simone
> -
> Simone Severini
> http://www.iqc.ca/~sseverin
According to the theorem of Helly-Radon the four circles in question do 
have
a non-empty intersection iff any three of them do have a non-empty
intersection.
Maybe this is useful for you to determine a necessary and sufficient
condition for a,b,alpha and beta.
Of course if they do have a non-empty intersection then any two of them do
have a non-empty intersection which immediately yields ||a+b||+||a-b|| <= 
2.
Hope this helps
Michael
===
Subject: Two papers published by Geometry & Topology
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
Two papers have been published by Geometry & Topology
(1) Geometry & Topology 11 (2007) 867-887
Blocking light in compact Riemannian manifolds
by Jean-Francois Lafont and Benjamin Schmidt
URL: http://www.msp.warwick.ac.uk/gt/2007/11/p016.xhtml
DOI: 10.2140/gt.2007.11.867
(2) Geometry & Topology 11 (2007) 889-937
Representations of the quantum Teichmueller
space and invariants of surface diffeomorphisms
by Francis Bonahon and Xiaobo Liu
URL: http://www.msp.warwick.ac.uk/gt/2007/11/p017.xhtml
DOI: 10.2140/gt.2007.11.889
Abstracts follow
(1) Blocking light in compact Riemannian manifolds
by Jean-Francois Lafont and Benjamin Schmidt
We study compact Riemannian manifolds (M,g) for which the light from
any given point x in M can be shaded away from any other point y in
M by finitely many point shades in M. Compact flat Riemannian
manifolds are known to have this finite blocking property. We
conjecture that amongst compact Riemannian manifolds this finite
blocking property characterizes the flat metrics. Using entropy
considerations, we verify this conjecture amongst metrics with
nonpositive sectional curvatures. Using the same approach, K Burns
and E Gutkin have independently obtained this result. Additionally,
we show that compact quotients of Euclidean buildings have the finite
blocking property.
On the positive curvature side, we conjecture that compact Riemannian
manifolds with the same blocking properties as compact rank one
symmetric spaces are necessarily isometric to a compact rank one
symmetric space. We include some results providing evidence for this
conjecture.
(2) Representations of the quantum Teichmueller
space and invariants of surface diffeomorphisms
by Francis Bonahon and Xiaobo Liu
We investigate the representation theory of the polynomial core T_S^q
of the quantum Teichmueuller space of a punctured surface S. This is a
purely algebraic object, closely related to the combinatorics of the
simplicial complex of ideal cell decompositions of S. Our main result
is that irreducible finite-dimensional representations of T_S^q are
classified, up to finitely many choices, by group homomorphisms from
the fundamental group pi_1(S) to the isometry group of the hyperbolic
3-space H^3. We exploit this connection between algebra and hyperbolic
geometry to exhibit invariants of diffeomorphisms of S.
===
Subject: Re: disproof of Riemann Hypothesis?
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
> Your results looks fine. But the main definition of delta is not
> introduced later - after (18), I think you point on (36) and on that
> the author give attantion to (11) and (20) - but on (10) and (11).
Delta is $A^*/log n$, so it is defined by A*, i.e. in (20). In (11) it
is only
announced: ... where A* is a ... number to be further specified.
I don't mean (36) because it is deduced correctly from (20) and below.
> the main idea of the author was to show, that we need that delta to
> proof RH, but there is no delta, so we can not proof RH after all (it
> is not a disproof but a 'not proveable' - statement).
No, not so. He wants to show that if we proved RH then there would
exist such a delta (what would lead to a contradiction).
===
Subject: This Week's Finds in Mathematical Physics (Week 252)
Also available as http://math.ucr.edu/home/baez/week252.html
May 27, 2007
This Week's Finds in Mathematical Physics (Week 252)
John Baez
Today I want to tell you about the electromagnetic snake at the center
of our Galaxy, and continue the Tale of Groupoidification.
But first, the long-range weather forecast. There's a 40% chance of
rain on Neptune in 8 billion years! More precisely, that's the chance
these authors give for the formation of an ocean on Neptune when its
interior cools down enough:
1) Sloane J. Wiktorowica and Andrew P. Ingersoll, Liquid water oceans
in ice giants, available as astro-ph/0609723.
