The conformal characters
HJE
The conformal characters
Antoine Bourget 0 1 3
Jan Troost 0 1 2
Paris 0 1
France 0 1
0 CNRS, PSL Research University , Sorbonne Universites
1 Calle Federico Garc a Lorca , 8, 33007 Oviedo , Spain
2 Departement de Physique, Ecole Normale Superieure
3 Department of Physics, Universidad de Oviedo
We revisit the study of the multiplets of the conformal algebra in any dimension. The theory of highest weight representations is reviewed in the context of the BernsteinGelfandGelfand category of modules. The KazhdanLusztig polynomials code the relation between the Verma modules and the irreducible modules in the category and are the key to the characters of the conformal multiplets (whether nite dimensional, in  nite dimensional, unitary or nonunitary). We discuss the representation theory and review in full generality which representations are unitarizable. The mathematical theory that allows for both the general treatment of characters and the full analysis of unitarity is made accessible. A good understanding of the mathematics of conformal multiplets renders the treatment of all highest weight representations in any dimension uniform, and provides an overarching comprehension of casebycase results. Unitary highest weight representations and their characters are classi ed and computed in terms of data associated to cosets of the Weyl group of the conformal algebra. An executive summary is provided, as well as lookup tables up to and including rank four.
Conformal and W Symmetry; Conformal Field Theory; Supersymmetry and
Duality
2.1
2.2
2.3
3.1
3.2
3.3
4.1
4.2
1 Introduction 2
Warmup: the so(2; 3) algebra
The representations of the so(3) algebra
The representations of the so(2; 3) algebra
The unitary representations
2.4 In physics conventions
3 The characters of irreducible representations
The KazhdanLusztig theory
The KazhdanLusztig polynomials
The nitedimensional representations
3.4 Examples
4 The unitary conformal multiplets
Useful concepts
4.3 The algebras Dk = so(2; 2k
6 The unitary conformal characters
6.1
In odd spacetime dimension
6.2 In even spacetime dimension
7 Summary and comparison with the physics literature
7.1
The executive summary
7.2 A brief comparison to the physics literature
8 Apologia
A The Lie algebra conventions
A.1 The Lie algebra Bk
A.2 The Lie algebra Dk
B The structure of real simple Lie algebras
B.1
The structure theory B.2 The classi cation of real simple Lie algebras B.3 The classi cation of Hermitian symmetric pairs
C The character tables for integral unitary weights
{ i {
Introduction
Quantum
eld theory is one of the most successful tools of theoretical physics. It is
ubiquitous in our understanding of physical phenomena from the smallest to the largest scales.
Conformal quantum
eld theories can be viewed as simpli ed quantum
eld theories that
arise at very low or very high energy, or at critical points. Their symmetry algebra is
enlarged. Relativistic conformal eld theories allow for a symmetry algebra which includes
the conformal algebra so(2; n) in spacetime dimension (n
It is useful to gather the spectrum of a physical theory in terms of multiplets of the
symmetry algebra. Hence it is crucial to study the representation theory of the conformal
algebra so(2; n). In physical theories, often only highest weight representations will arise.
Moreover, in unitary theories, these representations are required to be unitarizable. Thus,
the study of the unitary highest weight representations of so(2; n) has been an integral part
of the physics literature of the last fty years.
Importantly, before physicists classi ed conformal multiplets in all cases of their
interest, the mathematics literature yielded an overarching insight into the generic case,
providing a complete classi cation of conformal multiplets, with proof. In particular, the
representation theory will reduce to a theory of Weyl groups, and numbers associated to
pairs of Weyl group elements. That provides an e cient coding of otherwise lengthy
manipulations of conformal algebra generators. In most cases, the mathematics literature
precedes the physics literature, which is indicative of the fact that physicists have found
the mathematics literature hard to read. We intend to bridge this unfortunate gap in the
treatment of this central problem in quantum
eld theory by providing a physicist's guide
to the mathematics literature. Our treatment will be practical yet generic, referring to the
relevant mathematics books and papers for the complete proofs while still providing the
conformal eld theorist with all the necessary tools to reconstruct a particular result using
general principles only.
Bridges between the mathematics literature and the physics literature have been
constructed previously. We refer e.g. to [1] for the exploitation of the generic classi cation of
unitary multiplets, and to [2] for a review of the salient properties of parabolic
KazhdanLusztig polynomials relevant to conformal multiplets
nitely represented on the compact
subalgebra of the conformal algebra. We will provide a considerably more complete
treatment, and hopefully a more accessible bridge. From the e ort we invested in identifying
and marrying the mathematics and physics literature, we concluded that an introduction
for physicists to the intricate mathematics of conformal multiplets remained overdue. It
is possible to extend the scope of this work to include representations of superalgebras,
but in that situation additional subtleties arise  in particular, the Weyl group geometry
alone does not determine completely the representation theory  we plan to discuss this
in future work.
The structure of the paper is as follows. In section 2, we treat a warmup example,
namely the so(2; 3) conformal algebra in three spacetime dimensions. We compute the
characters of all irreducible highest weight multiplets, and gently introduce some of the
mathematics necessary to understand the structure of the representation theory. We also
{ 1 {
identify all multiplets that are unitarizable, and write out their characters in both a
mathematical and physical language. After the warmup section, we introduce the advanced
mathematics to treat the generic case. In section 3, we summarize how to compute
characters of all highest weight representations of the algebras so(2; n) with n arbitrary. This
discussion will include
nite dimensional, in nite dimensional, unitary and nonunitary
representations. We review how the multiplicities of the irreducible modules in the Verma
modules are given by the evaluation of KazhdanLusztig polynomials at argument equal
to one, and how the inversion of the decomposition
xes the irreducible characters. It will
be su cient to do calculations in the Weyl group (and Hecke algebra) of the conformal
algebra to understand the full structure of the representation theory. In order to apply
the formulas, we gather data on the Weyl groups of the Bk and Dk algebras, and the
corresponding KazhdanLusztig polynomials.
In section 4, we review how to identify the unitarizable representations among all those
studied in section 3. We will implicitly make use of necessary and su cient inequalities
on the quantum numbers which are elegantly derived in the mathematics literature. In
section 5 we exploit the speci c features of unitary representations to simplify the generic
KazhdanLusztig theory, and factorize a compact subalgebra Weyl group, which leads to
the study of parabolic KazhdanLusztig theory. That allows us to compute all unitary
highest weight characters in section 6.
Hasty readers can jump directly to section 7 where they will nd an executive summary,
with references to an appendix containing low rank unitary character tables.
2
Warmup: the so(2; 3) algebra
In this section, before facing the representation theory of the so(2; n) algebras in all its
complexity, we focus on the conformal algebra so(2; 3). We review the conformal multiplets
which have a highest weight. We determine the structure of the irreducible representations,
and also which irreducible highest weight representations are unitarizable. Our analysis
is phrased in the mathematical language of the category of highest weight modules, and
introduces a number of useful mathematical concepts. These serve as a warmup for the
introduction of more advanced concepts in section 3. References for proofs of the statements
in this section are mostly postponed to section 3 as well.
2.1
The representations of the so(3) algebra
We draw inspiration from the highest weight representation theory of the simplest Lie
algebra so(3) = su(2), generated by three generators, so(3) = hJ1; J2; J3i. Its representation
theory is obtained by rst choosing a Cartan subalgebra h = hJ3i, as well as raising and
lowering operators J
= J1
iJ2. Then, we pick a highest weight eigenvector of the Cartan
generator J3 with eigenvalue , called the highest weight. The highest weight vector is by
de nition annihilated by the raising operator. We then act on it with the lowering
operator, generating new vectors, which generate a representation of so(3). As is wellknown,
if
2= Z 0 (in a given normalization), the process never stops and the representation is
in nite dimensional. On the other hand, when
2 Z 0, we can consistently de ne a
+ 1
{ 2 {
dimensional (irreducible) representation, with a lowest weight vector which is annihilated
by the lowering operator. In this subsection, we formalize these wellknown facts in the
language of modules, which is used in the rest of the paper.
We de ne a Verma module M
with weight
as the representation of g where no
constraint is imposed beyond the relations of the Lie algebra. This means that the character
[M ] of the Verma module M is given by
w
so the character of the general irreducible module can be rewritten
where the lowering operator has eigenvalue
2. In particular, a Verma module is always
in nitedimensional. It may happen that a Verma module contains other Verma modules.
Here, this happens only when
2 Z 0, where the Verma module M
2 is included in the
Verma module M . In that case, we can construct the quotient module M =M
2, which
is nitedimensional and irreducible. We call this irreducible module L , and its character is
M is irreducible, and therefore we set L = M .
is not a positive integer, the Verma module
As we will see later, it is natural to introduce the Weyl vector
= 1 of the so(3) Lie
algebra, and the Weyl group W = f1; 1g. We further introduce the dot action of the
Weyl group on the weight space through the formula
(2.1)
HJEP04(218)5
(2.3)
(2.4)
(2.5)
2For the exploration of other objects in the category of highest weight modules of so(3) in a physical
context, see [3].
{ 3 {
We perform a similar analysis for the highest weight representations of the so(2; 3) = B2
algebra. The choice of real form of the algebra does not matter at this stage, but we
must come back to this point when we consider the question of unitarity. We choose a
Cartan subalgebra in the compact subalgebra so(2)
so(3) of so(2; 3), corresponding to
the dilatation operator and a spin component. The Verma module M
with highestweight
generically has character
generalizing (2.1), where the product over negative roots makes sure that we take into
account the free action of the lowering operators on the highest weight state. Depending
on the weight , the module M
may be reducible, and the character of the irreducible
module L
will di er from the Verma character (2.6). This can happen only if another
Verma module M is a strict submodule of M for some weight .
Integral regular weights.
