Classes/characters of reflection groups
Chevie.Chars
Chevie.Chars.CharTable
Chevie.Chars.J_induction_table
Chevie.Chars.charinfo
Chevie.Chars.charnames
Chevie.Chars.classinfo
Chevie.Chars.classnames
Chevie.Chars.conjPerm
Chevie.Chars.detPerm
Chevie.Chars.fakedegree
Chevie.Chars.fakedegrees
Chevie.Chars.induction_table
Chevie.Chars.j_induction_table
Chevie.Chars.on_chars
Chevie.Chars.representation
Chevie.Chars.representations
Chevie.Chars.schur_functor
Chevie.Chars
β ModuleCharacters and conjugacy classes of complex reflection groups.
The CharTable
of a finite complex reflection group W
is computed using the decomposition of W
in irreducible groups (see refltype
). For each irreducible group the character table is either computed using recursive formulas for the infinite series, or read into the system from a library file for the exceptional types. Thus, character tables can be obtained quickly even for very large groups (e.g., Eβ). Similar remarks apply for conjugacy classes.
The conjugacy classes and irreducible characters of irreducible finite complex reflection groups have canonical labelings by certain combinatorial objects; these labelings are used in the tables we give. For the classes, these are partitions or partition tuples for the infinite series, or, for exceptional Coxeter groups, Carter's admissible diagrams Carter1972; for other primitive complex reflection groups we just use words in the generators to specify the classes. For the characters, these are again partitions or partition tuples for the infinite series, and for the others they are pairs of two integers (d,e)
where d
is the degree of the character and e
is the smallest symmetric power of the reflection representation containing the given character as a constituent (the b
-invariant of the character). This information is given by the functions classinfo
and charinfo
. When you display the character table, the canonical labelings for classes and characters are displayed.
A typical example is coxgroup(:A,n)
, the symmetric group πβββ
where classes and characters are parameterized by partitions of n+1
(this is also the case for coxsym(n+1)
).
julia> W=coxgroup(:A,3)
Aβ
julia> CharTable(W)
CharTable(Aβ)
ββββββ¬ββββββββββββββββββ
β β1111 211 22 31 4β
ββββββΌββββββββββββββββββ€
β1111β 1 -1 1 1 -1β
β211 β 3 -1 -1 . 1β
β22 β 2 . 2 -1 .β
β31 β 3 1 -1 . -1β
β4 β 1 1 1 1 1β
ββββββ΄ββββββββββββββββββ
julia> W=coxgroup(:G,2)
Gβ
julia> ct=CharTable(W)
CharTable(Gβ)
βββββββ¬βββββββββββββββββββββ
β βAβ AΜβ Aβ Gβ Aβ Aβ+AΜββ
βββββββΌβββββββββββββββββββββ€
βΟβββ β 1 1 1 1 1 1β
βΟβββ β 1 -1 -1 1 1 1β
βΟβ²ββββ 1 1 -1 -1 1 -1β
βΟβ³ββββ 1 -1 1 -1 1 -1β
βΟβββ β 2 . . 1 -1 -2β
βΟβββ β 2 . . -1 -1 2β
βββββββ΄βββββββββββββββββββββ
julia> ct.charnames
6-element Vector{String}:
"\phi_{1,0}"
"\phi_{1,6}"
"\phi_{1,3}'"
"\phi_{1,3}''"
"\phi_{2,1}"
"\phi_{2,2}"
julia> ct.classnames
6-element Vector{String}:
"A_0"
"\tilde A_1"
"A_1"
"G_2"
"A_2"
"A_1+\tilde A_1"
Reflection groups have fake degrees (see fakedegrees
), whose valuation and degree give two integers b,B
for each irreducible character of W
. For spetsial groups (which include finite Coxeter groups), the valuation and degree of the generic degrees of the Hecke algebra give two more integers a,A
(for Coxeter groups see Carter1985, Ch.11 for more details). These integers are also used in the operations of truncated induction, see j_induction_table
and J_induction_table
.
Iwahori-Hecke algebras and cyclotomic Hecke algebras also have character tables, see the corresponding chapters.
We now describe for each type our conventions for labeling the classes and characters.
Type Aβ
(nβ₯0
). In this case we have W β
πβββ
. The classes and characters are labelled by partitions of n+1
. The partition labelling a class is the cycle type of the elements in that class; the representative in '.classtext' is the concatenation of the words corresponding to each part, where the word for a part i
is the product of i-1
consecutive generators (starting one higher than the last generator used for the previous parts). The partition labelling a character describes the type of the Young subgroup such that the trivial character induced from this subgroup contains that character with multiplicity 1
and such that every other character occurring in this induced character has a higher a
-value. Thus, the sign character is labelled by the partition (1βΏβΊΒΉ)
and the trivial character by the partition (n+1)
. The character of the reflection representation of W
is labelled by (n,1)
.
