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Hello,

Is there any software for calculating Green polynomials (of type A)? Or, at least, where can I find tables of Green polynomials? Also, I would be interested in some formulas for Green polynomials in simple cases.

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3 Answers

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I am not really an expert but it looks like the Morita paper Decomposition of Green polynomials of type A and Springer modules for hooks and rectangles could be of some help here. In particular, at page 483 one finds an explicit formula for these polynomials for one of the simplest special cases. This paper also gives an explicit formula for the Green polynomials for an arbitrary hook (see Proposition 8).

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Macdonald's book Symmetric Functions and Hall Polynomials has a section on Green polynomials. They are easy to compute using Stembridge's SF package for Maple, available at http://www.math.lsa.umich.edu/~jrs/maple.html#SF. For example, to compute (in Macdonald's notation) $X_{(2,1)}^\lambda$, write

Zee:=mu->zee(mu,0,t):

add_basis(HL,Zee):

toHL(p2*p1);

The output is $$ (-1+t^3)HL_{1,1,1}+tHL_{2,1}+HL_3, $$ so $X_{(2,1)}^{(1,1,1)}=t^3-1$, $X_{(2,1)}^{(2,1)}=t$, $X_{(2,1)}^{(3)}=1$.

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I'm not a specialist either but am aware of some specific sources. There is a lot of related literature, sometimes with examples, as well as algorthmic methods (Green, Lusztig, Shoji, and others): see R.W. Carter's 1985 book Finite Groups of Lie Type. I'm not sure what is currently available in the way of software, but some finite group theorists studying character theory have developed sophisticated programs like CHEVIE.

One of the earliest computer-assisted projects was carried out at Warwick by W.M. Beynon and N. Spaltenstein, resulting in "Computer Centre Report No. 23" (1982), The computation of Green functions of finite Chevalley groups of type $E_n (n=6,7,8)$. My own copy of this 173 page mimeographed document (consisting mainly of tables) was recently borrowed for a few hours by a visitor to make his own copy locally, since it's long been out of print.

ADDED: For type A the original source is J.A. Green's 1955 paper in Trans. AMS, which includes some small examples. The combinatorial framework is further developed in I.G. Macdonald's book Symmetric Functions and Hall Polynomials (2nd ed., 1993); see especially III.7, with examples, and Chapter IV.

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