Right now, even though Neptune is named after the Roman god of seas
and has a nice blue appearance, it's mighty dry - at least on top.
Spectroscopy detects no water at all in its upper atmosphere! But
that's consistent with the presence of water down below, since water
vapor is a lot heavier than hydrogen and helium, which make up most of
Neptune's upper atmosphere, and the pull of gravity there is mighty
fierce.
In fact, scientists believe Neptune has a core of molten metal and
rock surrounded by more rock, methane ice, ammonia ice, and water
ice - all solid due to high pressures. The overall density of Neptune
makes sense if there's a lot of water down there. But, Wiktorowica
and Ingersoll argue that the planet can't have an ocean of liquid water.
Surprisingly, this is because Neptune is too *hot* - even though its
upper atmosphere is a chilly 50 kelvin! I think their argument goes
roughly like this, though I don't understand it well. If you fell
down through the atmosphere of Neptune, you'd find that it gets
hotter and hotter as you go down, and moister and moister too - but
always too hot, given the amount of moisture, for liquid water to be
more stable than water vapor.
However, they also indulge in some predictions about the far future!
First let's set the stage:
In 1.1 billion years the Sun will become 10% brighter than now, and
the Earth's atmosphere will dry out.
In 3 billion years the Andromeda Galaxy will collide with our galaxy.
Many solar systems will be destroyed.
In 3.5 billion years the Sun will become 40% brighter than today. If
the Earth is still orbiting the sun, its oceans will evaporate.
In 5.4 billion years from now the Sun's core will run out of hydrogen.
It will enter its first red giant phase, becoming 1.6 times bigger and
2.2 times brighter than today.
In 6.5 billion years from now the Sun will become a full-fledged red
giant, 170 times bigger and 2400 times brighter than today. The
Republican Party will finally admit the existence of global warming,
but point out that it's not human-caused.
In 6.7 billion years from now the Sun will start fusing helium and
shrink back down to 10 times bigger and 40 times brighter than today.
In 6.8 billion years from now the Sun will runs out of helium. Being
too small to start fusing carbon and oxygen, it'll enter a second red
giant phase, growing 180 times bigger and 3000 times brighter than today.
But then, about 6.9 billion years from now, the Sun will start pulsating,
ejecting half of its mass in the form of solar wind! It'll become what
they call a planetary nebula. Eventually only its inner core will be
left. In week223 I quoted Bruce Balick's eloquent description:
The remnant Sun will rise as a dot of intense light, no larger
than Venus, more brilliant than 100 present Suns, and an intensely
hot blue-white color hotter than any welder's torch. Light from
the fiendish blue pinprick will braise the Earth and tear apart
its surface molecules and atoms. A new but very thin atmosphere
of free electrons will form as the Earth's surface turns to dust.
This is where Wiktorovika and Ingersoll *begin* their story. So far,
Neptune will have warmed up a lot - assuming for the sake of argument
that it wasn't thrown out of the Solar System when the Milky Way hit
Andromeda. But when the Sun loses mass, Neptune will either collide
with Uranus, be ejected from the Solar System, or assume a stable orbit
about twice its current size.
In the latter two scenarios, Neptune will chill out, at least when the
remnant Sun cools down and becomes a white dwarf. When the surface
temperature of Neptune reaches 30 kelvin, they estimate it has a 41.5%
plus or minus 4.2% chance of forming oceans.
So, they say: Billions of years from now, after the Sun has gone,
Neptune may therefore become the only object in the Solar system
with liquid water oceans.
This sounds nice - but don't buy your beachfront property there yet.
First, they don't study how the atmospheric composition of Neptune
may change when the Sun gets 3000 times brighter than now! Maybe this
will *help* Neptune form oceans. After all, light gases like molecular
hydrogen and helium, which dominate Neptune's upper atmosphere now, are
more likely than water vapor to be driven off into outer space when it
gets hot. But, they don't even check to see if Neptune will have
*any* atmosphere left after this era.
Second, the estimate of a 41.5% plus or minus 4.2% chance seems
strangely precise, given the uncertainties involved. The error
bars should probably have big error bars themselves!
