Splints of root systems for special Lie subalgebras

0.2 MB·English

by  Vladimir Lyakhovsky

Splints of root systems for special Lie subalgebras
5
1 V.D. Lyakhovsky1, A.A. Nazarov1,2, P.I. Kakin1
0
2 1 Department of High-energy and elementary particle physics,

n St Petersburg StateUniversity
a
198904, Saint-Petersburg, Russia,
J
2 e-mail: [email protected]
0
2

] Abstract
T
R Splint is a decomposition of root system into union of root systems.
Splint of root system for simple Lie algebra appears naturally in stud-
.
h ies of (regular) embeddings of reductive subalgebras. Splint can be used
t
a to construct branching rules. We consider special embedding of Lie sub-
m algebra to Lie algebra. We classify projections of algebra root systems
using extended Dynkin diagrams and single out the conditions of splint
[
appearance and coincidence of branching coefficients with weight multi-

1 plicities. While such a coincidence is not very common it is connected
v with Gelfand-Tsetlin basis.
9
0
8 1 Introduction
4
0
ThenotionofsplintwasintroducedbyDavidRichterinthepaper[Richter, 2012].
.
1 Splint is the decomposition of root system into disjoint union of images of two
0
ormoreembeddingsofsomeotherrootsystems. Embeddingφofarootsystem
5
∆ into a root system ∆ is a bijective map of roots of ∆ to a (proper) subset
1 1 1
: of ∆ that commutes with vector composition law in ∆1 and ∆.
v
i
X φ:∆1 −→∆
r
a φ◦(α+β)=φ◦α+φ◦β, α,β ∈∆
1
Note that the image Im(φ) is not required to inherit the root system prop-
erties except the addition rules equivalent to the addition rules in ∆ (for pre-
1
images). If an embedding φ preserves the angles between the roots it is called
“metric”. Two embeddings φ and φ can splinter ∆ when the latter can be
1 2
presented as a disjoint union of images Im(φ ) and Im(φ ).
1 2
Why one would consider non angle-preservingmaps of root systems? Addi-
tive properties of root system determine the structure of Verma module:
Mµ =U(g) ⊗ Dµ(b )={(E−α1)n1...(E−αs)ns|v i}ni=0,1,...
U(b+) + µ αi∈∆+
1
Here b is the Borel subalgebra of g, Dµ its one-dimensional representation,
+
|vµi is the highest weight vector and E−αj are lowering operators that are in
correspondence with positive roots α ∈∆+. Weyl character formula expresses
j
characterof irreducible representationas a combinationof charactersof Verma
modules:
ε(w)ew(µ+ρ)−ρ
chLµ = Pw∈W = ε(w)chMw(µ+ρ)−ρ
ε(w)ewρ−ρ X
Pw∈W w∈W
HerethesumisoverelementsoftheWeylgroupandtheiractionsw⊲µ=w(µ+
ρ)−ρ depend on angles between roots. But we can rewrite Weyl characterfor-
mula representing Weyl group elements as the products of reflections s ,α∈S
α
in hyperplanes orthogonalto simple roots: w =s ·s .... Reflections act on
α1 α2
rootsystembypermutations,sothecompositionofsuchanactionwithembed-
ding φ is easily obtained. The orbit of Weyl groups action w⊲µ,w ∈ W, can
be constructedby subtractionsofrootsfromthe highestweightµ. Let’s denote
Dynkinlabelsofµby(µ ,...µ ). Thenµ−µ α ,µ−µ α ,...,µ−µ α ,α ∈S,
1 r 1 1 2 2 r r i
are the points of the orbit that can be obtained by elementary reflections
s ,α ∈S. Next set of points that are obtained by two consecutive reflections
αi i
s ·s ⊲µ are obtained as the subtraction µ−µ α −µ (s α ). Continuing
αi αj i i j αi j
this way we constructthe image ofsingularelement Ψ= ε(w)ew(µ+ρ)−ρ
Pw∈W
after the embedding φ [Laykhovsky and Nazarov, 2012]. The multiplicities in
the characterofVerma module are unchangedby the embedding since they are
determinedbyadditivepropertiesofthe roots. Soweseethatweightmultiplic-
ities of irreducible modules are also preserved by the embedding.
