Frobenius-Schur indicators for semisimple Lie algebras
FROBENIUS-SCHUR INDICATORS FOR SEMISIMPLE LIE
ALGEBRAS
MOHAMMAD ABU-HAMED AND SHLOMO GELAKI
1. Introduction
Classically Frobenius-Schur indicators were defined for irreducible representa-
tions of finite groups over the field of complex numbers. The interest in doing so
came from the second indicator which determines whether an irreducible represen-
tation is real, complex or quaternionic. Namely, a classical theorem of Frobenius
and Schur asserts that an irreducible representation is real, complex or quaternionic
if and only if its second indicator is 1, 0 or −1, respectively (see e.g. [S]). However,
no representation-theoretic interpretation of the higher indicators is known.
Recently, Frobenius-Schur indicators of irreducible representations of complex
semisimple finite dimensional (quasi-)Hopf algebras H were defined by Linchenko
and Montgomery [LM] and Mason and Ng [MN] (see also [KSZ]), generalizing
the definition in the group case. The values of the mth indicator are cyclotomic
integers in Qm. Moreover, an analog of the Frobenius-Schur theorem on the second
indicator was proved, and in general it has been shown that the indicators carry
rich information on H , as well as on its representation category (see also [NS2]).
In fact, one can generalize the definition of Frobenius-Schur indicators to simple
objects of any semisimple tensor categories which admit a pivotal structure (=
tensor isomorphism id →∗∗), thus showing in particular that the indicators are
categorical invariants (see e.g. [FGSV], [NS1]).
The category of finite dimensional representations of a finite dimensional complex
semisimple Lie algebra is a pivotal semisimple tensor category, and hence one can
define the Frobenius-Schur indicators of its simple objects. The second indicator
was already defined and known to be nonzero if and only if the simple representation
is self-dual, and 1 or −1 if and only if the representation is orthogonal or symplectic,
respectively. Furthermore, Tits gave an explicit formula for it in representation-
theoretic terms (see Section 3).
The purpose of this paper is to study Frobenius-Schur indicators (of all degrees)
for semisimple Lie algebras. More specifically to find a closed formula for the
indicators in representation-theoretic terms and deduce its asymptotical behavior.
In particular we obtain that the indicators take integer values.
The organization of the paper is as follows.
Section 2 is devoted to preliminaries. We recall some basic definitions and facts
from Lie theory which we need (e.g. the Weyl integration formula). Next we
define the mth Frobenius-Schur indicator of the representation categories of finite
dimensional complex semisimple Lie algebras.
Date: February 11, 2007.
http://arxiv.org/abs/0704.0165v1
2 MOHAMMAD ABU-HAMED AND SHLOMO GELAKI
In section 3 we recall the properties of the second indicator. For the benefit of
the reader we also give a proof of Tits’ theorem.
Section 4 is dedicated to the proof of our main results. In 4.1 we prove the
formula for the mth Frobenius-Schur indicator νm, m ≥ 2, which is given by the
following theorem.
Theorem 1.1. Let g be a finite dimensional complex semisimple Lie algebra. Let
V (λ) be an irreducible representation of g with highest weight λ, W the Weyl group
of g, ρ the half sum of positive roots, and V (λ)
ρ−σ·ρ
the weight space of the
weight ρ−σ·ρ
where m ≥ 2 is an integer. Then the mth Frobenius-Schur indicator
νm(V (λ)) of V (λ) is given by
νm(V (λ)) =
sn(σ) dimV (λ)
ρ− σ · ρ
Our proof of Theorem 1.1 is analytic. Namely, we work with the equivalent
representation category of the associated simply connected Lie group and use the
Weyl integration formula to obtain our formula.
Next, in 4.2 we prove the following corollary of Theorem 1.1.
Corollary 1.2. For large enough m, νm(V (λ)) = dimV (λ)[0] (which is not zero if
and only if λ belongs to the root lattice). In particular for the classical Lie algebras
sl(n,C), so(2n,C), so(2n + 1,C) and sp(2n,C), νm(V (λ)) = dimV (λ)[0] for m
greater or equal to 2n− 1, 4n− 5, 4n− 3 and 2n+ 1, respectively.
Finally in 4.3 we use our formula and Kostant’s theorem to compute explicitly
the Frobenius-Schur indicators for the representation category of sl(3,C). More
specifically, we prove:
Theorem 1.3. Let V (a, b) be an irreducible representation of sl(3,C). Then
(1) ν2(V (a, b)) = 1 if a = b, and ν2(V (a, b)) = 0 if a 6= b.