Type Bβ
(nβ₯2
). In this case W=W(Bβ)
is isomorphic to the wreath product of the cyclic group of order 2
with the symmetric group πβ
. Hence the classes and characters are parameterized by pairs of partitions such that the total sum of their parts equals n
. The pair corresponding to a class describes the signed cycle type for the elements in that class, as in Carter1972. We use the convention that if (Ξ»,ΞΌ)
is such a pair then Ξ»
corresponds to the positive and ΞΌ
to the negative cycles. Thus, (1βΏ,-)
and (-,1βΏ)
label respectively the trivial class and the class of the longest element.
The pair corresponding to an irreducible character is determined via Clifford theory, as follows. We have a semidirect product decomposition W(Bβ)=N β πβ
where N
is the standard n
-dimensional π½ββΏ
-vector space. For a,b β₯ 0
such that n=a+b
let $Ξ·_{a,b}$ be the irreducible character of N
which takes value 1
on the first a
standard basis vectors and value -1
on the last b
standard basis vectors of N
. Then the inertia subgroup of $Ξ·_{a,b}$ has the form $T_{a,b}=N.(π_a Γ π_b)$ and we can extend $Ξ·_{a,b}$ trivially to an irreducible character $Ξ·Μ_{a,b}$ of $T_{a,b}$. Let Ξ±
and Ξ²
be partitions of a
and b
, respectively. We take the tensor product of the corresponding irreducible characters of π_a
and π_b
and regard this as an irreducible character of $T_{a,b}$. Multiplying this character with $Ξ·Μ_{a,b}$ and inducing to W(Bβ)
yields an irreducible character $Ο= Ο_{(Ξ±,Ξ²)}$ of W(Bβ)
. This defines the correspondence between irreducible characters and pairs of partitions as above.
For example, the pair ((n),-)
labels the trivial character and (-,(1βΏ))
labels the sign character. The character of the natural reflection representation is labeled by ((n-1),(1))
.
Type Dβ
(nβ₯4
). In this case W=W(Dβ)
can be embedded as a subgroup of index 2
into the Coxeter group W(Bβ)
. The intersection of a class of W(Bβ)
with W(Dβ)
is either empty or a single class in W(Dβ)
or splits up into two classes in W(Dβ)
. This also leads to a parameterization of the classes of W(Dβ)
by pairs of partitions (Ξ»,ΞΌ)
as before but where the number of parts of ΞΌ
is even and where there are two classes of this type if ΞΌ
is empty and all parts of Ξ»
are even. In the latter case we denote the two classes in W(Dβ)
by (Ξ»,+)
and (Ξ»,-)
, where we use the convention that the class labeled by (Ξ»,+)
contains a representative which can be written as a word in {sβ,sβ,β¦,sβ}
and (Ξ»,-)
contains a representative which can be written as a word in {sβ,sβ, β¦,sβ}
.
By Clifford theory the restriction of an irreducible character of W(Bβ)
to W(Dβ)
is either irreducible or splits up into two irreducible components. Let (Ξ±,Ξ²)
be a pair of partitions with total sum of parts equal to n
. If Ξ±!=Ξ²
then the restrictions of the irreducible characters of W(Bβ)
labeled by (Ξ±,Ξ²)
and (Ξ²,Ξ±)
are irreducible and equal. If Ξ±=Ξ²
then the restriction of the character labeled by (Ξ±,Ξ±)
splits into two irreducible components which we denote by (Ξ±,+)
and (Ξ±,-)
. Note that this can only happen if n
is even. In order to fix the notation we use a result of Stembridge1989 which describes the value of the difference of these two characters on a class of the form (Ξ»,+)
in terms of the character values of the symmetric group π_{n/2}
. Recall that it is implicit in the notation (Ξ»,+)
that all parts of Ξ»
are even. Let Ξ»'
be the partition of n/2
obtained by dividing each part by 2
. Then the value of Ο_{(Ξ±,-)}-Ο_{(Ξ±,+)}
on an element in the class (Ξ»,+)
is given by 2^{k(Ξ»)}
times the value of the irreducible character of π_{n/2}
labeled by Ξ±
on the class of cycle type Ξ»'
. (Here, k(Ξ»)
denotes the number of non-zero parts of Ξ»
.)
The labels for the trivial, the sign and the natural reflection character are the same as for W(Bβ)
, since these characters are restrictions of the corresponding characters of W(Bβ)
.
The groups G(d,1,n)
. They are isomorphic to the wreath product of the cyclic group of order d
with the symmetric group πβ
. Hence the classes and characters are parameterized by d
-tuples of partitions such that the total sum of their parts equals n
. The words chosen as representatives of the classes are, when d>2
, computed in a slightly different way than for Bβ
, in order to agree with the words on which Ram and Halverson compute the characters of the Hecke algebra. First the parts of the d
partitions are merged in one big partition and sorted in increasing order. Then, to a part i
coming from the j
-th partition is associated the word (l+1β¦1β¦ l+1)Κ²β»ΒΉl+2β¦l+i
where l
is the highest generator used to express the previous part.
