Unipotent characters
Chevie.Families
Chevie.Uch
Chevie.Families.Family
Chevie.Uch.UnipotentCharacters
Base.:*
Base.conj
Base.show
Chevie.ComplexR.degrees
Chevie.Families.FamiliesClassical
Chevie.Families.cospecial
Chevie.Families.drinfeld_double
Chevie.Families.family_imprimitive
Chevie.Families.fourier
Chevie.Families.fusion_algebra
Chevie.Families.ndrinfeld_double
Chevie.Families.special
Chevie.Lusztig.lusztig_induction_table
Chevie.Uch.CycPoldegrees
Chevie.Uch.almost_character
Chevie.Uch.deligne_lusztigCharTable
Chevie.Uch.deligne_lusztig_character
Chevie.Uch.deligne_lusztig_lefschetz
Chevie.Uch.lusztig_induce
Chevie.Uch.lusztig_restrict
Chevie.Uch.on_unipotents
Chevie.Uch.unipotent_character
CyclotomicNumbers.galois
PermGroups.Perms.invpermute
Chevie.Uch
β ModuleLet $π$ be a connected reductive group over the algebraic closure of a finite field $π½_q$, defined over $π½_q$ with corresponding Frobenius automorphism $F$. We want to study the irreducible characters of $π ^F$. More generally we consider $π ^F$ where $F$ is an isogeny of $π$ such that a power is a Frobenius (this covers the Suzuki and Ree groups).
If $π$ is an $F$-stable maximal torus of $π$, and $π$ is a (not necessarily $F$-stable) Borel subgroup containing $π$, we define the Deligne-Lusztig variety $X_π=\{gπ β π /π β£ gπ β© F(gπ )β β \}$. This variety affords a natural action of $π ^F$ on the left, so the corresponding Deligne-Lusztig virtual module $H^*_c(X_π):=βα΅’ (-1)β± Hβ±_c(X_π,βΜ _β)$ also. The (virtual) character of this module is the Deligne-Lusztig character $R_π ^π (1)$; the notation reflects the theorem that this character does not depend on the choice of $π$. This character can be parameterized by an $F$-conjugacy class of $W$: if $πββπβ$ is an $F$-stable pair, there is an unique $wβ W=N_π (πβ)/πβ$ such that the triple $(π,π,F)$ is $π$-conjugate to $(πβ,πβ,wF)$. We will thus denote $R_w$ for $R_π^π (1)$; this character depends only on the $F$-class of $w$.
The unipotent characters of $π ^F$ are the irreducible constituents of the $R_w$. In a similar way that the Jordan decomposition shows that the unipotent classes are a building block for describing the conjugacy classes of a reductive group, Lusztig has defined a Jordan decomposition of characters where the unipotent characters are the building block. The unipotent characters are parameterized by combinatorial data that Lusztig has defined just from the coset $WΟ$, where Ο
is the finite order automorphism of $X(πβ)$ such that $F=qΟ$. Thus, from our viewpoint, unipotent characters are objects combinatorially attached to a Coxeter coset.
A subset of the unipotent characters, the principal series unipotent characters, can be described in a more elementary way. They are the constituents of $Rβ$, or equivalently the characters of the virtual module $H^*_c(X_{π β})$, where $X_{π β}$ is the discrete variety $(π /πβ)^F$; this virtual module reduces to the actual module $βΜ _β[(π /πβ) ^F]$. Thus the Deligne-Lusztig induction $R_{πβ}^π (1)$ reduces to Harish-Chandra induction, defined as follows: let $π =π β π$ be an $F$-stable Levi decomposition of an $F$-stable parabolic subgroup of $π$. Then the Harish-Chandra induced $R_π^π$ of a character $Ο$ of $π^F$ is the character $Ind_{π^F}^{π ^F}ΟΜ$, where $ΟΜ$ is the lift to $π^F$ of $Ο$ via the quotient $π^F/π ^F=π^F$; Harish-Chandra induction is a particular case of Lusztig induction, which is defined when $π$ is not $F$-stable using the variety $X_π =\{ gπ βπ /π β£ gπ β© F(gπ )β β \}$, and gives for an $π^F$-module a virtual $π ^F$-module. Like ordinary induction, these functors have adjoint functors going from representations of $π ^F$ to representations (resp. virtual representations) of $π^F$ called Harish-Chandra restriction (resp. Lusztig restriction).
The commuting algebra of $π^F$-endomorphisms of $Rβ=R_{πβ}^π(1)$ is an Iwahori-Hecke algebra for $W^Ο$, with parameters some powers of q
; the parameters are all equal to q
when $W^Ο=W$. Thus principal series unipotent characters are parametrized by characters of $W^Ο$.
To understand the decomposition of more general $R_w$, and thus parameterize unipotent characters, is is useful to introduce another set of class functions which are parameterized by irreducible characters of the coset $WΟ$. If $Ο$ is such a character, we define the associated almost character by: $Rα΅ͺ=|W|β»ΒΉβ_{wβ W}Ο(wΟ) R_w$. The name reflects that these class function are close to irreducible characters. They satisfy $β¨Rα΅ͺ, R_Οβ©_{π^F}=Ξ΄_{Ο,Ο}$; for the linear and unitary group they are actually unipotent characters (up to sign in the latter case). They are in general the sum (with rational coefficients) of a small number of unipotent characters in the same Lusztig family, see Families
. The degree of $Rα΅ͺ$ is a polynomial in $q$ equal to the fake degree of the character $Ο$ of $WΟ$ (see fakedegree
).
We now describe the parameterization of unipotent characters when $W^Ο=W$, thus when the coset $WΟ$ identifies with $W$ (the situation is similar but a bit more difficult to describe in general). The (rectangular) matrix of scalar products $β¨Ο, Rα΅ͺβ©_{π ^F}$, 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 π
corresponds to a block of the Hecke algebra over a ring called the Rouquier ring. To π
Lusztig associates a small group $Ξ$ (not bigger than $(β€/2)βΏ$, or $πα΅’$ for $iβ€5$) such that the unipotent characters in the family are parameterized by the pairs $(x,ΞΈ)$ taken up to $Ξ$-conjugacy, where $xβΞ$ and $ΞΈ$ is an irreducible character of $C_Ξ(x)$. Further, the elements of π
themselves are parameterized by a subset of such pairs, and Lusztig defines a pairing between such pairs which computes the scalar product $β¨Ο, Rα΅ͺβ©_{π^F}$, called the Lusztig Fourier matrix. For more details see drinfeld_double
.
