Combinatorics of irreducible Gelfand-Tsetlin sl(3)-modules
In this paper we present an explicit description of all irreducible sl(3)-modules which admit a Gelfand-Tsetlin tableaux realization with respect to the standard Gelfand-Tsetlin subalgebra.
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2012
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| Cite this: | Combinatorics of irreducible Gelfand-Tsetlin sl(3)-modules / L.E. Ramirez // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 276–296. — Бібліогр.: 11 назв. — англ. |
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| citation_txt | Combinatorics of irreducible Gelfand-Tsetlin sl(3)-modules / L.E. Ramirez // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 276–296. — Бібліогр.: 11 назв. — англ. |
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| description | In this paper we present an explicit description of all irreducible sl(3)-modules which admit a Gelfand-Tsetlin tableaux realization with respect to the standard Gelfand-Tsetlin subalgebra.
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h.Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 14 (2012). Number 2. pp. 276 – 296
c© Journal “Algebra and Discrete Mathematics”
Combinatorics
of irreducible Gelfand-Tsetlin sl(3)-modules
Luis Enrique Ramirez1
Communicated by V. M. Futorny
Abstract. In this paper we present an explicit description of
all irreducible sl(3)-modules which admit a Gelfand-Tsetlin tableaux
realization with respect to the standard Gelfand-Tsetlin subalgebra.
Introduction
In the present paper we will describe irreducible sl(3)-modules in a
certain full subcategory of the category of Gelfand-Tsetlin modules (we
will abbreviate Gelfand-Tsetlin by GT); namely the category GTT of
GT-modules that admit a tableaux realization with respect to a GT-
subalgebra [9]. This description provides a realization similar to the sl(2)
case (in the latter it is always possible to choose a basis of eigenvectors
with respect to a Cartan subalgebra and write explicit formulas for the
action of the generators of sl(2)).
Following [9]; we say that an sl(n)-module V admits a tableaux
realization with respect to a GT-subalgebra Γ provided V decomposes as
V =
⊕
ξ∈Γ∗ Vξ where
Vξ := {v ∈ V : ∃k ∈ N such that (t − ξ(t))kv = 0 ∀t ∈ Γ},
1The author is supported by the CNPq grant (processo 142407/2009-7)
2010 MSC: 17B35, 17B37, 17B67, 16D60, 16D90, 16D70, 81R10.
Key words and phrases: Gelfand-Tsetlin modules, weight modules, Gelfand-
Tsetlin basis.
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L. E. Ramirez 277
dim(Vξ) ≤ 1 for all ξ ∈ Γ∗ and the action of the generators of sl(n) is
given by the GT-formulas ([7], [11]). It was shown in [2] that in sl(3),
for any irreducible GT-module V , dim(Vξ) ≤ 2 for all ξ ∈ Γ∗. Moreover,
there are explicit examples of GT-modules with dim(Vξ) = 2 for some
ξ ∈ Supp(V ). Hence GTT is a proper subcategory of GT.
In sections 1 and 2 we give the definitions and notations that we will
use throughout the paper. The section 3 is devoted to the description of
a basis for the irreducible sl(3)-modules in GTT which is the main result
of the paper. As a direct consequence of this description it is possible to
give simple conditions for a tableau such that the associated irreducible
module has bounded weight multiplicities or 1-dimensional weight spaces.
In section 4 we use the results of section 3 to answer when a highest
weight sl(3)-module admits a tableaux realization (with respect to some
GT-subalgebra). Finally, in the section 5 we give a characterization of the
irreducible sl(3)-modules in GTT which are Harish-Chandra modules.
1. Gelfand-Tsetlin modules
Let n ∈ N fixed; for k ∈ {1, 2, . . . , n} denotes by gk := gl(k); Uk :=
U(gk) the universal enveloping algebra of gk and Zk := Z(gk) the center
of gk; let also g := gn and U := U(g).
If {Eij} denotes the canonical basis of g, we have a natural identi-
fication between gk and the subalgebra of g generated by the matrices
{Eij}i,j=1,...,k; i.e. consider gi as a subalgebra of gi+1 with respect to the
upper left corner embedding.
a11 | a12 | a13 | . . . | a1n
__ | | | |
a21 a22 | a23 | . . . | a2n
__ _ __ | | |
a31 a32 a33 | . . . | a3n
__ _ __ _ __ | |
. . .
. . .
. . .
. . . | ...
__ _ __ _ __ _ __ |
an1 an2 an3 . . . ann
The chain of inclusions: g1 ⊂ g2 ⊂ · · · ⊂ gn = g induces a chain of
inclusions of the corresponding enveloping algebras.
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278 Combinatorics of irreducible sl(3)-modules
Definition 1. Let Γ the subalgebra of U generated by {Zk : k = 1, . . . , n};
this subalgebra is called standard Gelfand-Tsetlin subalgebra of U
[3].