Firstly, we consider for simplicity an integral weight : for
each root
the product h ; _i satis es h ; _i 2 Z. The weight space is twodimensional,
and the position of
with respect to the negative Weyl vector
is characterized by the
sign of the integers h
i
+ ; _ . This de nes eight (shifted) Weyl chambers, as shown in
gure 1. One can label the chambers with elements of the Weyl group W , associating
the identity element to the chamber that contains the weight
2 , as in
gure 1. On
the Weyl group, one can de ne a partial order, the Bruhat order [4]. This order can be
summarized in a Bruhat graph represented in gure 2, as well as on gure 1. In a minimal
representation of a Weyl group element in terms of simple re ections, the length of the
element is equal to the number of simple re ections. Any integral weight in the interior
of one of the (shifted) Weyl chambers can be written in a unique way as
= w
, where
w 2 W and
is antidominant, meaning that h
+ ; _i 2= Z>0 for each positive root .
The partial Bruhat order is instrumental in our understanding of the structure of Verma
modules [5]. Indeed, for an integral weight lying in the interior of the antidominant Weyl
chamber, and any Weyl group element w, we have that the irreducible module (and
character) can be understood in terms of the Verma modules (and characters) associated to the
same antidominant weight, and Weyl group elements smaller than w in the Bruhat order:
X
w0 w
[Lw ] =
bw0;w[Mw0 ]
bw0;w = ( 1)`(w) `(w0) ;
for some integer coe cients bw0;w. In the case of the algebra so(2; 3), these coe cients are
particularly simple  and it is mostly here that we exploit that we restrict to the example
of so(3; 2) in our warmup section. The coe cients bw0;w for the so(2; 3) algebra are given by
where `(w) is the length of the Weyl group element w, which can be read from the Bruhat
graph [4] (see gures 1 and 2). The dotted Weyl group action is still given by the formula
(2.7)
(2.8)
{ 4 {
re ections si, their Bruhat order, the simple roots i and the integral weight lattice. The red lines
correspond to singular weights, and delimitate the shifted Weyl chambers. The intersections of gray
lines correspond to integral weights.
w
. We have restricted to integral weights in the interior of a Weyl
chamber  those are called regular. We turn to an example.
Firstly, let us introduce the parameterization of roots and weights in terms
of an orthonormal basis ei (described in detail in appendix A). The simple roots are
1 = e1
weight
e2 and
2 = e2, see
gure 1. The so(3; 2) weights are denoted ( 1; 2) for a
=
1e1 +
2e2. Let us then consider the weight
= ( 1; 2). It sits inside the
Weyl chamber labeled by the Weyl group element w = s2s1s2 of length three. For this
example, the formula (2.7) gives rise to the character
[L( 1;2)] = [M( 1;2)] [M( 1; 3)] [M( 2;2)] + [M( 2; 3)] + [M( 4;0)] [M( 4; 1)] :
(2.9)
Integral singular weights.
The formula (2.7) provides the character of any irreducible
highestweight module with highest weight in the interior of a Weyl chamber, i.e. away
from the red lines in gure 1. Now we focus on singular integral weights, which are the
{ 5 {
s2
` = 4
to the s2 re ection, and we use the same color for elements of W connected by this re ection: they
contribute to the same module M c, see equation (2.10).
integral weights
such that h + ; _i = 0 for at least one root . They lie on a red line
in gure 1. The rule here is as follows: consider all the Weyl group elements that label the
Weyl chambers of which the closure contains , and pick the smallest such group element
w according to the Bruhat order. We can then write again
= w
for an antidominant
weight , and the character formula (2.7) remains true.
Example. Let us consider the weight
= ( 32 ; 12 ). This is an integral weight, but it is
singular. It belongs to the closure of the Weyl chambers labeled by the Weyl group elements
s2s1 and s2s1s2 of length two and three respectively. The smallest of these two elements
is w = s2s1, and therefore one writes
= ( 32 ; 12 ) = w ( 52
; 12 ). We then compute
[L( 32 ; 12 )] = [M( 32 ; 12 )] [M( 52 ; 12 )] [M( 32 ; 32 )] + [M( 52 ; 12 )] = [M( 32 ; 12 )] [M( 32 ; 32 )] :
The cancellation between Verma module characters occurs because we are studying a
representation with singular highest weight.
Nonintegral weights. Finally, we extend our computation to nonintegral weights. For
an arbitrary weight , we construct the set
[ ] of roots
that satisfy h ; _
i 2 Z. To
get a grasp on
[ ], we compute this scalar product for all positive roots, with as before
= 1e1 +
2e2. See table 1. A priori, since there are four positive roots we have 24 = 16
con gurations to consider, but consistency restricts this number to 7 con gurations, which
are listed in table 1. One observes that [ ] is a root system, and its Weyl group W[ ] will
play the role that the Weyl group W of the whole algebra played in the integral case. The
root system
[ ] determines the integrality class of . In this low rank case, the integer
coe cients bw;w0 again simplify to a sign depending on the length of the elements in the group
W[ ]. The character formula takes the form (2.7), but where the sum is restricted to the
Weyl group elements W[ ] and the length function is inside this group. In this manner, we
have found the characters of all irreducible highest weight representations of the B2 algebra.
{ 6 {
HJEP04(218)5
1
2
2
1
of the B2 algebra, the scalar product of the roots with the weights
= 1e1 + 2e2 as well as the root systems [ ] they give rise to.
= ( 12 ; 0). The integrality class is D2, and the
associated Weyl group has four elements, W[ ] = f1; s2; s1s2s1; s1s2s1s2g, using the notations of
gure 1. The weight
lies in the chamber of the longest element s1s2s1s2, so the irreducible
character with highest weight
is
[L( 12 ;0)] = [M( 12 ;0)] [M( 12 ; 1)] [M( 52 ;0)] + [M( 52 ; 1)] :
2.3
The unitary representations
As we review in full generality in section 4, only a subset of the irreducible modules L are
unitarizable. We say that a weight is unitary if the corresponding irreducible module L is
unitarizable. In this context, it is important that we consider the real form so(2; 3) of the
complex B2 algebra. Manifestly, this is a noncompact real form, and therefore nontrivial
unitary representations will be in nitedimensional. As we recall in section 4, in the case
of the algebra so(2; 3), the result of the identi cation of unitary weights is as represented
in gure 3, where the unitary weights are painted in blue.
A second observation is that for all unitary weights we have that 2 2 2 N. Thus, from
the point of view of table 1, the unitary weights correspond to the third, sixth or seventh
cases, i.e. with root systems [ ] = A1; D2 or B2. This corresponds to the fact that the
compact subalgebra su(2) = so(3)
so(2; 3) is nitely represented in a unitary highest weight
representation. In other words, for unitary irreducible modules the only source of in
nitedimensionality is the noncompact part of the algebra. We exploit this fact to write more
compact formulas for the characters. Firstly, we introduce notations that re ect this desire.
For a unitary weight, let us de ne a module M c which is the quotient of two Verma
M c = M =Ms2 :
[M c] = [M ] [Ms2 ] :
{ 7 {
modules:
of the module M c is
This is sensible because of the restriction on unitary weights. Accordingly, the character
Thus, we have already divided out a Verma module that is guaranteed to be a submodule
because of the fact that the compact algebra is nitely represented. Using this notation, we
can write down the characters of all irreducible unitary representations of so(2; 3) as follows:
(2.10)
(2.11)
algebra so(2; 3). The green circles correspond to dominant weights.
0) The highest weight
= 0 corresponds to the trivial representation, and we simply
have
we nd that the compact subtraction is the end of the story
1) For highest weights
which fall in one of the following categories:
2 1 2= Z (the A1 case)
2 1 2 Z and
1
2 2= Z (the D2 case) and
1
NorthWest chamber of D2)
32 (the weight is in the
is integral (the B2 case) and 1
2
2 (the WestNorth chamber of B2)
=
[L ] = [M c] :
in one of the two following categories, we nd a further subtraction:
2a) If 2 1 2 Z and
the NorthEast chamber of D2)  this category contains only two weights,
2b) If
is integral (the B2 case) and 1 >
2
2 and 1 <
32 (the NorthWest
chamber of B2)
[L ] = [M c] [M(cs1s2s1) ] :
[L ] = [M c] [M(cs2s1s2) ] :
These results comprise all characters of unitary irreducible highest weight representations
of the conformal algebra so(3; 2). In the next subsection, we render more manifest the
physical content of these results.
2.4
In physics conventions
Early physics references classifying the unitary representations of the so(3; 2) algebra and
their characters are [6{8] and [9]. The algebra so(2; 3) admits a basis made of three so(3)
spins J1;2;3, three translations P1;2;3, three special conformal transformations K1;2;3 and the
dilatation operator D. In order to de ne the Verma modules, we declare two operators to be
in the Cartan subalgebra, which we choose to be the spin component J3 and the dilatation
operator D which are in a compact subalgebra. We pick four raising operators (J+ and
K1;2;3) and four lowering operators (J
and P1;2;3). We consider highestweight modules,
so the weights
will consist of eigenvalues ( E; j) of ( D; J3). In these conventions, closer
to traditions in physics, the above generic Verma module characters translate into
where the fugacity x keeps track of the conformal dimension of the states, while the fugacity
s codes (twice) the 3component of the spin. The characters with respect to the su(2)
compact subalgebra read
(2.14)
(2.15)
HJEP04(218)5
(2.16)
(2.17)
(2.18)
k=0;1;:::
with the usual spin j character [Ls2uj(2)] of the representation of the su(2) subalgebra
de ned by
On the lower blue line in gure 3, we nd the trivial representation with ground state
energy and spin ( E =
1; j =
2) = (0; 0), the singleton ( E; j) = ( 1=2; 0) as well as
the other scalars ( E <
1=2; 0). On the second line, we have the singleton ( 1; 1=2),
{ 9 {
as well as the other spinors ( E <
1; 1=2). The other representations are the generic
( E
j
1; j) representations. See e.g. [9] for an early summary.
For the weights of type 1) in subsection 2.3, which include the generic scalar, spinor
and higher spin representations we nd the characters
For the weights of type 2), we have for the singletons (type 2a))
and for the other extremal representations (type 2b)  note that for those, j
1),
These calculations exhaust the characters of unitary highest weight representations of
so(3; 2), and are in agreement with the physics literature.
Summary remarks.
The warmup example of the threedimensional conformal algebra
is illuminating in multiple respects. It identi es the crucial role of the Weyl vector and
the Weyl group for all irreducible characters, as well as the role of the compact subalgebra
in the simpli cation of the unitary characters. It also motivated that we need to come to
terms with at least two more advanced mathematical concepts: the rst is the multiplicity
of the Verma modules in the characters of irreducible modules, and the second is the
generic classi cation of unitary highest weight representations. The generic treatment of
these points requires further levels of abstraction.
