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Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset \mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to $\mathfrak{g}$. Let $S(\mathfrak{h}^*)$ be the ring of polynomial functions on $\mathfrak{h}$. The Weyl group $W$ acts on $\mathfrak{h}$, and this action extends to an action of $W$ on $S(\mathfrak{h}^*)$. It is a well-known fact that the space of Weyl group invariants $S(\mathfrak{h}^*)^W$ is generated by $r$ algebraically independent homogeneous generators, where $r$ is the dimension of $\mathfrak{h}$ (equivalently, the rank of $\mathfrak{g}$). The degrees of the generators are uniquely determined, though the actual generators themselves are not.

The degrees of the generators for $S(\mathfrak{h}^*)^W$ are well-known and can be found, for example, in Humphreys' book "Reflection groups and Coxeter groups" (Section 3.7). When $\mathfrak{g}$ is of classical type (ABCD), it is also not hard to find explicit examples of generators for $S(\mathfrak{h}^*)^W$ (loc. cit. Section 3.12).

Where, if anywhere, can I find explicit examples of generators for $S(\mathfrak{h}^*)^W$ when $\mathfrak{g}$ is of exceptional type, specifically, for types $E_7$ and/or $E_8$?

I have found explicit examples for types $E_6$ and $F_4$ in a paper by Masaru Takeuchi (On Pontrjagin classes of compact symmetric spaces, J. Fac. Sci. Univ. Tokyo Sect. I 9 1962 313--328 (1962)). I have probably also come across examples for type $G_2$, though I don't recall where at this moment. But I have been unable to find anything for types $E_7$ or $E_8$.

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For $G_2$ you can look here: arxiv.org/abs/hep-th/9306062 –  José Figueroa-O'Farrill Sep 3 '10 at 12:50
Thanks for this detailed question post. This is the only place on the web where I can find definition of fundamental polynomial invariants of weyl groups, which is needed in order to understand Macdonald extension of Dyson constant term conjecture for root systems other than $A_n$. –  John Jiang Feb 7 '12 at 19:46
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6 Answers

up vote 11 down vote accepted

In Physics these are called the "Casimir operators" and googling this gives the following paper: F. Berdjis and E. Beslmüller Casimir operators for $F_4$, $E_6$, $E_7$ and $E_8$. For the case of $G_2$, see the paper in my comment above: Casimir operators for the exceptional group $G_2$ by A.M. Bincer and K. Riesselmann.

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Keeping in mind that a generating set of invariant polynomials having the required degrees is not unique, various computations have been recorded in the literature. Those I was aware of before 1990 are listed in the references to section 3.12 of my book, but there may have been others I overlooked. A later paper of interest discusses "canonical" choices of generators, with details for the classical cases as well as dihedral groups (including $G_2$):

MR1469638 (98j:13007) 13A50 (20F55), Iwasaki, Katsunori (J-KYUS), Basic invariants of finite reflection groups. J. Algebra 195 (1997), no. 2, 538–547.

Physicists usually look for very explicit expressions, though their notation and approach may be hard for mathematicians to decipher. Their interest comes from the direction of Casimir operators as Jose points out in his literature citations. But those operators live in the center of the universal enveloping algebra, which by Harish-Chandra is isomorphic to the Weyl group invariants asked about here. The complication is that expressions for Casimir operators get much more elaborate-looking in terms of the Lie algebra notation. (Also, the reflection group theory shows that polynomial invariants and degrees play a uniform role even in non-crystallographic cases like $H_3$ and $H_4$ as well as dihedral groups which are not Weyl groups.)

ADDED: I'd emphasize that writing suitable generators (Casimir operators) for the center of $U(\mathfrak{g})$ should involve a choice of PBW or other basis, though the initial approach might not start with such a basis but rather with the Killing form. However this is done (non-uniquely), it takes some care to realize from these operators a set of basic polynomial invariants for the Weyl group. The latter calculation by itself can be done much more straightforwardly, though hardly anyone has taken the trouble to write down (for example) a basic invariant polynomial in 8 variables of degree 30 for $E_8$. The 1988 paper by M.L. Mehta in Communications in Algebra seems to be a good attempt at giving a comprehensive treatment. Unfortunately, the journal itself is not so easy to access, and my own copy of the paper photographed from typescript by the journal is barely readable.

I have had less success in deciphering the physics literature, which may or may not all be mathematically reliable. In particular, I haven't yet reached any conclusions about what is in the JMP paper cited by Jose. (That journal is sometimes quite useful but can also be quite frustrating to extract information from for mathematical purposes.) My only experience has been with the literature on finite (mostly real) reflection groups and their invariants, where the degrees themselves are most important for most applications. One concrete source I should mention is the added Chapter 7 in the second edition of Grove-Benson Finite Reflection Groups (GTM 99, Springer, 1985). Their book was first developed as an advanced undergraduate text, then expanded somewhat, and gives more details than my book --- where for instance I left the computation of basic invariants for dihedral groups as an exercise.