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 $GLβ$ for $n>1$, and at most one in other classical groups). The $π^F$-endomorphism algebra of an Harish-Chandra induced $R_{π^F}^{π^F}Ξ»$, where $Ξ»$ is a cuspidal unipotent character turns out to be a Hecke algebra associated to the group $W_{π^F}(π^F):=N_{π^F}(π)/π$, which turns out to be a Coxeter group. Thus another parameterization is by triples $(π,Ξ»,Ο)$, where $Ξ»$ is a cuspidal unipotent character of $π^F$ and $Ο$ is an irreducible character of the relative group $W_{π^F}(π^F)$. Such characters are said to belong to the Harish-Chandra series determined by $(π,Ξ»)$.
A final piece of information attached to unipotent characters is the eigenvalues of Frobenius. Let $Fα΅$ be the smallest power of the isogeny $F$ which is a split Frobenius (that is, $Fα΅$ is a Frobenius and $Οα΅=1$). Then $Fα΅$ acts naturally on Deligne-Lusztig varieties and thus on the corresponding virtual modules, and commutes to the action of $π^F$; thus for a given unipotent character $Ο$, a submodule of the virtual module which affords $Ο$ affords a single eigenvalue $ΞΌ$ of $Fα΅$. Results of Lusztig and Digne-Michel show that this eigenvalue is of the form $qα΅α΅Ξ»α΅¨$ where $2aββ€$ and $λᡨ$ is a root of unity which depends only on $Ο$ and not the considered module. This $λᡨ$ is called the eigenvalue of Frobenius attached to $Ο$. Unipotent characters in the Harish-Chandra series of a pair $(π,Ξ»)$ have the same eigenvalue of Frobenius as $Ξ»$.
This package contains tables of all this information, and can compute Harish-Chandra and Lusztig induction of unipotent characters and almost characters. We illustrate this on some examples:
julia> W=coxgroup(:G,2)
Gβ
julia> uc=UnipotentCharacters(W)
UnipotentCharacters(Gβ)
βββββββββ¬βββββββββββββββββββββββββββββββββββββββββββββββββββββ
βΞ³ βnβ Deg(Ξ³) Feg Symbol Fr(Ξ³) labelβ
βββββββββΌβββββββββββββββββββββββββββββββββββββββββββββββββββββ€
βΟβββ β 1 1 1 (0,0,0,0,0,2) 1 β
βΟβββ β 2 qβΆ qβΆ (01,01,01,01,01,12) 1 β
βΟβ²βββ β 3 qΞ¦βΞ¦β/3 qΒ³ (0,0,1+) 1 (1,Ο)β
βΟβ³βββ β 4 qΞ¦βΞ¦β/3 qΒ³ (0,0,1-) 1 (gβ,1)β
βΟβββ β 5 qΞ¦βΒ²Ξ¦β/6 qΞ¦β (0,0,0,0,1,1) 1 (1,1)β
βΟβββ β 6 qΞ¦βΒ²Ξ¦β/2 qΒ²Ξ¦β (0,0,0,1,0,1) 1 (gβ,1)β
βGβ[-1] β 7 qΞ¦βΒ²Ξ¦β/2 0 (01,0,01,,0,) -1 (gβ,Ξ΅)β
βGβ[1] β 8 qΞ¦βΒ²Ξ¦β/6 0 (01,01,0,,,0) 1 (1,Ξ΅)β
βGβ[ΞΆβ] β 9 qΞ¦βΒ²Ξ¦βΒ²/3 0 (01,0,0,01,,) ΞΆβ (gβ,ΞΆβ)β
βGβ[ΞΆβΒ²]β10 qΞ¦βΒ²Ξ¦βΒ²/3 0 (01,01,,0,0,) ΞΆβΒ² (gβ,ΞΆβΒ²)β
βββββββββ΄βββββββββββββββββββββββββββββββββββββββββββββββββββββ
The first column gives the name of the unipotent character, derived from its Harish-Chandra classification; the first 6 characters are in the principal series so are named by characters of W
. The last 4 are cuspidal, and named by the corresponding eigenvalue of Frobenius, which is displayed in the fourth column. For classical groups, the Harish-Chandra data can be synthesized combinatorially to give a symbol.
The first two characters are each in a Lusztig family by themselves. The last eight are in a family associated to the group Ξ=πβ
: the last column shows the parameters (x,ΞΈ)
. The third 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
(extended by 0
outside the principal series).
One can get more information on the Lusztig Fourier matrix of the big family by asking
julia> uc.families[1]
Family(D(π β),[5, 6, 4, 3, 8, 7, 9, 10])
Drinfeld double of π β, Lusztigβ²s version
ββββββββββ¬βββββββββββββββββββββββββββββββββββββββββββββββββββββ
βlabel βeigen β
ββββββββββΌβββββββββββββββββββββββββββββββββββββββββββββββββββββ€
β(1,1) β 1 1//6 1//2 1//3 1//3 1//6 1//2 1//3 1//3β
β(gβ,1) β 1 1//2 1//2 . . -1//2 -1//2 . .β
β(gβ,1) β 1 1//3 . 2//3 -1//3 1//3 . -1//3 -1//3β
β(1,Ο) β 1 1//3 . -1//3 2//3 1//3 . -1//3 -1//3β
β(1,Ξ΅) β 1 1//6 -1//2 1//3 1//3 1//6 -1//2 1//3 1//3β
β(gβ,Ξ΅) β -1 1//2 -1//2 . . -1//2 1//2 . .β
β(gβ,ΞΆβ) β ΞΆβ 1//3 . -1//3 -1//3 1//3 . 2//3 -1//3β
β(gβ,ΞΆβΒ²)β ΞΆβΒ² 1//3 . -1//3 -1//3 1//3 . -1//3 2//3β
ββββββββββ΄βββββββββββββββββββββββββββββββββββββββββββββββββββββ
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.