The d
-tuple corresponding to an irreducible character is determined via Clifford theory in a similar way than for the Bβ
case. The identity character has the first partition with one part equal n
and the other ones empty. The character of the reflection representations has the first two partitions with one part equal respectively to n-1
and to 1
, and the other partitions empty.
The groups G(de,e,n)
. They are normal subgroups of index e
in G(de,1,n)
. The quotient is cyclic, generated by the image g
of the first generator of G(de,1,n)
. The classes are parameterized as the classes of G(de,e,n)
with an extra information for a component of a class which splits.
According to Hugues1985, a class C
of G(de,1,n)
parameterized by a de
-partition $(Sβ,β¦,S_{de-1})$ is in G(de,e,n)
if e
divides $βα΅’ i β_{pβ Sα΅’}p$. It splits in d
classes for the largest d
dividing e
and all parts of all Sα΅’
and such that Sα΅’
is empty if d
does not divide i
. If w
is in C
then 'gβ± w gβ»β±' for 'i in 0:d-1' are representatives of the classes of G(de,e,n)
which meet C
. They are described by appending the integer i
to the label for C
.
The characters are described by Clifford theory. We make g
act on labels for characters of G(de,1,n)
. The action of g
permutes circularly by d
the partitions in the de
-tuple. A character has same restriction to G(de,e,n)
as its transform by g
. The number of irreducible components of its restriction is equal to the order k
of its stabilizer under powers of g
. We encode a character of G(de,e,n)
by first, choosing the smallest for lexicographical order label of a character whose restriction contains it; then this label is periodic with a motive repeated k
times; we represent the character by one of these motives, to which we append E(k)β±
for 'i in 0:k-1' to describe which component of the restriction we choose.
Types Gβ
and Fβ
. The matrices of character values and the orderings and labelings of the irreducible characters are exactly the same as in Carter1985, p.412/413: in type Gβ
the character Οβ,β'
takes the value -1 on the reflection associated to the long simple root; in type Fβ
, the characters Οβ,ββ'
, Οβ,β'
, Οβ,β'
, Οβ,β'
and Οβ,β'
occur in the induced of the identity from the Aβ
corresponding to the short simple roots; the pairs (Οβ,ββ'
, Οβ,ββ³
) and (Οβ,β'
, Οβ,ββ³
) are related by tensoring by sign; and finally Οβ,ββ³
is the exterior square of the reflection representation. Note, however, that we put the long root at the left of the Dynkin diagrams to be in accordance with the conventions in Lusztig1985, (4.8) and (4.10).
The classes are labeled by Carter's admissible diagrams Carter1972. A character is labeled by a pair (d,b)
where d
denotes the degree and b
the corresponding b
-invariant. If there are several characters with the same pair (d,b)
we attach a prime to them, as in Carter1985.
Types Eβ,Eβ,Eβ
. The character tables are obtained by specialization of those of the Hecke algebra. The classes are labeled by Carter's admissible diagrams Carter1972. A character is labeled by the pair (d,b)
where d
denotes the degree and b
is the corresponding b
-invariant. For these types, this gives a unique labeling of the characters.
Non-crystallographic types Iβ(m)
, Hβ
, Hβ
. In these cases we do not have canonical labelings for the classes. We label them by reduced expressions.
Each character for type Hβ
is uniquely determined by the pair (d,b)
where d
is the degree and b
the corresponding b
-invariant. For type Hβ
there are just two characters (those of degree 30
) for which the corresponding pairs are the same. These two characters are nevertheless distinguished by their fake degrees: the character Οββ,ββ'
has fake degree qΒΉβ°+qΒΉΒ²+
higher terms, while Οββ,βββ³
has fake degree qΒΉβ°+qΒΉβ΄+
higher terms. The characters in the table for type Hβ
are ordered in the same way as in Alvis and Lusztig1982.
Finally, the characters of degree 2
for type Iβ(m)
are ordered as follows. The matrix representations affording the characters of degree 2
are given by: $Ο_j : sβsβ β¦ \begin{pmatrix}\zeta_m^j&0\\0&\zeta_m^{-j}\end{pmatrix}, sββ¦\begin{pmatrix}0&1\\1&0\end{pmatrix},$ where 1 β€ j β€ β(m-1)/2β
. The reflection representation is Οβ
. The characters in the table are ordered by listing first the characters of degree 1 and then Οβ,Οβ,β¦
.