3
The characters of irreducible representations
In this section, we explain how to write the characters of irreducible modules in terms of
the characters of Verma modules for an arbitrary semisimple complex Lie algebra g. Since
the full mathematical solution to this problem is available, but may be hard to read, or
even identify, we provide a very brief guide to the history and literature.
Important early contributions to the understanding of the category of modules with
highest weight are [10] and [11]. The generic solution to the character calculation is based
on the KazhdanLuzstig conjecture [12] proven in [13, 14]. The book [5] makes the
mathematics signi cantly more accessible. Furthermore, to understand the unitary characters
the parabolic KazhdanLusztig polynomials [15] are instrumental, in particular as
pertaining to Hermitian symmetric spaces [16]. The parabolic polynomials were computed in [17]
and in more technical detail in [18]. The nal step in summarizing the literature requires
the use of translation functors [5], and the resulting
nal formulation is most easily read
in [19] and [20]. We refer to the book [5] as well as to the summary [20] for further history.
3.1
In this subsection, we brie y remind the reader of basic concepts in Lie algebra and
representation theory. See e.g. [21{23] for gentler introductions. Let g be a semisimple complex
Lie algebra, with Cartan subalgebra h. We denote the set of roots of g by , the subset of
positive roots by
+ and by
s is the subset of simple roots. The Weyl group is W , the
Weyl vector , and we de ne the dot action
w
group is denoted W[ ]. The Bruhat order on W[ ] is consistent with the Bruhat order on
W, and the parity of the length functions agree [4].
We will use a handy parameterization for the weights [5]. A weight is called3
antidominant if for all
dominant if for all
2
+
+
, h + ; _i 2= Z>0;
2
, h + ; _i 2 Z>0.
Both of these subsets of weights are highly restrictive, and in particular, their union does
not include all weights. Note also that our de nition of dominant makes all dominant
weights integral. For any weight
2 h , there is a unique antidominant weight in the dot
. Therefore, any weight
can be written in a unique way as
with
antidominant and w 2 W[ ] of minimal length. The minimal length requirement
ensures that the decomposition (3.2) is unique.
Given a weight
2 h , we focus our attention on two modules, which are both
highestweight modules with highest weight . The rst one is the Verma module M . It is de ned
as the module generated from a highest weight state by the action of all lowering operators.4
Its character [M ] follows from the de nition,
We introduce the simple module L (also called the irreducible module), which is the unique
simple quotient of M . Writing down the character of the module L is a central task in
this paper.
Given an antidominant weight
[Lw ] of the irreducible module Lw
2 h , our goal is to understand how the character
decomposes into characters of Verma modules [M ].
3We warn the reader that some authors use di erent de nitions for these concepts.
4More precisely, the relevant object here is the universal enveloping algebra U(g), which can be thought
as g with an associative product such that the Lie bracket is given by the commutator. We start with
the onedimensional (h
according to the linear form
n+)module C
(where the raising operators n
+ give zero and the Cartan h acts
), and form the tensor product with U(g), M ( ) = U(g) U(h n+) C .
= w
(3.2)
(3.3)
Only weights
of the form
where PwW0;[w] (1) are coe cients, and we have factored out the sign contribution of the
length di erence `(w; w0) = `(w)
Lusztig polynomials PwW0;[w] (q) associated to the Weyl group W[ ] and two elements w0 and
w of the group W[ ], evaluated at q = 1. In the next subsection, we give an algorithm to
compute these polynomials. Note that we have presented a crucial property of the theory
of representations and characters, namely that the coe cients only depend on the relevant
Weyl group [5]. This property was surmised early and proven late in the development
of the theory. It implies that extensive manipulations of Lie algebra generators can be
summarized in the more e cient combinatorics of the Weyl group only.
`(w0). The coe cients PwW0;[w] (1) are the
Kazhdan
The KazhdanLusztig polynomials
We review one algorithm to compute the KazhdanLusztig polynomials for Coxeter groups
(which includes all Weyl groups that we encounter) [4]. Firstly, one computes the Bruhat
partial order, that we denote by
. Secondly, we proceed as follows. Let x; w be two
elements of the Coxeter group W . We are ultimately interested in the KazhdanLusztig
polynomial Px;w(q), but the algorithm requires to compute as well an auxiliary integer
denoted (x; w).
If x
If x
If x = w, set Px;w(q) = 1 and (x; w) = 0.
w, set Px;w(q) = 0 and (x; w) = 0.
w and x 6= w, then let s be a simple re ection such that `(sw) < `(w). Let
c = 0 if x
sx, and c = 1 otherwise. Then set (see the core of the existence proof
provided in [4], section 7.11)
Px;w(q) = q1 cPsx;sw(q) + qcPx;sw(q)
(z; sw)q(`(w) `(z))=2Px;z(q)
(3.5)
X
where the sum runs over those z 2 W such that z
sw and sz
z. Finally, de ne5
(x; w) = Coe cient of q(`(w) `(x) 1)=2 in Px;w(q) :
(3.6)
Using the algorithm, we can compute all the KazhdanLusztig polynomials for the Weyl
groups W appearing in the character formula (3.4). Thus, the proof [13, 14] of the
KazhdanLusztig conjecture [12] solves the problem of determining all characters of highest weight
representations of semisimple Lie algebras.
5In particular, note that if the degree of the polynomial Px;w(q) is strictly less than (`(w) `(x) 1)=2,
then (x; w) = 0.
HJEP04(218)5
The reader may nd comfort in recovering the Weyl character formula for nitedimensional
irreducible representations as a particular case of the vast generalization (3.4). The
irreducible representation L of the simple Lie algebra g is nitedimensional if and only if its
highest weight
is dominant (see subsection 3.1).
Let the weight
be dominant. Then we can write the weight
in the form
= w
with the weight
antidominant and w the longest element of the Weyl group. For all
elements x in the Weyl group W , the KazhdanLusztig polynomial Px;w (q) trivializes to
Px;w (q) = 1 [5]. Therefore, for nite dimensional representations, the generic character
formula (3.4) simpli es to
which includes a sum over the whole Weyl group. Intuitively, the further the highest weight
is from antidominance, the bigger the character sum. For nite representations, the sum
has the maximal number of terms.
A remark on some singular integral weights.
According to our de nition, a
dominant weight can not be singular. In fact, the integral weights located in the dominant
shifted Weyl chamber (those that satisfy h
+ ; _
i 2 Z 0 for all positive roots
) are
split into two categories: the dominant weights and the singular weights. An interesting
consequence of the general formula (3.4) is that the character of an irreducible module L( )
where
belongs to the second category vanishes. This property is useful in simplifying
character formulas.
3.4
Examples
The generic character formula captures (among others) the character of all highest weight
representations of the conformal algebras so(2; n). In the rest of the paper, we will mainly
be interested in the unitary highest weight representations, which are a small subclass of
all highest weight representations. These are the representations most evidently relevant
in physical theories. Nevertheless, nonunitary representations can play a role in unitary
theories with gauge symmetries, or in nonunitary theories of relevance to physics.
Therefore, we want to make the point that the mathematical formalism that we reviewed also
readily computes the characters of this much more general set of representations. To stress
that point, we compute an example character which involves a slightly more complicated
KazhdanLusztig polynomial.
A B3 example.
The Weyl group of B3 has 48 elements. They are arranged in ten levels,
depending on the number of simple Weyl re ections that occur in their reduced expression.
See gure 4. Since there are 48 Weyl chambers, and a proliferation of walls and weights of
various singular types, we do not provide a complete catalogue of characters. The results
are straightforward to obtain, but unwieldy to present. We only provide a avour of what
such a catalogue looks like.
contribute to the same module M c (see section 5). The compact subgroup WJ is isomorphic to the
Weyl group of B2, and one can check that the subset of elements in each given color is isomorphic
to gure 2.
To discern the features of the catalogue, it is su cient to analyze the geometry of
the chambers, the walls, and the corners. The positive root system
e3; e3; e2; e1; e1
e3; e1 + e2; e1 + e3; e2 + e3g of the algebra so(5; 2) can be divided into
subsystems in various ways. If the set of roots orthogonal to the weight
+
is empty, we
are in a chamber. If it is nonempty, we are on at least one wall. We have nine walls, given
by the equations i = j , i =
j and i = 0. We have weights living on a single wall,
weights living in the corner of two walls, in the corner of three, in the corner of four or on
the intersection of the nine walls. This provides us with a rst glimpse of the structure of
+ = fe1
e2; e2
the catalogue.
Next, we want to clarify the di culty of computing the KazhdanLusztig polynomials.
While laborious, the di culty remains well within reach of ancient computers. The most
complicated KazhdanLusztig polynomial turns out to be P1;s2s3s2s1s2s3s2 (q) = q2 + q + 1
(and it arises for a single other combination of Weyl group elements as well). At q = 1,
this will give rise to a triple multiplicity for a Verma module character in the character
sum formula. An example weight for which we need this polynomial is produced by acting
with w = s2s3s2s1s2s3s2 on an antidominant weight. Thus, we give the following example
entry in the catalogue.
Example. Consider the weight (s2s3s2s1s2s3s2) ( 2 ) = ( 1; 1; 1). We apply the
general procedure outlined in this section using a symbolic manipulation program, and
nd the character:
+ [M( 3;0;2)] + [M( 3;1; 2)] [M( 3;1;1)]
+ [M( 2; 4; 2)] [M( 2; 4;1)] [M( 2; 3; 3)]
+ [M( 2; 3;2)] + [M( 2;0; 3)] [M( 2;1; 2)]
[M( 1; 4; 1)] + [M( 1; 4;0)] + [M( 1; 2; 3)]
[M( 1; 2;2)] [M( 1; 1; 3)] + [M( 1;1; 1)] :
(3.8)
Note the multiplicities of the Verma modules, which go up to three, even in this low rank
example. Proceeding in this fashion, one can imagine lling out systematically the thick
catalogue of character formulas. The reader who is so inclined will surely nd the tables
to be constructed shortly equally mesmerizing.
