Let G be a connected reductive group defined over the algebraic closure of a finite field 𝔽q, with corresponding Frobenius automorphism F, or more generally let F be an isogeny of G such that a power is a Frobenius (this covers the Suzuki and Ree groups).
If T is an F-stable maximal torus of G, and B is a (not necessarily F-stable) Borel subgroup containing T, we define the Deligne-Lusztig variety XB={ gB∈G/B | gB∩ F(gB) ≠∅}. This variety has a natural action of GF on the left, so the corresponding Deligne-Lusztig virtual module ∑i (-1)i Hic(XB,ℚl) also. The character of this virtual module is the Deligne-Lusztig character RTG(1); the notation reflects the fact that one can prove that this character does not depend on the choice of B. Actually, this character is parameterized by an F-conjugacy class of W: if T0⊂B0 is an F-stable pair, there is an unique w∈ W=NG(T0)/T0 such that the triple (T,B,F) is G-conjugate to (T0,B0,wF). In this case we denote Rw for RTG(1); it depends only on the F-class of w.
The unipotent characters of GF are the irreducible constituents of the Rw. In a similar way that the unipotent classes are a building block for describing the conjugacy classes of a reductive group, the unipotent characters are a building block for the irreducible characters of a reductive group. They can be parameterized by combinatorial data that Lusztig has attached just to the coset Wφ, where φ is the finite order automorphism of X(T0) such that F=qφ. Thus, from the viewpoint of CHEVIE, they are objects combinatorially attached to a Coxeter coset.
A subset of the unipotent characters, the principal series unipotent characters, can be described in an elementary way. They are the constituents of R1, or equivalently the characters of the virtual module defined by the cohomology of XB0, which is the discrete variety (G/B0)F; the virtual module reduces to the actual module ℚl[(G/B0)F]. Thus the Deligne-Lusztig induction RT0G(1) reduces to Harish-Chandra induction, defined as follows: let P=U⋊L be an F-stable Levi decomposition of an F-stable parabolic subgroup of G. Then the Harish-Chandra induced RLG of a character χ of LF is the character IndPFGF~χ, where ~χ is the lift to PF of χ via the quotient PF/UF=LF; Harish-Chandra induction is a particular case of Lusztig induction, which is defined when P is not F-stable using the variety XU={ gU∈G/U | gU∩ F(gU) ≠∅}, and gives for an LF-module a virtual GF-module. Like ordinary induction, these functors have adjoint functors going from representations of GF to representations (resp. virtual representations) of LF called Harish-Chandra restriction (resp. Lusztig restriction).
The commuting algebra of GF-endomorphisms of RT0G(1) is an Iwahori-Hecke algebra for Wφ, with parameters which are some powers of q; they are all equal to q when Wφ=W. Thus principal series unipotent characters correspond to characters of Wφ.
To understand the decomposition of Deligne-Lusztig characters, and thus unipotent characters, is is useful to introduce another set of class functions which are parameterized by irreducible characters of the coset Wφ. If χ is such a character, we define the associated almost character by: Rχ=|W|-1∑w∈ Wχ(wφ) Rw. The reason to the name is that these class function are close to irreducible characters: they satisfy ⟨ Rχ, Rψ⟩GF=δχ,ψ; for the linear and unitary group they are actually unipotent characters (up to sign in the latter case). They are in general sum (with rational coefficients) of a small number of unipotent characters in the same Lusztig family (see Families of unipotent characters). The degree of Rχ is a polynomial in q equal to the fake degree of the character χ of Wφ (see Functions for Reflection cosets).
We now describe the parameterization of unipotent characters when Wφ=W, thus when the coset Wφ identifies with W (the situation is similar but a bit more difficult to describe in general). The (rectangular) matrix of scalar products ⟨ ρ, Rχ⟩GF, when characters of W and unipotent characters are arranged in the right order, is block-diagonal with rather small blocks which are called Lusztig families.
For the characters of W a family F corresponds to a block of the Hecke algebra over a ring called the Rouquier ring. To F Lusztig associates a small group Γ (not bigger than (ℤ/2)n, or Si for i ≤ 5) such that the unipotent characters in the family are parameterized by the pairs (x,θ) taken up to Γ-conjugacy, where x∈Γ and θ is an irreducible character of CΓ(x). Further, the elements of F themselves are parameterized by a subset of such pairs, and Lusztig defines a pairing between such pairs which computes the scalar product ⟨ ρ, Rχ⟩GF. For more details see DrinfeldDouble.
A second parameterization of unipotent character is via Harish-Chandra series. A character is called cuspidal if all its proper Harish-Chandra restrictions vanish. There are few cuspidal unipotent characters (none in linear groups, and at most one in other classical groups). The GF-endomorphism algebra of an Harish-Chandra induced ℝLFGFλ, where λ is a cuspidal unipotent character turns out to be a Hecke algebra associated to the group WGF(LF):=NGF(L)/L, which turns out to be a Coxeter group. Thus another parameterization is by triples (L,λ,φ), where λ is a cuspidal unipotent character of LF and φ is an irreducible character of the relative group WGF(LF). Such characters are said to belong to the Harish-Chandra series determined by (L,λ).
A final piece of information attached to unipotent characters is the eigenvalues of Frobenius. Let Fδ be the smallest power of the isogeny F which is a split Frobenius (that is, Fδ is a Frobenius and φδ=1). Then Fδ acts naturally on Deligne-Lusztig varieties and thus on the corresponding virtual modules, and commutes to the action of GF; thus for a given unipotent character ρ, a submodule of the virtual module which affords ρ affords a single eigenvalue μ of Fδ. Results of Lusztig and Digne-Michel show that this eigenvalue is of the form qaδλρ where 2a∈ℤ and λρ is a root of unity which depends only on ρ and not the considered module. This λρ is called the eigenvalue of Frobenius attached to ρ. Unipotent characters in the Harish-Chandra series of a pair (L,λ) have the same eigenvalue of Frobenius as λ.