Rootsystemofregularsubalgebraiscontainedintherootsystemofalgebra
so splint is useful in computation of branching coefficients.
In the paper [Laykhovsky and Nazarov, 2012] it was proven that the ex-
istence of splint leads to the coincidence of branching coefficients with weight
multiplicitiesundercertainconditions. WedenoteLiealgebrabygandconsider
it’s subalgebra a. If a is a regular subalgebra, its root system ∆ is contained
a
in ∆ . Branchingcoefficients b(µ) appear in decompositionof irreducible repre-
g ν
sentation Lµ of g to the sum of irreducible representations of a:
g
Lµ = b(µ)Lν (1)
g M ν a
ν
Assume that root system ∆ splinters as ∆ = ∆ ∪ φ(∆ ), where φ is
g g a s
an embedding of root system ∆ of some semisimple Lie algebra s. Then
s
branching coefficients b(µ) for the reduction L(µ) coincide with weight multi-
ν g↓a
plicities m(µ˜) in s-representations provided certain technical condition holds
ν
[Laykhovsky and Nazarov, 2012]. Highestweightµ˜ of s-representationis calcu-
lated from Dynkin labels of highest weight µ of g-module.
In the next section 2 we review the classification of splints for simple root
systems and results for branching coefficients. Then we move to the study of
special embeddings which is the main subject of the present paper. We con-
siderspecialembeddings ofLie subalgebrasintoa Lie algebra. Inthis caseroot
2
system of the subalgebra is not contained in the root system of the algebra.
So the original motivation for splints is not applicable. But we can consider
the projection of root system of the algebra on the root space of the subalge-
bra. Such a projection is not a root system anymore, but it satisfies milder
conditions (Section 3). It is possible to classify most projections using Dynkin
diagrams augmented with multiplicities. We then define splint for such ’weak’
rootsystemsandstatetheconditionsofitsappearanceandimplicationsforthe
calculation of branching coefficients (Section 4).
Use of representation theory of the subalgebra allows us to classify all the
splintsfortheprojectionsofalgebrarootsystem. We alsoapplythismethodto
metric splints and regular subalgebras and get a unified treatment. We obtain
Gelfand-Tsetlin rules for regular and special embeddings this way.
In conclusion 4 we discuss the cases when the projection of the root system
does not fall into the classification mentioned above.
2 Splints of root systems and regular subalge-
bras
The notionand classificationofsplints for simple rootsystems were introduced
in the paper [Richter, 2012].
Consider a simple Lie algebra g and its regular subalgebra a֒→g such that
a is a reductive subalgebra a⊂g with correlated root spaces: h∗ ⊂ h∗. Let as
a g
be a semisimple summand of a, this means that a=as⊕u(1)⊕u(1)⊕.... We
shall consider as to be a proper regular subalgebra and a to be the maximal
subalgebra with as fixed that is the rank r of a is equal to that of g.
Computationofbranchingcoefficientsreliesonroots∆ \∆ [Lyakhovsky and Nazarov,2011a]
g a
sosplint is naturally connectedto branchingcoefficients for regularsubalgebra.
Only three types ofsplints areinjective and thus are naturally connectedto
branching [Richter, 2012]:
type ∆ ∆ ∆
a s
(i) G A A
2 2 2
F D D
4 4 4
(ii) B (r ≥2) D ⊕rA (2)
r r 1
(∗) C (r ≥3) ⊕rA D
r 1 r
(iii) A (r ≥2) A ⊕u(1) ⊕rA
r r−1 1
B A ⊕u(1) A
2 1 2
Each row in the table gives a splint (∆ ,∆ ) of the simple root system ∆.
a s
In the first two types both ∆ and ∆ are embedded metrically. Stems in the
a s
first type splints are equivalent and in the second are not. In the third type
splints only∆ isembedded metrically. The summands u(1)areaddedto keep
a
r =r. Thisdoes notchangethe principlepropertiesofbranchingbutmakesit
a
possible to use the multiplicities of s -modules without further projecting their
weights.