Primitive complex reflection groups Gβ
to Gββ
. The groups Gββ=Hβ
, Gββ=Fβ
, Gββ=Hβ
are exceptional Coxeter groups and have been explained above. Similarly for the other groups labels for characters consist primarily of the pair (d,b)
where d
denotes the degree and b
is the corresponding b
-invariant. This is sufficient for Gβ
, Gββ
, Gββ
and Gββ
. For other groups there are pairs or triples of characters which have the same (d,b)
value. We disambiguate these according to the conventions of Malle2000 for Gββ, Gββ, Gββ, Gββ
and Gββ
:
- For
Gββ
:
The fake degree of Οβ,β
'
(resp. Οβ,ββ'
, Οβ,ββ³
) has smaller degree that of Οβ,β
β³
(resp. Οβ,βββ³
, Οβ,β'
). The characters Οβ
,ββ
'
and Οβ
,β'
occur with multiplicity 1 in the induced from the trivial character of the parabolic subgroup of type Aβ
generated by the first and third generator (it is asserted mistakenly in Malle2000 that Οβ
,ββ³
does not occur in this induced; it occurs with multiplicity 2).
- For
Gββ
:
The character Οβ,βββ΄
is the exterior square of Οβ,β
; its complex conjugate is Οβ,βββ
. The character Οββ
,ββ³
occurs in Οβ,ββΟβ,β
; the character Οββ
,βββ³
is tensored by the sign character from Οββ
,ββ³
. Finally Οβ,ββ'
occurs in the induced from the trivial character of the standard parabolic subgroup of type Aβ
generated by the first, second and fourth generators.
- For
Gββ
:
The characters Οββ
,β'
, Οββ
,ββ'
and Οββ
,ββ³
occur in Οβ,ββΟββ,β
; the character Οββ,ββ'
is complex conjugate of Οββ,β
; the character Οββ
,ββ'
is tensored by sign of Οββ
,β'
. The two terms of maximal degree of the fakedegree of Οββ,ββ'
are qβ΅β°+qβ΄βΆ
while for Οββ,βββ³
they are qβ΅β°+2qβ΄βΆ
.
- For
Gββ
:
The two terms of maximal degree of the fakedegree of Οββ,β'
are qΒ²βΈ+qΒ²βΆ
while for Οββ,ββ³
they are qΒ²βΈ+qΒ²β΄
. The terms of maximal degree of the fakedegree of Οββ,β
'
are qΒ³ΒΉ+qΒ²βΉ
while for Οββ,β
β³
they are qΒ³ΒΉ+2qΒ²βΉ
. The character Οββ,ββ'
is tensored by sign of Οββ,β'
and Οββ,ββ'
is tensored by sign of Οββ,β
'
.
- For
Gββ
:
The character Οββ,ββ'
occurs in Οβ,ββΟββ
,ββ
. The character Οββ,β'
is rational. The character Οββ,ββ³
occurs in Οβ,ββΟββ
,ββ
. The character Οββ,ββ
'
is rational.The character Οββ,ββ
β³
is tensored by the determinant character of Οββ,ββ³
. The character Οβ
ββ,ββ'
is rational. The character Οβ
ββ,βββ΄
occurs in Οβ,ββΟβββ,ββ
. The character Οβββ,ββ'
occurs in Οβ,ββΟβββ,ββ
. The character Οβββ,βββ³
occurs in Οβ,ββΟβββ,ββ
. The character Οβ
ββ,ββ'
occurs in Οβ,ββΟβββ
,ββ
. The character Οβββ
,β'
is complex conjugate of Οβββ
,β
, and Οβββ,ββ'
is complex conjugate of Οβββ,ββ
. The character Οβββ,ββ'
is complex conjugate of Οβββ,ββ
. Finally Οβββ,ββ'
occurs in induced from the trivial character of the standard parabolic subgroup of type Aβ
generated by the generators of Gββ
with the third one omitted.
For the groups Gβ
and Gβ
we adopt the following conventions. For Gβ
they are compatible with those of MalleRouquier2003 and BroueMalleMichel2014.
- For
Gβ
:
We let W=complex_reflection_group(5)
, so the generators are W(1)
and W(2)
.
The character Οβ,β'
(resp. Οβ,ββ'
, Οβ,β'
) takes the value 1
(resp. ΞΆβ
, -ΞΆβ
) on W(1)
. The character Οβ,ββ³
is complex conjugate to Οβ,ββ
, and the character Οβ,β'
is complex conjugate to Οβ,β'
. The character Οβ,β
β³
is complex conjugate to Οβ,β
; Οβ,β
'
takes the value -1
on W(1)
. The character Οβ,β'
is complex conjugate to Οβ,β
'
.
- For
Gβ
:
We let W=complex_reflection_group(7)
, so the generators are W(1)
, W(2)
and W(3)
.