4
The unitary conformal multiplets
In section 3 we exhibited how to compute the structure and character of any highest weight
representation of the conformal algebra so(2; n). In this section, we determine which of the
highest weight conformal multiplets are unitary. Those multiplets are the representation
theoretic building blocks of unitary conformal eld theories. The mathematical analysis of
the unitarizability of the representations of the conformal algebra ts into a more general
framework, which we recall brie y.
Firstly, let G be a simply connected and connected simple Lie group, and K a closed
maximal subgroup. Then, the group G admits a nontrivial unitary highest weight
module precisely when (G; K) is a Hermitian symmetric pair [24, 25]. Hermitian symmetric
pairs have been classi ed [16]. See appendix B for a summary of the relevant
structure theory of real simple Lie groups, and [26] for a complete treatment.
The
conformal group G = SO(2; n) satis es the condition, with the maximal compact subgroup
Spin(n). The techniques used to classify the unitary highest weight
representations for such groups include the identi cation of weights of null vectors and the
degeneration of the contravariant form on the Verma module [27, 28].
The full classi cation of the unitary highest weight modules of the conformal algebras
was obtained in [27]. It is based on an exploitation of necessary and su cient inequalities
satis ed by unitary representations. These were derived in full generality in [29]. The
analysis of physicists of level one and level two constraints on unitary representations can be
viewed as a partial analysis of the necessary conditions. In this section, we demonstrate that
it su ces to decipher the earlier and more complete mathematical classi cation results to
recuperate in a uniform manner the results in the physics literature. We provide a glimpse
of the concepts that underlie the classi cation result, illustrate the general analysis in the
example of B2 = so(2; 3), and then recall the full classi cation of the unitary highest weight
multiplets for the Bk = so(2; 2k
1) and Dk = so(2; 2k
2) algebras. A physics reference in the same vein is [1].
We again consider highest weight modules based on a highest weight state with respect
to a Borel subalgebra b of the complexi ed Lie algebra. The elements h in the Cartan
subalgebra h act as (h) where
is the highest weight. The span of the compact root
system
c has codimension one in the dual h of the Cartan subalgebra [26]. We de ne
to be the maximal noncompact root [26]. The classi cation theorem of [27] introduces
a class of weights, which we generically write , which are
compact subalgebra k is nitely represented) and which satisfy
c+ dominant (because the
where
is the maximal noncompact positive root of the conformal algebra.6
We also
introduce an element
of the weight space which satis es that it is orthogonal to all
compact roots as well as the normalization
h
+ ; i = 0 ;
h ; i
2h ; i = 1 :
(4.1)
(4.2)
Then the highest weights corresponding to unitarizable representations all lie on the lines
=
+ z
where z is a real number. See gure 5.
There is a halfline of unitary representations ending at a point which is generically
at a positive value of z, depending on the algebra g and the weight . Then, there are
further points where unitary representations can occur, taking values in an equally spaced
set, with a spacing which depends on the algebra only. There is an endpoint to this discrete
set. The calculation of the three constants (called A( ), B( ) and C( ) in gure 5) that
determine this set proceeds via the introduction of auxiliary root systems.
Indeed, we want to bring to the fore how singular the weight
is with respect to the
compact root system. To that end, we de ne the subset c( ) of compact roots orthogonal
6In [27], the weights
are called 0.
C( )
highest weights lie on lines of the form
=
+ z , and the gure represents the values of z 2 R
that give unitary weights. On a given line, there is a semiin nite line of highest weights which
is allowed, and then an equally spaced set of discrete allowed values, starting at the end of the
halfline, and ending after a nite number of steps.
to . We then de ne the new root system f
g [ c( ) and decompose it into simple root
subsystems. The simple root system which contains the maximal noncompact root
is
baptized Q( ). Exceptionally, we will make use of a second root system R( ), de ned as
follows. If the root system
has two root lengths and there is a short root not orthogonal
to the system Q( ) and such that h ; i=h ; i = 1, then we adjoin the short root to
Q( ). The simple component containing
of the resulting root system is named R( ).
These root systems can be algorithmically determined from the weight , and they allow
for the calculations of the three constants, which in turn determine all the unitary highest
weight conformal multiplets. The calculations are performed explicitly in [27]. We review
the results of the calculations in subsections 4.2 and 4.3.
4.2
The algebras Bk = so(2; 2k
1)
In this subsection, we generalize the example of the so(2; 3) algebra to include all conformal
algebras so(2; 2k
1) associated to a spacetimes of odd dimension 2k 1. We list all unitary
highest weight representations [27]. We again use the conventions of appendix A. Firstly, we
review the set c+ of positive compact roots, as well as the set
n+ of positive noncompact
roots [26]:
HJEP04(218)5
c+ = fei
ej j2
ej j2
i < j
j
ng [ fe1g :
ng [ fej j2
j
ng
The highest noncompact root
coincides with the highest root of the algebra. When we include the highest root of the
algebra in the Dynkin diagram, we obtain the a ne untwisted Dynkin diagram. The Weyl
vector
for the Bk algebra is
= (k
1=2; k
3=2; : : : ; 1=2) :
We parameterize the weights
in terms of their components in the basis ei of
orthonormal vectors and demand that the components corresponding to the compact subalgebra
= e1 + e2
(4.3)
(4.4)
(4.5)
so(2k
1) are dominant integral weights:
i
2
j 2 Z
3
2 k 2 Z :
i < j
k
We moreover need the 1 component of the weight
to be tuned such that the weight
is orthogonal to the maximal noncompact root , which implies
We moreover parameterize the line on which the unitaries with highest weight
terms of the normalized orthogonal vector
HJEP04(218)5
The root systems Q( ) and R( ) that measure the degenerate nature of the anker
will either be the full conformal algebra so(2; n) or su(1; p) with p smaller than
the rank of the conformal algebra. We distinguish three cases for the root systems Q
and R [27]. The
rst case is labelled by an extra integer p that satis es 1
Case (I,p) corresponds to root system Q = su(1; p) = R with anker weights
p < k.
obeying
2 =
2 =
3 =
=
p+1 for p
k
1. Case II corresponds to Q = so(2; 2k
1) = R and
= 0. Case III is exceptional in that it has a root system Q = su(1; k
1) that
di ers from the root system R = so(2; 2k
1). The weight satis es 2 =
= 1=2.
The theorem of [27] states that the highest weight irreducible module with highest
weight
=
+ z is unitarizable if the module is trivial, or if the highest weight obeys the
inequalities
z
z
p
k
for case (I,p) and
1=2 for cases II and III :
Preparing for a physicist's energetic lowest weight perspective, we denote the rst
compoby 1 =
E. We summarize the unitarity conditions for so(2; 2k
1)
nent of the weight
in tables 2 and 3.
4.3
The algebras Dk = so(2; 2k
2)
In this subsection, we list the unitary highest weight modules of the conformal algebra in
even dimensions. We again present highlights of the classi cation theorem proven in detail
in [27]. The nal result in all even dimensions can also be summarized very succinctly.
The ground work is layed by noting that the set of compact positive roots c+ and the
set of noncompact positive roots
n+ is given for the Dk algebra by
c+ = fei
ej j2
ej j2
i < j
j
kg :
kg
(4.6)
(4.7)
lie in
(4.8)
(4.9)
(4.10)
(I,1)
2
(I,p)
p
k
2
(I,k
1)
II
III
1
0
1
0
1
0
1
0
1
2
2
2
2
2
3
3
3
3
3
p
p
p
p
p + 1
p + 1
p + 1
p + 1
p + 1
k
k
k
k
k
1
1
1
1
1
k
k
k
k
k
noncompact root , equal to the a ne root. The noncircled black nodes are the roots that are
orthogonal to
. Because of the constraints on , the root 1 can never be orthogonal to
. The
small black dot means that the root satis es h ; i = 12
. The root system Q( ) is the one generated
by the big black dots, and the root system R( ) is the one generated by all the black dots.
Type (I,p) II III
De nition
Unitarity constraint
2 =
2 =
=
=
p+1 >
p+2
k 2= f0; 12 g
(1
p
k
2)
(p = k
1)
2 =
2 =
=
=
k = 0
k = 1=2
E
E
2k
k
E
1). Here the ( 2; : : : ; k) are either all integers or all
halfintegers, and 2
k
0. We use interchangeably the notation E =
2 +
2
p
32 or E = 0
k
1
1
HJEP04(218)5
again coincides with the highest root for this algebra
and the Weyl vector is
We use the following parameterization of the anchor weight
and the orthogonal pointer
is dominant in its compact components:
i
2
j 2 N
3
2 k 2 Z
i < j
The root systems Q( ) and R( ) are always equal since all roots have equal length. There
are again three cases to distinguish, but two of them are related by the outer automorphism
2). The latter acts on the weight components by ipping the sign of the nal
component k. This symmetry of our classi cation problem reduces the number of cases
to two, namely the root systems su(1; p) with p
k
1, and the root system so(2; 2k 2). The nal statement is that the representation is unitarizable if and only if
z
z
p
k
for case (I,p)
1
for case II ;
with the exception of z = 2k
3 in case II, which corresponds to the trivial representation
2). We now determine in which case we are, depending on the weight .
The root system Q( ) is simple, and contains at least the maximal noncompact root
= e1 + e2. Thus, we consider rst whether the compact roots containing an e2 term
belong to the root system Q( ). Given the constraints on the weights i of the
nite
dimensional representation of so(2k
2), this is the case if and only if 2 =
3. If these
entries are not equal, then the root system Q( ) corresponds to the rank one noncompact
algebra su(1; 1). When 2 =
3, we attach one further node. We continue in this manner,
and
nd that when we have consecutive components 2 =
3 =
=
p =
p+1 equal,
then the noncompact algebra is su(1; p). When we reach the end of the chain, we have the
case 2 = j kj 6= 0 with algebra su(1; k
1). Finally, we have the exceptional case 2 = 0
for which the root system Q( ) corresponds to the full algebra so(2; 2k
2). Thus, for each
weight , we have found the root system Q( ). We can then summarize all unitary highest
weight representations. We again declare 1 =
E, and we run through all possible cases.
We list the results for so(2; 2k
2) in tables 4 and 5.
