It’s based on the computer algebra package LiE, written by M. A. A. van Leeuwen, A. M. Cohen and B. Lisser in the early 90’s. They chose to implement a proprietary scripting language as a wrapper for all the fancy mathematical algorithms. While this language is useful for interactive computations and short scripts, python is more expressive and powerful — definitely what you want when exploring your favorite exceptional group.

A Fun Example

Here’s an example of using lie to do a calculation that’s near and dear to every high energy theorist’s heart. We’ll show how the 10 + 5bar + 1 representation of SU(5) contains a single standard model generation. First we’ll fire up python and import the lie module.

Python 3.1 (r31:73578, Jun 27 2009, 21:49:46)
>>> from lie import *

Let’s let g be the GUT group SU(5) (A4 in Cartan’s classifiation).

>>> g = A4; g.pprint()
'SU(5)'

We’ll let the representation r be a single GUT generation: 10 + 5bar + 1 of SU(5).

>>> r = g.fund().dual() + g.alt_tensor(2) + g.trivial()
>>> r.pprint()
'(1) + (5b) + (10)'

Here, we’ll check that SU(2)×SU(3) is a subgroup of SU(5). Then we’ll get the restriction matrix for SU(2)×SU(3) in SU(5), and add the U(1) part by hand:

>>> [h.pprint() for h in g.max_subgrps()]
['SU(4)', 'SO(5)', 'SU(2)xSU(3)']
>>> m = g.res_mat(A1*A2)
>>> m = m.transpose().concat(mat([[-2,1,-1,2]])).transpose()

Finally, we decompose r under the group SU(2)×SU(3)×U(1). The first two numbers are the SU(2) and SU(3) rep dimensions, and the third is the U(1) charge (which differs from the conventional hypercharge by a factor of 6).

>>> r.branch(A1*A2*T1, m).pprint()
'(1,1,0) + (1,1,6) + (1,3b,-4) + (1,3b,2) + (2,1,-3) + (2,3,1)'

These are indeed the representations of a generation of standard model fermions (written as left-handed Weyl spinors). In order, we have: a heavy neutrino, the positron, the up quark, the down quark, the lepton doublet, and the left-handed quark doublet. Yay!

Background

I decided I needed something like lie last summer while working on Superconformal Flavor Simplified with David Poland. Since none of the mathematical tools that physicists commonly use (e.g. Mathematica) had any routines for Lie group representation theory, I was delighted to stumble upon LiE, which implemented basically all the algorithms I wanted. However, the authors’ choice to create their own scripting language was a little silly, and definitely not future-proof. For instance, while LiE implements lists of numbers (vectors) and lists of lists of numbers (matrices), it doesn’t know anything about lists of lists of lists of numbers. This might seem insignificant, but it was frustrating for two reasons:

• Every real programming language on earth implements lists of lists of lists.
• I needed lists of lists of lists.

LiE uses a list of integers to represent the highest weight vector of a representation, so if I want lists of representations, like in a model, I need lists of lists. And if I want lists of models, I need lists of lists of lists. Turns out I was trying to write a whole paper about lists of models.

Future Work

I developed lie to the point where it was capable of doing what I needed for Superconformal Flavor Simplified. There’s definitely some debugging and restructuring to be done. For instance, the sourcecode currently includes the entire lexer and parser from the original LiE, just because I haven’t bothered to extract them from everything else (I originally took the approach of trying to modify the LiE source as little as possible). It’s a work in progress, but lie has already been useful to me, so I figure others might benefit from it too.

So check out the source, give it a whirl, and if it doesn’t do what you want, help me make it better!

Actually, the snarXiv only generates tantalizing titles and abstracts at the moment, while the arXiv delivers matching papers as well. Details of the implementation are below.^[2] I’m the author, and I don’t remember exactly why I decided to do this. I did already have the framework lying around from a previous project, and I swear I spent more time doing research last weekend than implementing snarXiv.org.