(2) ν3(V (a, b)) = 1 +min{a, b}.
(3) For m > 3 we have, νm(V (a, b)) = 1 + min{a, b} if (a, b) is in the root
lattice and νm(V (a, b)) = 0 otherwise.
Acknowledgments. This research was supported by the Israel Science Foundation
(grant No. 125/05).
2. Preliminaries
Throughout let g be a finite dimensional complex semisimple Lie algebra of rank
r, ( , ) its Killing form, h a Cartan subalgebra (CSA) of g, Φ the root system
corresponding to h, ∆ a fixed base, {h1, ..., hr} the corresponding coroot system,
and W the Weyl group.
Let λ ∈ h∗ be a dominant integral weight (i.e. λ(hi) is a nonnegative integer
for all i), V (λ) the finite dimensional irreducible representation of g with highest
weight λ and Π(λ) the set of integral weights occurring in V (λ); it is a finite
set which is invariant under the action of the Weyl group. For µ ∈ Π(λ), let
mλ(µ) = dimV (λ)[µ] be the multiplicity of µ in V (λ). Recall that the multiplicities
are invariant under the Weyl group action. Let ρ =
α (half sum of positive
roots); it is a strongly dominant integral weight.
FROBENIUS-SCHUR INDICATORS FOR SEMISIMPLE LIE ALGEBRAS 3
Let us recall Kostant’s theorem on the multiplicities of weights (for a proof see
[Hu]). Let µ ∈ h∗ and define p(µ) to be the number of sets of non-negative integers
{kα|α ≻ 0} for which µ =
kαα (p is called the Kostant’s partition function).
Of course, p(µ) = 0 if µ is not in the root lattice.
Theorem 2.1. (Kostant) Let λ be a dominant weight and µ ∈ Π(λ). Then the
multiplicities of V (λ) are given by the formula
mλ(µ) =
sn(σ)p(σ(λ + ρ)− µ− ρ).
Let gc be the compact real form of g, and G the corresponding simply connected
compact matrix Lie group with Lie algebra gc. It is known that Rep(g), Rep(gc)
and Rep(G) are equivalent symmetric tensor categories.
Let t be a CSA of gc; it corresponds to a maximal torus T of G. Then h = t⊕ it.
It is known that α(h) is purely imaginary for all h ∈ t and α ∈ Φ. If t∗ denotes
the space of real-valued linear functionals on t, then the roots are contained in
it∗ ⊂ h∗. It is then convenient to introduce the real roots, which are simply 1
times
the ordinary roots, the real coroots hα which are the elements of t corresponding
to the elements 2α
(α,α)
where α is a real root, and the real weights of an irreducible
representation of G. An element µ of t∗ is said to be integral if µ(hα) ∈ Z for each
real coroot hα. The real weights of any finite dimensional representation of g are
integral. (See [Ha].)
The Weyl denominator is the function Aρ : T −→ C given by
Aρ(t) = Aρ(e
sn(ω)ei(ω·ρ)(h).
Theorem 2.2. (Weyl integration formula) Let G be a simply connected compact
Lie group. Let f be a continuous class function on G, dg the normalized Haar
measure on G, and dt the normalized Haar measure on T . Then
f(g)dg =
f(t)|Aρ(t)|
Let us now define the Frobenius-Schur indicators of an irreducible representation
of g.
Definition 2.3. Let V be an irreducible representation of g and m ≥ 2 be an inte-
ger. The mth Frobenius-Schur indicator of V is the number νm(V ) = tr(c|(V ⊗m)g),
where c is the cyclic automorphism of V ⊗m given by v1⊗· · ·⊗vm 7→ vm⊗· · ·⊗vm−1.
Remark 2.4. In fact, as we mentioned in the introduction, the indicators can be
defined categorically. Applying the categorical definition to Rep(g) yields the above
definition, while applying it to Rep(G) yields tr(c|(V ⊗m)G). Since the indicators of
V regarded as a g-module coincide with the indicators of V regarded as a G-module
we have νm(V ) = tr(c|(V ⊗m)G).