(4.11)
(4.12)
(4.13)
(4.14)
(4.15)
k
2
(I,k
1)
(I,k
1)'
II
0
1
0
1
0
1
0
1
0
1
2
2
2
2
2
...
...
...
...
...
p
p
p
p
p + 1
p + 1
p + 1
p + 1
p + 1
...
...
...
...
...
k
k
k
k
k
k
2
k
k
2
k
k
2
k
k
2
k
1
1
1
1
The noncircled black nodes are the roots that are orthogonal to . Because of the conventions for
, the root 1 can never be orthogonal to .
Type (I,p) II
2 =
=
p+1 > j p+2j
2 =
2 =
=
=
k 6= 0
k 6= 0
(1
p
k
2)
for (I,k
for (I,k
1)
1)'
2 =
=
k = 0
De nition
Unitarity constraint
E
E
k 1
2k
k
3 +
2
p
2 or E = 0
j kj, all the di erences
i+1 2 Z for i = 2; : : : ; k
1 and 2 k 2 Z. We use interchangeably the notation E =
1
HJEP04(218)5
The Weyl group cosets
In this section, we combine the results of sections 3 and 4 to compute the characters of the
unitary irreducible representations of the conformal algebras in various dimensions. We
already saw how the generic representation theory boils down to Weyl group theory and
the calculation of KazhdanLusztig polynomials (evaluated at one). We will further show
that in the case of unitary representations, the Weyl group combinatorics can be simpli ed
by performing an e cient (parabolic) decomposition. The paper [2] summarizes some of
the features that we exhibit in detail. See also [30, 31] for a related approach, based on
the exact sequences of [32].
Consider the conformal algebra so(2; n) in n dimensions and its Weyl group W . The
this Weyl group WJ . Finally, we construct the set W J
Weyl group is generated by the simple re ections S = fs1; : : : ; sn+1g. The Weyl group of
the compact subalgebra so(n) is generated by the re ections J = fs2; : : : ; sn+1g. We call
W by taking, in each equivalence
class of WJ nW , the element of minimal length. Then, each w 2 W can be written in a
unique way as
w = wJ wJ ;
wJ 2 WJ ; wJ
2 W J :
PwJ;i;w1j =
1 i
j
0 otherwise:
wi
n + 2
wj
otherwise :
7In the last step, following Proposition 3.4 in [15], we have introduced the standard notation for parabolic
KL polynomials. In the context of our paper, we can take the last equation as their de nition. Since the
notation matches the mathematics literature, comparison is easier.
8
>0
>
>
>:1
PwJ;i;w1j = <
1 + qj n 1
j
2n
1 ; 1
i
2n
j
In particular, the longest element w of W decomposes as w ;J wJ , where w ;J is the longest
element of WJ .
The highest weights of unitary irreducible representations of the conformal algebras
are dominant in the compact direction. Let
= w
be such a weight, with w 2 W and
antidominant. Because
is dominant in the compact direction, the parabolic
decomposition of w reads w = w ;J wJ . Let v = vJ vJ
that [15]7
2 W . We want to evaluate Pv;w. We have
Pv;w = PvJ vJ ;w ;J wJ = Pw ;J vJ ;w ;J wJ = PvJJ; ;w1J :
We see that for an element w of the form w = w ;J wJ , the polynomial Pv;w depends only
on the representatives of v and w in W J . For that reason, it is necessary to study the
structure of W J . The Bruhat order in W J is given in
Using the notations of the gure, we have that [17, 18]
gure 6 for the conformal algebras.
(5.1)
(5.2)
(5.3)
(5.4)
.
.
.
wk+2
wk 1
.
.
.
w1
w2k
wk+1
wk
.
.
.
We note the drastic simpli cation in the complexity of the KazhdanLusztig polynomials.
They can be explicitly computed, and when evaluated at one, they equal no more than two.
These preliminaries allow us to simplify the character formula. Let
be a unitary
weight. It can be written
= ( 1; 2; : : : ; k)

{z
c
}
where c is a dominant integral weight of the compact subalgebra. As such, it is the highest
weight of a
nite dimensional representation of the compact algebra k, whose character is
denoted by [Lk ]. This generalizes equation (2.18). Then we can introduce the generalized
Verma modules M c, de ned from those
nitedimensional representations of the compact
subalgebra by induction to the full algebra. The characters are related by
(5.5)
(5.6)
(5.7)
and the Weyl character formula gives
2 n+
z
[Lk ]
X ( 1)`(w)[Mw ] :
w2WJ
Our goal is now to express the irreducible characters [L ] in terms of the induced characters
(5.8)
( 1)`(w ;J wJ ;wJ0 w0J )
PwJ0 w0J ;w ;J wJ (1) MwJ0 w0J
( 1)`(w ;J )+`(wJ ) `(wJ0 ) `(w0J )Pw ;J w0J ;w ;J wJ (1) MwJ0 w0J
( 1)`(w ;J )+`(wJ ) `(w0J )Pw ;J w0J ;w ;J wJ (1) M wc ;J w0J
h
:
[M c]. We have, for
integral and unitary,
X
X
X
X
wJ0 2WJ w0J 2W J
wJ0 2WJ w0J 2W J
X
w0J 2W J
We have reduced the sum over the Weyl group, which contains respectively 2
2k k! elements for Dk and Bk, to a sum over the 2k elements of W J . Thus, unitary highest
weight conformal representation theory has been reduced to the analysis of Weyl group
k 1 k! and
parabolic cosets, and their associated KazhdanLusztig polynomials. All the ingredients in
the character formula can be explicitly computed.
6
The unitary conformal characters
In this section, we apply the schemes of sections 3 and 4 to systematically calculate all
characters of unitary highest weight representations of the conformal algebras so(2; n).
To facilitate the calculations, we give in tables 7 and 6 the values of the parabolic coset
representatives wi in terms of simple re ections si. We number the simple roots as in the
Dynkin diagrams of tables 2 and 4, and denote by si the re ection through the simple root
i. Finally, we turn to the longest element8 of WJ , which is constructed from the longest
element of the subgroups Bk 1 and Dk 1. We give them as k
k matrices acting on the
orthonormal basis ei of the dual h of the Cartan subalgebra, introduced in appendix A:9
w ;J =
(Diag(+1; 1; : : : ; 1; 1)
Diag(+1; 1; : : : ; 1; +1)
for Bk and Dk odd
for Dk even :
(6.1)
We rst treat odd spacetime dimensions, and then even spacetime dimensions.
6.1
In odd spacetime dimension
Consider the so(2; 2k
1) conformal algebra. We know that the unitary weights fall into
k + 1 categories (see table 2). For these unitary weights, we compute the possible Weyl
groups W[ ].
8The longest element of the Weyl group of a simple Lie algebra can be obtained in terms of simple
re ections as follows [4, 21, 33]. Color the nodes of the Dynkin diagrams in white and black in such a
way that no two dots of the same color are connected. Let wblack (respectively wwhite) be the product
of the simple re ections associated to the simple roots painted in black (respectively white). The longest
element of the Weyl group is w
= (wblackwwhite)h=2 where h is the Coxeter number. Since
w induces
an automorphism of the Dynkin diagram, we have w =
1 for algebras other than Ak, Dk and E6. In the
case of Dk, one
nds that w
1 if k is even, and for k odd
w exchanges the last two simple roots.
9The dots represent minus ones.
wi (1
2
2
3
3
Inequality
i
1
1
w2k+1 i = s1 : : : sk 1sksk 1 : : : si
k
= w ;J wi
with
antidominant. We recall that
in Bk is antidominant if and only if 1
HJEP04(218)5
There are k2 positive roots: k short roots of the form
= ei, for which h ; _i = 2 i,
and k2
k long roots ei
weight, we saw that ( 2; : : : ; k) 2 Z
for 2
i < j
k and ei for 2
i < j
k for which h ; _i = i
[ (Z + 12 )k 1. Therefore the (k
j . For a unitary
1)2 roots ei
ej
k satisfy h ; _
i 2 Z. We have to examine the
remaining roots e1 and e1
ei. This leads to the three following possibilities, which de ne
what we call the integrality class of the weight :
i+1
i+1
1
1
Condition
2 2= 21 Z
2 2 Z
2 2 21 + Z Bk 1
A1 (halfintegral)
W[ ] (Integrality class)
Bk 1 (nonintegral)
Bk (integral)
(6.2)
We see that in addition to the integral case (where W[ ] = Bk), there are two other
integrality classes to consider, which have W[ ] = Bk 1
A1 and W[ ] = Bk 1.
We
examine them in turn in the following paragraphs.
In the nonintegral case, W[ ] reduces to the parabolic Weyl group WJ , so we just have
In the halfintegral case, we have to take into account a possible re ection with respect
to the A1 root, which in our notation is e1. The two A1 Weyl chambers are delimited by
the wall 1 = 12
k. From this, we deduce that
If 1
If 1 > 12
1
2
k, then [L ] = [M c].
k, we have to remove a correction, corresponding to the dot image of
under the e1 re ection. Using the notation (5.5), this gives the character formula
[L ] = [M(c 1; c)] [M(c1 2k 1; c)] :
(6.3)
Finally, the integral case is the most complicated one. We apply the KazhdanLusztig
formula (3.4), where the antidominant weight
is needed. If the weight
is singular,
several di erent pairs (w; ) can a priori be used in the parameterization (3.2), but for
equation (3.4) to be valid, we need to choose the Weyl group element w of minimal length.
Moreover we know that w 2 w ;J W J . In table 6, we have gathered for each such element
wi (1
1)
wk
wk+1
k
1)
2
2
= w ;J wj
inant. We recall that a weight
in Dk is antidominant if and only if 1
2
i
Inequality
k
+ k
w2k+1 i = s1 : : : sk 1sksk 2sk 3 : : : si
1
i+1
w the inequalities that w
satis es, with
antidominant. This means that we select the
lowest value of i such that
satis es the inequality associated to the coset representative
wi in table 6. In the table, in order to write the inequalities in a more compact way, we
use the shifted notation
:=
+ :
Then, combining equations (5.8) and (5.3) gives the result.