Suggested Uses for the snarXiv^[3]

• If you’re a graduate student, gloomily read through the abstracts, thinking to yourself that you don’t understand papers on the real arXiv any better.
• If you’re a post-doc, reload until you find something to work on.
• If you’re a professor, get really excited when a paper claims to solve the hierarchy problem, the little hierarchy problem, the mu problem, and the confinement problem. Then experience profound disappointment.
• If you’re a famous physicist, keep reloading until you see your name on something, then claim credit for it.
• Everyone else should play arXiv vs. snarXiv.^[4]

Context-Free Grammars

The snarXiv is based on a context free grammar (CFG) — basically a set of rules for computer-generated mad libs.^[5] Each rule in a CFG consists of a term, and a set of choices for how to make that term. The choices can contain text, or other terms, or even refer recursively to the term being defined. The CFG syntax used on the snarXiv is a collection of statements “term ::= choices”, where choices is a list of possibilities separated by “|”. Some possibilities are just text, but the ones that look like “<newterm>” are directions to go find the definition for newterm and fill it in. For instance, the following grammar

nounphrase ::= <noun> | <adj> <adj> <noun> | super <nounphrase>
noun ::= apple | pear | mailman
adj ::= smelly | chartreuse | enormous

can produce nounphrases like “apple,” “enormous smelly mailman,” or “super super smelly chartreuse mailman.” The snarxiv’s grammar is 622 lines long, and ends like this:

...
morecomments ::= <smallinteger> figures | JHEP style | Latex file
  | no figures | BibTeX | JHEP3 | typos corrected
  | <nzdigit> tables | added refs | minor changes
  | minor corrections | published in PRD
  | reference added | pdflatex
  | based on a talk given on <physicistname>'s <nzdigit>0th birthday
  | talk presented at the international <pluralphysconcept> workshop
comments ::= <smallinteger> pages | <comments>, <morecomments>

primarysubj ::= High Energy Physics - Theory (hep-th)|
                High Energy Physics - Phenomenology (hep-ph)|
secondarysubj ::= Nuclear Theory (nucl-th)|
      Cosmology and Extragalactic Astrophysics (astro-ph.CO)|
      General Relativity and Quantum Cosmology (gr-qc)|
      Statistical Mechanics (cond-mat.stat-mech)
papersubjects ::= <primarysubj> | <papersubjects>; <secondarysubj>

paper ::= <title> \\ <authors> \\ <comments> \\ <papersubjects> \\ <abstract>
...

The coolest and most natural thing to do with a CFG is exploit recursiveness as much as possible. The more recursion built in, the less predictable and richer the output. For instance, the following definition of a “space” has three rules: space, singspace, pluralspace, which refer recursively to each other in many different ways, allowing for a huge number of possibilities.

space ::= <pluralspace> | <singspace> | <mathspace>

singspace ::= a <spacetype> | a <spaceadj> <spacetype>
   | <properspacename> | <spaceadj> <properspacename>
   | <mathspace> | <mathspace>
   | a <bundletype> bundle over <space>
   | <singspace> fibered over <singspace>
   | the moduli space of <pluralspace>
   | a <spacetype> <spaceproperty>
   | the <spacepart> of <space>
   | a <group> <groupaction> of <singspace>
   | the near horizon geometry of <singspace>
pluralspace ::= <spacetype>s | <spaceadj> <spacetype>s
   | <n> copies of <mathspace>
   | <pluralspace> fibered over <space>
   | <spacetype>s <spaceproperty>
   | <bundletype> bundles over <space>
   | moduli spaces of <pluralspace>
   | <group> <groupaction>s of <pluralspace>

Of course, there’s also a danger that in a very small number of cases the output might be a little pathological. The nounphrase example above, for instance, can produce any phrase of the form “super super … super enormous pear.” The snarXiv similarly occasionally mentions QFTs living on “the moduli space of moduli spaces of moduli spaces of moduli spaces of moduli spaces of SU(3) bundles over elliptically fibered Enriques surfaces.” Too much recursion can also quickly lead to exponentially long abstracts, which are even harder to read all the way through than the usual ones on the arXiv.

