Let W be a finite Coxeter group, corresponding to a root system R in an Euclidean space V. Let S be the set of generators corresponding to some system of fundamental roots in R. Then {wsw^{-1} mid w in W, s in S} is the set of all reflections in W. A reflection subgroup H of W is a subgroup generated by the reflections it contains. If H is generated by reflections with respect to a subset of the fundamental roots it is called a standard parabolic subgroup of W. A parabolic subgroup is a subgroup H of W which is conjugate to some standard parabolic subgroup. A parabolic subgroup H is characterized among all reflection subgroups of W by the fact that the set of all roots in R for which the corresponding reflection lies in H is closed under (real) linear combinations.
It is a theorem discovered by Deodhar Deo89 and Dyer Dye90
independently at the same time that a reflection subgroup H has a
canonical set of fundamental roots even if it is not parabolic: Let Q
be the set of all roots for which the corresponding reflection lies in
H; then a fundamental system of roots for H is given by the positive
roots t in Q such that the set of roots whose sign is changed by the
reflection with root t meets Q in the single element t. This is
used by the routine ReflectionSubgroup
to determine the root system Q
of a reflection subgroup H. This function takes the original record
for W and a list of indices for the roots as input.
gap> W := CoxeterGroup( "G", 2 ); CoxeterGroup("G", 2) gap> W.roots[4]; [ 1, 2 ] gap> H := ReflectionSubgroup( W, [ 2, 4 ] ); ReflectionSubgroup(CoxeterGroup("G", 2), [ 2, 3 ]) gap> PrintDynkinDiagram( H ); ~A2 2 - 3 # not a parabolic subgroup
We also see that the result of the above algorithm is that W.roots[2]
and W.roots[3]
form a system of simple roots in H.
The line containing the Dynkin diagram of H introduces a convention:
we use the notation "~A"
to denote a root subsystem of type "A"
generated by short roots.
The record for the subgroup contains additional components the most
important of which is rootInclusion
which gives the positions of the
roots in Q in the original root system R:
gap> H.rootInclusion; [ 2, 3, 4, 8, 9, 10 ]The inverse map is stored in
H.rootRestriction
. If H is a standard
parabolic subgroup of W then the length function on H (with respect
to its set of generators) is the restriction of the length function on
W. This need not no longer be true for arbitrary reflection subgroups
of W:
gap> CoxeterLength( W, H.generators[2] ); 3 gap> CoxeterLength( H, H.generators[2] ); 1
In GAP, the Coxeter group W is represented as a permutation group on
R. Consequently, a reflection subgroup H subseteq W is a
permutation subgroup, i.e., its elements are represented as permutations
of the roots of the it parent group. This has to be kept in mind when
working with reduced expressions and functions like CoxeterWord
, and
PermCoxeterWord
.
Reduced words in simple reflections of H:
gap> el := CoxeterWords( H ); [ [ ], [ 2 ], [ 3 ], [ 2, 3 ], [ 3, 2 ], [ 2, 3, 2 ] ]
Reduced words in the generators of H:
gap> el1 := List( el, x -> H.rootRestriction{ x } ); [ [ ], [ 1 ], [ 2 ], [ 1, 2 ], [ 2, 1 ], [ 1, 2, 1 ] ]
Permutations on the roots of W:
gap> el2 := List( el, x -> PermCoxeterWord( H, x ) ); [ (), ( 1, 5)( 2, 8)( 3, 4)( 7,11)( 9,10), ( 1,12)( 2, 4)( 3, 9)( 6, 7)( 8,10), ( 1, 5,12)( 2,10, 3)( 4, 9, 8)( 6, 7,11), ( 1,12, 5)( 2, 3,10)( 4, 8, 9)( 6,11, 7), ( 2, 9)( 3, 8)( 4,10)( 5,12)( 6,11) ]
Reduced words in the generators of W:
gap> List( el2, x -> CoxeterWord( W, x ) ); [ [ ], [ 2 ], [ 1, 2, 1 ], [ 2, 1, 2, 1 ], [ 1, 2, 1, 2 ], [ 2, 1, 2, 1, 2 ] ]
Another basic result about reflection subgroups is that each coset of H in W contains a unique element of minimal length. Since a coset is a subset of W, the length of elements is taken with respect to the roots of W.) See ReducedRightCosetRepresentatives.