CHEVIE contains table of all this information, and can compute Harish-Chandra and Lusztig induction of unipotent characters and almost characters. We illustrate the information on some examples:
gap> W:=CoxeterGroup("G",2);
CoxeterGroup("G",2)
gap> uc:=UnipotentCharacters(W);
UnipotentCharacters( G2 )
gap> Display(uc);
Unipotent characters for G2
gamma | Deg(gamma) FakeDegree Fr(gamma) Label
________________________________________________________
phi{1,0} | 1 1 1
phi{1,6} | q^6 q^6 1
phi{1,3}' | 1/3qP3P6 q^3 1 (1,r)
phi{1,3}'' | 1/3qP3P6 q^3 1 (g3,1)
phi{2,1} | 1/6qP2^2P3 qP8 1 (1,1)
phi{2,2} | 1/2qP2^2P6 q^2P4 1 (g2,1)
G2[-1] | 1/2qP1^2P3 0 -1 (g2,eps)
G2[1] | 1/6qP1^2P6 0 1 (1,eps)
G2[E3] |1/3qP1^2P2^2 0 E3 (g3,E3)
G2[E3^2] |1/3qP1^2P2^2 0 E3^2 (g3,E3^2)
The first column gives the name of the unipotent character; the first 6 are in the principal series so are named according to the corresponding characters of W. The last 4 are cuspidal, and named by the corresponding eigenvalue of Frobenius, which is displayed in the fourth column. In general the names of the unipotent characters come from their parameterization by Harish-Chandra series; in addition, for classical groups, they are associated to symbols.
The first two characters are each in a family by themselves. The last eight are in a family associated to the group Γ=S3: the last column shows the parameters (x,θ). The second column shows the degree of the unipotent characters, which is transformed by the Lusztig Fourier matrix of the third column, which gives the degree of the corresponding almost character, or equivalently the fake degree of the corresponding character of W.
One can get more information on the Lusztig Fourier matrix of the big family by asking
gap> Display(uc.families[1]);
D(S3)
label |eigen
________________________________________________________
(1,1) | 1 1/6 1/2 1/3 1/3 1/6 1/2 1/3 1/3
(g2,1) | 1 1/2 1/2 0 0 -1/2 -1/2 0 0
(g3,1) | 1 1/3 0 2/3 -1/3 1/3 0 -1/3 -1/3
(1,r) | 1 1/3 0 -1/3 2/3 1/3 0 -1/3 -1/3
(1,eps) | 1 1/6 -1/2 1/3 1/3 1/6 -1/2 1/3 1/3
(g2,eps) | -1 1/2 -1/2 0 0 -1/2 1/2 0 0
(g3,E3) | E3 1/3 0 -1/3 -1/3 1/3 0 2/3 -1/3
(g3,E3^2) | E3^2 1/3 0 -1/3 -1/3 1/3 0 -1/3 2/3
One can do computations with individual unipotent characters. Here we construct the Coxeter torus, and then the identity character of this torus as a unipotent character.
gap> W:=CoxeterGroup("G",2);
CoxeterGroup("G",2)
gap> T:=ReflectionCoset(ReflectionSubgroup(W,[]),EltWord(W,[1,2]));
(q^2-q+1)
gap> u:=UnipotentCharacter(T,1);
[(q^2-q+1)]=<>
Then here are two ways to construct the Deligne-Lusztig character associated to the Coxeter torus:
gap> LusztigInduction(W,u);
[G2]=<phi{1,0}>+<phi{1,6}>-<phi{2,1}>+<G2[-1]>+<G2[E3]>+<G2[E3^2]>
gap> v:=DeligneLusztigCharacter(W,[1,2]);
[G2]=<phi{1,0}>+<phi{1,6}>-<phi{2,1}>+<G2[-1]>+<G2[E3]>+<G2[E3^2]>
gap> Degree(v);
q^6 + q^5 - q^4 - 2*q^3 - q^2 + q + 1
gap> v*v;
6
The last two lines ask for the degree of v, then for the scalar product of v with itself.
Finally we mention that CHEVIE can also provide unipotent characters of Spetses, as defined in BMM14. An example:
gap> Display(UnipotentCharacters(ComplexReflectionGroup(4)));
Unipotent characters for G4
gamma | Deg(gamma) FakeDegree Fr(gamma) Label
______________________________________________________________
phi{1,0} | 1 1 1
phi{1,4} | -ER(-3)/6q^4P"3P4P"6 q^4 1 1.-E3^2
phi{1,8} | ER(-3)/6q^4P'3P4P'6 q^8 1 -1.E3^2
phi{2,5} | 1/2q^4P2^2P6 q^5P4 1 1.E3^2
phi{2,3} |(3+ER(-3))/6qP"3P4P'6 q^3P4 1 1.E3^2
phi{2,1} |(3-ER(-3))/6qP'3P4P"6 qP4 1 1.E3
phi{3,2} | q^2P3P6 q^2P3P6 1
Z3:2 | -ER(-3)/3qP1P2P4 0 E3^2 E3.E3^2
Z3:11 | -ER(-3)/3q^4P1P2P4 0 E3^2 E3.-E3
G4 | -1/2q^4P1^2P3 0 -1 -E3^2.-1
UnipotentCharacters(W)
W should be a Coxeter group, a Coxeter Coset or a Spetses. The function gives back a record containing information about the unipotent characters of the associated algebraic group (or Spetses). This contains the following fields:
group:
charNames:
charSymbols:
harishChandra:
levi:l such that L corresponds to ReflectionSubgroup(W,l).
cuspidalName:
eigenvalue:
relativeType:
parameterExponents:
charNumbers:
families:
gap> W:=CoxeterGroup("Bsym",2);
CoxeterGroup("Bsym",2)
gap> WF:=CoxeterCoset(W,(1,2));
2Bsym2
gap> uc:=UnipotentCharacters(W);
UnipotentCharacters( Bsym2 )
gap> Display(uc);
Unipotent characters for Bsym2
gamma |Deg(gamma) FakeDegree Fr(gamma) Label
__________________________________________________
11. | 1/2qP4 q^2 1 +,-
1.1 | 1/2qP2^2 qP4 1 +,+
.11 | q^4 q^4 1
2. | 1 1 1
.2 | 1/2qP4 q^2 1 -,+
B2 | 1/2qP1^2 0 -1 -,-
gap> uc.harishChandra[1];
rec(
levi := [ ],
relativeType := [ rec(series := "B",
indices := [ 1, 2 ],
rank := 2) ],
eigenvalue := 1,
qEigen := 0,
parameterExponents := [ 1, 1 ],
charNumbers := [ 1, 2, 3, 4, 5 ],
cuspidalName := "" )
gap> uc.families[2];
Family("012",[1,2,5,6])
gap> Display(uc.families[2]);
012
label |eigen +,- +,+ -,+ -,-
________________________________
+,- | 1 1/2 1/2 -1/2 -1/2
+,+ | 1 1/2 1/2 1/2 1/2
-,+ | 1 -1/2 1/2 1/2 -1/2
-,- | -1 -1/2 1/2 -1/2 1/2
98.2 Operations for UnipotentCharacters
CharNames:
gap> uc:=UnipotentCharacters(CoxeterGroup("G",2));
UnipotentCharacters( G2 )
gap> CharNames(uc);
[ "phi{1,0}", "phi{1,6}", "phi{1,3}'", "phi{1,3}''", "phi{2,1}",
"phi{2,2}", "G2[-1]", "G2[1]", "G2[E3]", "G2[E3^2]" ]
gap> CharNames(uc,rec(TeX:=true));
[ "\\phi_{1,0}", "\\phi_{1,6}", "\\phi_{1,3}'", "\\phi_{1,3}''",
"\\phi_{2,1}", "\\phi_{2,2}", "G_2[-1]", "G_2[1]", "G_2[\\zeta_3]",
"G_2[\\zeta_3^2]" ]
Display:Display, a field items specifies which
columns are displayed. The possible values are
"n0":
"Name":
"Degree":
"FakeDegree":
"Eigenvalue":
"Symbol":
"Family":
"Signs":
The default value is
items:=["Name","Degree","FakeDegree","Eigenvalue","Family"]
This can be changed by setting the variable UnipotentCharactersOps.items
which holds this default value. In addition if the field byFamily is set,
the characters are displayed family by family instead of in index order.