The Guts

To get some actual output from the grammar definition, the most straightforward thing would be to write a script that reads in the grammar, and works its way down the tree, starting with the top term, filling in definitions recursively until it gets a block of text. Instead of using an external script, the snarXiv compiles each grammar into its own program, a technique that originated from a freshman CS project and evolved minimally from there — it’s less straightforward, not clearly better, but maybe a bit more fun. A perl script compiles the grammar file into OCaml code (snarxiv.ml):

type phrase = Str of string | Opts of phrase array array

let _ = Random.self_init ()

let randelt a = a.(Random.int (Array.length a))
let rec print phr = match phr with
  Str  s       -> print_string s
| Opts options ->
    let parts = randelt options in
    Array.iter print parts

(* Grammar definitions *)
let rec top = Opts [|
  [| paper;|];
|]

...

and comments = Opts [|
  [| smallinteger; Str " pages";|];
  [| comments; Str ", "; morecomments;|];
|]

and primarysubj = Opts [|
  [| Str "High Energy Physics - Theory (hep-th)";|];
  [| Str "High Energy Physics - Phenomenology (hep-ph)";|];
|]

and secondarysubj = Opts [|
  [| Str "Nuclear Theory (nucl-th)";|];
  [| Str "Cosmology and Extragalactic Astrophysics (astro-ph.CO)";|];
  [| Str "General Relativity and Quantum Cosmology (gr-qc)";|];
  [| Str "Statistical Mechanics (cond-mat.stat-mech)";|];
|]

and papersubjects = Opts [|
  [| primarysubj;|];
  [| papersubjects; Str "; "; secondarysubj;|];
|]

and paper = Opts [|
  [| title; Str " \\\\ "; authors; Str " \\\\ "; comments; Str " \\\\ "; papersubjects; Str " \\\\ "; abstract; Str " ";|];
|]

let _ = print top
let _ = print_string "\n"

And snarxiv.ml is now a specialized program that, when compiled and run, spits out a paper title and abstract. This setup is more elaborate than necessary, but OCaml is a lovely language for recursive structures, and the code is nice and simple. OCaml is also fast, allowing the snarXiv to generate papers even more swiftly than your favorite python script, or Ed Witten in the 80’s.

Other CFGs

A few years ago, the CFG-based CS paper generator SCIgen made a splash by getting one of their papers accepted to the conference SCI 2005. Their website has details, and links to some other random generators around the web.

1. For those who aren’t high energy physicists, and are still interested (though I can’t imagine who that would be), the “X” in arXiv or snarXiv is supposed to be a greek chi. We’re meant to pronounce them like archive (as in “archive of physics papers”) and snarchive (as in “snarky archive of physics papers”).
2. Please don’t sue me, arXiv.org, for stealing your CSS file and your beautiful color scheme. Also, Werner Heisenberg, if you’re still alive, please don’t sue me or my computer for libel.
3. If someone pretentious is annoying you, and you use the theorem generator instead, you could try something like this.
4. Also, pick up the unofficial arXiv vs. snarXiv wallpaper.
5. I first encountered these in freshman year of college in an assignment for CS51: Abstraction and Design in Computer Programming. We had to implement a CFG in LISP, and the cleverest won its author lunch at the faculty club. The eventual winner was my friend Matt Gline’s theorem generator, which has since been enhanced with LaTeX, commutative diagrams, ajax, and stuff like that.

Basically it is a result of static electricity created by friction as materials of dissimilar material strike against each other. In this case titanium/nickel blades moving through the air and dust. It occurs on the ground as well, but you don’t usually see it as much unless the aircraft is landing or taking off. The most common time is when fuel is being pumped. When large tankers are being fueled they must be grounded to prevent static electricity from discharging and creating explosions.

www.youtube.com/watch?v=bVwzJEhMmD8

“Print number 6517 and 6518 are smeared and does not define clarity.”