julia> W=coxgroup(:G,2)
Gβ
julia> T=spets(reflection_subgroup(W,Int[]),W(1,2))
Gβββ=Ξ¦β
julia> u=unipotent_character(T,1)
[Gβββ=Ξ¦β]:<Id>
To construct T
one could equivalently do
julia> T=torus(W,position_class(W,W(1,2)))
Gβββ=Ξ¦β
Then here are two ways to construct the Deligne-Lusztig character associated to the Coxeter torus:
julia> lusztig_induce(W,u)
[Gβ]:<Οβββ>+<Οβββ>-<Οβββ>+<Gβ[-1]>+<Gβ[ΞΆβ]>+<Gβ[ΞΆβΒ²]>
julia> v=deligne_lusztig_character(W,[1,2])
[Gβ]:<Οβββ>+<Οβββ>-<Οβββ>+<Gβ[-1]>+<Gβ[ΞΆβ]>+<Gβ[ΞΆβΒ²]>
julia> degree(v)
Pol{Int64}: qβΆ+qβ΅-qβ΄-2qΒ³-qΒ²+q+1
julia> 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 BroueMalleMichel2014. An example:
julia> UnipotentCharacters(complex_reflection_group(4))
UnipotentCharacters(Gβ)
βββββββ¬βββββββββββββββββββββββββββββββββββββββββ
βΞ³ βnβ Deg(Ξ³) Feg Fr(Ξ³) labelβ
βββββββΌβββββββββββββββββββββββββββββββββββββββββ€
βΟβββ β 1 1 1 1 β
βΟβββ β 2 -β-3qβ΄Ξ¦β³βΞ¦βΞ¦β³β/6 qβ΄ 1 1β§ΞΆββ
βΟβββ β 3 β-3qβ΄Ξ¦β²βΞ¦βΞ¦β²β/6 qβΈ 1 -1β§ΞΆβΒ²β
βΟβββ
β 4 qβ΄Ξ¦βΒ²Ξ¦β/2 qβ΅Ξ¦β 1 1β§ΞΆβΒ²β
βΟβββ β 5 -ΞΆββ-3qΞ¦β³βΞ¦βΞ¦β²β/3 qΒ³Ξ¦β 1 1β§ΞΆβΒ²β
βΟβββ β 6 ΞΆβΒ²β-3qΞ¦β²βΞ¦βΞ¦β³β/3 qΞ¦β 1 1β§ΞΆββ
βΟβββ β 7 qΒ²Ξ¦βΞ¦β qΒ²Ξ¦βΞ¦β 1 β
βZβ:2 β 8 -β-3qΞ¦βΞ¦βΞ¦β/3 0 ΞΆβΒ² ΞΆββ§ΞΆβΒ²β
βZβ:11β 9 -β-3qβ΄Ξ¦βΞ¦βΞ¦β/3 0 ΞΆβΒ² ΞΆββ§ΞΆββ΅β
βGβ β10 -qβ΄Ξ¦βΒ²Ξ¦β/2 0 -1 ΞΆββ§-1β
βββββββ΄βββββββββββββββββββββββββββββββββββββββββ
Chevie.Uch.UnipotentCharacters
β TypeUnipotentCharacters(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:
.harishChandra
: information about Harish-Chandra series of unipotent characters. This is itself a list of records, one for each pair (π,Ξ»)
of a Levi of an F
-stable parabolic subgroup and a cuspidal unipotent character of $π^F$. These records themselves have the following fields:
:levi
: a list 'l' such that π
corresponds to 'ReflectionSubgroup(W,l)'.
:cuspidalName
: the name of the unipotent cuspidal character lambda
.
:eigenvalue
: the eigenvalue of Frobenius for Ξ»
.
:relativeType
: the reflection type of $W_π (π)$;
:parameterExponents
: the $π ^F$-endomorphism algebra of $R_π^π (Ξ»)$ is a Hecke algebra for $W_π (π)$ with some parameters of the form $q^{a_s}$. This holds the list of exponents $a_s$.
:charNumbers
: the indices of the unipotent characters indexed by the irreducible characters of $W_π (π)$.
.almostHarishChandra
: information about Harish-Chandra series of unipotent character sheaves. This is identical to ΜharishChandra` for a split reductive group, and reflects the situation for the corresponding split group for a nonsplit group.
.families
: information about Lusztig families of unipotent characters. This is itself a list of records, one for each family. These records are described in the section about families below.
the following information is computed on demand from uc=UnipotentCharacters(W)
:
spets(uc)
: the reductive group W
.
julia> W=coxgroup(:Bsym,2)
Bsymβ
julia> WF=spets(W,Perm(1,2))
Β²Bsymβ
julia> uc=UnipotentCharacters(WF)
UnipotentCharacters(Β²Bsymβ)
ββββββββββ¬ββββββββββββββββββββββββββββββββββββββββββββββββββββββ
βΞ³ βnβ almostch Deg(Ξ³) Feg Symbol Fr(Ξ³) labelβ
ββββββββββΌββββββββββββββββββββββββββββββββββββββββββββββββββββββ€
β2 β 1 2. 1 1 (02,,0,0) 1 β
β11 β 2 .11 qβ΄ qβ΄ (012,1,01,01) 1 β
βΒ²Bβ[1,3]β 3 1.1 β2qΞ¦βΞ¦β/2 qΞ¦βΞ¦β (01,,1,0) ΞΆβΒ³ 1β
βΒ²Bβ[1,5]β 4 Bβ β2qΞ¦βΞ¦β/2 0 (01,,0,1) ΞΆββ΅ 2β
ββββββββββ΄ββββββββββββββββββββββββββββββββββββββββββββββββββββββ
julia> uc.families
3-element Vector{Family}:
Family(Cβ,[1])
Family(Cβ,[2])
Family(?4,[3, 4])
julia> uc.families[3]
Family(?4,[3, 4])
βββββββ¬βββββββββββββββββ
βlabelβeigen 1 2β
βββββββΌβββββββββββββββββ€
β1 β ΞΆβΒ³ β2/2 -β2/2β
β2 β -ΞΆβ β2/2 β2/2β
βββββββ΄βββββββββββββββββ
charnames(uc)
: the list of names of the unipotent characters. Using appropriate keywords, one can control the display in various ways.
julia> uc=UnipotentCharacters(coxgroup(:G,2));
julia> charnames(uc;limit=true)
10-element Vector{String}:
"Οβββ"
"Οβββ"
"Οβ²βββ"
"Οβ³βββ"
"Οβββ"
"Οβββ"
"Gβ[-1]"
"Gβ[1]"
"Gβ[ΞΆβ]"
"Gβ[ΞΆβΒ²]"
julia> charnames(uc;TeX=true)
10-element Vector{String}:
"\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]"
One can control the display of unipotent characters in various ways by IOContext
properties. In the display, the row labels are the names of the unipotent characters. The possible columns are numbered as follows:
- The index of the character in the list of unipotent characters.
- The degree of the unipotent character.
- The degree of the corresponding almost character.
- for imprimitive groups, the symbol attached to the unipotent character.
- The eigenvalue of Frobenius attached to the unipotent character.
- The parameter the character has in its Lusztig family.
Which columns are displayed can be controlled by the property :cols
(default [2,3,5,6] and 4 when applicable).
