tessel@tum.bot
Aug17-04, 11:27 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Mon, 16 Aug 2004, Marco Masi wrote:\n\n> There are so many different entropy-information measures today that I\n> would like to get somewhere a complete list of it (or at least the most\n> comprehnesive one).\n\nNot gonna happen, Marco :-(\n\nAs of a decade ago, there were tens of thousands of papers discussing\nhundreds of nominally distinct quantities called "entropies" in a huge\nvariety of contexts. (This is a genuine estimate, not a figure I made up\noff the top of my head---anyone who thinks it is too high hasn\'t looked at\nnearly enough journals/books! "MathSciNet" is the best place to start\npoking about, if your institution subscribes to this invaluable electronic\nZentralblatt, but you\'ll need to go through several iterations of adapting\nyour search terms according to what you\'ve already found.) Most of these\nhave some kind of recognizable relation to the various probabilistic\nentropies introduced by Shannon 1948, but many do not. There are many\nknown interrelationships between various of these quantities, but they are\noften rather subtle, and not always stated clearly/correctly in the\nliterature.\n\nExample: did you see the thread on "degrees of freedom"? If so, did you\nnotice the close connection with the well known combinatorial entropies\nintroduced by Boltzmann, having the form log(multinomial coefficient), and\nthus to Shannon\'s entropies (and a whole lot more)?\n\nIt turns out that in the context of general left actions by a group G,\ncertain "homogeneous G-sets" G/H, which Boltzmann called "complexions",\nform a "lattice" (in the sense of Garrett Birkhoff) with various nice\nproperties. Here, the subgroups H are nothing but the -pointwise-\nstabilizers in the sense of Galois, e.g.\n\n{x1,x2,x3}> = x1> cap x2> cap x3>\n\nwhere x> = {g in G: g.x = x}, and G/H denotes the set of left cosets of\nthe subgroup H. Similarly, of course, for right actions.\n\nA canonical example from algebraic geometry may help to quickly give some\nidea of how this works and what it means. Consider n dimensional real\nprojective space, where we can take G = GL(n+1,R) acting on one\ndimensional subspaces of R^(n+1), i.e. the projective action on the points\nof RP^n. Now the lattice can be drawn in the form of staggered vertical\n"columns", corresponding to "Young diagram" type sublattices, with the\nobvious "diagonal edges", somewhat like this:\n\no\n\\\n\\\noo\n|\\\n| \\\n| \\\no ooo\no |\\\n\\ | \\\n\\ oo ...\no\n|\\\n| \\\no ...\no\\\no \\\n...\n\nHere, each node is a "concept", a stabilizer-fixset pair in the sense of\nGalois. You can think of this lattice as listing all the geometrically\ndefined "configurations", and as organizing the information needed to\nsystematically and progressively define each configuration from simpler\nones. The edges correspond to quotienting, also to subgroups, and to\n"conditional complexions" which describe how much information we still\nneed to supply if we have already defined some simpler configuration. (I\nam not being very precise here, since for example both subgroups and\n-conjugacy classes- of subgroups play very prominent roles in the theory.)\nIn this example, the "bottom node" in the lattice is the diagonal subgroup\nD, which is analogous to the "Jacobson radical", and of course GL(n+1,R)/D\nis isomorphic to PGL(n+1,R). Notice that we can let n -> infinity,\nalthough we might then expect to obtain a significantly larger lattice.\n\nComing back to "progressive definition", one way of understanding the\nsignificance of "conditional complexions" is that, as Frobenius noticed,\nwe can "factor" motions of three points x,y,z like so: after an unknown\nelement of g has moved points around, in order to -restore- x,y,z to their\noriginal locations, we can try to proceed in stages:\n\nrestore x, while fixing x restore y, while fixing both x,y restore z.\n\nThis style of "factorization" does -not- respect group multiplication, but\nwe can fix this using "Frobenius cocyles". Another way to think about\nthese cocycles is to consider "quotients of homogeneous G-sets"\npictorially, using "Schreier diagrams", which are generalizations of\n"Cayley diagrams" and are defined in terms of the choice of some\n"generating set". Then, the Frobenius cocycles provide precisely the\ninformation needed to redraw the Schreier diagram defining the G-set X to\nsuggest (correctly) a fiber bundle over the quotient G-set X/~. In\nparticular, we always have G-epimorphisms (quotient maps)\n\nG/{x,y,z}> --->> G/{x,y}> --->> G/{x}>\n\ncorresponding to the Frobenius decomposition mentioned above. I am being\nsloppy again, but if you guess that there must be some relationship with\n"permutation puzzles" such as Rubik\'s cube, where again we have some\nchoice of generating set, you are correct! (Further relevant keywords\ninclude "wreath product", "Schreier tree", "computational group theory"\nand "cohomology of equivalence relations".)