3
Note that in the case of C -series (marked with the star in table (13)) met-
r
rical embedding ∆ →∆ does not lead to the appearance of regular subal-
Dr Cr
gebra a (See [Dynkin, 1952b]).
In the paper [Lyakhovsky and Nazarov,2011b] it was shown that
Property 2.1. There is a decomposition of a singular element of the algebra g
into singular elements of the subalgebra s:
Φµ = ǫ(w)w eµ+ρgφ e−µeΨµe , (3)
g X (cid:16) (cid:16) s(cid:17)(cid:17)
w∈Wa
where ǫ(w) is a determinant of the element w of the Weyl group W of the sub-
a
algebra a. The action of Weyl group on the weights is extended to the algebra of
formalexponentsbytherulew(eν)=ewν,w∈W. Hence,themultiplicityMµe
(s)νe
oftheweightν fromtheweightdiagram ofthealgebraswiththehighest weightµ˜
defines the breanching coefficient b(νµ) for the highest weight ν =(µ−φ(µ−ν)):
b(µ) =Mµe . e e(4)
(µ−φ(µe−νe)) (s)νe
3 Special embeddings and projections of root
system
Thestudyofspecialyembeddedsubalgebrastracesbacktofundamentalpapers
by Eugene Dynkin [Dynkin, 1952b, Dynkin, 1952a] where he called such sub-
algebras “S-subalgebras” to distinguish from regular or “R-subalgebras” that
have root systems obtained by dropping some roots of the algebra root sys-
tem. Thus regular subalgebras are easy to construct by dropping nodes from
extended Dynkin diagram of the algebra, while the case of special subalgebras
is more difficult. Complete classification for exceptional Lie algebras was ob-
tained recently in [Minchenko, 2006]. The algorithm of the special subalgebras
construction is available in GAP package but still requires manual intervention
[de Graaf, 2011].
Assume that Lie algebra g is simple. Let’s denote a subalgebra by a. We
denote corresponding Cartan subalgebras by h and h and identify them with
g a
the dual spaces h∗, h∗ using Killing forms of g and a.
g a
To construct an embedding a → g consider some representation Lν as a
a
subspaceofLie algebrag. Thenoneneeds tocheckthatthe generatorsofacan
be presented as linear combinations of generators of g. We can identify Cartan
subalgebrah ⊂a with dualspace h∗ using Killing form. Thenthe rootsystem
a a
∆ of g can be projected to h∗ using the expression of h -generators through
g a a
h -generators.
g
Thisprojectionisdescribedintheclassicalpapers[Dynkin, 1952b,Dynkin, 1952a]
by the following theorem:
Theorem 1. If a representation Lµ of the algebra g induces a representation
g
Lµ˜ on the subalgebra a then the weight system of Lµ˜ can be obtained from Lµ
a a g
by an orthogonal projection of h∗ on h∗.
g a
4
[Dynkin, 1952a].
Inrelationtotheadjointrepresentationofgthistheoremmeansthatthepro-
jectionoftherootsystem∆ toh∗ isaweightsystemofsomefinite-dimensional
g a
but not irreducible representation of a. Moreover this reducible representation
contains the adjoint representationof a. And if we denote such a projectionby
∆′ =π (∆ ), (5)
a g
then the roots of a will be contained in ∆′. Thus the system ∆′ can be a
root system but it might not be reduced and might contain some vectors with
multiplicities greater than one.
The following theorems [Dynkin, 1952b] also concern the properties of ∆′.
