Talk:Littelmann path model

I consider this to be a low priority article on the WP Math scale and am will not engage in tags-of-edit. I'll just list some of the salient features of the last version of the article that I had worked on, which have been subsequently discarded/replaced by Mathsci (in spite of his request for help?) Arcfrk (talk) 23:28, 11 September 2008 (UTC)

• Littlewood path model is usually formulated in terms of Lie algebras and modules over Lie algebras; it has no connection with Lie groups (compact or not) other than through the Lie correspondence
• In this context, it is important to note that modules L(λ) for which character formulas apply are finite-dimensional (and for other modules, the Lie correspondence breaks down)
• "Combinatorial model" is a common term that is more descriptive than "combinatorial device" and alike
• Mention right up in the first sentence that the model is named after Peter Littelmann
• The definition of L(λ) is, unsurprisingly, contained in the article highest weight module; it is that article, and not various (random?) other places on wikipedia dealing with simple modules and characters, that should be linked
• The theory of standard monomials is usually attributed to Lakshmibai, Musili (whose name got subsequently dropped) and Seshadri
• If the Lie algebras are denoted by fraktur letters like ${\mathfrak {g}}$ then the special Lie algebra should be ${\mathfrak {sl}}_{n}$
• "Levi component of a parabolic subalgebra" is commonly called "Levi subalgebra" in this context ("Levi component" implies that it is viewed as a factor)
• The Weyl character formula is the single most important formula in the subject, and should be explicitly quoted by its standard name, not masked behind a weasly expression (all other formulas mentioned right after it are, in fact, its corollaries); whether it was Weyl, Brauer, or even Elie Cartan or Killing who first found it is irrelevant for this article (it can be discussed at the main article devoted to it)
• I am familiar with adjectives "subtraction-free" and "combinatorial" for the type of formulas discussed here; "free of overcounting" is almost certainly OR
• The following description of the tensor product decomposition problem is far more complete and accurate than its substitute:
  • For two simple highest weight modules with dominant highest weights μ and ν, find the decomposition of their tensor product into simple summands(the generalized Littlewood–Richardson rule).
• Carefully thought through structure, including the division of the first section into "History and motivations" and "Connection with branching laws and tensor product decompositions"; I also think that general branching laws should be thoroughly discussed in a single place on wikipedia, perhaps, in a dedicated article (but not this one)
• Painstaking copyedit, so that the links use en-dashes rather than hyphens in situations like Kac–Moody algebras and Littlewood–Richardson rule, in accordance with MOS
• Copyedit for precise terminology ("weight multiplicity" as opposed to just "multiplicity", etc)

Arcfrk's comments

I incorporated all your reasonable proposals in the article: all were unsourced. It involved working simultaneously with 3 browser screens. In particular there is a very careful footnote with a precise citation referring to the work of Berenstein and Zelevinsky. What you wrote was imprecise, unsourced and not directly relevant to the article. Do you not not agree that just adding an unsourced sentence like this

. In the late 1980s and early 1990s, Arkady Berenstein and Andrey Zelevinsky found combinatorial expressions for the weight, tensor product, and branching multiplicities in terms of lattice points in certain convex lattice polytopes in certain cases.

is not a great way to edit wikipedia. Their main article appeared in 2001 and is now included in an extended footnote. Without sources it is WP:OR, an essay by Arcfrk. Mathsci (talk) 06:55, 12 September 2008 (UTC)

• Littlewood path model is usually formulated in terms of Lie algebras and modules over Lie algebras; it has no connection with Lie groups (compact or not) other than through the Lie correspondence

Who is Littlewood? Every complex semisimple Lie algebra is the complexification of the real Lie algebra of a compact semisimple Lie group. Almost any serious book on the subject goes through this. See Serre, Varadarajan, Hochschild, Fulton & Harris, Bourbaki, Dieudonne, Weyl. In the case of the branching rule, it's much easier to say branching to a closed subgroup containing a maximal torus. Of course it could be that these facts are not part of Arcfrk's expertise. D. E. Littlewood's book on representation theory contains no reference to Lie algebras, only classical groups, some compact; the eponymous rule is mentioned there; a complete proof was given much later by Lascoux and Schutzenberger.

• In this context, it is important to note that modules L(λ) for which character formulas apply are finite-dimensional (and for other modules, the Lie correspondence breaks down)

In Arcfrk's version, L(λ) appears without having been defined. The same applied to ${\mathfrak {g}}$. I made the change to gothic letters, which is normal in Lie algebra theory.