\n\nTo get part of the point here, at least according to Klein/Lie/Cartan and\ntheir heirs, you should compare the lattice above, for the projective\naction by G = GL(n+1,R), with the very different lattices we obtain for\nvarious n dimensional Kleinian geometries, e.g G = A(n) for affine\ngeometry, G = E(n) for euclidean distance geometry, etc.\n\nGoing back to general discussion of group actions and their relationship\nwith "entropies", in certain special cases (e.g. smooth Lie group\nactions, as for the examples just mentioned), it is possible to assign\n"dimensions" to the complexions, and then these dimensions behave formally\njust like Shannon\'s entropies. IOW, the edges in the lattice of concepts\ncan be assigned positive integers describing the "degrees of freedom"\ninvolved in various geometric configurations. This recovers the classic\nnotion of "degrees of freedom" as used in algebraic geometry, for example\nSchubert calculus. In the case of actions by a -finite- group, e.g.\nfinite projective groups like the finite simple group PGL(2,7), the\nlogarithm of the index [G:H] also behaves just like Shannon entropies, and\nindeed we can recover Boltzmann entropies.\n\nAs an amusing historical footnote in the great tradition of E. T. Bell, we\nhave here an immediate connection with another classical notion, the "Marx\nmatrix", so called because it was allegedly communicated by a visiting\nscholar called Marx in a letter from the mathematician Engels to the\nmathematician Burnside, who, possibly by mistake, anglicized the name\n"Marx" to "Marks". It has been said that this incident inspired James\nJoyce to write "three quarks..." and some other stuff. These\nrevolutionary endeavours ended badly, but something good came of it--- the\n"table of marks" [sic] was what Georg Frobenius was working on when he\nstumbled over something now regarded as much more important, namely the\ntheory of linear representations of finite groups and their characters! If\nyou know about this stuff, the table of marks is an analog of the\n"character table", with "conjugacy classes of subgroups" replacing\n"conjugacy classes of elements". It turns out to be more difficult to\nwork with the table of marks than with the character table, but this is\noften worth doing despite the inconvenience, so symbolic computation\npackages like GAP will compute both the character table and the table of\nmarks for a given sufficiently small permutation group, in case you want\nto play around with your favorite finite group actions (dihedral and\ncyclic groups are good places to start). BTW, if you have encountered the\n"Moebius inversion formula", the Marx matrix provides a classical example\nof Moebius inversion, as generalized by Rota.\n\nIf I might mention a third special case of great interest: for\n"oligomorphic actions" we have a natural topology ("topology of pointwise\nconvergence") on the symmetric group, and a natural connection with the\nso-called "first order homogeneous relational theories" discussed in\nmathematical logic. (Keywords include "Fraisse theory", "Polya counting",\n"Cameron cycle index", "Joyal cycle index" of a "combinatorial\nspecies/structor", "molecular decomposition", and "wreath product"--- this\nis all part of the general categorical theory take on the "generating\nfunction" approach to combinatorics.) A canonical example here is the\nfamous "universal random graph" of Erdos and Renyi. BTW, Renyi may\nalready be known to you for his pioneering work on Shannonian information\ntheory; his "Renyi entropies" are among the better known "entropic\nquantities", and have found employment in "chaotic dynamics"/"nonlinear\nscience".\n\nWell, as you can guess because both "symmetry" and "information" are such\n"universally relevant" concepts in mathematics, there\'s a lot more to say,\nand in fact we have already extensively discussed some of this stuff here\nin the past, e.g. you can look for a thread called "symmetry and\ninformation" for the basics on the Galois/Klein/Lie/Cartan way of\nunderstanding "information", as well as for threads on "concept theory",\nrandom graphs, invariant theory/Coxeter stuff/reflection groups/Young\ndiagrams/Schubert calculus. See also various of John Baez\'s "This Week"\npostings, including some recent ones discussing the ideal/variety\ncorrespondence in algebraic geometry, which is closely analogous to the\nsubgroup/fixset correspondence which appears above.