Finally, the field chars can be set, indicating which characters are to be
displayed in which order.
gap> W:=CoxeterGroup("B",2);
CoxeterGroup("B",2)
gap> uc:=UnipotentCharacters(W);
UnipotentCharacters( B2 )
gap> Display(uc);
Unipotent characters for B2
gamma |Deg(gamma) FakeDegree Fr(gamma) Label
_____________________________________________
11. | 1/2qP4 q^2 1 +,-
1.1 | 1/2qP2^2 qP4 1 +,+
.11 | q^4 q^4 1
2. | 1 1 1
.2 | 1/2qP4 q^2 1 -,+
B2 | 1/2qP1^2 0 -1 -,-
gap> Display(uc,rec(byFamily:=true));
Unipotent characters for B2
gamma |Deg(gamma) FakeDegree Fr(gamma) Label
_____________________________________________
*.11 | q^4 q^4 1
_____________________________________________
11. | 1/2qP4 q^2 1 +,-
*1.1 | 1/2qP2^2 qP4 1 +,+
.2 | 1/2qP4 q^2 1 -,+
B2 | 1/2qP1^2 0 -1 -,-
_____________________________________________
*2. | 1 1 1
gap> Display(uc,rec(items:=["n0","Name","Symbol"]));
Unipotent characters for B2
n^0 |gamma Symbol
____________________
1 | 11. (12,0)
2 | 1.1 (02,1)
3 | .11 (012,12)
4 | 2. (2,)
5 | .2 (01,2)
6 | B2 (012,)
UnipotentCharacter(W,l)
Constructs an object representing the unipotent character of the algebraic group associated to the Coxeter group or Coxeter coset W which is specified by l. There are 3 possibilities for l: if it is an integer, the l-th unipotent character of W is returned. If it is a string, the unipotent character of W whose name is l is returned. Finally, l can be a list of length the number of unipotent characters of W, which specifies the coefficient to give to each.
gap> W:=CoxeterGroup("G",2);
CoxeterGroup("G",2)
gap> u:=UnipotentCharacter(W,7);
[G2]=<G2[-1]>
gap> v:=UnipotentCharacter(W,"G2[E3]");
[G2]=<G2[E3]>
gap> w:=UnipotentCharacter(W,[1,0,0,-1,0,0,2,0,0,1]);
[G2]=<phi{1,0}>-<phi{1,3}''>+2<G2[-1]>+<G2[E3^2]>
98.4 Operations for Unipotent Characters
+:
-:
*:We go on from examples of the previous section:
gap> u+v;
[G2]=<G2[-1]>+<G2[E3]>
gap> w-2*u;
[G2]=<phi{1,0}>-<phi{1,3}''>+<G2[E3^2]>
gap> w*w;
7
Degree:
gap> Degree(w);
q^5 - q^4 - q^3 - q^2 + q + 1
gap> Degree(u+v);
(5/6)*q^5 + (-1/2)*q^4 + (-2/3)*q^3 + (-1/2)*q^2 + (5/6)*q
String and Print:CHEVIE.PrintUniChars. It is a record; if the field
short is bound (the default) they are printed in a compact form. If the
field long is bound, they are printed one character per line:
gap> CHEVIE.PrintUniChars:=rec(long:=true);
rec(
long := true )
gap> w;
[G2]=
<phi{1,0}> 1
<phi{1,6}> 0
<phi{1,3}'> 0
<phi{1,3}''> -1
<phi{2,1}> 0
<phi{2,2}> 0
<G2[-1]> 2
<G2[1]> 0
<G2[E3]> 0
<G2[E3^2]> 1
gap> CHEVIE.PrintUniChars:=rec(short:=true);;
Frobenius( WF ):WF is a Coxeter coset associated to the Coxeter
group W, the function Frobenius(WF) returns a function which does the
corresponding automorphism on the unipotent characters
gap> W:=CoxeterGroup("D",4);WF:=CoxeterCoset(W,(1,2,4));
CoxeterGroup("D",4)
3D4
gap> u:=UnipotentCharacter(W,2);
[D4]=<11->
gap> Frobenius(WF)(u);
[D4]=<.211>
gap> Frobenius(WF)(u,-1);
[D4]=<11+>
UnipotentDegrees(W,q)
Returns the list of degrees of the unipotent characters of the finite reductive group (or Spetses) with Weyl group (or Spetsial reflection group) W, evaluated at q.
gap> W:=CoxeterGroup("G",2);
CoxeterGroup("G",2)
gap> q:=Indeterminate(Rationals);;q.name:="q";;
gap> UnipotentDegrees(W,q);
[ q^0, q^6, (1/3)*q^5 + (1/3)*q^3 + (1/3)*q,
(1/3)*q^5 + (1/3)*q^3 + (1/3)*q, (1/6)*q^5 + (1/2)*q^4 + (2/3)*q^
3 + (1/2)*q^2 + (1/6)*q, (1/2)*q^5 + (1/2)*q^4 + (1/2)*q^2 + (1/
2)*q, (1/2)*q^5 + (-1/2)*q^4 + (-1/2)*q^2 + (1/2)*q,
(1/6)*q^5 + (-1/2)*q^4 + (2/3)*q^3 + (-1/2)*q^2 + (1/6)*q,
(1/3)*q^5 + (-2/3)*q^3 + (1/3)*q, (1/3)*q^5 + (-2/3)*q^3 + (1/3)*q ]
For a non-rational Spetses, Indeterminate(Cyclotomics) would be more
appropriate.