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You've written 8 books! math.umass.edu/~jeh/pub/book.html . I am guessing that the relevant one is Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, 29, Cambridge Univ. Press, 1990 , but it might be nice to confirm this. –  David Speyer Sep 3 '10 at 14:34
Jim, but if you restrict the Casimir operators (in the centre of the universal enveloping Lie algebra) to the Cartan subalgebra, by which I mean simply setting to zero the non-Cartan generators, don't you get the polynomials guaranteed by the Harish-Chandra isomorphism? –  José Figueroa-O'Farrill Sep 3 '10 at 17:55
@Jose: I guess my point was that it's more work to get explicit Casimir operators (in terms of a given PBW basis of the Lie algebra) than to compute basic invariants for $W$ (try rank 1). There is also a subtle $\rho$-twist, so restriction to $\mathfrak{h}$ (in the PBW basis) won't usually give a homogeneous invariant. The methods of Chevalley and Harish-Chandra are less direct than just restricting invariant polynomials on $\mathfrak{g}$ to $\mathfrak{h}$. Anyway, computing Casimir operators involves some overkill and doesn't apply to all reflection groups. –  Jim Humphreys Sep 3 '10 at 19:03
@Jim: thanks for the clarification. The lack of homogeneity is something which I have encountered in the past in explicit calculations. (I did some work on Casimir algebras in the context of conformal field theory years ago.) So the upshot is that some invariant polynomials can be read off from the papers I quoted in my answer, but perhaps not in the most convenient form. –  José Figueroa-O'Farrill Sep 4 '10 at 2:08
I believe the paper by Berdjis and Beslmüller that José cites directly gives homogeneous elements of $S(\mathfrak{h^*})$, so no further restriction or $\rho$-twist is necessary. –  Christopher Drupieski Sep 4 '10 at 15:35
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Like Jim, I've had some trouble deciphering some of the older references in the physics literature. Instead, I've recently come across a paper that I probably should have found in my initial reference search. The paper "Invariant polynomials of Weyl groups and applications to the centres of universal enveloping algebras" by C.Y. Lee (Canad. J. Math., Vol. XXVI, No. 3, 1974, pp.583-592) gives for each Lie type an explicit formula for computing generators for the ring $S(\mathfrak{h})^W$. One can then apply the $W$-invariant isomorphism $S(\mathfrak{h}) \rightarrow S(\mathfrak{h}^*)$ coming from the Killing form to obtain an explicit set of generators for $S(\mathfrak{h}^*)^W$.

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Somehow this paper escaped my attention around 1989 when I was writing my book. In fact I must have had a copy of the paper (not in my files now) when I wrote a short review for Math Reviews: MR0338100 (49 #2866). Mea culpa. It's part of a 1973 Simon Fraser thesis. This non-unique C.Y. Lee had one more paper (1974), on branching rules. The E7,E8 polynomials were checked at the time for algebraic independence by an early computer. I'll give it another look now. I hope it's all correct. There seems to be no citation trail. –  Jim Humphreys Sep 11 '10 at 17:38
P.S. If Lee's paper is correct and sufficiently detailed, it probably does answer your original question. But with computations of any complexity it's always reassuring to double-check the methodology. Since the set of polynomials won't be unique, it's tricky to make direct comparisons with other literature already cited. –  Jim Humphreys Sep 11 '10 at 17:39
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In the paper “Flat Bases of Invariant Polynomials and P-matrices of E7 and E8”, V. Talamini, Journal of Mathematical Physics 51, 023520-1-023520-20 (2010 AIP), I reported explicitly the bases of invariant polynomials for the weyl groups E7 and E8, written in terms of the 7 or 8 variables, respectively (not in the article itself but in a file at FTP directory /epaps/journ_math_phys/E-JMAPAQ-51-060912/ at ftp.aip.org). Of the many bases of invariant polynomials that one can write some are called flat and in the article are reported the flat bases, but frome these one may recover any other basis. The basis transformation connecting the flat bases reported in the article with the bases suggested by Mehta in Commun. in Algebra (1988) are also given in the article. V. Talamini

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For the record, these invariants (or rather, the ideals of positive-degree invariants) also come up in the Borel presentation of the cohomology ring of the flag manifold G/B, so one can find generators whenever people have computed these rings. For instance, the preprint http://arxiv4.library.cornell.edu/pdf/0911.4793v1 gives completely explicit polynomials for E7 in Proposition 2.1 and for E8 in Lemma 2.3. I can't vouch for their correctness -- a nontrivial computational matter, as Jim remarked -- but the calculations are written out in some detail.

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@Jim, thanks for pointing out the back-referral to the Mehta paper. I originally thought this one had a more explicit list of polynomials than the others, but on closer inspection of the Lee paper, I certainly agree with you and Christopher. –  Dave Anderson Sep 10 '10 at 23:42
This approach gets very complicated, because it has a much broader agenda. But it's a useful reminder that the Borel picture of cohomology has motivated closer study of the Weyl group invariants. I also have to look into this preprint, but it has lots of heavy prerequisites and can't include non-crystallographic finite reflection groups. (It also refers back to the very short paper of M.L. Mehta mentioned earlier. The even older paper by C.Y. Lee pointed out by Christopher could be a better source. But proposed sets of basic invariants have to be shown algebraically independent.) –  Jim Humphreys Sep 11 '10 at 11:15
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Appendices 1 and 2 to the preprint http://arxiv.org/pdf/alg-geom/9202002v1.pdf by Katz-Morrison, now published in J. Alg. Geom., give explicit generating sets in type $E_6$ and $E_7$. There is also some information on basic invariants in type $E_8$ in Appendix 0, but no explicit formulae (for the obvious reason).

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