\n\n> Browsing on the internet I couldn\'t find much.\n\nDidn\'t you find this site?\n\nhttp://www.math.psu.edu/gunesch/entropy.html\n\nAs the welcome page says: "The purpose of these pages is to promote the\nappreciation, understanding, and applications of entropy in its many\nforms"--- i.e., just what you want, I should think. See in particular the\n"suggested reading".\n\n(As you have probably guessed, "tessel@tum" is a dumbed-down student\nversion of the original author of these pages; they are now hosted by a\ncolleague at PSU.)\n\nHmm.... it seems the "journal page" has disappeared from Roland\'s pages,\nbut I assume you know that the IEEE Transactions on Information Theory is\na gold mine of information on information :-/ Another journal you should\ndefinitely look over is Ergodic Theory and Dynamical Systems. These\njournal focuses upon quantities defined in terms of probability theory,\nbut they are one of the obvious places to start (especially since the\nquantitity you asked about is evidently formulated in terms of probability\ntheory).\n\nHmmm... a zillion years ago I had something called something like\n"Mathematical Concepts of Entropy", but by now this would by now be\nhopelessly out of date. I am pretty sure, however, that noone else has\neven -attempted- anything like a comprehensive survey, because about a\ndecade ago I did enough reading (some of which was summarized in that\npaper) to see that even a modestly annotated bibliography would be book\nlength, and anything like a genuine survey would need a multivolume\nmonograph. By now, the situation must be even more daunting!\n\nAnyway, questions about the variety of entropy measures which have been\nintroduced since Shannon 1948 arise in this group on a regular basis, so\nwe\'ve discussed some of them extensively on many previous occasions. So,\nyou can also google for some of these past threads.\n\n> Can someone give me a link for this? Or alternatively can someone tell\n> me if the following corresponds to some information theoretic object:\n> \\frac{1}{1-\\beta} * [(1+ \\alpha \\log \\sum_{i} p_{i}^{q} )^{\\alpha} - 1]\n> with \\alpha, \\beta, q real parametres.\n\nCan you rewrite this in more ASCII-friendly format? Bearing in mind that\n(p^q)^alpha = p^[q alpha}, do you mean this?:\n\n1 [ ( { } ) ]\n------- [ 1 + beta log ( sum_i { p_i^(q alpha) - 1 } ) ]\n1- beta [ ( { } ) ]\n\nCan you give some context? Is the sum finite? Are the p_i probabilities?\nIf so, of what? Do alpha beta serve here as parameters? IOW, are you in\neffect defining a family of quantities H[alpha,beta;X,p], where X is a\nfinite set and p is a probability distribution on X? Do alpha, beta arise\nas Lagrange multipliers?\n\nAnswers to these and similar questions would almost certainly help us\npoint you toward some relevant stuff. Alas, by the time you can reply to\nthis post, I myself will probably be incommunicado, but you can look to\nsee if I reappear here in October and ask again then, if you are still\ninterested but if no-one else here was able to help you.\n\n"T. Essel" (hiding somewhere in cyberspace)\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Mon, 16 Aug 2004, Marco Masi wrote:
> There are so many different entropy-information measures today that I
> would like to get somewhere a complete list of it (or at least the most
> comprehnesive one).
Not gonna happen, Marco :-(
As of a[/itex] decade ago, there were tens of thousands of papers discussing
hundreds of nominally distinct quantities called "entropies" in a huge
variety of contexts. (This is a genuine estimate, not a figure I made up
off the top of my head---anyone who thinks it is too high hasn't looked at
nearly enough journals/books! "MathSciNet" is the best place to start
poking about, if your institution subscribes to this invaluable electronic
Zentralblatt, but you'll need to go through several iterations of adapting
your search terms according to what you've already found.) Most of these
have some kind of recognizable relation to the various probabilistic
entropies introduced by Shannon 1948, but many do not. There are many
known interrelationships between various of these quantities, but they are
often rather subtle, and not always stated clearly/correctly in the
literature.
Example: did you see the thread on "degrees of freedom"? If so, did you
notice the close connection with the well known combinatorial entropies
introduced by Boltzmann, having the form log(multinomial coefficient), and
thus to Shannon's entropies (and a whole lot more)?