In even spacetime dimension
Secondly, we perform the same analysis for even spacetime dimension. The algebra
i < j
k. For a root
ej , we have h ; _i = i
j . A unitary weight has i
j 2 Z for 2
i < j
k,
so only two con gurations are possible:
Condition
W[ ] (Integrality class)
1
1
2 2= Z
2 2 Z
Dk 1 (non integral)
Dk (integral)
In the nonintegral case, we have [L ] = [M c]. In the integral case, one has to examine the
inequalities satis ed by
. Again, we pick the smallest wi such that the corresponding
inequality in table 7 is satis ed,10 and use formula (5.8). The compact notation
:=
is also used. Let us study a few explicit examples.
Consider the weight
= ( 1; 0; 0) in so(2; 4). It is unitary of type II.
Moreover, we have
= (1; 1; 0), which satis es the inequalities for w5 and w6 but not the other
10Since the order is only partial, this could be ill de ned if the inequalities of wk and wk+1 could both be
satis ed, and not those for wi, i
k
1. However, one checks that the inequalities of wk and wk+1 imply
1 = k = 0, and therefore the inequality of wk 1 would be satis ed. Thus, there really is no ambiguity.
j kj
j 1j
j 1j
HJEP04(218)5
with
j kj
j kj.
antidom(6.4)
(6.5)
wi. So we must write
= w ;J w5
with
= ( 3; 2; 0). Using the polynomials (5.4),
we obtain
( 3; 0; 0). The compact part of the weights ( 2; 0; 1) is singular, so the corresponding
module is trivial and disappears in the character formula, as per the remark at the end of
section 3.3. We conclude
[L( 1;0;0)] = [M(c 1;0;0)] [M(c 3;0;0)] :
(6.7)
HJEP04(218)5
Note that this result arises from cancelling terms that contain multiplicities larger than one.
Example. Similarly, the weight
= ( 2; 0; 0; 0) in so(2; 6) is associated to the coset
representative w6, and we have
[L ] = [M wc ;J w6 ] [M wc ;J w5 ] [M wc ;J w4 ] + [M wc ;J w3 ]
2[M wc ;J w2 ] + 2[M wc ;J w1 ] :
This reduces to
[L( 2;0;0;0)] = [M(c 2;0;0;0)] [M(c 4;0;0;0)] :
(6.8)
Again, multiplicities larger than one (and therefore nontrivial KazhdanLusztig
polynomials) play an intermediate role.
Conclusion.
We conclude that the calculation of the characters of all highest weight
unitary representations of the conformal algebra in any dimension is straightforward using
the mathematical technology. Deciphering su ces.
7
dimension
E =
A large physics literature exploring the representation theory of the conformal algebras
so(2; n) is available. The literature concentrates on unitary representations. These were
classi ed in three dimensions [6, 9] and in four dimensions [34]. See also the more general
treatment in [35]. The paper [1] identi es the unitary representations in arbitrary
dimensions, based on the earlier mathematical treatment in [27] which we reviewed in section 4
and which we summarized in tables 3 and 5. Character formulas were computed in many
instances. The most general treatment across dimensions is [36].11
In this section, we translate the uniform mathematical results of section 6 into a
notation more frequently used by physicists in order to make both the mathematics and the
physics literature more accessible. We again identify the energy E, equal to the conformal
of the ground state, with minus the rst component of the highest weight,
1. Moreover, the compact subalgebra so(n) describes space rotations, and we switch
to spin labels (j1; : : : ; j[n=2]) to describe the highest weights of the rotation algebra,12
= ( 1; 2; : : : ; 1+[n=2]) = ( E; j1; : : : ; j[n=2]) :

M{azth
Ph{yzsics
}
(7.1)
See the remark in footnote 15.
Our convention is uniform across dimensions.
11However, the physics literature has not always been entirely accurate, even in the better of resources.
12The spin labels are closer but not yet identical to the most common spin labels in the physics literature.
7.1
Firstly though, we summarize the results of sections 4, 5 and 6 in an e ective algorithm that
can be used to compute the conformal character  and indeed the module decomposition
 for any unitary weight. The irreducible conformal character with highest weight
is
denoted [L ], and we will obtain an expression in terms of the Verma modules characters
[M ] de ned in equation (3.3).
The procedure runs as follows. Let
= ( 1; : : : ; k) be a weight in Bk or Dk.
1. Determine whether it is unitary or not using tables 3 and 5. If the weight is not
HJEP04(218)5
unitary, then the character is given by the general KazhdanLusztig formula (3.4). To
obtain the character, one needs to compute generic KazhdanLusztig polynomials. If
the weight is unitary, then a simpli cation of the generic formula occurs, as explained
in step two.
is unitary, determine its integrality class using (6.2) for a conformal
algebra of type B, and (6.5) for an algebra of type D.
If the integrality class is Bk 1 or Dk 1, then [L ] = [M c]. (See equation (5.6)
for the character [M c] of the Verma module induced from an irreducible
representation of the compact subalgebra.)
If the integrality class is Bk 1
when 1 > 12
k, and [L ] = [M c] otherwise.
A1, then [L ] = [M(c 1; c)]
[M(c1 2k 1; c)]
If the integrality class is Bk or Dk, then look for the lowest wi in gure 6 such
that
de ne
satis es the corresponding inequality in tables 7 and 6,13 and
= (w ;J wi) 1
. The irreducible character is then given by
2k
j=1
X( 1)`(wi) `(wj)bji[M wc ;J wj ] ;
where the length function ` is the height in gure 6, and the multiplicities bji
are obtained by evaluating expressions (5.3) and (5.4) at q = 1, i.e. for Bk,
bji =
(
1 j
i
0 otherwise;
bji = <
8
>0
>
wj
wi
>
>:1 otherwise :
2 k + 2
i
2k
1 ; 1
j
2k
i
(7.2)
(7.3)
(7.4)
and for Dk,
13We recall that = (k
action is de ned by w
given in equation (6.1).
12 ; k
32 ; : : : ; 12 ) for Bk and
= (k
1; k
2; : : : ; 0) for Dk. Moreover, the dot
. Finally, the longest element of the parabolic Weyl group w ;J is
Example. Before we delve into the exhaustive treatment of the low dimensions, we
illustrate how the algorithm allows to e ectively compute the character of any highestweight
irreducible representation in any dimension.
Consider the weight
= ( 8; 2; 2; 2; 2; 1) in so(2; 10). This algebra is of type Dk with
k = 6. First, we check that this weight is unitary. We have E =
1 = 8, and we observe in
table 5 that the unitary constraint is of type (I,4) and reads E
12 3 + 2 4 = 7, which is
satis ed. To determine the integrality class, we look at table (6.5), and since 1
2 =
is integer, we are in the integral case, called Dk. Hence we are instructed to look in table 7
for the smallest wi, in the order given by
gure 6, such that the corresponding inequality
holds. For that, we rst compute
= ( 3; 6; 5; 4; 3; 1). The inequalities for w4 and w5 are
both satis ed, but because w4 is smaller than w5, we pick w4. Now we compute . First,
note that w ;J = Diag(1; 1; 1; 1; 1; 1), and w4 = s1s2s3 is a cyclic permutation of
the four rst entries of a weight. So
= ( 11; 9; 7; 5; 4; 1). Finally, the coe cients
bji are nonvanishing only for j = 1; 2; 3; 4, so we compute the action of w ;J wj for these
values of j on : w ;J w1
( 9; 2; 2; 1; 2; 1), w ;J w4
conclude that
= ( 11; 1; 1; 1; 2; 1), w ;J w2
= ( 10; 2; 1; 1; 2; 1), w ;J w1
=
= ( 8; 2; 2; 2; 2; 1). Then, reading the lengths on gure 6, we
[L( 8;2;2;2;2;1)] = [M(c 8;2;2;2;2;1)] [M(c 9;2;2;1;2;1)]+[M(c 10;2;1;1;2;1)] [M(c 11;1;1;1;2;1)] : (7.5)
In appendix C, we execute the procedure, and explicitly write down the results of
formula (7.2) for dimensions up to and including seven, for all integral unitary weights.
A brief comparison to the physics literature
While the formalism we presented is e cient, it may be bene cial to make an explicit
comparison to results in the literature. We kick o the brief comparison in three dimensions.
Three dimensions: so(2; 3).
We write the highest weights
= ( E; j) in terms of the
energy E and spin j of the representation. The unitarity condition of table 3 becomes,
1
with j 2 2 Z 0:
Type (I,1) is E
Type II is E
Type III is E
j + 1 for j 6= 0; 12 ;
12 or E = 0 for j = 0;
1 for j = 12 .
If 2E 2= Z, [L ] = [M c].
For a unitary weight, we then look at the integrality classes:
[L( E;j)] = [M(c E;j)] otherwise.
If E + j 2 21 + Z, then [L( E;j)] = [M(c E;j)]
[M(c (3 E);j)] if E < 32
, and
w1
w2
w4
j
j
E
E
2
2
E + 1
j
E + 1
1
2
1
2
(7.6)
We look for the smallest i such that the wi condition is satis ed by , and read the
character in
gure 6. For reference, the results are listed in table (C.1), for each
possible value of i. In this way we recover the results of section 2.
HJEP04(218)5
Four dimensions: so(2; 4). In four dimensions, there are three types of unitary weights
= ( E; j1; j2). Table 5 gives, for j1
jj2j and j1
j2 2 Z:14
Type (I,1) is j1 > jj2j and E
j1 + 2;
Type (I,2) is j1 =
j2 6= 0 and E
j1 + 1;
Type II is j1 = j2 = 0 and E
1 or E = 0.
There are two integrality classes of unitary weights, namely D3 and D2, depending on
whether E
j1 2 Z, or not. In the non integral case, we have [L ] = [M c]. In the integral
case, we look in table 7 for the smallest i such that the appriopriate inequality is satis ed,
with
1 =
E + 2, 2 = j1 + 1 and
2 = j2. For each value of i, the character [L ] can
then be read in table (C.3). Thus, we recover the results of [36, 37].
Five dimensions: so(2; 5).