Third, it's worth admitting that the atmosphere of Neptune is a bit
mysterious. For example, nobody knows why it's bright blue. It's
probably because of methane - but Uranus also has methane in its
atmosphere, and it's not as blue. Also, despite being the coldest
planet in our Solar System, Neptune has the fastest winds: up to 2100
kilometers per hour, almost supersonic! Nobody knows what powers these
winds.
When the Voyager 2 spacecraft flew by Neptune in 1992, it saw a storm
system the size of Eurasia, which was dubbed The Great Dark Spot.
It seemed to resemble the Great Red Spot on Jupiter, which has been
around for at least 177 years. But when the Hubble Space Telescope
took another good look at Neptune in 1994, the Great Dark Spot was
gone! Meanwhile, other storms had formed.
So, the weather on Neptune is dynamic and poorly understood. Doing
forecasts for the next 8 billion years seems pretty risky... though fun.
Of course, liquid water oceans are nice if you're looking for life.
And while there probably isn't life on Neptune, there could be life on
similar planets in other solar systems. So far people have found 233
of these exoplanets, most of them heavier than Jupiter but close
to their suns - because such planets are the easiest to detect by how
they pull on their sun.
Here's a chart showing the masses and orbital radii of known exoplanets
2) P. R. Lawson, S. C. Unwin, and C. A. Beichman, Precursor Science for
Report at http://planetquest.jpl.nasa.gov/documents/RdMp272.pdf
For comparison, the letters V, E, M, J, S, U, N stand for planets in
our Solar System, not counting Mercury or the subsequently dethroned
Pluto.
As you'll see, there are lots of hot Jupiters - planets as big as
Jupiter, or even up to 200 times heavier, but closer to their sun than
we are to ours. Recently a postdoc at UCLA found evidence for water
in the atmosphere of one of these planets:
3) Travis Barman, Identification of absorption features in an extrasolar
planet atmosphere, available as arxiv:0704.1114.
This planet is only 7 million kilometers away from its yellow-white Sun,
much closer than Mercury is to ours. Its year is only 3.5 of our days!
It's bigger in size than Jupiter, but only 0.7 times as heavy. Its
surface temperature must be about 1000 kelvin. That's one really hot
Jupiter.
People have also been finding hot Neptunes. In fact, I read about a
nice one in the newspaper while writing this! It's called Gliese 436 b.
It's the size of Neptune and it's orbiting the red dwarf star Gliese
436, which is 33 light-years from Earth. This star is only 1% as bright
as our Sun - but the planet is so close that its year lasts less than
three of our days! So, its surface temperature is high: higher than
the melting point of lead.
However, because this planet is so big, the pressure down below can
still make water into a solid. In fact, its density suggests that
it's mainly made of ice!
I think this is the paper that triggered the newspaper reports:
4) M. Gillon et al, Detection of transits of the nearby hot Neptune
GJ 436 b, available as arxiv:0705.2219.
It seems hot Neptunes like this could have started as hot Jupiters
and then lost a lot of their atmosphere:
5) I. Baraffe, G. Chabrier, T. S. Barman, F. Selsis, F. Allard,
and P. H. Hauschildt, Hot-Jupiters and hot-Neptunes, a common origin?,
available at astro-ph/0505054.
There could also be cold Neptunes, perhaps with liquid water oceans.
We haven't seen these yet, but they'd be hard to see. So, while
Wiktorowica and Ingersoll's paper doesn't convince me about the future
of *our* Neptune, it raises some interesting possibilities.
Next: the snake at the center of our galaxy!
Gregory Benford is best known for his science fiction, which spans
the galaxy, but he's also an astrophysicist at U. C. Irvine. Recently
he spent a week at my school, U. C. Riverside, which has one of the
world's best SF libraries: the Eaton Collection. Since my wife is
involved in the SF program at the comparative literature department here,
he had dinner at our place one night. The conversation drifted all over
the place, with a heavy focus on fruit flies that have been bred to live
twice as long as usual. But when I asked him about his research, he
said he'd written some papers about enormous glowing filaments near
the center of the Milky Way.
I hadn't known about these! He said the biggest one could be a million
years old, perhaps formed by some energetic event, maybe a star falling
it:
6) Gregory Benford, The electromagnetic snake at the Galactic
center, http://www.ps.uci.edu/physics/news3/benford.html
I'll just quote a little:
Five years ago radio astronomy revealed the oddest and longest
filament yet discovered at our galactic center: a uniquely kinked
structure about 150 light years long and two to three light years
wide - the Snake. Its large kinks are its brightest parts. There
is energetic activity at one end and a supernova bubble at the other,
which the Snake appears to penetrate unharmed.