• "Combinatorial model" is a common term that is more descriptive than "combinatorial device" and alike

However it is not a model of the representation, that is misleading. The Lie algebra does not act on the rational vector space. The operators e_α and f_α, neither mentioned nor defined in any prior version of the article, are not directly related to the Serre generators of the Lie algebra. You put combinatorial model into the lede as a redlink: not such a smart idea.

• Mention right up in the first sentence that the model is named after Peter Littelmann

Thanks. that was an oversight in the rejigging of the lede and not deliberate. The model is often just called the path model, e.g. in the Seminaire Bourbaki by his former colleague Olivier Mathieu.

• The definition of L(λ) is, unsurprisingly, contained in the article highest weight module; it is that article, and not various (random?) other places on wikipedia dealing with simple modules and characters, that should be linked

Notation has to be defined in the article. This is Kac's notation for highest weight integrable module or representation. There is no standard notation in the literature or on wikipedia.

• The theory of standard monomials is usually attributed to Lakshmibai, Musili (whose name got subsequently dropped) and Seshadri

Littelmann only cites the articles of Lakshmibai and Seshadri in the sources as his motivation. Pointless to include a third author if he does not mention them. Until I included standard monomial bases, they did not appear in this article. This is not an Arcfrk WP:OR essay.

• If the Lie algebras are denoted by fraktur letters like ${\mathfrak {g}}$ then the special Lie algebra should be ${\mathfrak {sl}}_{n}$

Pedantry.

• "Levi component of a parabolic subalgebra" is commonly called "Levi subalgebra" in this context ("Levi component" implies that it is viewed as a factor)

Littelman's original use of Levi subalgebra was careless short hand. It is the Levi component of a parabolic subalgebra; and corrsponds to a maximal rank closed connected subgroup in the corresponding compact group picture. There is no such terminology in the case of symmetrisable Kac-Moody algebras.

• The Weyl character formula is the single most important formula in the subject, and should be explicitly quoted by its standard name, not masked behind a weasly expression (all other formulas mentioned right after it are, in fact, its corollaries); whether it was Weyl, Brauer, or even Elie Cartan or Killing who first found it is irrelevant for this article (it can be discussed at the main article devoted to it)

Brauer's 1925 thesis contained the character formula for the orthogonal groups, prior to Weyl's papers. This is made clear in the cited source, the second edition of Weyl's classical groups. Similarly the character formula for U(N), SU(N) or GL(N) is due to Schur in his 1901 thesis. What on earth are you talking about now?

• I am familiar with adjectives "subtraction-free" and "combinatorial" for the type of formulas discussed here; "free of overcounting" is almost certainly OR

Overcounting gets 13 hits on mathscinet, subtraction-free gets 11 hits.

• The following description of the tensor product decomposition problem is far more complete and accurate than its substitute:
  • For two simple highest weight modules with dominant highest weights μ and ν, find the decomposition of their tensor product into simple summands(the generalized Littlewood–Richardson rule).

Littelmann's precise rule, placed by me later in the article, is far more accurate than this.

• Carefully thought through structure, including the division of the first section into "History and motivations" and "Connection with branching laws and tensor product decompositions"; I also think that general branching laws should be thoroughly discussed in a single place on wikipedia, perhaps, in a dedicated article (but not this one)

The history section was garbled and wrong: it was a WP:OR essay obtained by juggling around the new sentences written by me abd including a source-less reference to Berenstein and Zelevinsky, irrelevant to the article. It gave a faulty idea of the genesis of crystal bases or the reason for Littelmann's own work.

• Painstaking copyedit, so that the links use en-dashes rather than hyphens in situations like Kac–Moody algebras and Littlewood–Richardson rule, in accordance with MOS

Why spoil Michael Hardy's fun?

• Copyedit for precise terminology ("weight multiplitity" as opposed to just "multiplicity", etc)

Usually clear from context.

Commentary added to unsigned comments by Arcfrk. Please sign your comments in future.

The article prior to my addition of correct mathematical content was one of the worst mathematics articles I've seen so far. I shall give four pieces of personal advice to Arcfrk on how to improve his mathematical contributions on wikipedia:

• Try not to pick fights about trivialities.
• Never add or discuss content without citing sources - that is WP:OR.
• Try to be more generous towards other editors who have clearly made major mathematical improvements to articles.
• Try not to act as a guru when dealing with other mathematicians.

Mathsci (talk) 06:17, 12 September 2008 (UTC)

Obviously, you are incapable of objective judgment — I don't see any benefit in arguing; I've simply restored the original (signed!) version of my comments. Arcfrk (talk) 22:13, 13 September 2008 (UTC)

"Obviously"? Cite your sources please. Pretty much everything you have mentioned is contradicted by the references of Weyl, Littlewood, Sundaram and King. Mathsci (talk) 01:14, 14 September 2008 (UTC)