It turns out that in the context of general left actions by a group G,
certain "homogeneous G-sets" G/H, which Boltzmann called "complexions",
form a "lattice" (in the sense of Garrett Birkhoff) with various nice
properties. Here, the subgroups H are nothing but the -pointwise-
stabilizers in the sense of Galois, e.g.
{x1,x2,x3}> = x1> cap x2> cap x3>
where x> = {g in G: g.x = x}, and G/H denotes the set of left cosets of
the subgroup H. Similarly, of course, for right actions.
A canonical example from algebraic geometry may help to quickly give some
idea of how this works and what it means. Consider n dimensional real
projective space, where we can take G = GL(n+1,R) acting on one
dimensional subspaces of R^(n+1), i.e. the projective action on the points
of RP^n. Now the lattice can be drawn in the form of staggered vertical
"columns", corresponding to "Young diagram" type sublattices, with the
obvious "diagonal edges", somewhat like this:
o
\
\
oo
|\
| \
| \
o ooo
o |\
\ | \\ oo ...
o
|\
| \
o ...
o\
o \
...
Here, each node is a "concept", a stabilizer-fixset pair in the sense of
Galois. You can think of this lattice as listing all the geometrically
defined "configurations", and as organizing the information needed to
systematically and progressively define each configuration from simpler
ones. The edges correspond to quotienting, also to subgroups, and to
"conditional complexions" which describe how much information we still
need to supply if we have already defined some simpler configuration. (I
am not being very precise here, since for example both subgroups and
-conjugacy classes- of subgroups play very prominent roles in the theory.)
In this example, the "bottom node" in the lattice is the diagonal subgroup
D, which is analogous to the "Jacobson radical", and of course GL(n+1,R)/D
is isomorphic to PGL(n+1,R). Notice that we can let n -> infinity,
although we might then expect to obtain a significantly larger lattice.
Coming back to "progressive definition", one way of understanding the
significance of "conditional complexions" is that, as Frobenius noticed,
we can "factor" motions of three points x,y,z like so: after an unknown
element of g has moved points around, in order to -restore- x,y,z to their
original locations, we can try to proceed in stages:
restore x, while fixing x restore y, while fixing both x,y restore z.
This style of "factorization" does -not- respect group multiplication, but
we can fix this using "Frobenius cocyles". Another way to think about
these cocycles is to consider "quotients of homogeneous G-sets"
pictorially, using "Schreier diagrams", which are generalizations of
"Cayley diagrams" and are defined in terms of the choice of some
"generating set". Then, the Frobenius cocycles provide precisely the
information needed to redraw the Schreier diagram defining the G-set X to
suggest (correctly) a fiber bundle over the quotient G-set X/~. In
particular, we always have G-epimorphisms (quotient maps)
G/{x,y,z}> --->> G/{x,y}> --->> G/{x}>
corresponding to the Frobenius decomposition mentioned above. I am being
sloppy again, but if you guess that there must be some relationship with
"permutation puzzles" such as Rubik's cube, where again we have some
choice of generating set, you are correct! (Further relevant keywords
include "wreath product", "Schreier tree", "computational group theory"
and "cohomology of equivalence relations".)
To get part of the point here, at least according to Klein/Lie/Cartan and
their heirs, you should compare the lattice above, for the projective
action by G = GL(n+1,R), with the very different lattices we obtain for
various n dimensional Kleinian geometries, e.g G = A(n) for affine
geometry, G = E(n) for euclidean distance geometry, etc.
Going back to general discussion of group actions and their relationship
with "entropies", in certain special cases (e.g. smooth Lie group
actions, as for the examples just mentioned), it is possible to assign
"dimensions" to the complexions, and then these dimensions behave formally
just like Shannon's entropies. IOW, the edges in the lattice of concepts
can be assigned positive integers describing the "degrees of freedom"
involved in various geometric configurations. This recovers the classic
notion of "degrees of freedom" as used in algebraic geometry, for example
Schubert calculus. In the case of actions by a -finite- group, e.g.
finite projective groups like the finite simple group PGL(2,7), the
logarithm of the index [G:H] also behaves just like Shannon entropies, and
indeed we can recover Boltzmann entropies.