In addition if ':byfamily=true', the characters are displayed family by family instead of in index order. Finally, the properties rows
and columnrepartition
of format
can be set, giving more tuning of the table.
julia> W=coxgroup(:B,2)
Bβ
julia> uc=UnipotentCharacters(W)
UnipotentCharacters(Bβ)
βββββ¬βββββββββββββββββββββββββββββββββββ
βΞ³ βnβ Deg(Ξ³) Feg Symbol Fr(Ξ³) labelβ
βββββΌβββββββββββββββββββββββββββββββββββ€
β11.β 1 qΞ¦β/2 qΒ² (12,0) 1 +,-β
β1.1β 2 qΞ¦βΒ²/2 qΞ¦β (02,1) 1 +,+β
β.11β 3 qβ΄ qβ΄ (012,12) 1 β
β2. β 4 1 1 (2,) 1 β
β.2 β 5 qΞ¦β/2 qΒ² (01,2) 1 -,+β
βBβ β 6 qΞ¦βΒ²/2 0 (012,) -1 -,-β
βββββ΄βββββββββββββββββββββββββββββββββββ
julia> xdisplay(uc;byfamily=true)
ββββββ¬βββββββββββββββββββββββββββββββββββ
βΞ³ βnβ Deg(Ξ³) Feg Symbol Fr(Ξ³) labelβ
ββββββΌβββββββββββββββββββββββββββββββββββ€
β11. β 1 qΞ¦β/2 qΒ² (12,0) 1 +,-β
β1.1Λ’β 2 qΞ¦βΒ²/2 qΞ¦β (02,1) 1 +,+β
β.2 β 5 qΞ¦β/2 qΒ² (01,2) 1 -,+β
βBβ β 6 qΞ¦βΒ²/2 0 (012,) -1 -,-β
ββββββΌβββββββββββββββββββββββββββββββββββ€
β2. β 4 1 1 (2,) 1 β
ββββββΌβββββββββββββββββββββββββββββββββββ€
β.11 β 3 qβ΄ qβ΄ (012,12) 1 β
ββββββ΄βββββββββββββββββββββββββββββββββββ
julia> xdisplay(uc;cols=[1,4])
UnipotentCharacters(Bβ)
βββββ¬ββββββββββββ
βΞ³ βnβ Symbolβ
βββββΌββββββββββββ€
β11.β 1 (12,0)β
β1.1β 2 (02,1)β
β.11β 3 (012,12)β
β2. β 4 (2,)β
β.2 β 5 (01,2)β
βBβ β 6 (012,)β
βββββ΄ββββββββββββ
Chevie.ComplexR.degrees
β Methoddegrees(uc::UnipotentCharacters,q=Pol())
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
.
julia> W=coxgroup(:G,2)
Gβ
julia> uc=UnipotentCharacters(W);
julia> degrees(uc)
10-element Vector{Pol{Rational{Int64}}}:
1
qβΆ
(1//3)qβ΅+(1//3)qΒ³+(1//3)q
(1//3)qβ΅+(1//3)qΒ³+(1//3)q
(1//6)qβ΅+(1//2)qβ΄+(2//3)qΒ³+(1//2)qΒ²+(1//6)q
(1//2)qβ΅+(1//2)qβ΄+(1//2)qΒ²+(1//2)q
(1//2)qβ΅+(-1//2)qβ΄+(-1//2)qΒ²+(1//2)q
(1//6)qβ΅+(-1//2)qβ΄+(2//3)qΒ³+(-1//2)qΒ²+(1//6)q
(1//3)qβ΅+(-2//3)qΒ³+(1//3)q
(1//3)qβ΅+(-2//3)qΒ³+(1//3)q
Chevie.Uch.CycPoldegrees
β FunctionCycPoldegrees(uc::UnipotentCharacters)
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 CycPol
s. It is faster than CycPol.(degrees(uc))
.
julia> W=coxgroup(:G,2)
Gβ
julia> CycPoldegrees(UnipotentCharacters(W))
10-element Vector{CycPol{Rational{Int64}}}:
1
qβΆ
qΞ¦βΞ¦β/3
qΞ¦βΞ¦β/3
qΞ¦βΒ²Ξ¦β/6
qΞ¦βΒ²Ξ¦β/2
qΞ¦βΒ²Ξ¦β/2
qΞ¦βΒ²Ξ¦β/6
qΞ¦βΒ²Ξ¦βΒ²/3
qΞ¦βΒ²Ξ¦βΒ²/3
Chevie.Uch.unipotent_character
β Functionunipotent_character(W,l)
or unichar(W,l)
Constructs an object representing the unipotent character specified by l
of the algebraic group associated to the Coxeter group or Coxeter coset specified by W
. 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 (where the names are as given by charnames(UnipotentCharacters(W))
). Finally, l
can be a list of length the number of unipotent characters of W
, which specifies the coefficient to give to each unipotent character.
julia> W=coxgroup(:G,2)
Gβ
julia> u=unichar(W,7)
[Gβ]:<Gβ[-1]>
julia> v=unichar(W,"G2[E3]")
[Gβ]:<Gβ[ΞΆβ]>
julia> w=unichar(W,[1,0,0,-1,0,0,2,0,0,1])
[Gβ]:<Οβββ>-<Οβ³βββ>+2<Gβ[-1]>+<Gβ[ΞΆβΒ²]>
julia> unichar(W,fourier(UnipotentCharacters(W))[3,:])
[Gβ]:2//3<Οβ²βββ>-1//3<Οβ³βββ>+1//3<Οβββ>+1//3<Gβ[1]>-1//3<Gβ[ΞΆβ]>-1//3<Gβ[ΞΆβΒ²]>
The last line shows the almost character associated to the 3rd unipotent character of W
.
some limited arithmetic is available on unipotent characters:
julia> coefficients(u) # so that u==unichar(W,coefficients(u))
10-element Vector{Int64}:
0
0
0
0
0
0
1
0
0
0
julia> w-2u
[Gβ]:<Οβββ>-<Οβ³βββ>+<Gβ[ΞΆβΒ²]>
julia> w*w # scalar product
7
julia> degree(w)
Pol{Int64}: qβ΅-qβ΄-qΒ³-qΒ²+q+1
Base.show
β MethodBase.show(io::IO,w::UniChar)
The formatting of unipotent characters is affected by IO property :compact . If true
(the default) they are printed in a compact form. Otherwise, they are printed one unipotent character per line:
julia> xdisplay(w;compact=false)
[Gβ]:
<Οβββ> 1
<Οβββ> 0
<Οβ²βββ> 0
<Οβ³βββ> -1
<Οβββ> 0
<Οβββ> 0
<Gβ[-1]> 2
<Gβ[1]> 0
<Gβ[ΞΆβ]> 0
<Gβ[ΞΆβΒ²]> 1
Chevie.Uch.deligne_lusztig_character
β Functiondeligne_lusztig_character(W,w)
or dlchar(W,w)
This function returns the Deligne-Lusztig character $R_π ^π (1)$ of the algebraic group π
associated to the Coxeter group or Coxeter coset W
. The torus π
can be specified in 3 ways: if w
is an integer, it represents the w
-th conjugacy class (or phi
-conjugacy class for a coset WΟ
) of W
. Otherwise w
can be a word or an element of W
, and it represents the class (or Ο
-class) of w
.