3. Tits’ theorem on the second indicator
Theorem 3.1. (See [B]) Let G be a compact Lie group. Let V be an irreducible
complex representation of G, and set ǫV =
χ(g2)dg. Then V is self dual if and
only if ǫV 6= 0. Furthermore, suppose V is self dual and let B be a (unique up to
4 MOHAMMAD ABU-HAMED AND SHLOMO GELAKI
scalar) G-invariant non-degenerate bilinear form on V . Then B is either symmetric
or skew-symmetric, and it is such if and only if ǫV = 1,−1, respectively.
Remark 3.2. In Proposition 4.4 we will prove that ǫV = ν2(V ) as defined above.
Historically ν2(V ) was defined by ǫV .
Example 3.3. Let us use Theorem 1.1 to calculate νm(V ) in the representation
category of sl(2,C). Let sl(2,C) = sp{h, x, y}, where h =
, x =
. The root system is Φ = {α,−α}, where α(h) = 2. The Weyl group
is W = {1, σα} , ρ =
α and σα(ρ) = −
α. Let V (n) = ⊕nj=0V [n − 2j] be
the irreducible representation of highest weight λ(h) = n with its weight space
decomposition. By Theorem 1.1,
νm(V (n)) = dimV (n)[0]− dimV (n)
Let m = 2. By the formula above, if n is odd, then dimV (n)[0] = 0 and
dimV (n)[α
] = 1. Hence ν2(V (n)) = −1. Similarly, if n is even, ν2(V (n)) = 1.
Consequently ν2(V (n)) = (−1)
n = (−1)λ(h).
For m ≥ 3,
is not an integer and hence νm(V (n)) = dimV (n)[0].
Therefore we have
νm(V (n)) =
1 if n is even
0 if n is odd
Let g = h
(⊕α∈Φgα) be the root space decomposition of g and ∆ = {α1, ..., αr}
a fixed base. Fix a standard set of generators for g: xi ∈ gαi , yi ∈ g−αi so that
[xi, yi] = hi. Let ρ̌ := 1/2
α∈Φ+ hα be the half sum of positive coroots.
Proposition 3.4. Let E := x1 + ...+ xr and H := 2ρ̌. Then there exist constants
a1, ..., ar such that the subalgebra P generated by H,E, F := a1y1 + ... + aryr is
isomorphic to sl(2,C).
The Lie subalgebra P ⊆ g is called a principal sl(2,C)-subalgebra of g (see [K]
or [D]).
Lemma 3.5. Let V = V (λ) be an irreducible representation of g. Let P be a
principal sl(2,C)-subalgebra of g. Consider V as a P-module. Then its highest
weight is λ(H), and it contains the irreducible sl(2,C)-representation V (λ(H)) with
multiplicity one.
Proof. Let v+ be a highest weight vector of V considered as a g-module. Then
obviously we have Hv+ = λ(H)v+ and Ev+ = 0. Hence v+ is a highest weight
vector with weight λ(H) for V considered as a P -module. Therefore we can write
V = V (λ(H))
V (nj). Now it remains to show that λ(H) > nj for any j. Let
V = V [λ] ⊕
V [µ] be the weight space decomposition of V as a g-module. It
is also a weight space decomposition of V considered as a P -module, so V [µ] is a
weight space of P with weight µ(H). Recall that µ = λ−
j=1 kjαj where kj ∈ Z
Note that λ(H) > µ(H) if and only if λ(H) > λ(H)−
j=1 kjαj(H) if and only if
j=1 kjαj(2ρ̆) > 0 if and only if
j=1 kj(αj , 2ρ) > 0. But 2ρ is strongly dominant,
i.e., (αj , 2ρ) > 0 for all 1 ≤ j ≤ r. The proof is complete. �
FROBENIUS-SCHUR INDICATORS FOR SEMISIMPLE LIE ALGEBRAS 5
Let ω0 ∈ W be the unique element sending ∆ to −∆.
Theorem 3.6. (Tits) Let V = V (λ) be a finite dimensional irreducible represen-
tation of g. If λ+ ω0λ 6= 0 then ν2(V ) = 0. Otherwise, ν2(V ) = (−1)
λ(2ρ̌).
Proof. It is known that the dual of V (λ) is V (−ω0λ), so if V (λ) is not self dual
(i.e., λ+ ω0λ 6= 0) then ν2(V ) = 0.
Suppose that V is self dual as a g-module. Then V admits a non-degenerate
g-invariant bilinear form, and we have to decide if it is symmetric or skew sym-
metric. To do so, consider the principal sl(2,C)-subalgebra P as in Lemma 3.5.