98.6 Ennola
Ennola(W)
Let W be an irreducible spetsial reflection group or coset, and z the
generator of the center of W, viewed as a root of unity. A property
checked case-by case is that, for a unipotent character γ with
polynomial generic degree \degγ(q) of the spets attached to W
(this spets is a finite reductive group, for W a Weyl group, in which
case z=-1 if -1 is in W), \degγ(zq) is equal to ±\deg
En(γ)(q) for another unipotent character En(γ), the Ennola
transform of γ. The function returns the permutation-with-signs done
by En on the unipotent degrees (as a permutation-with signs of
[1..Size(UnipotentCharacters(W))]). The argument W must be irreducible.
The permutation-with-signs is not uniquely determined by the degrees since two of them may be equal, but is uniquely determined by some additional axioms that we do not recall here.
gap> Ennola(RootDatum("3D4"));
(3,-4)(5,-5)(6,-6)(7,-8)
gap> Ennola(ComplexReflectionGroup(14));
(2,43,-14,16,41,34)(3,35,40,18,-11,42)(4,-37,25,-17,-26,-36)(5,-6,-79)\
(7,-7)(8,-74)(9,-73)(10,-52,13,31,-50,29)(12,53,15,32,-51,-30)(19,71,7\
0,21,67,68,20,69,72)(22,-39,27,-33,-28,-38)(23,24,-66,-23,-24,66)(44,4\
6,49,-44,-46,-49)(45,48,47,-45,-48,-47)(54,-63,-55,-57,62,-56)(58,-65,\
-59,-61,64,-60)(75,-77)(76,-76)(78,-78)
The last example shows that it may happen that the order of z-Ennola (here 18) is greater than the order of z (here 6); this is related to the presence of irrationalities in the character table of the spetsial Hecke algebra of W.
CycPolUnipotentDegrees(W)
Taking advantage that the degrees of unipotent characters of the finite
reductive group (or Spetses) with Weyl group (or Spetsial reflection group)
W are products of cyclotomic polynomials, this function returns these
degrees as a list of CycPols (see Cyclotomic polynomials).
gap> W:=CoxeterGroup("G",2);
CoxeterGroup("G",2)
gap> CycPolUnipotentDegrees(W);
[ 1, q^6, 1/3qP3P6, 1/3qP3P6, 1/6qP2^2P3, 1/2qP2^2P6, 1/2qP1^2P3,
1/6qP1^2P6, 1/3qP1^2P2^2, 1/3qP1^2P2^2 ]
DeligneLusztigCharacter(W,w)
This function returns the Deligne-Lusztig character RTG(1) of the algebraic group G associated to the Coxeter group or Coxeter coset W. The torus T can be specified in 3 ways: if w is an integer, it represents the w-th conjugacy class (or φ-conjugacy class for a coset) of W. Otherwise w can be a Coxeter word or a Coxeter element, and it represents the class (or φ-class) of that element.
gap> W:=CoxeterGroup("G",2);
CoxeterGroup("G",2)
gap> DeligneLusztigCharacter(W,3);
[G2]=<phi{1,0}>-<phi{1,6}>-<phi{1,3}'>+<phi{1,3}''>
gap> DeligneLusztigCharacter(W,W.1);
[G2]=<phi{1,0}>-<phi{1,6}>-<phi{1,3}'>+<phi{1,3}''>
gap> DeligneLusztigCharacter(W,[1]);
[G2]=<phi{1,0}>-<phi{1,6}>-<phi{1,3}'>+<phi{1,3}''>
gap> DeligneLusztigCharacter(W,[1,2]);
[G2]=<phi{1,0}>+<phi{1,6}>-<phi{2,1}>+<G2[-1]>+<G2[E3]>+<G2[E3^2]>
AlmostCharacter(W,i)
This function returns the i-th almost unipotent character of the
algebraic group G associated to the Coxeter group or Coxeter coset W.
If χ is the i-th irreducible character of W, the i-th almost
character is Rχ=W-1∑w∈ Wχ(w) RTwG(1), where
Tw is the maximal torus associated to the conjugacy class (or
φ-conjugacy class for a coset) of w.
gap> W:=CoxeterGroup("B",2);
CoxeterGroup("B",2)
gap> AlmostCharacter(W,3);
[B2]=<.11>
gap> AlmostCharacter(W,1);
[B2]=1/2<11.>+1/2<1.1>-1/2<.2>-1/2<B2>
LusztigInduction(W,u)
u should be a unipotent character of a parabolic subcoset of the Coxeter coset W. It represents a unipotent character λ of a Levi L of the algebraic group G attached to W. The program returns the Lusztig induced RLG(λ).
gap> W:=CoxeterGroup("G",2);;
gap> T:=CoxeterSubCoset(CoxeterCoset(W),[],W.1);
(q-1)(q+1)
gap> u:=UnipotentCharacter(T,1);
[(q-1)(q+1)]=<>
gap> LusztigInduction(CoxeterCoset(W),u);
[G2]=<phi{1,0}>-<phi{1,6}>-<phi{1,3}'>+<phi{1,3}''>
gap> DeligneLusztigCharacter(W,W.1);
[G2]=<phi{1,0}>-<phi{1,6}>-<phi{1,3}'>+<phi{1,3}''>
LusztigRestriction(R,u)
u should be a unipotent character of a parent Coxeter coset W of which R is a parabolic subcoset. It represents a unipotent character γ of the algebraic group G attached to W, while R represents a Levi subgroup L. The program returns the Lusztig restriction * RLG(γ).
gap> W:=CoxeterGroup("G",2);;
gap> T:=CoxeterSubCoset(CoxeterCoset(W),[],W.1);
(q-1)(q+1)
gap> u:=DeligneLusztigCharacter(W,W.1);
[G2]=<phi{1,0}>-<phi{1,6}>-<phi{1,3}'>+<phi{1,3}''>
gap> LusztigRestriction(T,u);
[(q-1)(q+1)]=4<>
gap> T:=CoxeterSubCoset(CoxeterCoset(W),[],W.2);
(q-1)(q+1)
gap> LusztigRestriction(T,u);
[(q-1)(q+1)]=0
LusztigInductionTable(R,W)
R should be a parabolic subgroup of the Coxeter group W or a parabolic
subcoset of the Coxeter coset W, in each case representing a Levi
subgroup L of the algebraic group G associated to W. The function
returns a table (modeled after InductionTable, see InductionTable)
representing the Lusztig induction RLG between unipotent
characters.
gap> W:=CoxeterGroup("B",3);;
gap> t:=Twistings(W,[1,3]);
[ ~A1xA1<3>.(q-1), ~A1xA1<3>.(q+1) ]
gap> Display(LusztigInductionTable(t[2],W));
Lusztig Induction from ~A1xA1<3>.(q+1) to B3
|11,11 11,2 2,11 2,2
___________________________
111. | 1 -1 -1 .