Remark 1. Zm is a polynomial algebra in m variables {cmk : k =
1, 2, . . . , m},
cmk =
∑
(i1,i2,...,ik)∈{1,...,m}k
Ei1i2Ei2i3 · · · Eiki1
and the algebra Γ is a maximal commutative polynomial subalgebra of
U(g) in n(n+1)
2 variables {cij : 1 ≤ j ≤ i ≤ n}.
Definition 2. Let M be a g-module; χ : Γ → C a homomorphism and
Mχ = {v ∈ M : ∃k ∈ N such that (g − χ(g))kv = 0 ∀g ∈ Γ}.
The module M is called Gelfand-Tsetlin module (respect to Γ) if
M =
⊕
χ∈Γ∗ Mχ and dim(Mχ) < ∞ for all χ ∈ Γ∗ [3].
Definition 3. An array of rows with complex entries {lij : 1 ≤ j ≤ i ≤ n}
as follows:
λn1 λn2 · · · λn,n−1 λnn
λn−1,1 · · · λn−1
n−1
· · · · · · · · ·
λ21 λ22
λ11
is called Gelfand-Tsetlin tableau. A Gelfand-Tsetlin tableau is called
standard if
λki −λk−1,i ∈ Z
≥0 and λk−1,i −λk,i+1 ∈ Z
≥0, for all1 ≤ i ≤ k ≤ n−1.
In the finite dimensional case we have the following classical result [7]:
Theorem 1. If L(λ) is a finite dimensional irreducible representation
of gl(n) of highest weight λ = (λ1, . . . , λn), there exist a bases {ξ[L]} of
L(λ) parameterized by all standard tableaux [L] with top row λn1 =
λ1, . . . , λnn = λn and the gl(n) generators acts by the formulas:
Ek,k+1(ξ[L]) = −
k∑
i=1
(
∏k+1
j=1(lki − lk+1,j)
∏k
j 6=i(lki − lkj)
)ξ[L+δki],
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L. E. Ramirez 279
Ek+1,k(ξ[L]) =
k∑
i=1
(
∏k−1
j=1(lki − lk−1,j)
∏k
j 6=i(lki − lkj)
)ξ[L−δki],
Ekk(ξ[L]) = (
k∑
i=1
λki −
k−1∑
i=1
λk−1,i)ξ[L],
Where lki = λki − i + 1, [L ± δki] is the tableau obtained by [L] adding
±1 to the ki position of [L]; and if ˜[L] is not standard, the vector ξ ˜[L]
would be zero. Moreover, the action of the generators of Γ in the basis
elements is given by:
cij(ξ[L]) = (
i∑
k=1
(lik + i)j
∏
s 6=k
(1 − 1
lik − lis
))ξ[L]
The formulas of the previous theorem are called Gelfand-Tsetlin
formulas for gl(n).
2. Gelfand-Tsetlin formulas for sl(3)
Remark 2. From now on we will prefer to use tableaux with entries lij
instead of λij because the formulas are symmetric with respect to the
lij ’s in the following sense. Let Ri denote the i-th row of the tableaux
[L] and Si the i-th symmetric group. We have a natural action of the
group S1 × S2 × · · · × Sn on the set of GT-tableaux (with entries lij):
(σ1, . . . , σn)([L]) is the tableau with the i-th row σi(Ri), for i = 1, 2, . . . , n.
Definition 4. The tableaux [L]1 and [L]2 are equivalent if there exist
(σ1, . . . , σn) such that (σ1, . . . , σn)([L]1) = [L]2 and we write in this case
[L]1 ≈ [L]2.
Remark 3. If we want to recover the tableaux with coefficients λij we
just need to remember the relations lki = λki − i + 1.
In the particular case of gl(3), let a, b, c, x, y, z ∈ C fixed complex
numbers; from now on we will use [L] to denote the fixed tableau;
a b c
[L]:= x y
z
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280 Combinatorics of irreducible sl(3)-modules
Then, the GT-formulas for the generators of gl(3) are:
E11([L]) = z[L] E12([L]) = −(x − z)(y − z)[L + δ11]
E22([L]) = (x + y + 1 − z)[L] E21([L]) = [L − δ11]
E33([L]) = (a + b + c + 2 − x − y)[L]
E32([L]) = (x−z)
(x−y) [L − δ21] − (y−z)
(x−y) [L − δ22]
E23([L]) = (a−x)(b−x)(c−x)
(x−y) [L + δ21] − (a−y)(b−y)(c−y)
(x−y) [L + δ22].