Theorem2. Every specialsubalgebraaofasemisimple Liealgebragisinteger,
i.e. the projections of roots of g to h∗ are linear combinations of simple roots of
a
a with integer coefficients [Dynkin, 1952b]
Theorem 3. If a is a semisimple subalgebra of a semisimple Lie algebra g
and the generator e corresponding to the root α of a is presented as a linear
α
combination e = e of g-generators corresponding to the roots β of g, then
α Pβ β
the roots β are projected into α, π (β)=α. [Dynkin, 1972, Dynkin, 1952b].
a
Fromthesetheoremsweseethattherootsystem∆ ismetricallyembedded
a
(in the sense of splint) into the projection ∆′. Moreover, the multiplicities of
some roots α∈∆ in this projection ∆′ are greater than one.
a
As an example we present a projection ∆′ of the root system of D to the
4
root system of the special subalgebra A (Fig. 1).
2
4 Splints for special embeddings
We want to classify all the cases when branching coefficients coincide with
weight multiplicities of some other algebra representations. Such a coinci-
dence is possible when the projection ∆′ of g root system admits a splint
∆′ = ϕ (∆ )∪ϕ (∆ ), where ϕ and ϕ are the embeddings of correspond-
a a s s a s
ing root systems. The embedding ϕ is metric and trivial. Moreover, the
a
projectionofsingularelementπ (Ψµ)shouldadmitadecompositionintolinear
a g
combination of singular elements of irreducible representations of s:
π Ψµ = κ eνΨµ˜, (6)
a(cid:0) g(cid:1) X ν s
ν
with integercoefficients κ , wherethe sum is overthe set ofweightsν that will
ν
be determined later. First we classify all the splints of the projection of a root
system into a union of simple root systems for special embeddings, and then
discuss the decomposition of singular elements.
Projectionofthegrootsystem∆′coincideswiththeprojectionoftheweight
diagram of the adjoint representation of algebra g with the exception of zero
5
1 1
3
3 3
1 1
3 3
3
1 1
Figure1: ProjectionofD (so(8))-rootsystemontotherootspaceofthespecial
4
subalgebra A (su(3)). Note that this roots system is G but with non-trivial
2 2
multiplicities.
weight. Theadjointrepresentationofgcontainsadjointrepresentationofaand
can be decomposed as
ad =ad ⊕Mχ, (7)
g a a
whereMχ isnotnecessaryirreduciblerepresentationofacalled“characteristic”
a
in [Dynkin, 1952b].
The weight diagram of Mχ coincides with the root system ∆ with the
a s
exception of zero weight. So we need to find all the representations for all
simple Lie algebras a, such that their weight diagrams are root systems after
theexclusionofzeroweight. Inordertodosowemustnote,thatalltheweights
of Mχ should have length not less than length of shortest root of a, since a is
a
an integer subalgebra of g 2. Moreover, Mχ should contain weights of at most
a
two different lengths.
If the root system ∆ contains roots α : α ⊥ β, ∀β ∈ ∆ , orthogonal to
g a
the root system of the subalgebra, one cannot proceed with the decomposi-
tion of singular element (6). The elements eν of π (Ψµ) should be augmented
a g
withthedimensionsofrepresentationsofalgebraa spannedbygeneratorscor-
⊥
responding to orthogonal roots [Lyakhovsky and Nazarov,2011a]. Then there
will be non-trivial multiplicities in the right hand side of (6) and no coinci-
dence of branching coefficients with weight multiplicities of s-representations.
One can write more cumbersome relation between multiplicities and branching
coefficients, but it is out of scope for the present paper.
The multiplicity of zero weight in the adjoint representation is equal to the
rank of the algebra. So if rankg−ranka = 1, the representation in question
6
Mχ must be multiplicity-free. If rankg−ranka > 1 only zero weight can have
a
non-trivial multiplicities.
Thesimplestclassofmultiplicity-freerepresentationsiscalled“stronglymul-
tiplicity free” [Lehrer and Zhang, 2006] and consists of multiplicity free rep-
resentations with weight systems admitting strict ordering ν < ν ⇔ ν =
1 2 1
ν +nα, where n∈N,α∈∆+ and for all ν ,ν either ν <ν or ν <ν .