In many applications it is useful to know the decomposition of the
irreducible characters of W when we restrict them from W to a
reflection subgroup H (defined as in Section~refReflection
subgroups). In order to apply the usual GAP functions for inducing
and restricting characters and computing scalar products, we need to know
the fusion map for the conjugacy classes of H into those of W. This
is done, as usual, with the GAP function FusionConjugacyClasses
,
which calls a special implementation for Coxeter groups. The
decomposition of induced characters into irreducibles then is a simple
matter of combining some functions which already exist in GAP. There
is an additional function, InductionTable
, which performs this job.
gap> W := CoxeterGroup( "G", 2 );; gap> W.roots[4]; [ 1, 2 ] gap> H := ReflectionSubgroup( W, [ 2, 4 ] );; gap> Display( InductionTable( H, W ) ); Induction from ~A2 into G2tt |
111 21 3 ______________________ phi_{1,0}tt |
. . 1 phi_{1,6}tt |
1 . . phi_{1,3}'tt |
. . 1 phi_{1,3}''tt |
1 . . phi_{2,1}tt |
. 1 . phi_{2,2}tt |
. 1 .
We have similar functions for the j-induction and the J-induction of characters. These operations are obtained by truncating the induced characters by using the a-invariants and b-invariants associated with jInductionTable for Macdonald-Lusztig-Spaltenstein induction and JInductionTable).
ReflectionSubgroup( W, r )
Returns the reflection subgroup of the real reflection group W generated by the reflections with roots specified by r. r is a list of indices specifying a subset of the roots of W.
A reflection subgroup H of W is a permutation subgroup, and otherwise has the same fields as a Coxeter group, with some new ones added which express the relationship with the parent W:
rootInclusion
:
parentN
:
rootRestriction
:2*H.parentN
with entries in
positions H.rootInclusion
bound to [1..2*H.N]
.
A Coxeter group which is not a subgroup actually also contains these
fields, set to the trivial values: rootInclusion = [ 1 .. 2*W.N ]
,
parentN = W.N
and rootRestriction = [ 1 .. 2*W.N ]
.
With these fields, you can, e.g., test if the i-th fundamental root of H root is left positive by the element w of H (given as a permutation of the roots of the parent) with
H.rootInclusion[ i ] ^ w <= H.parentN
Programs dealing with Coxeter groups should be written this way to work
for Coxeter subgroups (instead of just writing i^w <= H.N
).
ReflectionSubgroup
returns a subgroup of the parent group of the
argument (like the GAP function Subgroup
).
gap> W := CoxeterGroup( "F", 4 );; gap> H := ReflectionSubgroup( W, [ 1, 2, 11, 20 ] ); ReflectionSubgroup(CoxeterGroup("F", 4), [ 1, 2, 9, 16 ]) gap> CartanName( H ); "D4" # not a parabolic subgroup gap> H.rootRestriction; [ 1, 2,,, 5,,,, 3,, 6,,, 8,, 4,, 7,, 9,, 10, 11, 12, 13, 14,,, 17,,,, 15,, 18,,, 20,, 16,, 19,, 21,, 22, 23, 24 ] gap> ReflectionSubgroup( H, [ 1, 2, 6 ] ); ReflectionSubgroup(CoxeterGroup("F", 4), [ 1, 2, 3 ])
This function requires the package "chevie" (see RequirePackage).
All functions for Coxeter groups are actually defined for reflection
subgroups. The generators for the subgroups are labeled according to the
corresponding number of the root they represent in the parent group.
This affects the labeling given by all functions dealing with words and
generators, e.g., PrintDynkinDiagram
or PermCoxeterWord
.
gap> W := CoxeterGroup( "F", 4 ); CoxeterGroup("F", 4) gap> H := ReflectionSubgroup( W, [ 10, 11, 12 ] ); ReflectionSubgroup(CoxeterGroup("F", 4), [ 10, 11, 12 ]) gap> PrintDynkinDiagram( H ); C2 11 > 10 ~A1 12 gap> LongestCoxeterWord( H ); [ 10, 11, 10, 11, 12 ]
Also, as one may notice in the example above, there is one particularity
of the functions CartanType
, CartanName
and PrintDynkinDiagram
for
Coxeter subgroups: an irreducible subsystem which consists of short
roots in a system which has longer roots (i.e., type "B"
, "C"
,
"G"
or "F"
) is labeled as type "~A"
.
These functions require the package "chevie" (see RequirePackage).
ReducedInCoxeterCoset( W , w )
w is an automorphism of the Coxeter group W or of a parent group of
W, given as a permutation of the roots. ReducedInCoxeterCoset
returns
the unique element in the right coset W.w which sends all roots of W
to positive roots.
gap> W := CoxeterGroup("F", 4 );; gap> H := ReflectionSubgroup( W, [ 1, 2, 9, 16 ] );; gap> PrintDynkinDiagram( H ); D4 9 \ 1 - 16 / 2 gap> w := PermCoxeterWord( W, [ 3, 2, 3, 4, 3, 2 ] );; gap> f := ReducedInCoxeterCoset( H, w );; gap> CoxeterWord( W, f ); [ 4, 3 ] gap> H.rootInclusion{[ 1 ..4 ]}; [ 1, 2, 9, 16 ]
The triality automorphism of D_4:
gap> OnTuples( H.rootInclusion{[ 1 .. 4 ]}, f ); [ 1, 9, 16, 2 ]This function requires the package "chevie" (see RequirePackage).