julia> W=coxgroup(:G,2)
Gβ
julia> dlchar(W,3)
[Gβ]:<Οβββ>-<Οβββ>-<Οβ²βββ>+<Οβ³βββ>
julia> dlchar(W,W(1))
[Gβ]:<Οβββ>-<Οβββ>-<Οβ²βββ>+<Οβ³βββ>
julia> dlchar(W,[1])
[Gβ]:<Οβββ>-<Οβββ>-<Οβ²βββ>+<Οβ³βββ>
julia> dlchar(W,[1,2])
[Gβ]:<Οβββ>+<Οβββ>-<Οβββ>+<Gβ[-1]>+<Gβ[ΞΆβ]>+<Gβ[ΞΆβΒ²]>
Chevie.Uch.deligne_lusztigCharTable
β Functiondeligne_lusztigCharTable(W)
or dlCharTable(W)
for each conjugacy class of W
, gives the decomposition of R_{T_w}^G
in unipotent characters.
julia> dlCharTable(W)
6Γ10 Matrix{Int64}:
1 1 1 1 2 2 0 0 0 0
1 -1 1 -1 0 0 0 0 0 0
1 -1 -1 1 0 0 0 0 0 0
1 1 0 0 -1 0 1 0 1 1
1 1 0 0 0 -1 0 1 -1 -1
1 1 -1 -1 0 0 -2 -2 0 0
Chevie.Uch.almost_character
β Functionalmost_character(W,i)
or almostchar(W,i)
This function returns the i
-th almost unipotent character of the algebraic group π associated to the Coxeter group or Coxeter coset W
. If Ο
is the i
-th irreducible character of W
, the i
-th almost character is $R_Ο=Wβ»ΒΉβ_{wβ W} Ο(w) R_{π_w}^π (1)$ where $π_w$ is the maximal torus associated to the conjugacy class (or Ο
-conjugacy class for a coset) of w
.
julia> W=coxgroup(:B,2)
Bβ
julia> almostchar(W,3)
[Bβ]:<.11>
julia> almostchar(W,1)
[Bβ]:1//2<11.>+1//2<1.1>-1//2<.2>-1//2<Bβ>
Chevie.Uch.on_unipotents
β Functionon_unipotents(W,aut)
W
is a reflection group or reflection coset representing a finite reductive group $π ^F$, and aut
is an automorphism of $π ^F$ (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 $π ^F$ induced by aut
. This makes sense for Spetsial complex reflection groups and is implemented for them.
julia> WF=rootdatum("3D4")
Β³Dβ
julia> on_unipotents(Group(WF),WF.phi)
(1,7,2)(8,12,9)
Chevie.Uch.deligne_lusztig_lefschetz
β Functiondeligne_lusztig_lefschetz(h,m=0)
or dllefschetz(h,m=0)
Here h
is an element of a Hecke algebra associated to a Coxeter group W
or Coxeter coset WΟ
which itself is associated to an algebraic group π
. By DigneMichel1985, for $gβ π^F$, the number of fixed points of Fα΅
on the Deligne-Lusztig variety associated to the element wΟβWΟ
, have for m
divisible by a sufficently large integer d
, the form $β_Ο Ο_{(qα΅)}(T_wΟ)R_Ο(g)$ where Ο
runs over the irreducible characters of $WΟ$, where $R_Ο$ is the corresponding almost character, and where $Ο_{(qα΅)}$ is a character value of the Hecke algebra $H(WΟ,qα΅)$ of $WΟ$ with parameter qα΅
. This expression is called the Lefschetz character of the Deligne-Lusztig variety. If we consider qα΅
as an indeterminate x
, it can be seen as a sum of unipotent characters with coefficients character values of the generic Hecke algebra $H(WΟ,x)$. A more complicated formula involving the eigenvalues of Frobenius attached to unipotent characters applies for m
not prime to d
. The function returns this formula when a second parameter mβ 0
is given.
The function 'dllefschetz' 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.
julia> W=coxgroup(:A,2)
Aβ
julia> H=hecke(W,Pol(:q))
hecke(Aβ,q)
julia> T=Tbasis(H);
julia> dllefschetz(T(1,2))
[Aβ]:<111>-q<21>+qΒ²<3>
julia> dllefschetz((T(1)+T())*(T(2)+T()))
[Aβ]:q<21>+(qΒ²+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).
julia> H=hecke(spets(W,Perm(1,2)),Pol(:q)^2)
hecke(Β²Aβ,qΒ²)
julia> T=Tbasis(H);dllefschetz(T(1))
[Β²Aβ]:-<11>-q<Β²Aβ>+qΒ²<2>
Finally, there is a second form dllefschetz(H::HeckeAlgebra,w,i=0)
where the arguments are a Hecke algebra and an element of w
. This may be used for Spetses where we know the column of the CharTable
of H
for w
but not other columns of the spetsial Hecke algebra charcater table.
Chevie.Uch.lusztig_induce
β Functionlusztig_induce(W,u)
u
should be a unipotent character of a parabolic subcoset of the Coxeter coset W
. It represents a unipotent character Ξ»
of a Levi π
of the algebraic group π
attached to W
. The program returns the Lusztig induced $R_π^π(Ξ»)$.
julia> W=coxgroup(:G,2)
Gβ
julia> WF=spets(W)
Gβ
julia> T=subspets(WF,Int[],W(1))
Gβββ=Ξ¦βΞ¦β
julia> u=unichar(T,1)
[Gβββ=Ξ¦βΞ¦β]:<Id>
julia> lusztig_induce(WF,u)
[Gβ]:<Οβββ>-<Οβββ>-<Οβ²βββ>+<Οβ³βββ>
julia> dlchar(W,W(1))
[Gβ]:<Οβββ>-<Οβββ>-<Οβ²βββ>+<Οβ³βββ>
Chevie.Uch.lusztig_restrict
β Functionlusztig_restrict(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 Ξ³
of the algebraic group π
attached to W
, while R
represents a Levi subgroup L
. The program returns the Lusztig restriction $*R_π^π(Ξ³)$.
julia> W=coxgroup(:G,2)
Gβ
julia> WF=spets(W)
Gβ
julia> T=subspets(WF,Int[],W(1))
Gβββ=Ξ¦βΞ¦β
julia> u=dlchar(W,W(1))
[Gβ]:<Οβββ>-<Οβββ>-<Οβ²βββ>+<Οβ³βββ>
julia> lusztig_restrict(T,u)
[Gβββ=Ξ¦βΞ¦β]:4<Id>
julia> T=subspets(WF,Int[],W(2))
Gβββ=Ξ¦βΞ¦β
julia> lusztig_restrict(T,u)
[Gβββ=Ξ¦βΞ¦β]:0
Chevie.Lusztig.lusztig_induction_table
β Functionlusztig_induction_table(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 π
of the algebraic group π
associated to W
. The function returns an InductionTable
representing the Lusztig induction $R_π^π$ between unipotent characters.