How does nature form stable, long-lived magnetic structures which
display considerable polarization (about 60% at 10.55 GHz in the
Snake)? In 1988 I had modeled others of the dozens of filaments
seen uniquely at the galactic center in terms of an electrodynamic
view, in which currents set up coherent magnetic pinches. Such
self-organizing filaments can exist in laboratory plasmas for long
times; the galactic ones could be at least a million years old, as
estimated by the time that shear forces would disrupt them.
The electrodynamic view uses pinch forces of currents to form filaments,
driven by the E = v x B of conducting molecular clouds moving across a
strong milliGauss ambient, ordered field. A return current must then
flow at larger radii, making a closed loop which has a springy
flexibility, able to withstand the turbulent velocity fields known
near the galactic center. The picture then anticipates that aberrant
molecular clouds, moving contrary to the general galactic rotation,
should accompany each filament. This prediction has held up as more
filaments were found.
I'd like to learn more about these!
Back in week248, I mentioned some of the complex things that
electromagnetic fields and plasma do in the Sun. The center of the
Galaxy is another place where electromagnetohydrodynamics runs
rampant. There's a black hole there, of course, but also these
filaments, and a fairly strong magnetic field that contains about
4 x 10^{47} joules of energy within about 300 light years of the
galactic center. By comparison, a supernova emits a mere 10^{44}
joules. So, there's a lot of energy around....
The big picture here, created by Farhad Yusef-Zadeh and collaborators,
shows the galactic center quite nicely, as viewed in radio frequencies:
7) National Radio Astronomy Observatory, Origin of enigmatic galactic-
You can see some supernova remnants (SNRs), the region Sgr A which
contains the black hole at the galactic center, various nonthermal
radio filaments (NRFs), and the Snake. Some of these filaments
come from regions where stars are forming - that could be important.
role of turbulence in winding up the galactic magnetic field:
8) Stanislav Boldyrev and Farhad Yusef-Zadeh, Turbulent origin of
the galactic-center magnetic field: nonthermal radio filaments,
available as astro-ph/0512373.
It's a complicated stew. I don't hope to understand it, just admire it.
And finally: the Tale of Groupoidification! In week250 we reached
the point of seeing how spans of groupoids over a fixed group G
subsume the theory of G-sets and invariant relations between
these - which are traditionally studied using double cosets.
There is a lot more we could say about this. But, our most urgent
goal is to see how spans of groupoids act like twice categorified
matrices - matrices whose entries are not just *numbers*, and not
just *sets*, but *groupoids*! This will expose the secret
combinatorial underpinnings of a lot of fancy linear algebra. Once
we've categorified linear algebra this way, we'll be in a great
position to tackle fashionable topics like categorified quantum groups,
invariants of higher-dimensional knots, and the like.
But, we should restrain ourselves from charging ahead too fast!
Everything will hang together better if we lay the groundwork properly.
For this, it pays to re-examine the history of mathematics a bit.
If we're trying to understand linear algebra using groupoids, it
pays to ask: how did people connect linear algebra and group theory
in the first place?
This book is very helpful:
9) Charles W. Curtis, Pioneers of Representation Theory: Frobenius,
Providence, Rhode Island, 1999.
book in English on finite groups. It was called Theory of Groups of
Finite Order. In the preface, he explained why he studied finite
groups by letting them act as permutations of sets, but not as
linear transformations of vector spaces:
Cayley's dictum that a group is defined by means of the laws of
combination of its symbols would imply that, in dealing with the
theory of groups, no more concrete mode of representation should be
used than is absolutely necessary. It may then be asked why, in
a book that professes to leave all applications to one side, a
considerable space is devoted to substitution groups [permutation
groups], but other particular modes of representation, such as
groups of linear transformations, are not even referred to. My
answer to this question is that while, in the present state of our
knowledge, many results in the pure theory are arrived at most
readily by dealing with properties of substitution groups, it
would be difficult to find a result that could most directly be
obtained by the consideration of groups of linear transformations.