11.1 | -1 . 1 -1
1.11 | . . -1 .
.111 | -1 . . .
21. | . . . .
1.2 | 1 -1 . 1
2.1 | . 1 . .
.21 | . . . .
3. | . . . 1
.3 | . 1 1 -1
B2:2 | . . 1 -1
B2:11 | 1 -1 . .
DeligneLusztigLefschetz(h)
Here h is an element of a Hecke algebra associated to a Coxeter group W which itself is associated to an algebraic group G. By results of Digne-Michel, for g∈GF, the number of fixed points of Fm on the Deligne-Lusztig variety associated to the element wφ of the Coxeter coset Wφ, have, for m sufficiently divisible, the form ∑χ χqm(Twφ)Rχ(g) where χ runs over the irreducible characters of Wφ, where Rχ is the corresponding almost character, and where χqm is a character value of the Hecke algebra H(Wφ,qm) of Wφ with parameter qm. This expression is called the Lefschetz character of the Deligne-Lusztig variety. If we consider qm as an indeterminate x, it can be seen as a sum of unipotent characters with coefficients character values of the generic Hecke algebra H(Wφ,x).
The function DeligneLusztigLefschetz takes as argument a Hecke element
and returns the corresponding Lefschetz character. This is defined on the
whole of the Hecke algebra by linearity. The Lefschetz character of various
varieties related to Deligne-Lusztig varieties, like their completions or
desingularisation, can be obtained by taking the Lefschetz character at
various elements of the Hecke algebra.
gap> W:=CoxeterGroup("A",2);;
gap> q:=X(Rationals);;q.name:="q";;
gap> H:=Hecke(W,q);
Hecke(A2,q)
gap> T:=Basis(H,"T");
function ( arg ) ... end
gap> DeligneLusztigLefschetz(T(1,2));
[A2]=<111>-q<21>+q^2<3>
gap> DeligneLusztigLefschetz((T(1)+T())*(T(2)+T()));
[A2]=q<21>+(q^2+2q+1)<3>
The last line shows the Lefschetz character of the Samelson-Bott desingularisation of the Coxeter element Deligne-Lusztig variety.
We now show an example with a coset (corresponding to the unitary group).
gap> H:=Hecke(CoxeterCoset(W,(1,2)),q^2);
Hecke(2A2,q^2)
gap> T:=Basis(H,"T");
function ( arg ) ... end
gap> DeligneLusztigLefschetz(T(1));
[2A2]=-<11>-q<2A2>+q^2<2>
PermutationOnUnipotents(W,aut)
W is a reflection group or reflection coset representing a finite reductive group GF, and aut is an automorphism of GF (for W a permutation group, this can be given as a permutation of the roots). The function returns the permutation of the unipotent characters of GF induced by aut. This makes sense for Spetsial complex reflection groups and is implemented for them.
gap> WF:=RootDatum("3D4");
3D4
gap> PermutationOnUnipotents(Group(WF),WF.phi);
( 1, 7, 2)( 8,12, 9)
98.15 Families of unipotent characters
The blocks of the (rectangular) matrix ⟨ Rχ,ρ⟩GF when χ runs over Irr(W) and ρ runs over the unipotent characters, are called the Lusztig families. When G is split and W is a Coxeter group they correspond on the Irr(W) side to two-sided Kazhdan-Lusztig cells --- for split Spetses they correspond to Rouquier blocks of the Spetsial Hecke algebra. The matrix of scalar products ⟨ Rχ,ρ⟩GF can be completed to a square matrix ⟨ Aρ',ρ⟩GF where Aρ' are the characteristic functions of character sheaves on GF; this square matrix is called the Fourier matrix of the family.
The UnipotentCharacters record in CHEVIE contains a field .families,
a list of family records containing information on each family, including
the Fourier matrix. Here is an example.
gap> W:=CoxeterGroup("G",2);;
gap> uc:=UnipotentCharacters(W);
UnipotentCharacters( G2 )
gap> uc.families;
[ Family("D(S3)",[5,6,4,3,8,7,9,10]), Family("C1",[1]),
Family("C1",[2]) ]
gap> f:=last[1];
Family("D(S3)",[5,6,4,3,8,7,9,10])
gap> Display(f);
D(S3)
label |eigen
________________________________________________________
(1,1) | 1 1/6 1/2 1/3 1/3 1/6 1/2 1/3 1/3
(g2,1) | 1 1/2 1/2 0 0 -1/2 -1/2 0 0
(g3,1) | 1 1/3 0 2/3 -1/3 1/3 0 -1/3 -1/3
(1,r) | 1 1/3 0 -1/3 2/3 1/3 0 -1/3 -1/3
(1,eps) | 1 1/6 -1/2 1/3 1/3 1/6 -1/2 1/3 1/3
(g2,eps) | -1 1/2 -1/2 0 0 -1/2 1/2 0 0
(g3,E3) | E3 1/3 0 -1/3 -1/3 1/3 0 2/3 -1/3
(g3,E3^2) | E3^2 1/3 0 -1/3 -1/3 1/3 0 -1/3 2/3
gap> f.charNumbers;
[ 5, 6, 4, 3, 8, 7, 9, 10 ]
gap> CharNames(uc){f.charNumbers};
[ "phi{2,1}", "phi{2,2}", "phi{1,3}''", "phi{1,3}'", "G2[1]",
"G2[-1]", "G2[E3]", "G2[E3^2]" ]
The Fourier matrix is obtained by Fourier(f); the field f.charNumbers
holds the indices of the unipotent characters which are in the family. We
obtain the list of eigenvalues of Frobenius for these unipotent characters
by Eigenvalues(f). The Fourier matrix and vector of eigenvalues satisfy
the properties of fusion data, see below. The field f.charLabels is
what is displayed in the column labels when displaying the family. It
contains labels naturally attached to lines of the Fourier matrix. In the
case of reductive groups, the family is always attached to the
DrinfeldDouble of a small finite group and the .charLabels come from
this construction.