We distinguish the generic representations with E
3 + 2
and
2 >
3 (case I,1), the representations E
2 +
2 where 2 =
3 (case I,2) and
the representations with E
3=2, which are scalar (case II), or E = 0, and E
for the spinor (case III). The analysis runs along the lines of the analysis of the conformal
algebra so(3; 2) in three dimensions. We provide the explicit results for the integral unitary
weights in table (C.4). When the results can easily be compared, they coincide with [36].
The remark on singular weights in subsection 3.3 plays a role in interpreting the results
of [36] correctly.
Six dimensions: so(2; 6).
The analysis is as for the four dimensional conformal algebra.
We provide the explicit results for the integral unitary weights in table (C.5). When the
results of [38] can be unambiguously compared, they agree with ours.
Remark on the generic case.
Our treatment is generic, as is [36], but we carefully
keep track of possible multiple subtractions of Verma modules.15 As in [1], our analysis
has the advantage of being proven necessary and su cient in arbitrary dimension in regards
to unitarity.
14For comparison with most of the physics literature, one rede nes ~j1 = (j1 + j2)=2 and ~j2 = (j1
j2)=2.
15Historically, in the mathematical literature, this was not analyzed correctly [5]. In particular, the
otherwise important contribution by Verma [39] was mistaken on the possibly larger than one multiplicity
of Verma modules to be added or subtracted in the character formula. This has led to wrong claims in
the mathematics literature, which unfortunately have propagated to the physics literature (see appendix A
of [36]). It will be interesting to attempt to prove the character formulas of [36], using the techniques we
explained.
Our main aim was to provide physicists with an overview of the representation theory
of conformal multiplets. Highest weight representations make up a large category of
representations that is well understood. The minimal data to compute character formulas
for irreducible representations is coded in the Weyl group and the KazhdanLusztig
polynomials.
Mathematicians have also provided a complete analysis of the necessary and
su cient conditions for unitarity, using a more powerful version of the inequalities derived
in the physics literature. Moreover, unitarity restricts the highest weights such that the
combinatorial KazhdanLusztig calculations drastically simplify.
Secondly, by translating mathematics, we have added to the physics literature. We
explained how to systematically compute the characters of irreducible highest weight
representations even when they are not unitary.
We have stressed that the conditions for
unitarity are necessary and su cient, and that they can be formulated at arbitrary rank.
In our analysis, we have dealt systematically with both nonintegral as well as singular
weights. Moreover, we have provided a clear classi cation of all cases of unitary characters
in terms of coset representatives of a Weyl subgroup of the Weyl group of the conformal
algebra. Using our systematic insight, we provided lookup tables for unitary highest weight
representation characters for conformal algebras up to and including rank four. They are
guaranteed to be complete. Mostly, we hope these tables have gained in transparency.
Thirdly, these techniques can be re ned to apply to superconformal characters. We
plan to discuss the necessary extensions elsewhere.
Finally, we wrote this paper because we would have liked to read it.
Acknowledgments
We thank James Humphreys for his writing in general, and his correspondence in particular.
A.B. acknowledges support from the EU CIG grant UE14GT5LD2013618459, the
Asturias Government grant FC15GRUPIN14108 and Spanish Government grant
MINECO16FPA201563667P. J.T. thanks the High Energy Physics Theory Group of the University
of Oviedo for their warm hospitality and acknowledges support from the grant
ANR13BS050001. We thank Xavier Bekaert, Nicolas Boulanger and Hugh Osborn for comments
on the rst version of this paper.
A
The Lie algebra conventions
We use the parameterization of [26] for the roots and weights of the Bk and Dk simple Lie
algebras. We describe these conventions in detail. In both cases, the dual of the Cartan
subalgebra h is spanned by an orthonormal basis (e1; : : : ; ek). When we write a weight in
components, it is always understood that the coordinates are with respect to this basis.
A.1
The Lie algebra Bk
We have the set of roots
= f e
i ej ; eig and a choice of set of simple roots s = f i<k =
i ei+1; k = ekg. The fundamental weights can then be written as $i<k = e1 +
+ei and
$k = (e1 +
+ ek)=2. The Weyl vector
equals
1)e1 + (2k
3)e2 +
The Weyl group is WBn = Z2k o Sk and acts by permutations and sign changes of the
orthonormal vectors ei. The conformal algebra so(2; n) with n odd corresponds to a Bk
algebra of rank k = (n + 1)=2.
A.2
The Lie algebra Dk
i
(e1 +
= (k
The set of roots is
= f e
i
ej g while simple roots are collected in the set s = f i<k =
ei+1; k = ek 1 + ekg. The fundamental weights are $i<k 1 = e1 +
+ ei; $k 1 =
+ ek 1
1)e1 + (k
ek)=2; $k = (e1 +
+ ek)=2. The Weyl vector
comes out to be
HJEP04(218)5
2)e2 +
+ ek 1. The Weyl group is WDn = Z2
k 1 o Sk and acts by
permuting the vectors ei and an even number of sign changes. For the conformal algebra
so(2; n) with n even, we have a Dk algebra of rank k = (n + 2)=2.
B
The structure of real simple Lie algebras
We summarize results of the structure theory of semisimple real Lie algebras. We follow
the notation of [26] to which we must refer the reader for a complete exposition.16
B.1
The structure theory
Every complex semisimple algebra g has a compact real form. We denote the compact real
forms by su(n); so(n); sp(n) and e6;7;8; f4 and g2. The Killing form on a compact semisimple
Lie algebra is negative semide nite and nondegenerate.
Every real semisimple Lie algebra g0 has a Cartan involution , unique up to
conjugation. It is such that B (X; Y ) =
B(X; (Y )) is positive de nite, where B is the Killing
form. This involution
gives rise to an eigenspace decomposition
g0 = k0
p0
h0 = t0
a0
into eigenspaces of eigenvalues +1 and
1 respectively. In matrix realizations of Lie
algebras, the Cartan involution
can be de ned by (X) =
Xy, where the dagger stands
for the conjugate transpose. The Killing form on g0 is negative semide nite on k0 and
positive semide nite on p0.
Every Cartan subalgebra h0 of g0 is conjugate to a stable subalgebra, and we will
assume that we have picked a Cartan subalgebra h0 that is stable. We can then decompose
the Cartan subalgebra into subalgebras
with t0
k0 and a0
p0. The dimension of t0 is called the compact dimension of h0, and
the dimension of a0 is called the noncompact dimension. We say that a Cartan subalgebra
is maximally (non)compact if its (non)compact dimension is maximal.
16Our summary is mainly based on chapters VI on the Structure Theory of Semisimple Groups, and
chapter VII on the Advanced Structure Theory.
(B.1)
(B.2)
Given a stable Cartan subalgebra h0 = t0
a0, the roots of (g; h) are imaginary on
t0 and real on a0. As a consequence, we say that a root
2 h0 is real if it vanishes on t0,
and that it is imaginary if it vanishes on a0. Otherwise, the root is said to be complex.
We say that an imaginary root is compact if the associated root space is included in k, and
that it is noncompact if it is included in p.
To a real semisimple Lie algebra g0, we associate a Vogan diagram which is the Dynkin
diagram of its complexi cation g, adorned with additional data. For a maximally compact
choice of h0, there are no real roots. Since there are no real roots, we can pick a set of
positive roots such that ( +) =
. The Vogan diagram of the triple (g0; h0; +) is the
Dynkin diagram of
+ with 2element orbits of
made manifest, and with the 1element
orbits painted when corresponding to a noncompact simple root, and unpainted when
compact [26].
B.2
The classi cation of real simple Lie algebras
Firstly, there are the complex simple Lie algebras, considered as an algebra over the real
numbers. Secondly, there are the Lie algebras whose complexi cation is simple over the
complex numbers. These algebras always have a Vogan diagram with at most one simple
root painted. Amongst these diagrams, one can remove further equivalences. The resulting
classi cation of simple real Lie algebras is summarized e.g. in Theorem 6.105 in [26]. It
includes the noncompact forms so(p; q) of the special orthogonal algebras. The Vogan
1) is
1
(B.3)
(B.4)
They summarize all of the Lie algebra data of the real simple algebra.
B.3
The classi cation of Hermitian symmetric pairs
Unitary discrete highest weight representations only exist for algebras g0 that are part of a
Hermitian symmetric pair. This is because the Cartan subalgebra should be entirely within
the compact subalgebra k0 (as follows from analyzing unitarity within a Cartan subgroup
and the matrix realization of the Cartan involution ), which is equivalent to the Hermitian
symmetric pair condition. Hermitian symmetric spaces are coset spaces G=K (with G
a real group and K its maximal compact subgroup) which are Riemannian manifolds
with a compatible complex structure and on which the group G acts by holomorphic
transformations. A manifold X = G=K is Hermitian if and only if the center of K is a
onedimensional central torus. They were classi ed by Cartan [16], and fall into the list
recorded in table 8.17 Crucial to us is the entry so(2; n).
17Reference [26] table (7.147).
su(p; q)
so(2; n)
sp(n; R)
so (2n)
E III
E VII
su(p)
R
R
R
R
R
The character tables for integral unitary weights
We collect the tables of characters of integral unitary highest weight representations,
classi ed by their parabolic coset representative wi. See section 6. Some wi are not
associated with any unitary weight. In the following tables, they are signalled by an asterisk.
Moreover, the brackets around M c are omitted. As always, we use the notation (7.1) for
the weights.
B2 = so(2; 3)
w1
w2
w3
w4
M c
( E;j)
M c
M c
( E;j)
( j 2;E 2)
M(cE 3;j) + M c
( E;j) + M c
( j 2;1 E)
M(cE 3;j) + M c
( E;j) + M c
( j 2;1 E)
M(cj 1;1 E)
To illustrate how these tables can be used, let us recover the character of the trivial
representation L(0;0) of so(2; 3). This corresponds to the coset representative w4, and
we read in the table
[L(0;0)] =
[M(c 3;0)] + [M(c0;0)] + [M(c 2;1)] [M(c 1;1)] :
Using the explicit expression (2.17), we obtain [L(0;0)] = 1, as expected.