2 a 1 2 1 2 2 1
The list of strongly multiplicity free representations consists of (first) fun-
damental representations for series A ,B ,C , exceptional Lie algebra G (7-
r r r 2
dimensional representation) and all A representations.
1
Fundamental weights of the algebra A are shorter than its roots, but the
r
projections of g weights must be given by linear combinations of the roots of
the special subalgebra a with integer coefficients, so the first class of strongly
multiplicity free representations does not produce such a projection.
Nevertheless, the union of diagrams of two fundamental representations of
A with the root system of A produces weight diagram of G . This case
2 2 2
correspondstosplint∆ =ϕ (∆ )∪ϕ (∆ )whichisconnectedwithregular
G2 1 A2 2 A2
subalgebra A ⊂G (see Section 2).
2 2
FirstfundamentalrepresentationofB immediatelygivesusthesplint∆′ =
r
π ∆ =∆ ∪∆ correspondingtoGelfand-Tsetlinmultiplicity-
Br(cid:0) Dr+1(cid:1) Br A1+···+A1
free branching for the special embedding so(2r+1)→so(2r+2).
The length of the first fundamental weight of C is less than the length of
r
its short root, so this case can not correspond to an integer subalgebra.
Iftherearevectorsofdifferentlengthintheprojection∆′ onthesubalgebra
A , such a projection is not multiplicity free, and ∆ contains parallel roots.
1 s
The simplest case is the special embedding A → A with index 4 where ∆
1 2 s
is the root system BC . Such systems do not correspond to semisimple Lie
1
algebras, so we can not speak of a coincidence of branching coefficients for the
reduction g↓a with weight multiplicities of s-representations.
The weight diagram of the seven-dimensional representationof G together
2
with G root system form the projection of the root system B which cor-
2 3
responds to special embedding G → B . Branching coefficients in this case
2 3
coincide with weight multiplicities of representations of algebra s = A with
2
image of root system consisting of short roots of G .
2
The complete classification of multiplicity-free irreducible representations
was obtained in [Howe, 1995] (see also [Stembridge, 2003]). It consists of mi-
nuscule and quasi-minuscule representations.
The weight µ is minuscule if hµ,α∨i≤1 for all α∈∆+ and all the weights
of the irreducible representationLµ lie on the Weyl group orbit of µ. A weight
is quasi-minuscule if hµ,α∨i≤2 for all α∈∆+ and all non-zero weights lie on
the single Weyl group orbit.
Minuscule representations are well-studied and very useful in computations
of e.g. tensor products [Stembridge, 2003, Stembridge, 2001]. The minuscule
representationsareindexedbytheweightlatticemodulotherootlattice. There
isauniquequasi-minusculerepresentationthatisnotminusculeforeachsimple
Lie algebra. The multiplicity of the zero weight in quasi-minuscule representa-
tion is the number of short nodes of the Dynkin diagram.
7
Listofminusculerepresentationsconsistsoftensorpowersofvectorrepresen-
tationfortheseriesA ;spinrepresentationsfortheseriesB ;vectorrepresenta-
r r
tionsforC ;vectorandtwohalf-spinforD ;two27-dimensionalrepresentations
r r
for E and 56-dimensional representation of E .
6 7
Quasi-minuscule representations that are not minuscule are: adjoint rep-
resentation of A , vector representation for B , 2r2 −r −1-dimensional rep-
r r
resentation for C , adjoint representations for D ,E ,E ,E , 26-dimensional
r r 6 7 8
representation for F and 7-dimensional of G .
4 2
Classification of multiplicity-free irreducible highest weight modules Lµ in
[Howe, 1995, Stembridge, 2003] consists of the following classes:
• (1) µ is minuscule,
• (2) µ is quasi-minuscule and Lie algebra has only one short simple root,
• (3) Lie algebra is sp(6) and µ=ω , or
1
• (4) Lie algebra is sl(n+1) and µ=mω or µ=mω
1 n
We see that case (1) contains strongly multiplicity free modules of series
A and exceptional Lie algebra G , class (2) contains strongly multiplicity free
r 2
modules ofB andC , stronglymultiplicity free modules ofA areincluded in
r r 1
class (4).