As we want to restrict our attention to sl(3), we have to consider just
the tableaux such that E11([L]) + E22([L]) + E33([L]) = 0 that implies
a + b + c + 3 = 0; then the GT-formulas for the generators of sl(3) are
given by:
GT-formulas:
h1([L]) = (2z − (x + y + 1))[L]
h2([L]) = (2(x + y + 1) − z)[L]
E12([L]) = −(x − z)(y − z)[L + δ11]
E21([L]) = [L − δ11]
E32([L]) = (x−z)
(x−y) [L − δ21] − (y−z)
(x−y) [L − δ22]
E23([L]) =
(a − x)(b − x)(c − x)
(x − y)
[L + δ21]−
− (a − y)(b − y)(c − y)
(x − y)
[L + δ22].
Now we introduce some notation that will help us to simplify the
desired description.
Notation 1. Let
l31 l32 l33
[T ]= l21 l22
l11
be an arbitrary tableau, B1([T ]) := {l31 − l21, l32 − l21, l33 − l21} and
B2([T ]) := {l31−l22, l32−l22, l33−l22}. We consider the following functions:
• t0([T ]) := l21 − l22; t3([T ]) := l21 − l11; t−
3 ([T ]) := l22 − l11
• ti([T ]) := min{{Bi([T ]) ∩ Z
≥0} ∪ {+∞}}; i = 1, 2
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L. E. Ramirez 281
• t−
i ([T ]) := max{{Bi([T ]) ∩ Z
<0} ∪ {−∞}}; i = 1, 2.
In the cases that ti([T ]) = +∞ or t−
i ([T ]) = −∞ for some i = 1, 2, we
will write ti([T ]) /∈ Z or t−
i ([T ]) /∈ Z respectively.
From now on in order to simplify the notation we will write t0, t1, t2,
t−
1 , t−
2 , t3, t−
3 instead to t0([L]), t1([L]), t2([L]), t−
1 ([L]), t−
2 ([L]), t3([L]),
t−
3 ([L]), where [L] is the fixed tableau as before.
3. Description of irreducible GTT sl(3)-modules
Given a tableau [L], we can look at the set of all tableaux that can be
obtained with non-zero coefficients from [L] using the GT-formulas. It is
natural to ask what are the possible tableaux [L] that we can consider in
order to obtain an sl(3)-module structure on the vector space generated
by this set of tableaux.
The only problem (if we apply the GT-formulas) is a possibility of
zero denominators. Thus we have to restrict our attention to the lattice
of tableaux of [L]
Latt([L]) := {[L̃] : [L̃] is obtained from [L]
using GT-formulas and t0([L̃]) 6= 0}.
Here to obtain [L̃] from [L] using the GT-formulas means that there
exist X ∈ U(sl(3)) such that [L̃] appear with non-zero coefficient in
X([L]).
The following result from [3] implies the existence of some GT-modules
called generic Gelfand-Tsetlin modules.
Theorem 2. If t0, t1, t2, t−
1 , t−
2 , t3, t−
3 /∈ Z then, the C-vector space V[L]
generated by the set of vectors {ξ[L̃] : [L̃] ∈ Latt([L])} defines an irre-
ducible sl(3)-module with the action of sl(3) given by the GT-formulas.
Corollary 3. If t0 /∈ Z then, the C-vector space V[L] generated by the
set of vectors {ξ[L̃] : [L̃] ∈ Latt([L])} has a structure of GT sl(3)-module;
where the action of sl(3) is given by the GT-formulas.
Definition 5. We say that an sl(3)-module V admits a tableaux realiza-
tion with respect to a GT-subalgebra Γ provided that V is a GT-module
(with respect to Γ), dim(Vξ) ≤ 1 for all ξ ∈ Γ∗ and the action of the
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282 Combinatorics of irreducible sl(3)-modules
generators of sl(3) is given by the GT-formulas. Equivalently, a GT sl(3)-
module is said to have a tableaux realization if it is isomorphic to
V[T ] for some tableau [T ]. We say that a module is a GTT-module if it
admits a tableaux realization.
In this section we will describe explicitly bases of all irreducible sl(3)-
modules in GTT and then we will be able to calculate weight multiplicities
(with respect to the standard Cartan subalgebra of sl(3)) of this modules
in terms of the values of the constants t0, t1, t2, t−
1 , t−
2 , t3, t−
3 .
By the GT-formulas, we have that Latt([L]) ⊂ {[L]m,n,k : m, n, k ∈ Z}
where the tableaux [L]m,n,k is defined as:
a b c
[L]m,n,k:= x+m y+n
z+k
Then we can identify Latt([L]) with points of R3 with integer coordi-
nates to describe a basis of the module V[L].