The restriction of V to P has a unique copy of the largest representation of P
occurring in V , with highest weight λ(2ρ̌). We already proved that this represen-
tation has indicator (−1)λ(2ρ̆). Now we can use Theorem 3.1 to prove that V has
a symmetric (skew-symmetric) g-invariant form if and only if it has a symmetric
(skew-symmetric) P -invariant form. The first direction is obvious. Conversely, sup-
pose that V has a symmetric P -invariant form and suppose on the contrary that V
admits a skew-symmetric g-invariant form. Then if we restrict the bilinear g-form
to P we get that V has a skew-symmetric P -invariant form which is a contradiction.
Similar considerations are applied when V has a skew-symmetric P -invariant form.
We conclude that ν2(V ) = (−1)
λ(2ρ̆). �
4. The Main results
4.1. Proof of Theorem 1.1. Let G be the associated simply connected compact
Lie group. From now on we will consider V (λ) as a G-module. For convenience
set V = V (λ), N = V (λ)⊗m, and let π : G −→ GL(V ) be the irreducible represen-
tation.
The following lemma is easily derived from linear algebra.
Lemma 4.1. Let T ∈ End(V ) be a projection, W = ImT and S ∈ End(V ) an
operator preserving W. Then tr|W (S) = tr|V (S ◦ T ).
Proof. Fix a basis A = {w1, ..., wk} for W , and let Ã = {w1, ..., wk, wk+1, ..., wn}
be a completion to a basis for V . Let C = [S|W ]A be the matrix representing S|W
with respect to the basis A. Since T |W = idW and S(W ) ⊆ W we find out that
, [S]
, and hence [S]
. The lemma
follows easily now. �
Proposition 4.2. We have,
νm(V ) = tr|NG(c) =
tr|V (c ◦ π
⊗m(g))dg.
Proof. We follow the lines of the proof of the first formula for Frobenius-Schur
indicators in the Hopf case, given in Section 2.3 of [KSZ].
Set τ = π⊗m. Consider the operator
τ(g)dg : N −→ N . Let us first show that
the image of this operator is NG. Indeed, by the invariance of the Haar measure,
τ(g)vdg =
τ(hg)vdg =
τ(g)vdg for all h ∈ G and v ∈ N . Hence
τ(g)dg
⊆ NG.
Conversely, suppose that u ∈ NG, then
τ(g)udg =
udg = u
dg = u.
Hence NG ⊆ Im
τ(g)dg
and we are done.
6 MOHAMMAD ABU-HAMED AND SHLOMO GELAKI
In fact, the above shows also that the operator
τ(g)dg is a projection onto
Finally, c ∈ Aut(NG), so by Lemma 4.1,
tr|NG(c) = tr|N
τ(g)dg
tr|N (c ◦ τ(g))dg,
as claimed. �
The following lemma is a particular case of a lemma in Section 2.3 of [KSZ] and
its proof replicates the proof of that lemma.
Lemma 4.3. Let f1, ..., fm ∈ End(V ). Then,
tr|V ⊗m(c ◦ (f1 ⊗ ...⊗ fm)) = tr|V (f1 ◦ ... ◦ fm).
Proof. Let v1, ..., vn be a basis of V with dual basis v
1 , ..., v
n. For l = 1, ...,m, fl is
presented by the matrix
i,j=1
, where alij = (v
i , fl(vj)). Therefore, tr
i=1(v
i , fl(vi)). We now have
tr|V ⊗m(c ◦ (f1 ⊗ ...⊗ fm)) =
i1,...,im=1
(v∗i1 ⊗ v
⊗ ...⊗ v∗im , c(f1(vi1)⊗ f2(vi2)⊗ ...⊗ fm(vim))) =
i1,...,im=1
(v∗i1 , f2(vi2)) · · · (v
, fm(vim ))(v
, f1(vi1 )) =
i1,...,im=1
a2i1,i2a
i2,i3
· · · amim−1,ima
im,i1
= tr|V (f2 ◦ f3 ◦ · · · ◦ fm ◦ f1) =
tr|V (f1 ◦ f2 ◦ · · · ◦ fm),
as desired. �
Consequently we have the following proposition which is analogous to the finite
group case.
Proposition 4.4. Let χ be the irreducible character of V . Then
νm(V ) =
χ(gm)dg.
Proof. We follow the lines of the proof of the first formula for Frobenius-Schur
indicators in the Hopf case, given in Section 2.3 of [KSZ].