We do not need to consider minuscule representations for simply-laced Lie
algebras, since in this case length of the weights of representation Mχ is less
a
than length of a roots, that contradicts to a being an integer subalgebra. For
the series B we have already discussed minuscule spin representation, since it
r
is strongly multiplicity free. The last case in (1) is the vector representation
for C . It is 2r-dimensional, but the length of its highest weight ω is less than
r 1
the length of the short root of algebra C . So it can not be a characteristic
r
representation for the special subalgebra.
Case (2) consists of the series B and the exceptional root system G . The
r 2
embedding G → B was already discussed. The dimension of the adjoint
2 3
representationforthe seriesB is2r2+r andofquasi-minusculerepresentation
r
it is 2r+1,so the dimensionofthe algebrag shouldbe (r+1)(2r+1)with the
rankg = r+1. The only solution is g = D and we have already seen that
r+1
this case corresponds to strongly multiplicity free module and Gelfand-Tsetlin
basis.
Incase(3)thediagramoftherepresentationcontainsweightswiththelength
smaller than the length of short root, so it cannot be a projection of ∆ to an
g
integer subalgebra.
We need to consider representations of class (4) only for m = 1,2 since the
root system ∆ can have roots of at most two different lengths. In the case
s
m = 1 we get strongly multiplicity free modules of A which were discussed
r
above. And for algebra A and m = 2 the weight diagram of representation
2
L2ω1 coincides with the root system of exceptional Lie algebra G . This case
A2 2
appearsinthe specialembedding A →B . Butfor A ,r >2 there areno root
2 3 r
systems that coincide with weight diagram of L2ω1.
Ar
8
So far we’ve considered all the cases when rankg−ranka=1. To complete
our classification of splints we need the classification of all the representations
where multiplicities of all non-zero weights are equal to one. Fortunately, such
aclassificationwasobtainedin[Plotkin et al., 1998]. Itconsistsofmultiplicity-
free representations and adjoint representations of algebras A ,B ,C , F and
r r r 4
G .
2
In order to have the embedding a→g such that the adjoint representation
of g is decomposed into two adjoint representations of a (and possibly several
trivial representations) we need to satisfy following conditions:
• Denotebyn thetotalnumberofrootsin∆ andtheprojection∆′. Then
g g
n =2n . (8)
g a
• Denote rank of g by shorthand notation r . Then the dimension
g
dimg=n +r ≥2dima=2n +2r (9)
g g a a
Forthe adjointrepresentationofalgebraA weseethattheprojection∆′ of
r
therootsystem∆ shouldconsistofrootsofsubalgebraa=A withmultiplicity
g r
2. The total number of roots for A is r(r+1), so ∆′ and ∆ contain 2r(r+
r g
1) roots, since there are no roots orthogonal to ∆ . There are two possible
a
solutions for g: D and G for r = 2. But the dimension of D is (r +
r+1 2 r+1
1)(2r+1) while the dimension of A is r(r +2) and 2dimA > dimD , so
r r r+1
the adjoint representation of D cannot be decomposed as twice the adjoint
r+1
representationofA . There is no correspondingembedding A →D andno
r r r+1
splint. We see that conditions (8)(9) do not hold.
Let’s consider the embedding B → g such that the adjoint representation
r
of g is decomposedinto two adjoint representations of B . It is straightforward
r
tocheckthatthere is nosolutionsfor gsatisfyingthe (8)(9)forthe seriesA, B,
C, D and all exceptional simple Lie algebras.
Since the number of roots of root system C is the same as for B and we
r r
need to check the same cases, we see that there is no embedding for C too.
r
TheadjointrepresentationofG givesusthealgebraD asthesolutionofthe
2 4
constraints(8)(9). The embedding G →D exists, since there areembeddings
2 4
G →B and B →D , but the decomposition of adjoint representation of D
2 3 3 4 4
is different: ad = ad ⊕2Lω1. So this case does not produce splint for the
projection ∆′. D4 G2 G2
For the adjointrepresentationof F there is no solution that satisfies condi-
4
tions (8)(9).