Let m, n, k ∈ Z
≥0, applying the GT-formulas to [L] we see that:
1) [L + mδ21] appears in the decomposition of Em
23[L] with coefficient
m−1∏
i=0
(a − x − i)(b − x − i)(c − x − i)
(x − y + i)
2) [L − mδ21] appears in the decomposition of Em
32[L] with coefficient
m−1∏
i=0
(x − z − i)
(x − y − i)
3) [L + nδ22] appears in the decomposition of En
23[L] with coefficient
n−1∏
i=0
(a − y − i)(b − y − i)(c − x − i)
(x − y − i)
4) [L − nδ22] appears in the decomposition of En
32[L] with coefficient
n−1∏
i=0
(y − z − i)
(x − y + i)
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5) Ek
21([L]) = [L − kδ11]
6) Ek
12([L]) =
∏k−1
i=0 (x − z − i)(y − z − i)[L + kδ11]
As an immediate consequence of the above observation we have the
following lemma:
Lemma 4. For i = 1, 2, 3 denote by Ai the conditions ti /∈ Z
≥0 and by
A−
3 the condition t−
3 /∈ Z
≥0. Then the following statements hold:
1) [L + mδ21] ∈ Latt([L]) for all m ∈ Z
+ if A1 and t0 /∈ Z
<0.
2) [L − mδ21] ∈ Latt([L]) for all m ∈ Z
+ if A3 and t0 /∈ Z
>0.
3) [L − kδ11] ∈ Latt([L]) for all k ∈ Z
+.
4) [L + kδ11] ∈ Latt([L]) for all k ∈ Z
+ if A3 and A−
3 .
5) [L + nδ22] ∈ Latt([L]) for all m ∈ Z
+ if A2 and t0 /∈ Z
>0.
6) [L − nδ22] ∈ Latt([L]) for all m ∈ Z
+ if A−
3 and t0 /∈ Z
<0.
Now we will answer the following question: what conditions on the
entries of [L] guarantee that Latt([L]) = Z
3 (i.e. when Latt([L]) is the
largest possible)?
Definition 6. Given m, n, k ∈ Z, we say that the tableau [L]m,n,k, can
be obtained from [L] by the path r → s → t, with {r, s, t} = {1, 2, 3} if:
From [L] we can obtain [L](m,0,0) if r = 1 (respectively [L](0,n,0) if r = 2
and [L](0,0,k) if r = 3); from [L](m,0,0) we obtain [L](m,n,0) if s = 2 or
[L](m,0,k) if s = 3 (respectively from [L](0,n,0) we obtain [L](m,n,0) if s = 1
or [L](0,n,k) if s = 3 and from [L](0,0,k) we obtain [L](m,0,k) if s = 1 or
[L](0,n,k) if s = 2) and in the last step we obtain the tableau [L]m,n,k.
Example 1. [L]7,−1,4 is obtained from [L] by the path 3 → 1 → 2 means:
from [L] we obtain the tableau [L]0,0,4; with this tableau we obtain [L]7,0,4
and from this, we can obtain [L]7,−1,4.
Proposition 5. Latt([L]) = Z
3 if and only if
t1, t2, t3, t−
3 /∈ Z
≥0; t0 /∈ Z.
Proof:. (⇐) Let m, n, k ∈ Z. Using lemma 4 in each step of the path
indicated below it is possible to obtain the tableau [L]m,n,k. In each case
the path will depend of the ordered triple of signs of m, n, k as follows:
(+, +, +) (−, −, −) (+, −, +)
{
3 → 2 → 1; if n ≥ m
3 → 1 → 2; if m ≥ n
{
1 → 2 → 3; if m ≥ n
2 → 1 → 3; if n ≥ m
3 → 1 → 2
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284 Combinatorics of irreducible sl(3)-modules
(+, +, −) (−, −, +) (+, −, −)
{
1 → 2 → 3; if m ≥ n
2 → 1 → 3; if n ≥ m
{
1 → 2 → 3; if m ≥ n
2 → 1 → 3; if n ≥ m
1 → 2 → 3
(−, +, +) (−, +, −)
3 → 2 → 1 1 → 2 → 3
(⇒) Without loss of generality we can assume that some of these
constants are zero. Then we conclude:
1) If t1 = 0 then it is not possible to obtain [L]1,0,0 from [L].
2) If t2 = 0 then [L]0,1,0 /∈ Latt([L]).
3) If t3 = 0 or t−
3 = 0 then [L]0,0,1 /∈ Latt([L]) or [L]0,0,−1 /∈ Latt([L])
respectively.
Now we have enough information about Latt([L]) in order to describe
irreducible modules. For this we will use the following characterization.
The module V[L] is irreducible if and only if Latt([L̃]) = Latt([L]) for
all [L̃] ∈ Latt([L]).
Theorem 6. Let [L] be such that Latt([L]) = Z
3. Then V[L] is irreducible
if and only if
t0, t1, t−
1 , t2, t−
2 , t3, t−
3 /∈ Z.
Proof:. (⇐) Under the conditions it is possible to apply the GT-formulas
to any tableau in Latt([L]) and we never obtain zero coefficients. Then for
all [L̃] ∈ Latt([L]) we have Latt([L̃]) = Z
3 which implies V[L] irreducible.