julia> W=coxgroup(:B,3)
Bβ
julia> t=twistings(W,[1,3])
2-element Vector{Spets{FiniteCoxeterSubGroup{Perm{Int16},Int64}}}:
Bβββββ=AΜβΓAβΞ¦β
Bβββββ=AΜβΓAβΞ¦β
julia> lusztig_induction_table(t[2],W)
Lusztig induction from Bβββββ=AΜβΓAβΞ¦β to Bβ
βββββββ¬ββββββββββββββββββββββββ
β β11β 11 11β 2 2β 11 2β 2β
βββββββΌββββββββββββββββββββββββ€
β111. β 1 -1 -1 .β
β11.1 β -1 . 1 -1β
β1.11 β . . -1 .β
β.111 β -1 . . .β
β21. β . . . .β
β1.2 β 1 -1 . 1β
β2.1 β . 1 . .β
β.21 β . . . .β
β3. β . . . 1β
β.3 β . 1 1 -1β
βBβ:2 β . . 1 -1β
βBβ:11β 1 -1 . .β
βββββββ΄ββββββββββββββββββββββββ
Chevie.Families
β ModuleFamilies of unipotent characters
The blocks of the (rectangular) matrix $β¨Rα΅ͺ,Οβ©_{π ^F}$ when Ο
runs over Irr(W)
and Ο
runs over the unipotent characters, are called the Lusztig families. When π
is split and W
is a Coxeter group they correspond on the Irr(W)
side to two-sided Kazhdan-Lusztig cells β- for split Spetses they correspond to Rouquier blocks of the Spetsial Hecke algebra. The matrix of scalar products $β¨Rα΅ͺ,Οβ©_{π ^F}$ can be completed to a square matrix $β¨A_{Ο'},Οβ©_{π ^F}$ where $A_{Ο'}$ are the characteristic functions of character sheaves on $π ^F$; 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.
julia> W=coxgroup(:G,2)
Gβ
julia> uc=UnipotentCharacters(W);
julia> uc.families
3-element Vector{Family}:
Family(D(π β),[5, 6, 4, 3, 8, 7, 9, 10])
Family(Cβ,[1])
Family(Cβ,[2])
julia> uc.families[1]
Family(D(π β),[5, 6, 4, 3, 8, 7, 9, 10])
Drinfeld double of π β, Lusztigβ²s version
ββββββββββ¬βββββββββββββββββββββββββββββββββββββββββββββββββββββ
βlabel βeigen β
ββββββββββΌβββββββββββββββββββββββββββββββββββββββββββββββββββββ€
β(1,1) β 1 1//6 1//2 1//3 1//3 1//6 1//2 1//3 1//3β
β(gβ,1) β 1 1//2 1//2 . . -1//2 -1//2 . .β
β(gβ,1) β 1 1//3 . 2//3 -1//3 1//3 . -1//3 -1//3β
β(1,Ο) β 1 1//3 . -1//3 2//3 1//3 . -1//3 -1//3β
β(1,Ξ΅) β 1 1//6 -1//2 1//3 1//3 1//6 -1//2 1//3 1//3β
β(gβ,Ξ΅) β -1 1//2 -1//2 . . -1//2 1//2 . .β
β(gβ,ΞΆβ) β ΞΆβ 1//3 . -1//3 -1//3 1//3 . 2//3 -1//3β
β(gβ,ΞΆβΒ²)β ΞΆβΒ² 1//3 . -1//3 -1//3 1//3 . -1//3 2//3β
ββββββββββ΄βββββββββββββββββββββββββββββββββββββββββββββββββββββ
julia> charnames(uc)[uc.families[1].charNumbers]
8-element Vector{String}:
"phi2,1"
"phi2,2"
"phi1,3''"
"phi1,3'"
"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 "drinfeld_double" of a small finite group and the '.charLabels' come from this construction.
Chevie.Families.Family
β TypeFamily(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 struct Family
.
The other (optional) arguments add information to the family defined by the first argument. If given, the second argument becomes f.charNumbers
. If given, the third argument opt
is a Dict
whose keys are added to the resulting family.
If opt
has a key signs
, this 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.
julia> Family("C2")
Family(Cβ,4)
Drinfeld double D(β€/2)
ββββββββ¬βββββββββββββββββββββββββββββ
βlabel βeigen β
ββββββββΌβββββββββββββββββββββββββββββ€
β(1,1) β 1 1//2 1//2 1//2 1//2β
β(gβ,1)β 1 1//2 1//2 -1//2 -1//2β
β(1,Ξ΅) β 1 1//2 -1//2 1//2 -1//2β
β(gβ,Ξ΅)β -1 1//2 -1//2 -1//2 1//2β
ββββββββ΄βββββββββββββββββββββββββββββ
julia> Family("C2",4:7,Dict(:signs=>[1,-1,1,-1]))
Family(Cβ,4:7)
Drinfeld double D(β€/2)
ββββββββ¬βββββββββββββββββββββββββββββββββββ
βlabel βeigen signs β
ββββββββΌβββββββββββββββββββββββββββββββββββ€
β(1,1) β 1 1 1//2 -1//2 1//2 -1//2β
β(gβ,1)β 1 -1 -1//2 1//2 1//2 -1//2β
β(1,Ξ΅) β 1 1 1//2 1//2 1//2 1//2β
β(gβ,Ξ΅)β -1 -1 -1//2 -1//2 1//2 1//2β
ββββββββ΄βββββββββββββββββββββββββββββββββββ
CyclotomicNumbers.galois
β Methodgalois(f::Family,p::Int)
x->galois(x,p)
is applied to the Fourier matrix and eigenvalues of Frobenius of the family.
julia> f=UnipotentCharacters(complex_reflection_group(3,1,1)).families[2]
Family(0011,[4, 3, 2])
imprimitive family
βββββββ¬βββββββββββββββββββββββββββββββ
βlabelβeigen 1 2 3β
βββββββΌβββββββββββββββββββββββββββββββ€
β1 β ΞΆβΒ² β-3/3 β-3/3 -β-3/3β
β2 β 1 β-3/3 ΞΆβΒ²β-3/3 -ΞΆββ-3/3β
β3 β 1 -β-3/3 -ΞΆββ-3/3 ΞΆβΒ²β-3/3β
βββββββ΄βββββββββββββββββββββββββββββββ
julia> galois(f,-1)
Family(conj(0011),[4, 3, 2])
conj(imprimitive family)
βββββββ¬βββββββββββββββββββββββββββββββ
βlabelβeigen 1 2 3β
βββββββΌβββββββββββββββββββββββββββββββ€
β1 β ΞΆβ -β-3/3 -β-3/3 β-3/3β
β2 β 1 -β-3/3 -ΞΆββ-3/3 ΞΆβΒ²β-3/3β
β3 β 1 β-3/3 ΞΆβΒ²β-3/3 -ΞΆββ-3/3β
βββββββ΄βββββββββββββββββββββββββββββββ
PermGroups.Perms.invpermute
β Methodinvpermute(f::Family, p::Perm)
returns a copy of f
with the Fourier matrix, eigenvalues of Frobenius, :charLabelsβ¦
invpermuted by p
.