98.16 Family
Family(f [, charNumbers [, opt]])
This function creates a new family in two possible ways.
In the first case f is a string which denotes a family known to CHEVIE.
Examples are "S3", "S4", "S5" which denote the family obtained as the
Drinfeld double of the symmetric group on 3,4,5 elements, or "C2" which
denotes the Drinfeld double of the cyclic group of order 2.
In the second case f is already a family record.
The other (optional) arguments add information to the family record defined
by the first argument. If given, the second argument becomes the field
.charNumbers. If given, the third argument opt is a record whose fields
are added to the resulting family record.
If opt has a field signs, this field should be a list of 1 and -1,
and then the Fourier matrix is conjugated by the diagonal matrix of those
signs. This is used in Spetses to adjust the matrix to the choice of signs
of unipotent degrees.
gap> Display(Family("C2"));
C2
label |eigen
___________________________________
(1,1) | 1 1/2 1/2 1/2 1/2
(g2,1) | 1 1/2 1/2 -1/2 -1/2
(1,eps) | 1 1/2 -1/2 1/2 -1/2
(g2,eps) | -1 1/2 -1/2 -1/2 1/2
gap> Display(Family("C2",[4..7],rec(signs:=[1,-1,1,-1])));
C2
label |eigen signs
_________________________________________
(1,1) | 1 1 1/2 -1/2 1/2 -1/2
(g2,1) | 1 -1 -1/2 1/2 1/2 -1/2
(1,eps) | 1 1 1/2 1/2 1/2 1/2
(g2,eps) | -1 -1 -1/2 -1/2 1/2 1/2
Fourier(f):Eigenvalues(f):
String(f), Print(f):
Display(f):
Size(f):
f*g:ComplexConjugate(f):OnFamily(f,-1).
IsFamily(obj)
returns true if obj is a family, and false otherwise.
gap> List(UnipotentCharacters(ComplexReflectionGroup(4)).families,IsFamily);
[ true, true, true, true ]
OnFamily(f,p)
f should be a family. This function has two forms.
In the first form, p is a permutation, and the function returns a copy of
the family f with the Fourier matrix, eigenvalues of Frobenius,
.charLabels, etc... permuted by p.
In the second form, p is an integer and x->GaloisCyc(x,p) is applied
to the Fourier matrix and eigenvalues of Frobenius of the family.
gap> f:=UnipotentCharacters(ComplexReflectionGroup(3,1,1)).families[2];
Family("0011",[4,3,2])
gap> Display(f);
0011
label |eigen 1 2 3
_________________________________________________
1 | E3^2 ER(-3)/3 ER(-3)/3 -ER(-3)/3
2 | 1 ER(-3)/3 (3-ER(-3))/6 (3+ER(-3))/6
3 | 1 -ER(-3)/3 (3+ER(-3))/6 (3-ER(-3))/6
gap> Display(OnFamily(f,(1,2,3)));
0011
label |eigen 3 1 2
_________________________________________________
3 | 1 (3-ER(-3))/6 -ER(-3)/3 (3+ER(-3))/6
1 | E3^2 -ER(-3)/3 ER(-3)/3 ER(-3)/3
2 | 1 (3+ER(-3))/6 ER(-3)/3 (3-ER(-3))/6
gap> Display(OnFamily(f,-1));
'0011
label |eigen 1 2 3
_________________________________________________
1 | E3 -ER(-3)/3 -ER(-3)/3 ER(-3)/3
2 | 1 -ER(-3)/3 (3+ER(-3))/6 (3-ER(-3))/6
3 | 1 ER(-3)/3 (3-ER(-3))/6 (3+ER(-3))/6
FamiliesClassical(l)
The list l should be a list of symbols as returned by the function
Symbols, which classify the unipotent characters of groups of type "B",
"C" or "D". FamiliesClassical returns the list of families determined
by these symbols.
gap> FamiliesClassical(Symbols(3,1));
[ Family("0112233",[4]), Family("01123",[1,3,8]),
Family("013",[5,7,10]), Family("022",[6]), Family("112",[2]),
Family("3",[9]) ]
The above example shows the families of unipotent characters for the group B3.
FamilyImprimitive(S)
S should be a symbol for a unipotent characters of an imprimitive complex
reflection group G(e,1,n) or G(e,e,n). The function returns the family
of unipotent characters to which the character with symbol S belongs.
gap> FamilyImprimitive([[0,1],[1],[0]]);
Family("0011")
gap> Display(last);
0011
label |eigen 1 2 3
_________________________________________________
1 | E3^2 ER(-3)/3 -ER(-3)/3 ER(-3)/3
2 | 1 -ER(-3)/3 (3-ER(-3))/6 (3+ER(-3))/6
3 | 1 ER(-3)/3 (3+ER(-3))/6 (3-ER(-3))/6
DrinfeldDouble(g[,opt])
Given a (usually small) finite group Γ, Lusztig has associated a family (a Fourier matrix, a list of eigenvalues of Frobenius) which describes the representation ring of the Drinfeld double of the group algebra of Γ, and for some appropriate small groups describes a family of unipotent characters. We do not explain the details of this construction, but explain how its final result building Lusztig's Fourier matrix, and a variant of it that we use in Spetses, from Γ.
The elements of the family are in bijection with the set M(Γ) of pairs (x,χ) taken up to Γ-conjugacy, where x∈Γ and χ is an irreducible complex-valued character of CΓ(x). To such a pair ρ=(x,χ) is associated an eigenvalue of Frobenius defined by ωρ:=χ(x)/χ(1). Lusztig then defines a Fourier matrix T whose coefficient is given, for ρ=(x,χ) and ρ'=(x', χ'), by:
| Tρ,ρ':=#CΓ(x)-1 ∑ρ1=(x1,χ1)χ1(x)χ(y1) |
where the sum is over all pairs ρ1∈ M(Γ) which are Γ-conjugate to ρ' and such that y1∈ CΓ(x). This coefficient also represents the scalar product ⟨ρ,ρ'⟩GF of the corresponding unipotent characters.