ReducedRightCosetRepresentatives( W, H )
returns a list of reduced elements in the Coxeter group W which are
distinguished representatives for the right cosets of the reflection
subgroup H in W. The distinguished representative in the coset H.w
is the unique element in the coset which sends all roots of H to
positive roots (the element returned by ReducedInCoxeterCoset
). It is
also the element of minimal length in the coset. The representatives are
returned in order of increasing length.
gap> W := CoxeterGroup( "B", 3 );; gap> H := ReflectionSubgroup(W, [ 2, 3 ]);; gap> List( ReducedRightCosetRepresentatives( W, H ), > x-> CoxeterWord( W, x ) ); [ [ ], [ 1 ], [ 1, 2 ], [ 1, 2, 1 ], [ 1, 2, 3 ], [ 1, 2, 1, 3 ], [ 1, 2, 1, 3, 2 ], [ 1, 2, 1, 3, 2, 1 ] ]
This function requires the package "chevie" (see RequirePackage).
PermCosetsSubgroup( W, H )
returns the list of permutations induced by the standard generators of
the Coxeter group W on the cosets of the Coxeter subgroup H. The
cosets are in the order determined by the result of the function
ReducedRightCosetRepresentatives( W, H )
.
gap> W := CoxeterGroup( "F", 4 );; gap> PermCosetsSubgroup( W, ReflectionSubgroup( W, [ 1, 2, 3 ] ) ); [ ( 4, 5)( 6, 7)( 8,10)(16,18)(17,20)(19,21), ( 3, 4)( 7, 9)(10,12)(14,16)(15,17)(21,22), ( 2, 3)( 4, 6)( 5, 7)( 9,11)(12,14)(13,15)(17,19)(20,21)(22,23), ( 1, 2)( 6, 8)( 7,10)( 9,12)(11,13)(14,15)(16,17)(18,20)(23,24) ]
This function requires the package "chevie" (see RequirePackage).
jInductionTable( H, W )
computes the decomposition into irreducible characters of W of the
j-induced of the irreducible characters of H. The j-induced of
chi is the sum of the irreducible components of the induced of chi
which have same b-function (see LowestPowerFakeDegrees) as chi.
In the table the rows correspond to the characters of the parent group,
the columns to those of the subgroup. What is returned is actually a
record with several fields: scalar
contains the induction table
proper, and there is a Display
method. The other fields contain
labeling information taken from the character tables of H and W when
it exists.
gap> W := CoxeterGroup( "D", 4);; gap> H := ReflectionSubgroup( W, [ 1, 3 ] );; gap> Display( jInductionTable( H, W ) ); j-Induction from A2 into D4tt |
111 21 3 ________________ 11.+tt |
. . . 11.-tt |
. . . 1.111tt |
. . . .1111tt |
. . . 11.2tt |
. . . 1.21tt |
1 . . .211tt |
. . . 2.+tt |
. . . 2.-tt |
. . . .22tt |
. . . 1.3tt |
. 1 . .31tt |
. . . .4tt |
. . 1
This function requires the package "chevie" (see RequirePackage).
JInductionTable( H, W )
JInductionTable
computes the decomposition into irreducible characters
of W of the J-induced of the irreducible characters of H. The
J-induced of chi is the sum of the irreducible components of the
induced of chi which have same a-function (see
LowestPowerGenericDegrees) as chi. In the table the rows correspond
to the characters of the parent group, the columns to those of the
subgroup. What is returned is actually a record with several fields:
scalar
contains the induction table proper, and there is a Display
method. The other fields contain labeling information taken from the
character tables of H and W when it exists.
gap> W := CoxeterGroup( "D", 4 );; gap> H := ReflectionSubgroup( W, [ 1, 3 ] );; gap> Display( JInductionTable( H, W ) ); J-Induction from A2 into D4tt |
111 21 3 ________________ 11.+tt |
. . . 11.-tt |
. . . 1.111tt |
. . . .1111tt |
. . . 11.2tt |
1 . . 1.21tt |
1 . . .211tt |
. . . 2.+tt |
. . . 2.-tt |
. . . .22tt |
. . . 1.3tt |
. 1 . .31tt |
. . . .4tt |
. . 1
This function requires the package "chevie" (see RequirePackage).
Previous Up Next
Index
GAP 3.4.4