In short, he didn't see the point of representing groups on vector
spaces - at least as a tool in the pure theory of finite groups,
as opposed to their applications.
However, within months after this book was published, he discovered
the work of Georg Frobenius, who used linear algebra very effectively to
study groups! So, Burnside started using linear algebra in his own work
book in 1911, he'd changed his tune completely:
Very considerable advances in the theory of groups of finite
order have been made since the appearance of the first edition
of this book. In particular the theory of groups of linear
substitutions has been the subject of numerous and important
investigations by several writers; and the reason given in the
original preface for omitting any account of it no longer holds
good. In fact it is now more true to say that for further
advances in the abstract theory one must look largely to the
representation of a group as a group of linear transformations.
It's interesting to see exactly how representing finite groups on vector
spaces lets us understand them better. By now almost everyone agrees
this is true. But how much of the detailed machinery of linear algebra
is really needed? How much we could do purely combinatorially, using
just spans of groupoids?
I don't really know the full answer to this question. But, it quickly
leads us into the fascinating theory of Hecke operators, which will
play a big role in the Tale of Groupoidification. So, let's pursue it
a bit.
Suppose we have two guys, William and Georg, who are studying a finite
group G.
William says, I'm studying how G acts on sets.
Georg replies, I'm studying how it acts on complex vector spaces,
as linear transformations. Mere sets are feeble entities. I can do
anything you can do - but I have the tools of linear algebra to help me!
William says, But, you're paying a moral price. You're getting the
complex numbers - a complicated infinite thing - involved in what
should be a completely finitary and combinatorial subject: the study
of a finite group. Is this really necessary?
Georg replies, I don't know, but it's sure nice. For example,
suppose I have G acting on vector space V. Then I can always
break down V into a direct sum of subspaces preserved by G, which
can't themselves be broken down any further. In technical terms:
every representation of G is a direct sum of irreducible representations.
And, this decomposition is unique! It's very nice: it's a lot like
how every natural integer has a unique prime factorization.
William says, Yes, it's always fun when we can break things down
into 'atoms' that can't be further decomposed. It's very satisfying
to our reductionist instincts. But, I can do even better than you!
George raises an eyebrow. Oh?
Yeah, William says. Suppose I have our group G acting on a set S.
Then I can always break down S into a disjoint union of subsets
preserved by G, which can't themselves be broken down any further.
In technical terms: every action of G is a disjoint union of transitive
actions. And, this decomposition is unique!
Embarrassed, Georg replies, Oh, right - we heard that back in
week249.
I wasn't paying enough attention. But how is what you're doing better
than what I'm doing? It sounds just the same.
William hesitates. Well, first of all, a rather minor point, which I
can't
resist mentioning ... when you said your decomposition of representations
into irreducibles was unique, you were exaggerating a little. It's just
unique up to isomorphism, and not a canonical isomorphism either.
For example, if you have an irreducible representation of G on V,
there are lots of *different ways* to slice the direct sum V + V
into two copies of the representation V. It's a sort of floppy
business. On the other hand, when I have a transitive action of G on S,
there's exactly *one way* to chop the disjoint union S + S into two
copies of the G-set S. I just look at the two orbits.
Georg says, Hmm. This is actually a rather delicate point.
There's not really a *canonical* isomorphism in your case either,
since S may be isomorphic to itself in lots of ways, even as a G-set.
There's something in what you say, but it's a bit hard to make
precise, and it's certainly not enough to worry me.
Okay, but here's my more serious point. Given that I can break apart
any set on which G acts into 'atoms', my job is to classify those
atoms: the transitive G-sets. And there's a very nice classification!
Any subgroup H of G gives a transitive G-set, namely G/H, and all
transitive G-sets look like this. More precisely: isomorphism classes
of transitive G-sets correspond to conjugacy classes of subgroups of
G.
Even better, this has a nice meaning in terms of Klein geometry.
Any type of figure in Klein geometry gives a transitive G-set,
with H being the stabilizer of a chosen figure of that type.
You, on the other hand, need to classify irreducible representations
of G. And this is not so conceptual. What do these irreducible
representations *mean* in terms of the group G?
George replies, Well, there's one irreducible representation for
each conjugacy class in G...