julia> f=UnipotentCharacters(complex_reflection_group(3,1,1)).families[2]
Family(0011,[4, 3, 2])
imprimitive family
βββββββ¬βββββββββββββββββββββββββββββββ
βlabelβeigen 1 2 3β
βββββββΌβββββββββββββββββββββββββββββββ€
β1 β ΞΆβΒ² β-3/3 β-3/3 -β-3/3β
β2 β 1 β-3/3 ΞΆβΒ²β-3/3 -ΞΆββ-3/3β
β3 β 1 -β-3/3 -ΞΆββ-3/3 ΞΆβΒ²β-3/3β
βββββββ΄βββββββββββββββββββββββββββββββ
julia> invpermute(f,Perm(1,2,3))
Family(0011,[2, 4, 3])
Permuted((1,2,3),imprimitive family)
βββββββ¬βββββββββββββββββββββββββββββββ
βlabelβeigen 3 1 2β
βββββββΌβββββββββββββββββββββββββββββββ€
β3 β 1 ΞΆβΒ²β-3/3 -β-3/3 -ΞΆββ-3/3β
β1 β ΞΆβΒ² -β-3/3 β-3/3 β-3/3β
β2 β 1 -ΞΆββ-3/3 β-3/3 ΞΆβΒ²β-3/3β
βββββββ΄βββββββββββββββββββββββββββββββ
Base.conj
β Methodconj(f::Family)
: is a synonym for 'galois(f,-1)'.
Chevie.Families.fourier
β Functionfourier(uc::UnipotentCharacters)
the Lusztig Fourier matrix for uc
.
fourier(f::Family
: returns the Fourier matrix for the family f
.
Chevie.Families.drinfeld_double
β Functiondrinfeld_double(g;lu=false,pivotal=nothing)
Given a (usually small) finite group Ξ
, 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 Ξ
, 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 Ξ
.
The elements of the family are in bijection with the set π (Ξ)
of pairs (x,Ο)
taken up to Ξ
-conjugacy, where xβΞ
and Ο
is an irreducible complex-valued character of C_Ξ(x)
. To such a pair Ο=(x,Ο)
is associated an eigenvalue of Frobenius defined by $Ο_Ο:=Ο(x)/Ο(1)$. Lusztig then defines a Fourier matrix Sβ
whose coefficient is given, for Ο=(x,Ο)
and Ο'=(x', Ο')
, by:
$Sβ_{Ο,Ο'}:=|C_Ξ(x)β»ΒΉ|β_{Οβ=(xβ,Οβ)}Οβ(x)Ο(yβ)$
where the sum is over all pairs Οββπ (Ξ)
which are Ξ
-conjugate to Ο'
and such that $yββ C_Ξ(x)$. This coefficient also represents the scalar product $β¨Ο,Ο'β©_{π^F}$ of the corresponding unipotent characters.
A way to understand the formula for $Sβ_{Ο,Ο'}$ better is to consider another basis of the complex vector space with basis π (Ξ)
, indexed by the pairs (x,y)
taken up to Ξ
-conjugacy, where x
and y
are commuting elements of Ξ
. This basis is called the basis of Mellin transforms, and given by:
$(x,y)=β_{Οβ Irr(C_Ξ(x))}Ο(y)(x,Ο)$
In the basis of Mellin transforms, the linear map Sβ
is given by (x,y)β¦(xβ»ΒΉ,yβ»ΒΉ)
and the linear transformation T
which sends Ο
to Ο_ΟΟ
becomes (x,y)β¦(x,xy)
. These are particular cases of the permutation representation of GLβ(β€)
on the basis of Mellin transforms where $\begin{pmatrix}a&b\cr c&d\end{pmatrix}$ acts by (x,y)β¦(xα΅yα΅,xαΆyα΅)
.
Fourier matrices in finite reductive groups are given by the above matrix Sβ
. But for non-rational Spetses, we use a different matrix S
which in the basis of Mellin transforms is given by (x,y)β¦(yβ»ΒΉ,x)
. Equivalently, the formula $S_{Ο,Ο'}$ differs from the formula for $Sβ_{Ο,Ο'}$ in that there is no complex conjugation of Οβ
; thus the matrix S
is equal to Sβ
multiplied on the right by the permutation matrix which corresponds to (x,Ο)β¦(x,Ο)
. The advantage of the matrix S
over Sβ
is that the pair S,T
satisfies directly the axioms for a fusion algebra (see below); also the matrix S
is symmetric, while Sβ
is Hermitian.
Thus there are two variants of 'drinfeld_double`:
drinfeld_double(g;lu=false)
returns a family containing Lusztig's Fourier matrix Sβ
, and an extra field '.perm' containing the permutation of the indices induced by (x,Ο)β¦(x,Ο)
, which allows to recover S
, as well as an extra field `:lusztig', set to 'true'.
drinfeld_double(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
: the group Ξ
.
:charLabels
: a list of labels describing the pairs (x,Ο)
, and thus also specifying in which order they are taken.
:fourierMat
: the Fourier matrix (the matrix S
or Sβ
depending on the call).
:eigenvalues
: the eigenvalues of Frobenius.
:xy
: a list of pairs '[x,y]' which are representatives of the Ξ
-orbits of pairs of commuting elements.
:mellinLabels
: a list of labels describing the pairs '[x,y]'.
:mellin
: the base change matrix between the basis (x,Ο)
and the basis of Mellin transforms, so that |f.fourierMat^(f.mellin^-1)| is the permutation matrix (for (x,y)β¦(yβ»ΒΉ,x)
or (x,y)β¦(yβ»ΒΉ,xβ»ΒΉ)
depending on the call).