Another excellent find among my grandmother’s old documents was a set of hilariously informative instructions for nose-printing your dog.

When my father was young, his family owned an enormous Great Dane named Lady who turns out to have had a pedigree. Canadian National Live Stock Records show her mother’s name as “Duchess of Willowdale” and her father’s as “Dandy of Metheringham.” To register Lady herself with the Canadian Kennel Club, my grandfather had to send in a nose-print. I have no idea what the primary method for taking nose prints was, but it apparently failed, according to this letter from the Dept. of Agriculture:

                       February 6th, 1951.

Mr. W.K.G. Duffin,
R.R. 4,
Thorndale, Ont.

Dear Sir:

      We return two applications for your Great
Danes and the nose prints you provided. Print
number 6525 is incomplete as not enough of the
dogs nose was shown. Print number 6517 and 6518
are smeared and does not define clarity.

                 Yours truly,
                 CANADIAN NATIONAL LIVE STOCK RECORDS
                 R.G.T. HITCHMAN
                 DIRECTOR

Enclosed along with the note and the “incomplete” and “smeared” nose prints were the following helpful instructions…

Alternative Method of Taking Nose-Prints

For the information of those using this method of identification, we would report that the following method of procedure, in taking prints, has proven satisfactory:

1. A small amount of tube ink (see below) is placed on the smooth surface of a piece of wood, glass, or other smooth surfaced material and is spread evenly on such surface.
2. Excessive moisture is removed from the dog’s nose by cloth or other absorbent material.
3. Using tip of index finger, a small amount of ink is spread as evenly as possible across area of nose to be nose-printed. The amount of ink to be used and the extent to which the ink is spread by fingertip can best be determined by experiment.
4. Take an official nose-print form (or plain paper when experimenting) and grasp it firmly, at centre of its outer edges, by the fingertips of each hand, holding the form in a vertical position.
5. Person taking print (while dog’s head is firmly held by another person) takes position directly in front of dog, facing its nose and holding the form (still in vertical position) so that gummed side is toward holder of form.
6. Move form forward till contact is made, at centre of form, with nose, then carry outer edges of form slightly more forward so as to allow for curvature of dog’s nose.
7. As soon as print is taken, remove form by quick backward movement of hands.
8. Set print to one side until ink is thoroughly dry.

As a suggestion to those nose-printing such breeds as Boxers, Bulldogs, Pekes, etc., make fold in the form about one-third from bottom of form, and in taking print see that fold is directly opposite the lowest section of the nose to be printed. Straighten out form after print is taken.

Care should be exercised, when folding the form, to see that gummed edge folded back does not adhere to gummed section unto which it is folded.

Other Nose Print Ink Available

It has been found that excellent prints can be obtained by following the method outlined above, and using the same type of ink that was used, for nose-printing purposes, some years ago. In order to accommodate those who would wish to use such ink, arrangements have been made so that a two-ounce tube of this ink may be obtained from the Canadian Kennel Club, 667 Young Street, Toronto, Ont., on payment of 75¢. Official nose-print forms (30 for 25¢) can be obtained by forwarding order, with covering payment, to the Canadian Kennel Club. No C.O.D. orders, please.

Success

Presumably, thanks to these instructions, print number 6537 was more satisfactory.

Lady’s nose-print (via the “alternative method”)

Personally, I think it’s quite attractive. Broad strokes sweep diagonally towards the nostrils, where they fragment into little splotches and dots. Flecks of muzzle hair speckle the region below. Definitely Lady. I can see it now — my grandfather pressing Lady’s moist and inky nose to the official nose-print form. It sticks in the middle, but the outsides smear as she twists her head away, preferring to paint other things like furniture and squirrels.

Lady the Great Dane, with my father and aunt, summer 1953

Perhaps the quick brown fox jumps over the lazy dog. But probably my oxen will haul a dozen loads of gravel just as quickly.

Going through my grandmother’s old things last night, my father and aunt came across her typing textbook from secretarial school in England, 1934 to 1936. One of the exercises, about halfway through, includes a somewhat hilarious list of sentences using every letter of the alphabet.