At this, William pounces. You mean the *number* of isomorphism
classes of irreducible representations of G equals the *number*
of conjugacy classes in G! But as you know full well, there's no
god-given correspondence. You can't just take a conjugacy class in
G and cook up an irreducible representation, or irrep. So, you've
just made my point. You've shown how mysterious these irreps actually
are!
Georg replies, Actually in some cases there *is* a nice way to
read off irreps from conjugacy classes. For example, you can
do it for the symmetric groups S_n. But, I admit you can't in
general... or at least, I don't know how.
William laughs, So, I win!
Not at all! replies Georg. First, there are lots of theorems about
groups that I can prove using representations, which you can't prove
using actions on sets. For example, nobody knows how to prove that
every group with an odd number of elements is solvable without using
the tools of linear algebra.
William nods. I admit that linear algebra is very practical. But
just give us time! I proved back in 1904 that every group of size
p^a q^b is solvable if p and q are prime. To do it, I broke down
and used linear algebra. But then, in 1972, Helmut Bender found a
proof that avoids linear algebra.
Georg said, Okay, struggle on then. So far, without using linear
algebra, nobody can even prove my famous theorem on 'Frobenius groups'.
The details don't matter here: the point is, this is a result on
group actions, which seems to need linear algebra for its proof.
But if practicality won't sway you, maybe this conceptual argument
will. My atoms are more fine-grained than yours!
What do you mean? asks William.
You can decompose any action of G into 'atoms', namely transitive
G-sets. Similarly, I can decompose any representation of G into
one of my 'atoms', namely irreps. But, there's an obvious way to
turn G-sets into representations of G, and if we do this to one of
your atoms, we don't get one of my atoms! We can usually break it down
further! So, my atoms are smaller than yours.
How does this work, exactly?
It's easy, says George, getting a bit cocky. Say you have a group
G acting on a set S. Then I can form the vector space C[S] whose elements
are formal linear combinations of elements of S. In other words, it's
the vector space having S as basis. If we're feeling sloppy we can think
of guys in C[S] as functions on S that vanish except at finitely many
points. It's better to think of them as measures on S. But anyway:
since G acts on S, it acts linearly on C[S]!
So, any G-set gives a representation of G. But, even when G acts
transitively on S, its representation on C[S] is hardly ever irreducible.
William glowers. Oh yeah?
Yeah. Suppose for example that S is finite. Then the constant functions
on S form a 1-dimensional subspace of C[S] that's invariant under the
action of G. So, at the very least, we can break C[S] into two pieces.
Well, replies William defensively, That's pretty obvious. But it's
also not such a big deal. So you can break up any my transitive G-sets
into two of your irreps, one being the 'trivial' irrep. So what???
It wouldn't be a big deal if that's *all* that ever happened, says
Georg. In fact, we can break C[S] into precisely two irreps whenever
the action of G on S is 'doubly transitive' - meaning we can send any 
*pair*
of distinct elements of S to any other using some element of G. But,
there lots of transitive actions aren't doubly transitive! And usually,
one your atoms breaks down into a *bunch* of my atoms. In fact I'd like
to show you how this works, in detail, for the symmetric groups.
Maybe next week, says William. But, I see your point. Your atoms
are more atomic than my atoms.
Georg seems to have won the argument. But, William wouldn't have
conceded the point quite so fast, if he'd thought about invariant
relations!
The point is this. Suppose we have two G-sets, say X and Y. Any
G-set map from X to Y gives an intertwining operator from C[X] to C[Y].
But, even after taking linear combinations, there are typically
plenty of intertwining operators that don't arise this way. It's
these extra intertwining operators that let us chop representations
into smaller atoms.
But where do these extra intertwining operators come from? They come
from *invariant relations* between X and Y!
And, what are these extra intertwining operators called? In some
famous special cases, like in study of modular forms, they're called
Hecke operators. In some other famous special cases, like in
the study of symmetric groups, they form algebras called Hecke
algebras.
A lot of people don't even know that Hecke operators and Hecke
algebras are two faces of the same idea: getting intertwining
operators from invariant relations. But, we'll see this is true, once
we look at some examples.
I think I'll save those for future episodes. But if you've followed
the Tale so far, you can probably stand a few extra hints of where
we're going. Recall from week250 that invariant relations between
G-sets are just spans of groupoids equipped with some extra stuff.