(⇒) If t−
1 ∈ Z
<0 (respectively t−
2 or t−
3 ∈ Z
<0) then [L] /∈ Latt([L]−1,0,0)
(respectively [L] /∈ Latt([L]0,−1,0) or [L] /∈ Latt([L]0,0,−1)). Hence V[L] can
not be irreducible.
Remark 4. Note that it is possible to obtain Latt([L]) = Z
3 in the case
when some constants are negative integers, but not necessarily we obtain
an irreducible module.
Remark 5. To know a basis of the module generated by [L] it is enough
to know the values of the constants {t0, t1, t−
1 , t2, t−
2 , t3, t−
3 }. For some
subset A of {t0, t1, t−
1 , t2, t−
2 , t3, t−
3 }, the notation A ⊂ Z will means from
now on that A ⊂ Z and the complement of A in {t0, t1, t−
1 , t2, t−
2 , t3, t−
3 }
has empty intersection with Z. In particular t1, t2 ∈ Z
≥0 means that
{t0, t−
1 , t−
2 , t3, t−
3 } ⋂
Z = ∅.
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L. E. Ramirez 285
Proposition 7. Let [L] be a fixed tableau as before. Denote by V[L] the
sl(3)-module generated by Latt([L]) using the GT-formulas. Then
1) If t1 ∈ Z
≥0, V[L] is an irreducible module with bases parameterized
by Latt([L]) = {[L]m,n,k : m ≤ t1}.
2) If t2 ∈ Z
≥0, V[L] is an irreducible module with bases parameterized
by Latt([L]) = {[L]m,n,k : n ≤ t2}.
3) If t3 ∈ Z
≥0, V[L] is an irreducible module with bases parameterized
by Latt([L]) = {[L]m,n,k : k − m ≤ t3}.
4) If t−
1 ∈ Z
<0, the irreducible module that contains [L] can be obtain
as a quotient module of V[L]; and the bases is parameterized by the
set of tableaux {[L]m,n,k : m > t−
1 }.
5) If t−
1 ∈ Z
<0, the irreducible module that contains [L] can be obtain
as a quotient module of V[L]; and the bases is parameterized by the
set of tableaux {[L]m,n,k : n > t−
2 }.
6) If t3 ∈ Z
<0, the irreducible module that contains [L] can be obtain
as a quotient module of V[L]; and the bases is parameterized by the
set of tableaux {[L]m,n,k : k − m > t3}.
Proof. The cases 1, 2, 3 are obvious from the GT-formulas and the irre-
ducibility is guaranteed by the Theorem 6. In each of the cases 4, 5, 6 we
can apply the GT-formulas and obtain in Latt([L]) a tableaux [L̃] that
satisfies t1([L̃]) ∈ Z
≥0 (respectively t2([L̃]) ∈ Z
≥0 or t3([L̃]) ∈ Z
≥0). Then
the irreducible module that contains [L] is isomorphic to the quotient
module V[L]/V
[L̃]
.
Corollary 8. Using Proposition 7 we can characterize the set of tableaux
that parameterizes a basis of the irreducible module that contains [L] as
follows:
1) For t1 ∈ Z
≥0; {[L]m,n,k : t1([L]m,n,k) ∈ Z
≥0}.
2) For t2 ∈ Z
≥0; {[L]m,n,k : t2([L]m,n,k) ∈ Z
≥0}.
3) For t3 ∈ Z
≥0; {[L]m,n,k : t3([L]m,n,k) ∈ Z
≥0}.
4) For t−
1 ∈ Z
<0; {[L]m,n,k : t−
1 ([L]m,n,k) ∈ Z
<0}.
5) For t−
2 ∈ Z
<0; {[L]m,n,k : t−
2 ([L]m,n,k) ∈ Z
<0}.
6) For t−
3 ∈ Z
<0; {[L]m,n,k : t−
3 ([L]m,n,k) ∈ Z
<0}.
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286 Combinatorics of irreducible sl(3)-modules
Corollary 9. If A denotes the set {t1, t2, t−
1 , t−
2 , t3, t−
3 }, [L] satisfies the
conditions A1 := A
⋂
Z
≥0 and A2 := A
⋂
Z
<0 and those conditions implies
t0 6= 0; then, a base for the irreducible module that contains [L] can be
parameterized by:
{[L]m,n,k : A1([L]m,n,k) ∈ Z
≥0 and A2([L]m,n,k) ∈ Z
<0}
Definition 7. For each tableau [T ] satisfying the conditions of corollary
9 we will denote by I[T ] the irreducible sl(3)-module generated by [T ]
with the basis parameterized by the set of tableaux described as before.
This basis we will be denote by B[T ].
We can take advantage of knowing these bases to calculate the weights
dimensions of modules with tableaux realization.
If we want to know the action of h1 and h2 in the module I[L] it is
enough to describe the action of h1 and h2 in tableaux of type [L]m,n,k.