The characters Οβ,β'
and Οβ,ββ'
take the value 1
on W(2)
. The character Οβ,ββ³
is complex conjugate to Οβ,ββ
and Οβ,β'
is complex conjugate to Οβ,β'
. The characters Οβ,ββ'
and Οβ,ββ'
take the value ΞΆβ
on W(2)
. The character Οβ,βββ³
is complex conjugate to Οβ,ββ
and Οβ,ββ'
is complex conjugate to Οβ,ββ'
. The character Οβ,β'
takes the value -ΞΆβ
on W(2)
and Οβ,ββ'
takes the value -1
on W(2)
. The characters Οβ,βββ³
, Οβ,β
β³
, Οβ,ββ΄
and Οβ,β
are Galois conjugate, as well as the characters Οβ,β'
, Οβ,ββ'
, Οβ,ββ'
and Οβ,β
'
. The character Οβ,β'
is complex conjugate to Οβ,ββ
and Οβ,ββ΄
is complex conjugate to Οβ,β'
.
Finally, for the remaining groups Gβ, Gβ
to Gββ, Gββ
to Gββ
, Gββ
, Gββ
, Gββ
and Gββ
there are only pairs of characters with same value (d,b)
. We give labels uniformly to these characters by applying in order the following rules :
If the two characters have different fake degrees, label
Ο_{d,b}'
the one whose fake degree is minimal for the lexicographic order of polynomials (starting with the highest term).For the not yet labeled pairs, if only one of the two characters has the property that in its Galois orbit at least one character is distinguished by its
(d,b)
-invariant, label itΟ_{d,b}'
.For the not yet labeled pairs, if the minimum of the
(d,b)
-value (for the lexicographic order(d,b)
) in the Galois orbits of the two character is different, labelΟ_{d,b}'
the character with the minimal minimum.We define now a new invariant for characters: consider all the pairs of irreducible characters
Ο
andΟ
uniquely determined by their(d,b)
-invariant such thatΟ
occurs with non-zero multiplicitym
inΟβΟ
. We definet(Ο)
to be the minimal (for lexicographic order) possible list(d(Ο),b(Ο),d(Ο),b(Ο),m)
.
For the not yet labeled pairs, if the t-invariants are different, label Ο_{d,b}'
the character with the minimal t
-invariant.
After applying the last rule all the pairs will be labelled for the considered groups. The labelling obtained is compatible for Gββ
, Gββ
, Gββ
and Gββ
with that of Malle2000 and for Gβ
with that described in MalleRouquier2003.
We should emphasize that for all groups with a few exceptions, the parameters for characters do not depend on any non-canonical choice. The exceptions are G(de,e,n)
with e>1
, and Gβ
, Gβ
, Gββ
, Gββ
, Gββ
and Gββ
, groups which admit outer automorphisms preserving the set of reflections, so choices of a particular value on a particular generator must be made for characters which are not invariant by these automorphisms.
Labels for the classes. For the exceptional complex reflection groups, the labels for the classes represent the decomposition of a representative of the class as a product of generators, with the additional conventions that 'z' represents the generator of the center and for well-generated groups 'c' represents a Coxeter element (a product of the generators which is a regular element for the highest reflection degree).
Chevie.Chars.CharTable
β TypeCharTable is a structure to hold character tables of groups and Hecke algebras
Chevie.Chars.on_chars
β Functionon_chars(G,aut)
aut
is an automorphism of the group G
(for a permutation group, this could be given as a permutation normalizing G
). The result is the permutation of the indices of the irreducible characters induced by aut
.
julia> WF=rootdatum("3D4")
Β³Dβ
julia> on_chars(Group(WF),WF.phi)
(1,2,7)(8,9,12)
Chevie.Chars.charinfo
β Functioncharinfo(W)
returns information about the irreducible characters of the finite reflection group or Spets W
. The result is an object with various entries describing properties of the irreducible characters of W
. This object prints at the Repl or in Pluto or Jupyter as a table synthesizing most information.
A field not in the table is .charparams
: it contains parameters for the irreducible characters. A parameter is a list with one item for each irreducible component of W
(as given by refltype
). For an irreducible W
see the helpstring for Chars
for what are the parameters.
julia> charinfo(coxgroup(:G,2)).charparams
6-element Vector{Vector{Vector{Int64}}}:
[[1, 0]]
[[1, 6]]
[[1, 3, 1]]
[[1, 3, 2]]
[[2, 1]]
[[2, 2]]
julia> charinfo(coxgroup(:G,2))
ββββ¬βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
βn0β name ext b B a A spaltenstein lusztig symbolβ
ββββΌβββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ€
β1 β Οβββ Id 0 0 0 0 1 1 (0,0,0,0,0,2)β
β2 β Οβββ det 6 6 6 6 Ξ΅ Ξ΅ (01,01,01,01,01,12)β
β3 βΟβ²βββ 3 3 1 5 Ξ΅β Ξ΅β² (0,0,1+)β
β4 βΟβ³βββ 3 3 1 5 Ξ΅_c Ξ΅β³ (0,0,1-)β
β5 β Οβββ ΞΒΉ 1 5 1 5 ΞΈβ² ΞΈβ² (0,0,0,0,1,1)β
β6 β Οβββ 2 4 1 5 ΞΈβ³ ΞΈβ³ (0,0,0,1,0,1)β
ββββ΄βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
In the table printed at the Repl, the columns reflect various fields of charinfo
. The column name
reflects the field .charnames
, a name computed from .charparams
. This is the same as charnames(io,W)
where here io
being the Repl has the property :limit
true.