The complete classificationofsplints forspecialembeddings ∆′ =π (∆ )≡
a g
∆ ∪∆ where g,a are simple and s is semisimple is:
a s
• B →D
n n+1
• G →B
2 3
9
Note that in the splints for special embeddings there is no case where ∆ is
s
embeddednon-metricallywhichisadirectresultofourexhaustiveclassification
of suitable characteristic representations of the subalgebra a.
Having obtained the complete classification of splints of the projections of
the rootsystems for specialembeddings, we need to check whether such splints
lead to the coincidence of branching coefficients with weight multiplicities in
modules of the algebra s.
Let’sconsiderthefirstcase-theembeddingB →D . Therootsystemof
n n+1
D hasexactlytwosimplerootsthataredifferentfromthesimplerootsofB .
n+1 n
Inthenotationof[Bourbaki, 2002]theyareα =e −e andα =e +e
n n n+1 n+1 n n+1
while αBnn = en. This difference leads to the crucial difference in the Weyl
groups of the algebras: while W is a semidirect product of the group of
Bn
permutation e →e and the group of the change of sign e →(±1) e , W
i j i i i Dn+1
is the same but for the additional condition on the groupof the change of sign:
(±) =1.Since theprojectionofD onB actsas(e ,e ,...,e ,e )→
Qi i n+1 n 1 2 n n+1
(e ,e ,...,e ,0), the singular element Φ(µ) which is the orbit of W will
1 2 n Dn+1 Dn+1
become a composition of the orbits of W after the projection.
Bn
Indeed,ifµ+ρ =(a ,a ,...,a )inthestandartbasis(e ,e ,...,e ),
Dn+1 1 2 n+1 1 2 n+1
thentheWeylgroupW willactonitbypermutatinga andchangingtheir
Dn+1 i
signs. After the projection the last coefficient in every element will be cut.
Among these elements will be groups in which all elements will have the same
setofa standinginarbitraryorderandhavinganarbitrarysign. Thesegroups
i
will be similar to the orbits of the Weyl group W but for the singular milti-
Bn
plicity. A singular miltiplicity is a determinant ε(w) of the element w of the
Weyl group. Because of the additionalcondition onthe Weyl groupW the
Dn+1
change of the signs of a doesn’t change the singular multiplicity. That’s not
i
true for the Weyl group W , thus exactly half of the elements of each newly
Bn
formed orbit Φ˜ν˜ of W will have the wrong singular multiplicity. Moreover,
Bn Bn
these orbits can be rewritten in the following form:
Φ˜νB˜n = X ε(w)(1+sen)weν˜+ρBn. (10)
w∈WDn
This decomposition is possible because the orbit of the Weyl group of D
n
coincides withthe half ofΦ˜ν˜ thathas correctsingularmultiplicities and other
Bn
half can be obtained by the action of the element s of W .
en Bn
Thus,the projection(Φ(µ) )′ consistsof(n+1)quasi-orbits(with“wrong”
Dn+1
signs) of the Weyl group W . It means that there are (n+1) weights in the
Bn
fundamentalWeylchamberC¯ ofB . Itcanbeeasilyshownthattheseweights
Bn n
are (a ,a ,...,a ) and weights obtained from (a ,a ,...,a ) by consequent
1 2 n 1 2 n
subtraction of µ e starting with µ e , where µ is Dynkin labels of µ plus
i i n n i
1. As it turns out, if one were to add to this set the required weights with
corresponding singular multiplicities to construct the singular element of s =
A +A +···+A as was done in the Introduction of this paper, all additional
1 1 1
weightwouldlieontheboundariesofthefundamentalWeylchamberC¯ . This
Bn
fact is easily provedby observing that the action of one of s on the weights
αBn
i
10