Fake Polynomials:

The information given below is related to the calculations described in the paper On singular Calogero-Moser spaces (arXiv).

The code used to calculate the data in the table Table (3.3) of the above paper is given here (pdf).

For a quick explanation of what "fake polynomials" are, click here.

Below, for each of the exceptional, irreducible complex reflection groups G4 - G37 (as labelled in [ST]), we give its character table and the fake polynomial associated to each irreducible character of the group. We also give the remainder on division of the Poincaré polynomial of the coinvariant ring by a certain multiple of each fake polynomial (see section 3.2 of (pdf) for an explanation).

We also give the same information for some of the groups of smaller rank belonging to the infinite series G(m,p,n) here.

The data was produced using the computer algebra package MAGMA [MAG], the corresponding code is available here. This data can also be calculated using the package Chevie [CHE] which is part of GAP [GAP]. Code to do this is given below. Note that the fake polynomials for G19 take a long time to calculate (~2 hours on a desktop pc) using the magma code. Chevie will calculate them much faster.

Group Character Table Fake Polynomials Remainder on division
G4
table
fakes
remainder
G5
table
fakes
remainder
G6
table
fakes
remainder
G7
table
fakes
remainder
G8
table
fakes
remainder
G9
table
fakes
remainder
G10
table
fakes
remainder
G11
table
fakes
remainder
G12
table
fakes
remainder
G13
table
fakes
remainder
G14
table
fakes
remainder
G15
table
fakes
remainder
G16
table
fakes
remainder
G17
table
fakes
remainder
G18
table
fakes
remainder
G19
table
fakes
remainder
G20
table
fakes
remainder
G21
table
fakes
remainder
G22
table
fakes
remainder
G23
table
fakes
remainder
G24
table
fakes
remainder
G25
table
fakes
remainder
G26
table
fakes
remainder
G27
table
fakes
remainder
G28
table
fakes
remainder
G29
table
fakes
remainder
G30
table
fakes
remainder
G31
table
fakes
remainder
G32
table
fakes
remainder
G33
table
fakes
remainder
G34
table
fakes
remainder
G35
table
fakes
remainder
G36
table
fakes
remainder
G37
table
fakes
remainder

The groups G35,G36,G37,G28,G23 and G30 correspond the Weyl groups with root system E6, E7, E8, F4, H3 and H4 respectively.

 

Some examples from the series G(m,p,n)

Group G(m,p,n) Character Table Fake polynomials Remainder on divison
G(1,1,2)
table
fakes
remainder
G(2,1,2)
table
fakes
remainder
G(2,2,2)
table
fakes
remainder
G(3,1,2)
table
fakes
remainder
G(3,3,2)
table
fakes
remainder
G(4,1,2)
table
fakes
remainder
G(4,2,2)
table
fakes
remainder
G(4,4,2)
table
fakes
remainder
G(6,1,2)
table
fakes
remainder
G(6,2,2)
table
fakes
remainder
G(6,3,2)
table
fakes
remainder
G(6,6,2)
table
fakes
remainder
G(1,1,3)
table
fakes
remainder
G(2,1,3)
table
fakes
remainder
G(2,2,3)
table
fakes
remainder
G(3,1,3)
table
fakes
remainder
G(3,3,3)
table
fakes
remainder
G(4,1,3)
table
fakes
remainder
G(4,2,3)
table
fakes
remainder
G(4,4,3)
table
fakes
remainder
G(6,1,3)
table
fakes
remainder
G(6,2,3)
table
fakes
remainder
G(6,3,3)
table
fakes
remainder
G(6,6,3)
table
fakes
remainder
G(1,1,4)
table
fakes
remainder
G(2,1,4)
table
fakes
remainder
G(2,2,4)
table
fakes
remainder
G(3,1,4)
table
fakes
remainder
G(3,3,4)
table
fakes
remainder
G(4,1,4)
table
fakes
remainder
G(4,2,4)
table
fakes
remainder
G(4,4,4)
table
fakes
remainder
G(6,1,4)
table
fakes
remainder
G(6,2,4)
table
fakes
remainder
G(6,3,4)
table
fakes
remainder
G(6,6,4)
table
fakes
remainder
G(1,1,5)
table
fakes
remainder
G(2,1,5)
table
fakes
remainder
G(2,2,5)
table
fakes
remainder
G(3,1,5)
table
fakes
remainder
G(3,3,5)
table
fakes
remainder
G(4,1,5)
table
fakes
remainder
G(4,2,5)
table
fakes
remainder
G(4,4,5)
table
fakes
remainder

 

Chevie Code

To load the Chevie package type:

RequirePackage( "chevie" );
q := X( Cyclotomics );; q.name := "q";;

Given a complex reflection group G type:

FakeDegrees(G,q);

to get a list of the corresponding fake polynomials.

Then the following code allows you to divide a nonzero polynomial p by a sufficient power of t so that the degree zero coefficient of the resulting polynomial is nonzero.

PolynomialCoefficients:= function(f)
local coes,g,d,c,i;
coes := [];
g := f;
while Degree(g) > 0 do
d := Degree(g);
c := LeadingCoefficient(g);
coes[d+1] := c;
g := g - c*q^d;
od;
coes[1] := LeadingCoefficient(g);
for i in [1..Length(coes)] do
if IsBound(coes[i]) = false then
coes[i] := 0;
fi;
od;
return coes;
end;

TrailingTerm:= function(f)
local coes,p,i;
coes := PolynomialCoefficients(f);
p := PositionProperty(coes, i -> not( i = 0));
return coes[p]*q^(p-1);
end;

Given a complex reflection group G, set O := Size(G); then the following code produces a list of the Poincare polynomial of G mod the "reduced" fake polynomials.

fakedegrees := FakeDegrees(G,q);
N := Length(fakedegrees);
exponents := ReflectionDegrees(G);
PoincarePolynomial := Product(List([1..Length(exponents)], i -> ((1-q^exponents[i]) / (1-q))));
remainders := List([1..N], i -> EuclideanRemainder(TrailingTerm(fakedegrees[i])*PoincarePolynomial,fakedegrees[i]));

References:

[ST] : G.C. Shephard and J.A. Todd, Finte unitary reflection groups, Canad. J. Math., 6, 274 - 304, (1954).

[MAG] : MAGMA is described in W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput., 24, 235-265 (1997). The Magma homepage is at http://magma.maths.usyd.edu.au/magma/

[GAP] : Martin Schönert et.al. GAP -- Groups, Algorithms, and Programming -- version 3 release 4 patchlevel 4. Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 1997.

[CHE] : M. Geck, G. Hiss, F. Lübeck, G. Malle, and G. Pfeiffer. CHEVIE: A system for computing and processing generic character tables for finite groups of Lie type, Weyl groups and Hecke algebras. Appl. Algebra Engrg. Comm. Comput., 7, 175 - 210, 1996. The CHEVIE homepage is at http://www.math.rwth-aachen.de/~CHEVIE