As an amusing historical footnote in the great tradition of E. T. Bell, we
have here an immediate connection with another classical notion, the "Marx
matrix", so called because it was allegedly communicated by a visiting
scholar called Marx in a letter from the mathematician Engels to the
mathematician Burnside, who, possibly by mistake, anglicized the name
"Marx" to "Marks". It has been said that this incident inspired James
Joyce to write "three quarks..." and some other stuff. These
revolutionary endeavours ended badly, but something good came of it--- the
"table of marks" [sic] was what Georg Frobenius was working on when he
stumbled over something now regarded as much more important, namely the
theory of linear representations of finite groups and their characters! If
you know about this stuff, the table of marks is an analog of the
"character table", with "conjugacy classes of subgroups" replacing
"conjugacy classes of elements". It turns out to be more difficult to
work with the table of marks than with the character table, but this is
often worth doing despite the inconvenience, so symbolic computation
packages like GAP will compute both the character table and the table of
marks for a given sufficiently small permutation group, in case you want
to play around with your favorite finite group actions (dihedral and
cyclic groups are good places to start). BTW, if you have encountered the
"Moebius inversion formula", the Marx matrix provides a classical example
of Moebius inversion, as generalized by Rota.
If I might mention a third special case of great interest: for
"oligomorphic actions" we have a natural topology ("topology of pointwise
convergence") on the symmetric group, and a natural connection with the
so-called "first order homogeneous relational theories" discussed in
mathematical logic. (Keywords include "Fraisse theory", "Polya counting",
"Cameron cycle index", "Joyal cycle index" of a "combinatorial
species/structor", "molecular decomposition", and "wreath product"--- this
is all part of the general categorical theory take on the "generating
function" approach to combinatorics.) A canonical example here is the
famous "universal random graph" of Erdos and Renyi. BTW, Renyi may
already be known to you for his pioneering work on Shannonian information
theory; his "Renyi entropies" are among the better known "entropic
quantities", and have found employment in "chaotic dynamics"/"nonlinear
science".
Well, as you can guess because both "symmetry" and "information" are such
"universally relevant" concepts in mathematics, there's a lot more to say,
and in fact we have already extensively discussed some of this stuff here
in the past, e.g. you can look for a thread called "symmetry and
information" for the basics on the Galois/Klein/Lie/Cartan way of
understanding "information", as well as for threads on "concept theory",
random graphs, invariant theory/Coxeter stuff/reflection groups/Young
diagrams/Schubert calculus. See also various of John Baez's "This Week"
postings, including some recent ones discussing the ideal/variety
correspondence in algebraic geometry, which is closely analogous to the
subgroup/fixset correspondence which appears above.
> Browsing on the internet I couldn't find much.
Didn't you find this site?
http://www.math.psu.edu/gunesch/entropy.html
As the welcome page says: "The purpose of these pages is to promote the
appreciation, understanding, and applications of entropy in its many
forms"--- i.e., just what you want, I should think. See in particular the
"suggested reading".
(As you have probably guessed, "tessel@tum" is a dumbed-down student
version of the original author of these pages; they are now hosted by a
colleague at PSU.)
Hmm.... it seems the "journal page" has disappeared from Roland's pages,
but I assume you know that the IEEE Transactions on Information Theory is
a gold mine of information on information :-/ Another journal you should
definitely look over is Ergodic Theory and Dynamical Systems. These
journal focuses upon quantities defined in terms of probability theory,
but they are one of the obvious places to start (especially since the
quantitity you asked about is evidently formulated in terms of probability
theory).
Hmmm... a zillion years ago I had something called something like
"Mathematical Concepts of Entropy", but by now this would by now be
hopelessly out of date. I am pretty sure, however, that noone else has
even -attempted- anything like a comprehensive survey, because about a
decade ago I did enough reading (some of which was summarized in that
paper) to see that even a modestly annotated bibliography would be book
length, and anything like a genuine survey would need a multivolume
monograph. By now, the situation must be even more daunting!
Anyway, questions about the variety of entropy measures which have been
introduced since Shannon 1948 arise in this group on a regular basis, so
we've discussed some of them extensively on many previous occasions. So,
you can also google for some of these past threads.
> Can someone give me a link for this? Or alternatively can someone tell
> me if the following corresponds to some information theoretic object:
> [itex]\frac{1}{1-\beta} * [(1+ \alpha \log \sum_{i} p_{i}^{q} )^{\alpha} - 1]
> with \alpha, \beta, q real parametres.