So, invariant relations between G-sets are just a warmup for the more
general, and simpler, theory of spans of groupoids. I said back in
week248 that spans of groupoids give linear operators. What I'm
trying to say now is that these linear operators are a massive
generalization - but also a simplification - of what people call
Hecke operators.
Finally, for students trying to learn a little basic category theory, I'd
like to cast the argument between William and Georg into the language of
categories, just to help you practice your vocabulary.
A G-set is the same as a functor
A: G -> Set
where we think of G as a 1-object category. There's a category of
G-sets, namely
hom(G,Set)
This has functors A: G -> Set as objects, and natural transformations
between these as morphisms. Usually the objects are called G-sets,
and the morphisms are called maps of G-sets.
We can also play this whole game with the category of vector spaces
replacing the category of sets. A representation of G is the same as
a functor
A: G -> Vect
As before, there's a category of such things, namely
hom(G,Vect)
This has functors A: G -> Vect as objects, and natural transformations
between these as morphisms. Now the objects are called representations
of G and the morphisms are called intertwining operators.
We could let any groupoid take the place of the group G. We could also
let any other category take the place of Set or Vect.
Part of what William and Georg were debating was: how similar are
hom(G,Set) and hom(G,Vect)? How are they related?
First of all, there's a functor
F: Set -> Vect
sending each set S to the vector space C[S] with that set as basis. So,
given an action of G on a set:
A: G -> Set
we can compose it with F and get a representation of G:
FA: G -> Vect
This kind of representation is called a permutation representation.
And, this trick gives a functor from G-sets to representations of G:
hom(G,Set) -> hom(G,Vect)
A |-> FA
If this functor were an equivalence of categories, it would have to
be essentially surjective, full and faithful. But, not every
representation of G is isomorphic to a permutation representation!
In other words, the functor
hom(G,Set) -> hom(G,Vect)
is not essentially surjective.
Moreover, not every intertwining operator between permutation
representations comes from a map between their underlying G-sets!
In other words, the functor
hom(G,Set) -> hom(G,Vect)
is not full.
But, given two different maps from one G-set to another, they give
different intertwining operators. So, at least our functor is faithful.
Maps of G-sets are a special case of invariant relations. So, to get a
category that more closely resembles hom(G,Vect), while remaining purely
combinatorial, we can replace hom(G,Set) by the category with G-sets as
objects and invariant binary relations as morphisms. This is the basic
idea of Hecke operators.
Or, even better, we can try a weak 2-category, with
groupoids over G as objects
spans of groupoids over G as morphisms
maps between spans of groupoids over G as 2-morphisms
This is where groupoidification comes into its own.
-----------------------------------------------------------------------
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twfcontents.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
===
Subject: Statistica Sinica discussion paper meeting
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
You are cordially invited to the Statistica Sinica Discussion Paper
meeting, jointly held with the 2007 Taipei International Statistical
Symposium & ICSA International Conference from June 25-27 at Academia
Sinica:
10:10am-12:10pm, June 26, 1st Conference Room, Activity Center
Modeling and Predicting Probability with Outlooks
by Mendel Fygenson, University of Southern California, USA
With invited discussions by
Cheng-Der Fuh, Academia Sinica, Taiwan
Chao Agnes Hsiung, National Health Research Institutes, Taiwan
Peter McCullagh, University of Chicago, USA
Steve Portnoy, University of Illinois, USA
For downloading the discussion paper and submitting your contributed
discussion, please visit the journal site at
http://www.stat.sinica.edu.tw/statistica/
For registration of the conference and other details, please complete
and return the registration form at
http://www.stat.sinica.edu.tw/2007symp/regist-note3.htm
The speaker and discussants will also give invited talks at the
Symposium on Recent Development of Statistics in Biological
Sciences, organized by the Department of Biostatistics and
Bioinformatics & Office of Biomedical Informatics, NHRI from June
28-29. For more details, please visit
http://www.tbi.org.tw/symposium/RDBS_2007/speakers.html
Please also let Suru Chang know (suru@nhri.org.tw) if you have
questions about registration of the NHRI symposium.
Karen Li, on behalf of the co-editors
Editorial Assistant
Statistica Sinica