It follows immediately from Proposition 4.2 and Lemma 4.3 that
νm(V ) =
tr|N (c ◦ π
⊗m(g))dg =
tr|N (c ◦ (π(g) ⊗ ...⊗ π(g))dg =
tr|V (π(g) ◦ .. ◦ π(g))dg =
χ(gm)dg.
FROBENIUS-SCHUR INDICATORS FOR SEMISIMPLE LIE ALGEBRAS 7
Recall the integral real elements which are those elements µ of t∗ for which
2(µ,α)
(α,α)
is an integer for any simple real root α. For each real integral element µ, there is a
function µ̃ on T given by
µ̃(eh) = eiµ(h)
for all h in t. Functions of this form are called torus characters and they have the
following property.
Lemma 4.5.
µ̃(t)dt =
eiµ(h)deh =
1 µ = 0,
0 otherwise.
Proof. Suppose that µ 6= 0, then there exists t0 ∈ t such that µ̃(t0) 6= 1. Therefore
µ̃(t)dt =
µ̃(t0t)dt = µ̃(t0)
µ̃(t)dt,
hence
µ̃(t)dt = 0. �
Let χ be the character of V . Before we begin the proof of Theorem 1.1, recall
that if t = eh ∈ T then for all t ∈ T ,
(1) χ(t) = χ(eh) =
µ∈Π(V )
dim(V [µ])eiµ(h).
We can now prove our main result.
Proof of Theorem 1.1: By Proposition 4.4 and the Weyl integration formula we
have,
(2) νm(V ) =
χ(gm)dg =
χ(tm)|Aρ(t)|
On the other hand,
(3) χ(tm) = χ(emh) =
µ∈Π(V )
dim(V [µ])eimµ(h).
Hence by (2) and (3) we have,
(4) νm(V ) =
µ∈Π(V )
dimV [µ]
eimµ(h)|Aρ(e
h)|2deh.
Now let us calculate the last integral. We have
eimµ(h)|Aρ(e
h)|2deh =
eimµ(h)Aρ(e
h)Aρ(eh)de
eimµ(h)
ω∈W sn(ω)e
i(ω·ρ)(h)
τ∈W sn(τ)e
−i(τ ·ρ)(h)
deh =
ω,τ∈W
sn(ωτ)
ei(mµ+ω·ρ−τ ·ρ)(h)deh.
But from Lemma 4.5 we have
ei(mµ+ω·ρ−τ ·ρ)(h)deh =
1 if mµ+ ω · ρ− τ · ρ = 0
0 otherwise.
8 MOHAMMAD ABU-HAMED AND SHLOMO GELAKI
Hence (4) becomes,
νm(V ) =
ω,τ∈W
τ·ρ−ω·ρ
sn(ωτ)dimV [µ]
ω,τ∈W
sn(ωτ)dimV
τ · ρ− ω · ρ
Since dimV [ζ] = dimV [τ · ζ] for all ζ ∈ Π(V ) and τ ∈ W , we can write,
νm(V ) =
ω,τ∈W
sn(ωτ)dimV
ρ− τ−1ω · ρ
Now if we fix ω ∈ W , substitute σ = τ−1ω and use the fact that sn(ωτ) =
sn(τ−1ω), we get
sn(ωτ)dimV
ρ− τ−1ω · ρ
sn(τ−1ω)dimV
ρ− τ−1ω · ρ
sn(σ)dimV
ρ− σ · ρ
Consequently,
νm(V ) =
ω,σ∈W
sn(σ)dimV
ρ− σ · ρ
sn(σ)dimV
ρ− σ · ρ
as desired. �
It may be interesting to state the following immediate consequence of Theorem
1.1 and Theorem 3.6.
Corollary 4.6. Let V (λ) be an irreducible self dual representation of g, then
sn(σ)dimV (λ)
ρ− σ · ρ
= (−1)λ(2ρ̌).
If V (λ) is not self dual, the sum equals 0.
4.2. Proof of Corollary 1.2. Since ρ is strongly dominant, σ · ρ = ρ only when
σ = 1. Write
νm(V ) = dimV [0] +
σ 6=1
sn(σ)dimV
ρ− σ · ρ
We wish to show that for large enough m, ρ−σ·ρ
is not a weight of V when σ 6= 1.