The column ext
shows the exterior powers of the reflection representation. It corresponds to the field .extrefl
which is present only if W
is irreducible. Otherwise, only two items are shown in the column: Id
corresponds to the field .positionId
and shows the trivial character. det
corresponds to the field .positionDet
and shows the determinant character (for Coxeter groups the sign character). When W
is irreducible, the characters marked Ξβ±
are the i
-th exterior power of the reflection representation. They are irreducible by a theorem of Steinberg.
The column b
shows the field .b
listing for each character the valuation of the fake degree, and the column B
shows the field .B
, the degree of the fake degree.
The columns a
and A
only appear for Spetsial groups. They correspond then to the fields .a
and .A
, and contain respectively the valuation and the degree of the generic degree of the character (in the one-parameter Hecke algebra hecke(W,Pol())
for W
).
For irreducible groups, the table shows sometimes additional columns, corresponding to a field of the same name.
for Fβ
, the column kondo
gives the labeling of the characters given by Kondo, also used in Lusztig1985, (4.10).
for Eβ, Eβ, Eβ
the column frame
gives the labeling of the characters given by Frame, also used in Lusztig1985, (4.11), (4.12), and (4.13).
for Gβ
the column spaltenstein
gives the labeling of the characters given by Spaltenstein.
for G(de,e,2)
even e
and d>1
, the column malle
gives the parameters for the characters used in Malle1996.
If W
is irreducible spetsial and imprimitive, the column 'symbol(corresponding to the field
.charSymbols`) shows the symbol attached to the corresponding unipotent caracter.
If W isa Spets
, the column restr.
(corresponding to the field .charRestrictions
) gives the number of the corresponding character of Group(W)
.
Finally, the field .hgal
contains the permutation of the characters resulting from a Galois action on the characters of H=hecke(W,Pol()^e)
where e
is the order of the center of W
. H
splits by taking v
an e
-th root of Pol()
, and .hgal
records the permutation effected by the Galois action v->E(e)*v
(charinfo
does not have the key :hgal
if this permutation is trivial). .hgal*conj
, where conj
is the complex conjugaison, is the Opdam involution.
julia> charinfo(complex_reflection_group(24))
ββββ¬ββββββββββββββββββββββ
βn0β name ext b B a Aβ
ββββΌββββββββββββββββββββββ€
β1 β Οβββ Id 0 0 0 0β
β2 βΟββββ det 21 21 21 21β
β3 β Οβββ 8 18 8 20β
β4 β Οβββ ΞΒΉ 1 11 1 13β
β5 βΟββββ ΞΒ² 10 20 8 20β
β6 β Οβββ 3 13 1 13β
β7 β Οβββ 2 12 1 13β
β8 β Οβββ 9 19 8 20β
β9 β Οβββ 6 18 6 18β
β10β Οβββ 3 15 3 15β
β11β Οβββ 4 16 4 17β
β12β Οβββ
5 17 4 17β
ββββ΄ββββββββββββββββββββββ
hgal=(11,12)
Chevie.Chars.charnames
β Methodcharnames(ComplexReflectionGroup or Spets;options...)
charnames(io::IO,ComplexReflectionGroup or Spets)
returns the list of character names for the reflection group or Spets W
. The options may imply alternative names in certain cases, or a different formatting of names in general. They can be specified by IO
attributes if giving an IO
as argument.
julia> W=coxgroup(:G,2)
Gβ
julia> charnames(W;limit=true)
6-element Vector{String}:
"Οβββ"
"Οβββ"
"Οβ²βββ"
"Οβ³βββ"
"Οβββ"
"Οβββ"
julia> charnames(W;TeX=true)
6-element Vector{String}:
"\phi_{1,0}"
"\phi_{1,6}"
"\phi_{1,3}'"
"\phi_{1,3}''"
"\phi_{2,1}"
"\phi_{2,2}"
julia> charnames(W;spaltenstein=true,limit=true)
6-element Vector{String}:
"1"
"Ξ΅"
"Ξ΅β"
"Ξ΅_c"
"ΞΈβ²"
"ΞΈβ³"
julia> charnames(W;spaltenstein=true,TeX=true)
6-element Vector{String}:
"1"
"\varepsilon"
"\varepsilon_l"
"\varepsilon_c"
"\theta'"
"\theta''"
The last two commands show the character names used by Spaltenstein and Lusztig when describing the Springer correspondence.
Chevie.Chars.classnames
β Functionclassnames(W;options...)
classnames(io::IO,W)
returns the list of class names for the reflection group W
. The optional options are IOContext attributes which can give alternative names in certain cases, or a different formatting of names in general. They can be specified by giving an IO as argument.