The Clark’s College System of Rhythmic Touch Typewriting begins with an introduction to the home row, with diagrams,

The British Empire Key-board

And proceeds through a collection of exercises like this one:

Exercise VII.

(Two Fingers on Two Rows of Keys.)

(Right Hand Only.)

Upper Two Rows.

1st and 2nd fingers.

ykykykykykykykykykykykykykykykykykykykykykykyk

1st and 2nd fingers.

ukukukukukukukukukukukukukukukukukukukukukukuk

2nd and 4rd fingers.

ililililililililililililililililililililililil

3rd and 4th fingers.

o;o;o;o;o;o;o;o;o;o;o;o;o;o;o;o;o;o;o;o;o;o;o;

1st and 3rd fingers.

ululululululululululululululululululululululul

1st and 3rd fingers.

ylylylylylylylylylylylylylylylylylylylylylylyl

…

 

The combinations marked with an asterisk are those most frequently used. You must be very sure of these.

Students work gradually through more patterns, adding more and more letters, while the book admonishes “The object to aim at is perfect evenness of touch. If one letter prints more heavily than another you are typing badly,” and “Accuracy and style are everything. speed is nothing.” However, it recommends that students construct a cardboard model of a typewriter before attempting any of the exercises on an actual machine.

“The student is recommended to construct a model … upon which to practise before using the actual typewriter.”

I’m sure it was worth working through it all to get to this:

Instructions for Exercise XIX.

(Alphabetical Sentences)

We have now learnt to write the alphabet correctly, and may proceed to work on sentences which contain every letter in the alphabet in sequences other than that in which they occur in the alphabet itself.

Exercise XIX is a rather more difficult way of testing your knowledge of the key-board. Here you have sentences, each of which takes the fingers over every letter of the alphabet, but not in any recognised order.

Let the first 10 minutes of each lesson be devoted to a small portion of each of the Exercises XVII, XVIII, and XIX. The whole could be completed each week by taking three lines from each exercise at every lesson.

Great care must be taken when working through the alphabetical sentences. This will be your first acquaintance with connected matter, and you must try to preserve a uniform rate of typing.

Don’t type in spasmodic jerks—going quickly over the easy words only to slow down when the hard ones are encountered.

Each sentence must be practised until it can be done correctly, evenly and quickly.

CAPITALS.—Important.

All students, when the sentences can be properly accomplished as they stand, should practise them starting each word with a Capital, as in the case of a heading.

Exercise XIX.

(Alphabetical Sentences.)

• The quick brown fox jumps over the lazy dog.
• The boy was not lazy and jumped up quickly before the box gave way.
• My help squeezed in and joined the weavers again before six o’clock.
• They provoked a quarrel by zealously mixing a few black cats.
• You spoke about having a ten quart zinc box joined firmly forward.
• Whenever the black fox jumped the squirrel gazed very suspiciously.
• We dislike to exchange job lots of sizes varying from a quarter up.
• The black jury fixed up the question of prizes with the Government.
• Probably my oxen will haul a dozen loads of gravel just as quickly.
• The job requires extra pluck and zeal from every young wage earner.
• The judicious advocate will never forget that a good cause may be quickly lost by too much zeal.
• Dazzling jewels, pendant from her neck were thrown quite in the shadow by exceeding wit, which ever flashed from her brilliant mind.
• Old and quaint maxims filled the pages of that very curious book, on which, with quiet joy, he gazed for hours together.

The incredibly lifelike scenes are actually huge works of art, painted on the side of perfectly intact buildings.

Even that woman peering into the ruin above is not real. The paintings, which have fooled many, were created by John Pugh, who specialises in trompe l’oeil — or ‘trick of the eye’ — art.

Bobby McFerrin demonstrates the power of the pentatonic scale, using audience participation, at the event “Notes & Neurons: In Search of the Common Chorus”, from the 2009 World Science Festival, June 12, 2009.