Indeed, suppose that σ 6= 1. Recall that ρ− σ · ρ is an integral element, hence if we
fix some coroot hα, we have the following set of integers: Uα = {(ρ− σ · ρ)(hα)|σ ∈
W , σ 6= 1}. Therefore if we take mα = 1 + uα, where uα is the maximal element
of Uα, then
ρ−σ·ρ
/∈ Π(V ). Hence dimV
ρ−σ·ρ
= 0 for all σ 6= 1, and therefore
νm(V ) = dimV [0], for all m ≥ mα. �
Note that by the procedure of the above proof, m =: min{mα|α ∈ ∆} is a better
bound. Let us now give an explicit such lower bound.
FROBENIUS-SCHUR INDICATORS FOR SEMISIMPLE LIE ALGEBRAS 9
Lemma 4.7. If ω ∈ W then
ω · ρ = ρ−
−1(α)∈Φ−
In particular, sα(ρ) = ρ− α for α ∈ ∆.
Proof. Evidently, ω · ρ is half sum of the set {ω(α)|α ∈ Φ+}. Like Φ+, this is a
set of exactly half of the roots, containing each root or its negative but not both.
More precisely, this set is obtained from Φ+ by replacing each α ∈ Φ+ such that
ω−1 · α ∈ Φ− by its negative. Now,
ω · ρ = ρ−
−1(α)∈Φ−
is evident , and sα(ρ) = ρ− α is a special case since one shows that if α ∈ ∆ and
β ∈ Φ+, then either β = α or sα(β) ∈ Φ
Proposition 4.8. Let V be an irreducible representation of G. Then νm(V ) =
dimV [0] for all m ≥ M := minα∈∆{
|β(hα)|+ 1}.
Proof. Let h = hα be a simple coroot. For all 1 6= ω ∈ W we have,
|(ρ− ω · ρ)(h)| =
β∈Φ+,
−1(β)∈Φ−
β∈Φ+,
−1(β)∈Φ−
|β(h)| ≤
|β(h)|.
Therefore if we choose m =
β∈Φ+ |β(h)|+ 1 then
ρ−ω·ρ
(h) /∈ Z, namely, ρ−ω·ρ
not a weight. Consequently,
σ 6=1
sn(σ)dimV
ρ− σ · ρ
= 0, and we are done. �
Let us calculate the bound M defined in Proposition 4.8 for sl(n,C). Let the
Cartan subalgebra be the set of diagonal matrices in sl(n,C). Let the set of positive
roots be Φ+ = {βi,j|1 ≤ i < j ≤ n}, where βi,j(diag(a1, . . . , an)) = ai − aj . The
subset ∆ = {βi,i+1|1 ≤ i ≤ n− 1} is a base. With respect to this base the simple
coroots are {hi|1 ≤ i ≤ n− 1}, where hi is the matrix with 1 in the (i, i) position,
−1 in the (i+1, i+1) position and 0 elsewhere. Then, by an elementary calculation,
we get that for any simple coroot h,
1≤i<j≤n
|βi,j(h)|+ 1 = 2n− 1.
Consequently we obtain that M = 2n− 1.
Let us calculate the bound M defined in Proposition 4.8 for so(2n+ 1,C). Let
the Cartan subalgebra be the set of diagonal matrices in so(2n + 1,C). Let the
set of positive roots be Φ+ = {βi ± βj |1 ≤ i < j ≤ n} ∪ {βi|1 ≤ i ≤ n}, where
βi(hj) = δij . The subset ∆ = {βi − βi+1, βn|1 ≤ i ≤ n − 1} is a base. With
respect to this base the simple coroots are {hi − hi+1, 2hn|1 ≤ i ≤ n − 1}, where
hi is the matrix with 1 in the (i, i) position, −1 in the (n+ i, n+ i) position and 0
10 MOHAMMAD ABU-HAMED AND SHLOMO GELAKI
elsewhere. Then, by an elementary calculation, we get that for any simple coroot
h := hk − hk+1, 1 ≤ k ≤ n− 1, the sum
β∈Φ+ |β(h)|+ 1 equals
1≤i<j≤n
|(βi + βj)(h)|+
1≤i<j≤n
|(βi − βj)(h)| +
|βi(h)|+ 1 = 4n− 3,
while for the simple coroot h := 2hn it equals 4n− 1. Consequently we obtain that
M = 4n− 3.
Applying similar arguments to the other classical simple Lie algebras yields the
following result.