D3 = so(2; 4)
w1
w2
w3
w4
w5
w6
M c
( E;j1;j2)
M c
( j1 3; j2 1;2 E)
M c
( E;j1;j2)
( j1 3;E 3;j2)
M c
( E;j1;j2) + M c
M c
( E;j1;j2) + M c
M(cE 4;j1; j2) + M c
( E;j1;j2)
M(cE 4;j1; j2) + M c
( E;j1;j2)
M(cj2 2;j1;2 E)
M c
2M c
M c
( j1 3;1 E; j2)
( j1 3;1 E; j2)
( j1 3;j2 1;E 2)
( j2 2;j1;E 2)
M c
( j2 2;j1;E 2)
M(cj2 2;j1;2 E)
M(cj1 1;1 E;j2) + M c
( j2 2;1 E; j1 1) + M(cj2 2;1 E;j1+1)
M c
M c
+ M c
+M c
+ M c
+ M c
M c
M c
M c
M c
+ M c
+ M c
+ M c
M c
M c
M c
M c
M c
(E 5;j1;j2)
+M c
(E 5;j1;j2)
( j2 3;j1;2 E)
( j1 4;1 E;j2)
( j2 3;j1;2 E)
(j2 2;j1;2 E)
(E 5;j1;j2)
( j1 4;1 E;j2)
(j1 1;1 E;j2)
( j2 3;1 E;j1+1)
(j2 2;1 E;j1+1)
M c
+ M c
( j1 4;E 4;j2)
( j1 4;j2 1;E 3)
( j2 3;j1;E 3)
M c
( E;j1;j2;j3)
M c
( E;j1;j2;j3)
( j1 5;E 5;j2;j3)
( E;j1;j2;j3)
( j1 5;j2 1;E 4;j3)
( j2 4;j1;E 4;j3)
( E;j1;j2;j3)
( j1 5;j2 1;j3 1;E 3)
( j2 4;j1;j3 1;E 3)
( E;j1;j2;j3)
( j2 4;j1; j3 1;3 E)
w1
w2
w3
w4
w?
5
w6
D4 = so(2; 6)
w1
w2
w3
w4
w5
w6
w?
7
w8
M c
M c
M c
M c
2M c
M c
M c
M c
(C.4)
(C.5)
M c
+ M c
M c
M c
M c
M c
M c
+ M c
+ M c
M c
+M c
+ M c
M c
+ M c
+ M c
2M c
+ M c
M c
+ M c
M c
( j3 3;j1;j2;E 3)
(j3 3;j1;j2;3 E)
+ 2M c
( j1 5;j2 1;2 E; j3)
( j2 4;j1;2 E; j3)
( j3 3;j1;j2;E 3)
(j3 3;j1;j2;3 E)
M c
(E 6;j1;j2; j3)
( E;j1;j2;j3)
( j1 5;1 E;j2; j3)
( j2 4;j1;2 E; j3)
(j2 2;j1;2 E;j3)
( j3 3;j1;2 E; j2 1)
(E 6;j1;j2; j3)
( E;j1;j2;j3)
( j1 5;1 E;j2; j3)
M c
(j1 1;1 E;j2;j3)
+ M c
M c
( j2 4;1 E;j1+1; j3)
(j2 2;1 E;j1+1;j3)
( j3 3;1 E;j1+1; j2 1)
(j3 3;1 E;j1+1;j2+1)
w1
w2
w3
w4
w5
w?
6
w?
7
w8
M c
M c
M c
( j1 6;E 6;j2;j3)
M c
( E;j1;j2;j3) + M c
( j1 6;j2 1;E 5;j3)
( j2 5;j1;E 5;j3)
M c
M c
M c
( j1 6;j2 1;j3 1;E 4) + M c
( j2 5;j1;j3 1;E 4)
M c
( j3 4;j1;j2;E 4)
M(cE 7;j1;j2;j3) + M c
( E;j1;j2;j3) + M c
( j1 6;j2 1;j3 1;3 E)
M c
( j2 5;j1;j3 1;3 E) + M c
( j3 4;j1;j2;3 E)
M(cE 7;j1;j2;j3) + M c
M c
( E;j1;j2;j3)
( j1 6;j2 1;2 E;j3)
+M c
( j2 5;j1;2 E;j3) + M c
( j3 4;j1;j2;3 E)
M(cj3 3;j1;j2;3 E)
M(cE 7;j1;j2;j3) + M c
( E;j1;j2;j3) + M c
+M c
( j2 5;j1;2 E;j3)
M(cj2 2;j1;2 E;j3)
+M(cj3 3;j1;2 E;j2+1)
( j1 6;1 E;j2;j3)
M c
( j3 4;j1;2 E;j2+1)
M(cE 7;j1;j2;j3) + M c
( E;j1;j2;j3) + M c
( j1 6;1 E;j2;j3)
M(cj1 1;1 E;j2;j3)
M c
( j2 5;1 E;j1+1;j3) + M(cj2 2;1 E;j1+1;j3)
+M c
( j3 4;1 E;j1+1;j2+1)
M(cj3 3;1 E;j1+1;j2+1)
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
[1] S. Ferrara and C. Fronsdal, Conformal elds in higher dimensions, in Recent developments in
theoretical and experimental general relativity, gravitation and relativistic
eld theories.
Proceedings, 9th Marcel Grossmann Meeting, MG'9, Rome, Italy, July 2{8, 2000. Pts. AC,
pp. 508{527 (2000) [hepth/0006009] [INSPIRE].
JHEP 09 (2016) 070 [arXiv:1509.00428] [INSPIRE].
[2] J. Penedones, E. Trevisani and M. Yamazaki, Recursion Relations for Conformal Blocks,
[3] J. Troost, Models for modules: The story of O, J. Phys. A 45 (2012) 415202
[arXiv:1202.1935] [INSPIRE].
(1992).
American Mathematical Soc. (2008).
(1963) 901 [INSPIRE].
[4] J.E. Humphreys, Re ection groups and Coxeter groups, vol. 29, Cambridge University Press
[5] J.E. Humphreys, Representations of semisimple Lie algebras in the BGG category O, vol. 94,
[6] P.A.M. Dirac, A Remarkable representation of the 3 + 2 de Sitter group, J. Math. Phys. 4
Springer (1979), pp. 11{41.
Invent. Math. 64 (1981) 387.
(1935), pp. 116{162.
[12] D. Kazhdan and G. Lusztig, Representations of coxeter groups and hecke algebras, Invent.
[13] A. Beilinson and J. Bernstein, Localisation de gmodules, CR Acad. Sci. Paris 292 (1981) 15.
[14] J.L. Brylinski and M. Kashiwara, Kazhdanlusztig conjecture and holonomic systems,
[15] V.V. Deodhar, On some geometric aspects of bruhat orderings ii. the parabolic analogue of
kazhdanlusztig polynomials, J. Algebra 111 (1987) 483.
[16] E. Cartan, Sur les domaines bornes homogenes de l'espace den variables complexes, in
Abhandlungen aus dem mathematischen Seminar der Universitat Hamburg, vol. 11, Springer
[7] C. Fronsdal, Elementary Particles in a Curved Space, Rev. Mod. Phys. 37 (1965) 221
[8] N. Evans, Discrete series for the universal covering group of the 3 + 2 de sitter group,
[9] V. Dobrev and E. Sezgin, Spectrum and character formulae of so (3, 2) unitary
representations, in Di erential Geometry, Group Representations, and Quantization,
Springer (1991), pp. 227{238.
[10] I. Bernstein, I. Gelfand and S. Gelfand, Structure of representations generated by highest
[11] J.C. Jantzen, Moduln mit einem hochsten gewicht, in Moduln mit einem hochsten Gewicht,
[27] T. Enright, R. Howe and N. Wallach, A classi cation of unitary highest weight modules, in
Representation theory of reductive groups, Springer (1983), pp. 97{143.
[17] B.D. Boe, Kazhdanlusztig polynomials for hermitian symmetric spaces, Trans. Am. Math.
[18] F. Brenti, Parabolic KazhdanLusztig polynomials for Hermitian symmetric pairs, Trans.
Am. Math. Soc. 361 (2009) 1703.
weights over a ne lie algebras, math/9903123.
[19] M. Kashiwara and T. Tanisaki, Characters of irreducible modules with noncritical highest
[20] J.C. Jantzen, Character formulae from Hermann Weyl to the present, in Groups and
analysis, Lond. Math. Soc. Lect. Notes Ser. 354 (2008) 232.
[21] N. Bourbaki, Groupes et algebres de Lie. Chapitre iv{vi, Hermann, Paris (1968), Actualites
[22] J.E. Humphreys, Introduction to Lie algebras and representation theory, vol. 9, Springer
Scienti ques et Industrielles (1972).
Science & Business Media (2012).
Media (2013).
[23] A.L. Onishchik and E.B. Vinberg, Lie groups and algebraic groups, Springer (1990).
[24] B. HarishChandra, Representations of semisimple Lie groups: IV, Proc. Nat. Acad. Sci. 37
[25] B. HarishChandra, Representations of semisimple Lie groups. V, Am. J. Math 78 (1956) 1.
[26] A.W. Knapp, Lie groups beyond an introduction, vol. 140, Springer Science and Business
theory, JHEP 08 (2014) 113 [arXiv:1406.3542] [INSPIRE].
theoretical glance, JHEP 05 (2017) 081 [arXiv:1612.08166] [INSPIRE].
HJEP04(218)5
energy, Commun. Math. Phys. 55 (1977) 1 [INSPIRE].
Adv. Theor. Math. Phys. 2 (1998) 783 [hepth/9712074] [INSPIRE].
eld theory, J. Math. Phys. 47 (2006) 062303 [hepth/0508031] [INSPIRE].
sphere, JHEP 01 (2006) 160 [hepth/0501063] [INSPIRE].
AdS3, higher spins and AdS/CFT, Nucl. Phys. B 892 (2015) 211 [arXiv:1412.0489]
weight vectors , Funct. Anal. Appl 5 ( 1971 ) 1 [Funktsional . Anal. i Prilozhen . 5 ( 1971 ) 1].
J. Funct . Anal. 52 ( 1983 ) 385 .
Indian Acad. Sci 89 ( 1980 ) 1 . [31] T. Basile , X. Bekaert and N. Boulanger , Mixedsymmetry elds in de Sitter space: a group [32] O.V. Shaynkman , I. Yu . Tipunin and M.A. Vasiliev , Unfolded form of conformal equations in