Can you rewrite this in more ASCII-friendly format? Bearing in mind that
(p^q)^\alpha = p^[q \alpha}, do you mean this?:
1 [ ( { } ) ]
------- [ 1 + \beta log ( sum_i { p_i^(q \alpha) - 1 } ) ]1- \beta [ ( { } ) ]
Can you give some context? Is the sum finite? Are the p_i probabilities?
If so, of what? Do \alpha \beta serve here as parameters? IOW, are you in
effect defining a family of quantities H[\alpha,\beta;X,p], where X is a
finite set and p is a probability distribution on X? Do \alpha, \beta arise
as Lagrange multipliers?
Answers to these and similar questions would almost certainly help us
point you toward some relevant stuff. Alas, by the time you can reply to
this post, I myself will probably be incommunicado, but you can look to
see if I reappear here in October and ask again then, if you are still
interested but if no-one else here was able to help you.
"T. Essel" (hiding somewhere in cyberspace)
tessel@tum.bot
Aug18-04, 04:08 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Tue, 17 Aug 2004 I wrote:\n\n> Consider n dimensional real projective space, where we can take G =\n> GL(n+1,R) acting on one dimensional subspaces of R^(n+1), i.e. the\n> projective action on the points of RP^n. Now the lattice\n^\nstabilizer-fixset concept\n\n> lattice can be drawn in the form of staggered vertical "columns",\n> corresponding to "Young diagram" type sublattices, with the obvious\n> "diagonal edges", somewhat like this:\n\nOops, I ommitted the first node! Sorry---let me try again and say just a\nbit more. For the case of three dimensional real projective geometry, the\nconcept lattice begins like this:\n\no\n\\ 3\n\\\n*\n\\ 3\n\\\n**\n|\\\n1 | \\ 3\n| \\\n* ***\n* |\\\n\\ 1| \\\n3 \\ ** ...\n*\n|\\\n1 | \\\n* ...\n*\\\n* \\\n...\n\nwhere the reader will be able to draw in the fourth column after I say a\ntiny bit more. In this diagram,\n\n"o" denotes the concept "empty set",\n\n"*" denotes the concept "point",\n\n"**" denotes the concept "two distinct points",\n\n"*"\n* denotes the concept "line"\n\n"**"\n* denotes the concept "a line together with a point not on the line"\n\nand so forth. The integers labeling the edges show that in order to\nspecify the motion of a point in RP^3 under a projective transformation\nyou must name three real numbers, in order to specify the motion of two\npoints you must name six reals, to specify the motion of a line you must\nname seven reals, etc. The fourth and final column is the usual lattice\nof "four-asterisk" Young diagrams, and you connect this to the bit drawn\nabove in the obvious way using diagonal edges. If you draw it in (as a\ncheck, identify the node in the fourth column which corresponds to a pair\nof skew lines in RP^3) and then add up all the integers along any downward\npath starting at the top node, you should get fifteen, the dimension of\nPGL(4,R), the group of projective transformations on RP^3.\n\nNote that you need to name an extra real number to specify the motion of\nall points on a line L (naming seven real numbers in all) because while\nspecifying the motion of two distinct points P,P\' on L (recall that this\nrequires naming just six real numbers) specifies the -setwise- motion of\nthe line, in projective geometry you still have "one degree of freedom" in\nmoving the points on the line L -while keeping two points on L fixed-.\nIOW, the conditional complexion representing "the possible motions of L if\nyou know the motions of P and P\'" is one-dimensional.\n\nContrast this with three-dimensional affine geometry, where specifying the\nmotion of two points on L suffices to determine the motion of every point\non L. Klein would say that this reflects the fact that affine geometry is\n"more rigid" than projective geometry; euclidean distance geometry is in\nturn "more rigid" than affine geometry. The three concept lattices\n(euclidean, affine, projective) are very different; I worked these all out\nin detail in the past thread called "symmetry and information", but the\nassidious reader should have no difficulty in working them out now on a\nbit of scrap paper. Following Klein, we obtain a whole hierarchy of\n"three-dimensional" geometries in this way.\n\nRemember that a major point here is that the integer labels, which enable\nyou to keep track of "degrees of freedom", are just the pale shadow of the\nunderlying objects of study, the complexions and conditional complexions,\nwhich are homogeneous spaces which -directly represent- the various spaces\nof possible "motions" associated with our Kleinian geometries. The nice\nformal behavior of the entropies is an immediate consequence of the nice\nformal behavior of the underlying complexions. But for some purposes,\nkeeping track of only the dimensions ("degrees of freedom") isn\'t\nenough--- you need the complexions themselves. For example, dimensions\nalone can\'t possibly tell the difference between some closed Lie subgroup\nH and a discrete supergroup of H, c.f. "proper motions" versus "improper\nmotions". Less trivially, consider (for an action by a discrete group\nsuch as PGL(4,3), the Z_3 analog of the Lie group we were just discussing)\nthe restoration problem for some permutation puzzle.\n\n"T. Essel" (hiding somewhere in cyberspace)\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Tue, 17 Aug 2004 I wrote:
> Consider n dimensional real projective space, where we can take G =
> GL(n+1,R) acting on one dimensional subspaces of R^(n+1), i.e. the
> projective action on the points of RP^n. Now the lattice
^
stabilizer-fixset concept
> lattice can be drawn in the form of staggered vertical "columns",
> corresponding to "Young diagram" type sublattices, with the obvious
> "diagonal edges", somewhat like this:
Oops, I ommitted the first node! Sorry---let me try again and say just a
bit more. For the case of three dimensional real projective geometry, the
concept lattice begins like this:
o
\ 3
\
*
\ 3
\
**
|\
1 | \ 3| \* **** |\\ 1| \3 \ ** ...
*
|\
1 | \
* ...
*\
* \
...
where the reader will be able to draw in the fourth column after I say a
tiny bit more. In this diagram,
"o" denotes the concept "empty set",
"*" denotes the concept "point",
"**" denotes the concept "two distinct points",
"*"
* denotes the concept "line"
"**"
* denotes the concept "a line together with a point not on the line"
and so forth. The integers labeling the edges show that in order to
specify the motion of a point in RP^3 under a projective transformation
you must name three real numbers, in order to specify the motion of two
points you must name six reals, to specify the motion of a line you must
name seven reals, etc. The fourth and final column is the usual lattice
of "four-asterisk" Young diagrams, and you connect this to the bit drawn
above in the obvious way using diagonal edges. If you draw it in (as a
check, identify the node in the fourth column which corresponds to a pair
of skew lines in RP^3) and then add up all the integers along any downward
path starting at the top node, you should get fifteen, the dimension of
PGL(4,R), the group of projective transformations on RP^3.
Note that you need to name an extra real number to specify the motion of
all points on a line L (naming seven real numbers in all) because while
specifying the motion of two distinct points P,P' on L (recall that this
requires naming just six real numbers) specifies the -setwise- motion of
the line, in projective geometry you still have "one degree of freedom" in
moving the points on the line L -while keeping two points on L fixed-.
IOW, the conditional complexion representing "the possible motions of L if
you know the motions of P and P'" is one-dimensional.
Contrast this with three-dimensional affine geometry, where specifying the
motion of two points on L suffices to determine the motion of every point
on L. Klein would say that this reflects the fact that affine geometry is
"more rigid" than projective geometry; euclidean distance geometry is in
turn "more rigid" than affine geometry. The three concept lattices
(euclidean, affine, projective) are very different; I worked these all out
in detail in the past thread called "symmetry and information", but the
assidious reader should have no difficulty in working them out now on a
bit of scrap paper. Following Klein, we obtain a whole hierarchy of
"three-dimensional" geometries in this way.
Remember that a major point here is that the integer labels, which enable
you to keep track of "degrees of freedom", are just the pale shadow of the
underlying objects of study, the complexions and conditional complexions,
which are homogeneous spaces which -directly represent- the various spaces
of possible "motions" associated with our Kleinian geometries. The nice
formal behavior of the entropies is an immediate consequence of the nice
formal behavior of the underlying complexions. But for some purposes,
keeping track of only the dimensions ("degrees of freedom") isn't
enough--- you need the complexions themselves. For example, dimensions
alone can't possibly tell the difference between some closed Lie subgroup
H and a discrete supergroup of H, c.f. "proper motions" versus "improper
motions". Less trivially, consider (for an action by a discrete group
such as PGL(4,3), the Z_3 analog of the Lie group we were just discussing)
the restoration problem for some permutation puzzle.
"T. Essel" (hiding somewhere in cyberspace)
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