Proposition 4.9. The bound M for sl(n,C), so(2n,C), so(2n+1,C) and sp(2n,C)
is equal to 2n− 1, 4n− 5, 4n− 3 and 2n+ 1, respectively.
4.3. The proof of Theorem 1.3. Let h be the CSA of sl(3,C) generated by the
two elements h1 = diag(1,−1, 0) and h2 = diag(0, 1,−1). We will identify any
functional α on h with the pair (α(h1), α(h2)). Under this identification the six
roots of sl(3,C) are α1 = (2,−1), α2 = (−1, 2), α1 + α2 = (1, 1), −α1 = (−2, 1),
−α2 = (1,−2) and −α1 − α2 = (−1,−1). The roots α1 = (2,−1), α2 = (−1, 2)
form a base and the corresponding simple coroots are h1, h2, respectively.
Recall that if V = V (λ) is an irreducible representation of sl(3,C) of highest
weight λ, then λ is of the form (a, b) with a and b non-negative integers.
Recall thatW ∼= S3 and it acts on h by σ·diag(d1, d2, d3) = diag(dσ(1), dσ(2), dσ(3)).
Therefore, (12) · α1 = −α1, (12) · α2 = α1 + α2; (13) · α1 = −α2, (13) · α2 = −α1;
(23) · α1 = α1 + α2, (23) · α2 = −α2; (123) · α1 = −α1 − α2, (123) · α2 = α1; and
(132) · α1 = α2, (123) · α2 = −α1 − α2.
The half sum of positive roots is ρ = 1
(2α1 + 2α2) = α1 + α2. We have,
ρ− (12)ρ = (2,−1), ρ− (13)ρ = (2, 2), ρ− (23)ρ = (−1, 2), ρ− (123)ρ = (0, 3), and
ρ− (132)ρ = (3, 0).
Let m = 2. Considering our formula, we cancel all the summands which include
roots that one of their two components is not divisible by 2. Consequently we get
ν2(V ) = dimV [(0, 0)]− dimV [(1, 1)].
Recall that an irreducible representation V (a, b) is self dual if and only if a = b.
Since λ = (s, s) = sα1 + sα2, (λ, 2ρ̌) = (λ, 2h1 + 2h2) = (λ, 2h1) + (λ, 2h2) = 4s, it
follows from Tits’ theorem that
ν2(V (a, b)) =
0 a 6= b,
1 otherwise.
Similar considerations for m ≥ 3 yield,
ν3(V ) = dimV [(0, 0)] + dimV [(1, 0)] + dimV [(0, 1)]
νm≥4(V ) = dimV [(0, 0)].
In particular, if λ does not belong to the root lattice, νm≥4(V ) = 0.
We now calculate dimV [(0, 0)], dimV [(1, 0)] and dimV [(0, 1)]. Recall that for η ∈
h∗, p(η) ≥ 1 if and only if η belongs to the root lattice and η ≻ 0. If η = kα1 + lα2
with nonnegative integers k and l, then p(η) = 1 +min{k, l}. Write λ = kα1 + lα2
where k and l are real numbers and identify it with the pair (λ(h1), λ(h2)) = (a, b) =
(2k − l, 2l− k).
FROBENIUS-SCHUR INDICATORS FOR SEMISIMPLE LIE ALGEBRAS 11
Note that (0, 1) = 1
α2 and (1, 0) =
α2. Therefore by Kostant’s
formula (see Theorem 2.1),
dimV [(0, 0)] =
sn(ω)p((k + 1)ω · α1 + (l + 1)ω · α2 − α1 − α2),
dimV [(0, 1)] =
sn(ω)p
(k + 1)ω · α1 + (l + 1)ω · α2 −
dimV [(1, 0)] =
sn(ω)p
(k + 1)ω · α1 + (l + 1)ω · α2 −
It is straightforward to verify that in each of the three cases the surviving
terms correspond to ω = 1, (12), (23). For example, in the first case calculating
(k + 1)ω · α1 + (l + 1)ω · α2 − α1 − α2 for ω = (12), (13), (23), (123), (132), yields
(l − k − 1)α1 + lα2, −(l + 2)α1 − (k + 2)α2 (hence p = 0), kα1 + (k − l − 1)α2,
(l−k−1)α1− (k+2)α2 (hence p = 0), and −(l+2)α1+(k− l−1)α2 (hence p = 0),
respectively.