:special
: the index of the special element, which is (x,Ο)=(1,1)
.
julia> drinfeld_double(coxsym(3)) # needs "using GAP"
Family(drinfeld_double(coxsym(3)),8)
Drinfeld double D(coxsym(3))
βββββββββ¬βββββββββββββββββββββββββββββββββββββββββββββββββββββ
βlabel βeigen β
βββββββββΌβββββββββββββββββββββββββββββββββββββββββββββββββββββ€
β(1,1) β 1 1//6 1//3 1//6 -1//2 -1//2 1//3 1//3 1//3β
β(1,X.2)β 1 1//3 2//3 1//3 . . -1//3 -1//3 -1//3β
β(1,X.3)β 1 1//6 1//3 1//6 1//2 1//2 1//3 1//3 1//3β
β(21,1) β 1 -1//2 . 1//2 1//2 -1//2 . . .β
β(21,-1)β -1 -1//2 . 1//2 -1//2 1//2 . . .β
β(3,1) β 1 1//3 -1//3 1//3 . . 2//3 -1//3 -1//3β
β(3,ΞΆβ) β ΞΆβ 1//3 -1//3 1//3 . . -1//3 -1//3 2//3β
β(3,ΞΆβΒ²)β ΞΆβΒ² 1//3 -1//3 1//3 . . -1//3 2//3 -1//3β
βββββββββ΄βββββββββββββββββββββββββββββββββββββββββββββββββββββ
julia> drinfeld_double(coxsym(3);lu=true)
Family(Ldrinfeld_double(coxsym(3)),8)
Lusztigβ²sDrinfeld double D(coxsym(3))
βββββββββ¬βββββββββββββββββββββββββββββββββββββββββββββββββββββ
βlabel βeigen β
βββββββββΌβββββββββββββββββββββββββββββββββββββββββββββββββββββ€
β(1,1) β 1 1//6 1//3 1//6 -1//2 -1//2 1//3 1//3 1//3β
β(1,X.2)β 1 1//3 2//3 1//3 . . -1//3 -1//3 -1//3β
β(1,X.3)β 1 1//6 1//3 1//6 1//2 1//2 1//3 1//3 1//3β
β(21,1) β 1 -1//2 . 1//2 1//2 -1//2 . . .β
β(21,-1)β -1 -1//2 . 1//2 -1//2 1//2 . . .β
β(3,1) β 1 1//3 -1//3 1//3 . . 2//3 -1//3 -1//3β
β(3,ΞΆβ) β ΞΆβ 1//3 -1//3 1//3 . . -1//3 2//3 -1//3β
β(3,ΞΆβΒ²)β ΞΆβΒ² 1//3 -1//3 1//3 . . -1//3 -1//3 2//3β
βββββββββ΄βββββββββββββββββββββββββββββββββββββββββββββββββββββ
Chevie.Families.ndrinfeld_double
β Functionndrinfeld_double(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.
julia> Families.ndrinfeld_double(complex_reflection_group(5))
378
Chevie.Families.family_imprimitive
β Functionfamily_imprimitive(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 containing S
.
julia> family_imprimitive([[0,1],[1],[0]])
Family(0011,3)
imprimitive family
βββββββ¬βββββββββββββββββββββββββββββββ
βlabelβeigen 1 2 3β
βββββββΌβββββββββββββββββββββββββββββββ€
β1 β ΞΆβΒ² β-3/3 -β-3/3 β-3/3β
β2 β 1 -β-3/3 ΞΆβΒ²β-3/3 -ΞΆββ-3/3β
β3 β 1 β-3/3 -ΞΆββ-3/3 ΞΆβΒ²β-3/3β
βββββββ΄βββββββββββββββββββββββββββββββ
Chevie.Families.FamiliesClassical
β FunctionFamiliesClassical(l)
l
should be a list of symbols which classify the unipotent characters of a classical reductive group, like symbols(2,r)
for type Bα΅£
or Cα΅£
, or symbols(2,r,0)
for type Dα΅£
. The function returns the list of families determined by these symbols.
julia> FamiliesClassical(symbols(2,3)) # for a reductive group of type Bβ
6-element Vector{Family}:
Family(112,[2])
Family(022,[6])
Family(3,[9])
Family(01123,[1, 3, 8, 11])
Family(0112233,[4])
Family(013,[5, 7, 10, 12])
Base.:*
β Method<f>*<g>
: returns the tensor product of two families <f> and <g>; the Fourier matrix is the Kronecker product of the matrices for <f> and <g>, and the eigenvalues of Frobenius are the pairwise products.
Chevie.Families.special
β Functionspecial(f::Family)
the index of the special character in f
Chevie.Families.cospecial
β Functioncospecial(f::Family)
the index of the cospecial character in f
Chevie.Families.fusion_algebra
β Functionfusion_algebra(f::Family)
or fusion_algebra(S,special=1)
All the Fourier matrices S
in Chevie are unitary, that is Sβ»ΒΉ=conj(S)
, and have a special line s
(the line of index s=special(f)
for a family f
) such that no entry Sβ,α΅’
is equal to 0
. Further, they have the property that the sums Cα΅’,β±Ό,β=sumβ Sα΅’,β Sβ±Ό,β conj(Sβ,β)/Sβ,β
take integral values. Finally, S
has the property that complex conjugation does a permutation with signs Ο
of the lines of S
.
It follows that we can define a β€
-algebra A
as follows: it has a basis bα΅’
indexed by the lines of S
, and has a multiplication defined by the fact that the coefficient of bα΅’bβ±Ό
on bβ
is equal to Cα΅’,β±Ό,β
. This algebra can be specified by giving a family f
or just its Fourier matrix and the number of its special line.
A
is commutative, and has as unit the element bβ
; the basis Ο(bα΅’)is
dual to bα΅’
for the linear form (bα΅’,bβ±Ό)=Cα΅’,β±Ό,Οβββ`.
julia> W=complex_reflection_group(4)
Gβ
julia> uc=UnipotentCharacters(W);f=uc.families[4];
julia> A=fusion_algebra(fourier(f),1)
Fusion Algebra dim.5
julia> b=basis(A)
5-element Vector{AlgebraElt{Chevie.Families.FusionAlgebra, Int64}}:
Bβ
Bβ
Bβ
Bβ
Bβ
julia> b*permutedims(b)
5Γ5 Matrix{AlgebraElt{Chevie.Families.FusionAlgebra, Int64}}:
Bβ Bβ Bβ Bβ Bβ
Bβ -Bβ+Bβ
Bβ+Bβ Bβ-Bβ Bβ
Bβ Bβ+Bβ -Bβ+Bβ
-Bβ+Bβ Bβ
Bβ Bβ-Bβ -Bβ+Bβ Bβ+Bβ-Bβ
-Bβ
Bβ
Bβ Bβ -Bβ Bβ
julia> CharTable(A)
CharTable(Fusion Algebra dim.5)
βββ¬ββββββββββββββββββ
β β1 2 3 4 5β
βββΌββββββββββββββββββ€
β1β1 β-3 -β-3 2 -1β
β2β1 1 1 . 1β
β3β1 -1 -1 . 1β
β4β1 . . -1 -1β
β5β1 -β-3 β-3 2 -1β
βββ΄ββββββββββββββββββ