• h1([L]m,n,k) = (2(z + k) − (x + y + 1 + n + m))[L]m,n,k
• h2([L]m,n,k) = (2(x + y + 1 + n + m) − (z + k))[L]m,n,k.
Set λ
(1)
m,n,k := 2(z +k)− (x+y + 1 +n+m) and λ
(2)
m,n,k := 2(x+y + 1 +
n + m) − (z + k). Since x, y, z are fixed, we have a natural identification
between weights of the module I[L] and points in Z × Z as follows:
(λ
(1)
m,n,k, λ
(2)
m,n,k) ! (2k, 2(m + n)) ! (k, n + m) ! (α, β).
Theorem 10. For each (α, β) ∈ Z×Z the dimension of the weight space
(I[L])(2(z+α)−(x+y+1+β),2(x+y+1+β)−(z+α)) is equal to the cardinality of the
set
T(α,β) := {[L]t,β−t,α : t ∈ Z}
⋂
B[L]
Proof. It is sufficient to note that the vector associated with a tableaux
[L]m,n,k has weight (2(z + α) − (x + y + 1 + β), 2(x + y + 1 + β) − (z + α))
if and only if m + n = β and k = α.
Now we will describe explicitly bases and weight multiplicities of all
irreducible sl(3)-modules that admit a tableaux realization. To do that
we have to consider all possible combinations of conditions defining non-
isomorphic modules (some of these conditions define isomorphic modules
in the sense of the Definition 4; for instance, a module defined by a
tableau [L] satisfying the conditions t1 ∈ Z
≥0 is naturally isomorphic
to the module defined by the tableau σ([L]) where σ ∈ S1 × S2 × S3;
in particular to a module defined by a tableau satisfying the conditions
t2 ∈ Z
≥0).
First we consider the conditions that give infinite dimensional weight
spaces.
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Conditions B[L]
{Lm,n,k : m, n, k ∈ Z}
t2 ∈ Z
≥0 {Lm,n,k : n ≤ t2}
t3 ∈ Z
≥0 {Lm,n,k : k ≤ m + t3}
t−
1 ∈ Z
<0 {Lm,n,k : m > t−
1 }
t3 ∈ Z
<0 {Lm,n,k : m < k − t3}
t2, t3 ∈ Z
≥0 {Lm,n,k : k − t3 ≤ m; n ≤ t2}
t1 ∈ Z
≥0, t3 ∈ Z
<0 {Lm,n,k : m < k − t3; m ≤ t1}
t−
2 , t3 ∈ Z
<0 {Lm,n,k : m < k − t3; n > t−
2 }
t1 ∈ Z
≥0, t−
2 ∈ Z
<0 {Lm,n,k : m ≤ t1; n > t−
2 }
t3 ∈ Z
≥0, t−
1 ∈ Z
<0 {Lm,n,k : m ≥ k − t3; m > t−
1 }
t2, t3 ∈ Z
≥0; t−
1 ∈ Z
<0 {Lm,n,k : m ≥ k − t3; n ≤ t2; m > t−
1 }
t1 ∈ Z
≥0; t−
2 , t3 ∈ Z
<0 {Lm,n,k : m < k − t3; n > t−
2 ; m ≤ t1}
t3 ∈ Z
≥0, t−
3 ∈ Z
<0 ∗ {Lm,n,k : n + t−
3 < k ≤ m + t3}
In all other cases we have dim(V(λ(1),λ(2))) < ∞ for all weight space.