Therefore we have that dimV [(0, 0)] equals
b+ 2a
2b+ a
b− a− 3
2b+ a
b+ 2a
−b+ a− 3
dimV [(0, 1)] equals
b+ 2a− 1
2b+ a− 2
b− a− 4
2b+ a− 2
b+ 2a− 1
−b+ a− 5
and dimV [(1, 0)] equals
b+ 2a− 2
2b+ a− 1
b− a− 5
2b+ a− 1
b+ 2a− 2
−b+ a− 4
Now, modulo 3, exactly one of the following holds: 1) b+2a = 0 and 2b+a = 0 (in
this case λ belongs to the root lattice), 2) b+2a = 1 and 2b+a = 2 and 3) b+2a = 2
and 2b + a = 1. Hence by the above and elementary calculations, we obtain that
in the first case dimV [(0, 1)] = dimV [(1, 0)] = 0, in the second case dimV [(0, 0)] =
dimV [(1, 0)] = 0 and in the third case dimV [(0, 0)] = dimV [(0, 1)] = 0. Therefore,
in the first case ν3(V (a, b)) equals
1 +min
b+ 2a
2b+ a
1 +min
b− a− 3
2b+ a
1 +min
b+ 2a
−b+ a− 3
12 MOHAMMAD ABU-HAMED AND SHLOMO GELAKI
in the second case it equals
1 +min
b+ 2a− 1
2b+ a− 2
1 +min
b− a− 4
2b+ a− 2
1 +min
b+ 2a− 1
−b+ a− 5
and in the third case it equals
1 +min
b+ 2a− 2
2b+ a− 1
1 +min
b− a− 5
2b+ a− 1
1 +min
b+ 2a− 2
−b+ a− 4
Finally, it is easy to check that in each case the sum equals 1 + min{a, b}, as
claimed. This completes the proof of the theorem. �
References
[B] D. Bump, Lie Groups, Springer-Verlag NY, LLC, (2004).
[D] E. Dynkin, Semisimple subalgebras of semisimple Lie algebras (Russian) Mat.Sbornik
N.S. 30 (27) (1952) 349-462, English: AMS Translations 6 (1957), 111-244.
[FGSV] J. Fucs, C. Ganchev, K. Szlachányi, and P. Vescernyes, S4-symmetry of 6j-sympols and
Frobenius-Schur indicators in rigid monoidal C∗-categories. J.Math Phys. 40 (1999),
408-426.
[Ha] B. Hall, Lie groups, Lie algebras and representations, Springer-Verlag, Berlin-Heidelberg-
New York, (2006).
[Hu] J. Humphreys, Introdution to Lie algebras and representation theory, Springer-Verlag,
Berlin-Heidelberg-New York, (1972).
[K] B. Kostant, The principal three dimensional subgroup and betti numbers of complex
simple Lie group, Amer.J.Math. 81 (1959), 973-1032.
[KSZ] Y. Kashina, Y. Sommerhaeuser, and Y. Zhu, On higher Frobenius-Schur indicators,
Memoirs of the AMS 181, no 855 (2006).
[LM] V. Linchenko and S. Montgomery, A Frobenius-Schur theorem for Hopf algebras, Algebr.
Represent. Theory 3 (2000), no. 4, 347-355, Special issue dedicated to Klaus Roggenkamp
on the occasion of his 60th birthday.
[MN] G. Mason and S-H. Ng, Central invariants and Frobenius-Schur indicators for semi-
simple qusi-Hopf algebras, Adv. Math. 190 (2005), 161-195.
[NS1] S-H. Ng and P. Schauenburg, Higher Frobenius-Schur indicators for pivotal categories,
preprint arXiv:math,QA/0503167.
[NS2] S-H. Ng and P. Schauenburg, Central invariants and higher indicators for semisimple
quasi-Hopf algebras, Transactions of the AMS, to appear, arXiv:math,QA/0508140.
[S] J-P. Serre, Linear Representation of Finite Groups, Springer-Verlag, New York, (1977).
Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000,
Israel
E-mail address: mohammad@tx.technion.ac.il, mohammad.abu-hamed@weizmann.ac.il
Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000,
Israel
E-mail address: gelaki@math.technion.ac.il
	1. Introduction
	2. Preliminaries
	3. Tits' theorem on the second indicator
	4. The Main results
	4.1. Proof of Theorem ??
	4.2. Proof of Corollary ??
	4.3. The proof of Theorem ??
	References