A way to understand the formula for Tρ,ρ' better is to consider another basis of the complex vector space with basis M(Γ), indexed by the pairs (x,y) taken up to Γ-conjugacy, where x and y are commuting elements of Γ. This basis is called the basis of Mellin transforms, and given by:
| (x,y)=∑χ∈ Irr(CΓ(x))χ(y)(x,χ) |
In the basis of Mellin transforms, the linear map T is given by (x,y)→(x-1,y-1) and the linear transformation which sends ρ to ωρρ becomes (x,y)→(x,xy). These are particular cases of the permutation representation of GL2(ℤ) on the basis of Mellin transforms where (
| a | b |
| c | d |
acts by (x,y)→(xayb,xcyd).
Fourier matrices in finite reductive groups are given by the above matrix T. But for non-rational Spetses, we use a different matrix S which in the basis of Mellin transforms is given by (x,y)→(y-1,x). Equivalently, the formula Sρ,ρ' differs from the formula for Tρ,ρ' in that there is no complex conjugation of χ1; thus the matrix S is equal to T multiplied on the right by the permutation matrix which corresponds to (x,χ)→(x,χ). The advantage of the matrix S over T is that the pair S,Ω satisfies directly the axioms for a fusion algebra (see below); also the matrix S is symmetric, while T is Hermitian.
Thus there are two variants of DrinfeldDouble:
DrinfeldDouble(g,rec(lusztig:=true))
returns a family containing Lusztig's Fourier matrix T, and an extra
field .perm containing the permutation of the indices induced by
(x,χ)→(x,χ), which allows to recover S, as well as
an extra field .lusztig, set to true.
DrinfeldDouble(g)
returns a family with the matrix S, which does not have fields .lusztig
or .perm.
The family record f returned also has the fields:
.group:
.charLabels:
.fourierMat:
.eigenvalues:
.xy:[x,y] which are representatives of the
Γ-orbits of pairs of commuting elements.
.mellinLabels:[x,y].
.mellin:f.fourierMat^(f.mellin^-1) is the
permutation matrix (for (x,y)→(y-1,x) or
(x,y)→(y-1,x-1) depending on the call).
.special:
gap> f:=DrinfeldDouble(SymmetricGroup(3));
Family("D(Group((1,3),(2,3)))")
gap> Display(f);
D(Group((1,3),(2,3)))
label |eigen
_______________________________________________________
(Id,Id) | 1 1/6 1/6 1/3 1/2 1/2 1/3 1/3 1/3
(Id,X.2) | 1 1/6 1/6 1/3 -1/2 -1/2 1/3 1/3 1/3
(Id,X.3) | 1 1/3 1/3 2/3 0 0 -1/3 -1/3 -1/3
(2a,Id) | 1 1/2 -1/2 0 1/2 -1/2 0 0 0
(2a,X.2) | -1 1/2 -1/2 0 -1/2 1/2 0 0 0
(3a,Id) | 1 1/3 1/3 -1/3 0 0 2/3 -1/3 -1/3
(3a,X.2) | E3 1/3 1/3 -1/3 0 0 -1/3 -1/3 2/3
(3a,X.3) | E3^2 1/3 1/3 -1/3 0 0 -1/3 2/3 -1/3
gap> f:=DrinfeldDouble(SymmetricGroup(3),rec(lusztig:=true));
Family("LD(Group((1,3),(2,3)))")
gap> Display(f);
LD(Group((1,3),(2,3)))
label |eigen
_______________________________________________________
(Id,Id) | 1 1/6 1/6 1/3 1/2 1/2 1/3 1/3 1/3
(Id,X.2) | 1 1/6 1/6 1/3 -1/2 -1/2 1/3 1/3 1/3
(Id,X.3) | 1 1/3 1/3 2/3 0 0 -1/3 -1/3 -1/3
(2a,Id) | 1 1/2 -1/2 0 1/2 -1/2 0 0 0
(2a,X.2) | -1 1/2 -1/2 0 -1/2 1/2 0 0 0
(3a,Id) | 1 1/3 1/3 -1/3 0 0 2/3 -1/3 -1/3
(3a,X.2) | E3 1/3 1/3 -1/3 0 0 -1/3 2/3 -1/3
(3a,X.3) | E3^2 1/3 1/3 -1/3 0 0 -1/3 -1/3 2/3
NrDrinfeldDouble(g)
This function returns the number of elements that the family associated to the Drinfeld double of the group g would have, without computing it. The evident advantage is the speed.
gap> NrDrinfeldDouble(ComplexReflectionGroup(5));
378
FusionAlgebra(f)
The argument f should be a family, or the Fourier matrix of a family. All
the Fourier matrices S in CHEVIE are unitary, that is
S-1=tS, and have a special line s (the line of
index s=f.special for a family f) such that no entry Ss,i is
equal to 0. Further, they have the property that the sums
Ci,j,k:=∑l Si,l Sj,lSk,l/Ss,l take
integral values. Finally, S has the property that complex conjugation
does a permutation with signs σ of the lines of S.
It follows that we can define a ℤ-algebra A as follows: it has a basis bi indexed by the lines of S, and has a multiplication defined by the fact that the coefficient of bibj on bk is equal to Ci,j,k.
A is commutative, and has as unit the element bs; the basis σ(bi) is dual to bi for the linear form (bi,bj)=Ci,j,σ(s).
gap> W:=ComplexReflectionGroup(4);;uc:=UnipotentCharacters(W);
UnipotentCharacters( G4 )
gap> f:=uc.families[4];
Family("RZ/6^2[1,3]",[2,4,10,9,3])
gap> A:=FusionAlgebra(f);
Fusion algebra dim.5
gap> b:=A.basis;
[ T(1), T(2), T(3), T(4), T(5) ]
gap> List(b,x->x*b);
[ [ T(1), T(2), T(3), T(4), T(5) ],
[ T(2), -T(4)+T(5), T(1)+T(4), T(2)-T(3), T(3) ],
[ T(3), T(1)+T(4), -T(4)+T(5), -T(2)+T(3), T(2) ],
[ T(4), T(2)-T(3), -T(2)+T(3), T(1)+T(4)-T(5), -T(4) ],
[ T(5), T(3), T(2), -T(4), T(1) ] ]
gap> CharTable(A);
1 2 3 4 5
1 1 -ER(-3) ER(-3) 2 -1
2 1 1 1 . 1
3 1 -1 -1 . 1
4 1 . . -1 -1
5 1 ER(-3) -ER(-3) 2 -1
98.25 The d-Harish-Chandra series
d-Harish-Chandra series describe unipotent l-blocks of a finite reductive group G(\BFq) for l|Φd(q) (at least, when l is not too small which means mostly not a bad prime for G). Some of the facts stated below are still partly conjectural, we do not try to distinguish precisely what has been established and what is still conjectural.