Conditions B[L] Dimension of V(α,β)
t1, t3 ∈ Z≥0 k − t3 ≤ m ≤ t1
{
0 if α > t1 + t3
t1 + t3 − α + 1, if α ≤ t1 + t3
t1, t2 ∈ Z≥0
{
n ≤ t2;
m ≤ t1
{
0 if β > t1 + t2
t1 + t2 − β + 1, if β ≤ t1 + t2
t−
1 , t3 ∈ Z<0 t−
1 < m < k − t3
{
0 if α ≤ t−
1 + t3
α − t3 − t−
1 − 1, if α > t−
1 + t3
t−
1 , t−
2 ∈ Z<0
{
m > t−
1 ;
n > t−
2
{
0 if β ≤ t−
1
+ t−
2
β − t−
2
− t−
1
− 1, if β > t−
1
+ t−
2
t1, t2, t3 ∈ Z≥0
{
n ≤ t2;
k − t3 ≤ m ≤ t1
0 if β > t1 + t2
0 if α > t1 + t3
t1 + t2 − β + 1, if β − α ≥ t2 − t3
t1 + t3 − α + 1, if β − α ≤ t2 − t3{
t1 ∈ Z≥0;
t−
1 ∈ Z<0
t−
1 < m ≤ t1 t := t1 − t−
1
{
t3 ∈ Z≥0,
t−
2 ∈ Z<0
{
m ≥ k − t3;
n > t−
2
{
0 if β − α > t−
2 − t3
β − α − t−
2 + t3, if β − α ≤ t−
2 − t3{
t2 ∈ Z≥0,
t3 ∈ Z<0
{
m < k − t3;
n ≤ t2
{
0 if β − α ≥ t2 − t3
α − β + t2 − t3, if β − α < t2 − t3
{
t3 ∈ Z≥0;
t−
1 , t−
2 ∈ Z<0
m ≥ k − t3;
n > t−
2 ;
m > t−
1
0 if β ≤ t−
1 + t−
2 + 1
0 if β − α ≤ t−
2 − t3
β − α − t−
2 + t3, if α ≥ t−
1 + t3 + 1
β − t−
1 − t−
2 − 1, if α ≤ t−
1 + t3 + 1
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288 Combinatorics of irreducible sl(3)-modules
Conditions B[L] Dimension of V(α,β)
{
t2 ∈ Z≥0;
t−
1 , t3 ∈ Z<0
{
t−
1 < m < k − t3;
n ≤ t2
0 if α ≤ t3 + t−
1 + 1
0 if β − α ≥ t2 − t3
α − t3 − t−
1 − 1, if β ≤ t−
1 + t2 + 1
α − β − t3 + t2, if β ≥ t−
1 + t2 + 1
t−
1 , t−
2 , t3 ∈ Z<0
{
t−
1 < m < k − t3;
n > t−
2
0 if α ≤ t3 + t−
1
0 if β ≤ t−
1 + t−
2
β − t−
2 − t−
1 − 1, if β − α ≤ t−
2 − t3
α − t3 − t−
1 − 1, if β − α ≥ t−
2 − t3
{
t2, t3 ∈ Z≥0;
t−
2 ∈ Z<0
{
m ≥ k − t3;
t−
2 < n ≤ t2
0 if β − α ≤ t−
2 − t3
t := t2 − t−
2 , if β − α ≥ t2 − t3
β − α + t3 − t−
2 , if β − α ≤ t2 − t3
{
t1, t2, t3 ∈ Z≥0;
t−
1 ∈ Z<0
n ≤ t2;
t−
1 < m ≤ t1;
k − t3 ≤ m
0 if β > t1 + t2
0 if α > t1 + t3
t1 + t2 − β + 1, if β − α ≥ t2 − t3∧
β ≥ t2 + t−
1 + 1
t1 + t3 − α + 1, if β − α ≤ t2 − t3∧
α ≥ t3 + t−
1 + 1
t := t1 − t−
1 , if α ≤ t3 + t−
1 + 1∧
β ≤ t2 + t−
1 + 1
{
t2 ∈ Z≥0;
t−
2 , t3 ∈ Z<0
{
m < k − t3;
t−
2 < n ≤ t2
0 if β − α ≥ t2 − t3
t := t2 − t−
2 , if β − α ≤ t−
2 − t3
α − β − t3 + t2, if β − α ≥ t−
2 − t3
{
t2 ∈ Z≥0;
t−
2 , t−
1 ∈ Z<0
{
m > t−
1 ;
t−
2 < n ≤ t2
0 if β ≤ t−
1 + t−
2 + 1
t := t2 − t−
2 , if β ≥ t2 + t−
1 + 1
β − t−
1 − t−
2 − 1, if β ≤ t2 + t−
1 + 1
{
t1, t3 ∈ Z≥0;
t−
1 ∈ Z<0
{
m ≥ k − t3;
t−
1 < m ≤ t1
0 if α > t1 + t3
t := t1 − t−
1 , if α ≤ t−
1 + t3 + 1
t1 + t3 − α, if α ≥ t−
1 + t3 + 1
{
t1, t2 ∈ Z≥0;
t−
1 ∈ Z<0
{
n ≤ t2;
t−
1 < m ≤ t1
0 if β > t1 + t2
t := t1 − t−
1 , if β ≤ t−
1 + t2 + 1
t1 + t2 − β + 1, if β ≥ t−
1 + t2 + 1
{
t2 ∈ Z≥0;
t−
2 , t−
3 ∈ Z<0
{
n < k − t−
3 ;
t−
2 < n ≤ t2
0 if α ≤ t−
2 + t−
3 + 1