Chevie.Chars.classinfo
β Functionclassinfo(W)
returns information about the conjugacy classes of the finite reflection group or Spets W
. The result is an object with various entries describing properties of the conjugacy classes of W
. This object prints at the Repl or in Pluto or Jupyter as a table synthesizing most information.
A field not in the table is .classparams
, containing parameters for the conjugacy classes. Each parameter is a vector which has one item for each irreducible component of W
. For what are the parameters for an irreducible W
, see the helpstring of Chars
.
julia> classinfo(coxgroup(:A,2))
ββββ¬βββββββββββββββββββββββ
βn0βname length order wordβ
ββββΌβββββββββββββββββββββββ€
β1 β 111 1 1 .β
β2 β 21 3 2 1β
β3 β 3 2 3 12β
ββββ΄βββββββββββββββββββββββ
The table contains the columns:
name
, corresponding to the field.classnames
: strings describing the conjugacy classes, made out of the information in:classparams
.length
, corresponding to the field.classes
, is the number of elements in the conjugacy class.order
, corresponding to the field.orders
, is the order of elements in the conjugacy class.word
, corresponding to the field.classtext
, describes a word in the generators for the representatives of each conjugacy class. Each word is a list of integers where the generatorW(i)
is represented by the integeri
. For finite Coxeter groups, it is the same asword.(Ref(W),classreps(W))
, and each such representative is of minimal length in its conjugacy class and is a "very good" element in the sense of GeckMichel1997.
Chevie.Chars.fakedegree
β Functionfakedegree(W, Ο, q=Pol())
returns the fake degree (see fakedegrees
for a definition) of the character of parameter Ο (see charinfo(W).charparams
) of the reflection group W
, evaluated at q
.
julia> fakedegree(coxgroup(:A,2),[[2,1]],Pol(:q))
Pol{Cyc{Int64}}: qΒ²+q
Chevie.Chars.fakedegrees
β Functionfakedegrees(W, q=Pol())
returns a list holding the fake degrees of the reflection group W
on the vector space V
, evaluated at q
. These are the graded multiplicities of the irreducible characters of W
in the quotient SV/I
where SV
is the symmetric algebra of V
and I
is the ideal generated by the homogeneous invariants of positive degree in SV
. The ordering of the result corresponds to the ordering of the characters in charinfo(W)
.
julia> fakedegrees(coxgroup(:A,2),Pol(:q))
3-element Vector{Pol{Int64}}:
qΒ³
qΒ²+q
1
Chevie.Chars.representation
β Methodrepresentation(W,i)
returns, for the i
-th irreducible representation of the complex reflection group or Spets W
, a list of matrices images of the generating reflections of W
in a model of the representation (for Spets, the result is a NamedTuple
with fields gens
, a representation of Group(W)
, and F
, the matrix for W.phi
in the representation). This function is based on the classification, and is not yet fully implemented for Gββ
; 78 representations are missing out of 169, that is, representations of dimension β₯140, except half of those of dimensions 315, 420 and 840.
julia> representation(complex_reflection_group(24),3)
3-element Vector{Matrix{Cyc{Int64}}}:
[1 0 0; -1 -1 0; -1 0 -1]
[-1 0 -1; 0 -1 (1-β-7)/2; 0 0 1]
[-1 -1 0; 0 1 0; 0 (1+β-7)/2 -1]
Chevie.Chars.representations
β Methodrepresentations(W)
returns the list of representations of the complex reflection group or Spets W
(see representation
).
julia> representations(coxgroup(:B,2))
5-element Vector{Vector{Matrix{Int64}}}:
[[1;;], [-1;;]]
[[1 0; -1 -1], [1 2; 0 -1]]
[[-1;;], [-1;;]]
[[1;;], [1;;]]
[[-1;;], [1;;]]
Chevie.Chars.induction_table
β Functioninduction_table(u,g)
returns an object describing the decomposition of the irreducible characters of the subgroup u
induced to the group g
. At the repl or IJulia or Pluto, a table is displayed where the rows correspond to the characters of the parent group, and the columns to those of the subgroup. The returned object has a field scalar
which is a Matrix{Int}
containing the induction table, and the other fields contain labeling information taken from the character tables of u
and g
when it exists.