If (L,λ) is a d-cuspidal pair then the constituents of the
Lusztig induced RLG(λ) are called a d-Harish-Chandra
series; they form the unipotent part of an l-block of GF. It is
conjectured (and proven in some cases) that the GF-endomorphism
algebra of the l-adic cohomology of the variety \bX which defines the
Lusztig induction is a d-cyclotomic Hecke algebra HG(L,λ)
for the group WG(L,λ):=NG(L,λ)/L, which is a
complex reflection group --- here d-cyclotomic means that the parameters
of HG(L,λ) are monomials in q and that HG(L,λ)
specializes to the algebra of WG(L,λ) for q→E(d).
It follows that the decomposition of the Lusztig induction is of the form
| RLG(λ)=∑φ∈Irr(WG(L,λ))(-1)nφ φ(1)γφ, |
Series allows to explore a
d-Harish-Chandra series.
gap> W:=RootDatum("3D4");;
gap> l:=CuspidalPairs(W,3);
[ [ 3D4, 8 ], [ (q^2+q+1)^2, 1 ] ]
gap> s:=Series(W,l[2][1],l[2][2],3);;
gap> Display(s);
E(3)-series R^{3D4}_{(q^2+q+1)^2}() H_G(L,c)=Hecke(G4,[[E3q^2,E3,E3q]])
| Name WGLname eps family # ____________________________________
1 | phi{1,0} phi{1,0} 1 1
2 | phi{1,6} phi{1,4} 1 2
3 | phi{2,2} phi{1,8} -1 5
6 |phi{1,3}'' phi{2,5} 1 4
5 | phi{1,3}' phi{2,3} -1 3
7 | phi{2,1} phi{2,1} -1 5
4 | 3D4[1] phi{3,2} 1 5
Above we explore the 3-series corresponding to RTG(Id) where
G is the triality group and T is the torus of type (q2+q+1)2.
The group WG(T) is the complex reflection group G4. The displays
shows in the column Name the name of the unipotent characters
constituents of RTG(Id), and in the first column the number
of these characters in the list of unipotent characters. In the column
WGLname the name of the character φ of WG(T) corresponding to
the unipotent character γφ is shown; in the column eps we show
the sign (-1)nφ. Finally in the last column we show in which
family of unipotent characters is γφ.
The theory of d-Harish-Chandra series can be generalized to spetsial complex reflection groups using some axioms. We show below an example.
gap> W:=ComplexReflectionGroup(4);
ComplexReflectionGroup(4)
gap> l:=CuspidalPairs(W,3);
[ [ G4, 3 ], [ G4, 6 ], [ G4, 7 ], [ G4, 10 ], [ (q-1)(q-E3), 1 ] ]
gap> s:=Series(W,l[5][1],l[5][2],3);;
gap> Display(s);
E(3)-series R^{G4}_{(q-1)(q-E3)}() W_G(L,c)=Z6
| Name WGLname(mod 3) eps param family # _______________________________________________
1 |phi{1,0} 1 1 E3q^2 1
5 |phi{2,3} -E3^2 1 -E3q 2
2 |phi{1,4} E3 -1 E3 4
8 | Z3:2 -1 -1 -E3^2q 2
9 | Z3:11 E3^2 -1 E3^2 4
4 |phi{2,5} -E3 -1 -E3 4
Above we explore the 3-series corresponding to the trivial character of
the torus of type (q-1)(q-E(3)). For cyclic groups WG(L,λ)
we display the parameters in the table since they are associated to
characters of WG(L,λ). Finally the mention (mod 3) which
appears in the WGLname column means that in this case the axioms leave an
ambiguity in the correspondence between unipotent characters γφ
and characters φ (as well as with parameters): the correspondence is
known only up to a translation by 3 (in this case, the same as a global
multiplication of all φ by -1).
Finally, we should note that if the reflection group or coset W is not
defined over the integers, what counts is not cyclotomic polynomials but
factors of them over the field of definition of W. In this case, one
should not give as argument an integer d representing E(d) but specify
a root of unity with a fraction r/d representing E(d)^r. For instance,
in the above case we get a different answer with:
gap> CuspidalPairs(W,2/3);
[ [ G4, 2 ], [ G4, 5 ], [ G4, 7 ], [ G4, 10 ], [ (q-1)(q-E3^2), 1 ] ]
Series(W, L, cuspidal, d)
If the reflection coset or group W corresponds to the algebraic group G
and cuspidal to the d-cuspidal unipotent character λ of L,
constructs the d-series corresponding to RLG(λ).
If s is the result, it
is a record with the following fields and functions:
s.spets:
s.levi:
s.cuspidal:UnipotentCharacters(L).
s.d:d (represents a root of unity as a fraction in [0,1[).
s.WGL:
s.RLG:UnipotentCharacter given by RLG(λ).
s.eps:s.WGL the sign (-1)nφ in
the cohomology of the variety defining s.RLG of the corresponding
constituent γφ of s.RLG.
Degree(s):s.RLG, as a CycPol.
CharNumbers(s):UnipotentCharacters(W) of the
constituents of s.RLG.
Hecke(s):
Series has another form:
Series(W [,d [,ad]])
returns all the d-series of W corresponding to a d-eigenspace of dimension ad (default d=1, and if ad not given returns all d-series).
gap> W:=ComplexReflectionGroup(4);
ComplexReflectionGroup(4)
gap> Series(W,3,1);
[ E(3)-series R^{G4}_{(q-1)(q-E3)}() W_G(L,c)=Z6 ]
gap> s:=last[1];
E(3)-series R^{G4}_{(q-1)(q-E3)}() W_G(L,c)=Z6
gap> s.spets;
G4
gap> s.levi;
(q-1)(q-E3)
gap> s.cuspidal;
1
gap> s.d;
1/3
gap> Hecke(s);
Hecke(Z6,[[E3q^2,-E3q,E3,-E3^2q,E3^2,-E3]])
gap> Degree(s);
E3P1P2^2P"3P4P6
gap> s.RLG;
[G4]=<phi{1,0}>-<phi{1,4}>-<phi{2,5}>+<phi{2,3}>-<Z3:2>-<Z3:11>
gap> CharNumbers(s);
[ 1, 5, 2, 8, 9, 4 ]
gap> s.eps;
[ 1, 1, -1, -1, -1, -1 ]
gap> ReflectionName(s.WGL);
"Z6"
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