julia> g=Group([Perm(1,2),Perm(2,3),Perm(3,4)])
Group([(1,2),(2,3),(3,4)])
julia> u=Group( [ Perm(1,2), Perm(3,4) ])
Group([(1,2),(3,4)])
julia> induction_table(u,g) # needs "using GAP"
Induction table from Group((1,2),(3,4)) to Group((1,2),(2,3),(3,4))
βββββ¬ββββββββββββββββ
β βX.1 X.2 X.3 X.4β
βββββΌββββββββββββββββ€
βX.1β . 1 . .β
βX.2β . 1 1 1β
βX.3β 1 1 . .β
βX.4β 1 . 1 1β
βX.5β 1 . . .β
βββββ΄ββββββββββββββββ
julia> g=coxgroup(:G,2)
Gβ
julia> u=reflection_subgroup(g,[1,6])
Gββββ
β=Aβ
julia> t=induction_table(u,g)
Induction table from Gββββ
β=Aβ to Gβ
βββββββ¬βββββββββ
β β111 21 3β
βββββββΌβββββββββ€
βΟβββ β . . 1β
βΟβββ β 1 . .β
βΟβ²ββββ 1 . .β
βΟβ³ββββ . . 1β
βΟβββ β . 1 .β
βΟβββ β . 1 .β
βββββββ΄βββββββββ
IO
attributes can be transmitted to the table format method
julia> xdisplay(t;rows=[5],cols=[3,2])
Induction table from Gββββ
β=Aβ to Gβ
βββββββ¬βββββ
β β3 21β
βββββββΌβββββ€
βΟβββ β. 1β
βββββββ΄βββββ
It is also possible to TeX induction tables with xdisplay(t;TeX=true)
.
induction_table
also works for spets (reflection cosets).
Chevie.Chars.j_induction_table
β Functionj_induction_table(H, W)
computes the decomposition into irreducible characters of the reflection group W
of the j
-induced of the irreducible characters of the reflection subgroup H
. The j
-induced of Ο
is the sum of the irreducible components of the induced of Ο
which have same b
-function (see charinfo
) as Ο
. What is returned is an InductionTable
struct.
julia> W=coxgroup(:D,4)
Dβ
julia> H=reflection_subgroup(W,[1,3])
Dβββββ=AβΞ¦βΒ²
julia> j_induction_table(H,W)
j-induction table from Dβββββ=AβΞ¦βΒ² to Dβ
βββββββ¬βββββββββ
β β111 21 3β
βββββββΌβββββββββ€
β11+ β . . .β
β11- β . . .β
β1.111β . . .β
β.1111β . . .β
β11.2 β . . .β
β1.21 β 1 . .β
β.211 β . . .β
β2+ β . . .β
β2- β . . .β
β.22 β . . .β
β1.3 β . 1 .β
β.31 β . . .β
β.4 β . . 1β
βββββββ΄βββββββββ
Chevie.Chars.J_induction_table
β FunctionJ_induction_table(H, W)
computes the decomposition into irreducible characters of the reflection group W
of the J
-induced of the irreducible characters of the reflection subgroup H
. The J
-induced of Ο
is the sum of the irreducible components of the induced of Ο
which have same a
-function (see charinfo
) as Ο
. What is returned is an InductionTable
struct.
julia> W=coxgroup(:D,4)
Dβ
julia> H=reflection_subgroup(W,[1,3])
Dβββββ=AβΞ¦βΒ²
julia> J_induction_table(H,W)
J-induction table from Dβββββ=AβΞ¦βΒ² to Dβ
βββββββ¬βββββββββ
β β111 21 3β
βββββββΌβββββββββ€
β11+ β . . .β
β11- β . . .β
β1.111β . . .β
β.1111β . . .β
β11.2 β 1 . .β
β1.21 β 1 . .β
β.211 β . . .β
β2+ β . . .β
β2- β . . .β
β.22 β . . .β
β1.3 β . 1 .β
β.31 β . . .β
β.4 β . . 1β
βββββββ΄βββββββββ
Chevie.Chars.schur_functor
β Functionschur_functor(mat,l)
mat
should be a square matrix and l
a partition. The result is the Schur functor of the matrix mat
corresponding to partition l
; for example, if l==[n]
it returns the n-th symmetric power and if l==[1,1,1]
it returns the 3rd exterior power. The current algorithm (from Littlewood) is rather inefficient so it is quite slow for partitions of n where n>6
.
julia> m=cartan(:A,3)
3Γ3 Matrix{Int64}:
2 -1 0
-1 2 -1
0 -1 2
julia> schur_functor(m,[2,2])
6Γ6 Matrix{Rational{Int64}}:
9 -6 4 3//2 -2 1
-12 16 -16 -4 8 -4
4 -8 16 2 -8 4
12 -16 16 10 -16 12
-4 8 -16 -4 16 -12
1 -2 4 3//2 -6 9
julia-repl
Chevie.Chars.detPerm
β FunctiondetPerm(W)
return the permutation of the characters of the reflection group W
which is effected when tensoring by the determinant character (for Coxeter groups this is the sign character).
julia> W=coxgroup(:D,4)
Dβ
julia> detPerm(W)
(1,8)(2,9)(3,11)(4,13)(7,12)
Chevie.Chars.conjPerm
β FunctionconjPerm(W)
return the permutation of the characters of the group W
which is effected when taking the complex conjugate of the character table.
julia> W=complex_reflection_group(4)
Gβ
julia> conjPerm(W)
(2,3)(5,6)