t := t2 − t−
2 , if α ≥ t2 + t−
3 + 1
α − t−
3 − t−
2 − 1, if α ≤ t2 + t−
3 + 1
{
t1, t2 ∈ Z≥0;
t3 ∈ Z<0
m < k − t3;
n ≤ t2;
m ≤ t1
0 if β − α ≥ t2 − t3
0 if β > t2 + t1
t1 + t2 − β + 1, if α ≥ t1 + t3 + 1
α − β − t3 + t2, if α ≤ t1 + t3 + 1
{
t1, t3 ∈ Z≥0;
t−
2 ∈ Z<0
{
k − t3 ≤ m ≤ t1;
n > t−
2
0 if β − α ≤ t−
2 − t3
0 if α > t3 + t1
t1 + t3 − α + 1, if β ≥ t1 + t−
2 + 1
β − α − t−
2 + t3, if β ≤ t1 + t−
2 + 1
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Conditions B[L] Dimension of V(α,β)
{
t2 ∈ Z≥0;
t−
2 , t−
1 , t3 ∈ Z<0
{
t−
1 < m < k − t3;
t−
2 < n ≤ t2
0 if α ≤ t−
1 + t3 + 1
0 if β − α ≥ t2 − t3
0 if β ≤ t−
1 + t−
2 + 1
t := t2 − t−
2 , if β − α ≤ t−
2 + t3∧
β ≥ t2 + t−
1 + 1
α − β − t3 + t2, if β − α ≤ t−
2 + 1∧
β ≥ t2 + t−
1 + 1
α − t3 − t−
1 − 1, if β − α ≤ t−
2 + 1∧
β ≤ t2 + t−
1 + 1
β − t−
2 − t−
1 − 1, if β − α ≥ t−
2 + t3∧
β ≤ t2 + t−
1 + 1
{
t1, t2 ∈ Z≥0;
t−
1 , t−
3 ∈ Z<0
m ≤ t1;
t−
2 < n ≤ t2;
k ≤ m + t3
0 if α − β < t−
3 − t1 + 1
0 if β > t2 + t1
α − β + t1 − t−
3 , if α − β ≤ t−
3 − t−
1 ∧
α ≤ t−
3 + t2 + 1
t1 + t2 − β + 1, if β ≥ t−
1 + t2 + 1∧
α ≥ t2 + t−
3 + 1
t := t1 − t−
1 , if β ≤ t−
1 + t2 + 1∧
β − α ≤ t−
1 − t−
3
{
t2 ∈ Z≥0;
t−
2 , t−
1 , t−
3 ∈ Z<0
t−
1 < m;
t−
2 < n ≤ t2;
n < k − t−
3
0 if α ≤ t−
2 + t−
3 + 1
0 if β ≤ t−
1 + t−
2 + 1
t := t2 − t−
2 , if α ≥ t2 + t−
3 + 1∧
β ≥ t2 + t−
1 + 1
α − t−
3 − t−
2 − 1, if β − α ≥ t−
1 − t−
3 ∧
α ≤ t2 + t−
3 + 1
β − t−
2 − t−
1 − 1, if β − α ≤ t−
1 − t−
3 ∧
β ≤ t2 + t−
1 + 1
{
t1, t3 ∈ Z≥0;
t−
1 , t−
2 ∈ Z<0
t−
1 < m ≤ t1;
t−
2 < n;
k − t3 ≤ m
0 if α > t1 + t3
0 if β − α < t−
2 − t3 + 1
0 if β < t−
1 + t−
2
t := t1 − t−
1 , if β ≥ t1 + t−
2 − 1∧
α ≤ t3 + t−
1 + 1
β − α − t−
2 + t3, if β ≤ t−
2 + t1 + 1∧
α > t3 + t−
1
β − t−
2 − t−
1 − 1, if β ≤ t−
2 + t1 − 1∧
α ≤ t3 + t−
1 + 1
t1 + t3 − α + 1, if β ≥ t−
2 + t1 − 1∧
α > t3 + t−
1
{
t2, t3 ∈ Z≥0;
t−
1 , t−
2 ∈ Z<0
t−
1 < m;
t−
2 < n ≤ t2;
k − t3 ≤ m
0 if β ≤ t−
1 + t−
2 + 1
0 if β − α ≤ t−
2 − t3
t := t2 − t−
2 , if β ≥ t2 + t−
1 + 1∧
β − α ≥ t2 − t3
β − t−
2 − t−
1 − 1, if β ≤ t2 + t−
1 + 1∧
α ≤ t3 + t−
1 + 1
β − α − t3 − t−
2 , if β − α ≤ t2 − t3∧
α > t3 + t−
1
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290 Combinatorics of irreducible sl(3)-modules
Conditions B[L] Dimension of V(α,β)
{
t3 ∈ Z≥0;
t−
1 , t−
2 , t−
3 ∈ Z<0
t−
1 < m;
t−
2 < n;
n + t−
3 < k ≤ m + t3
0 if α ≤ t−
3 + t−
2 + 1
0 if β − α ≤ t−
2 − t3
0 if β ≤ t−
1 + t−
2 + 1
β − α − t−
2 + t3, if 2α − β ≥ t−
3 + t3 + 1∧
α ≥ t−
1 + t3 + 1
β − t−
2 − t−
1 − 1, if α ≤ t−
1 + t3 + 1∧
β − α ≤ t−
1 − t−
3
α − t−
3 − t−
2 − 1, if 2α − β > t−
3 + t3∧
α > t−
1 + t3
{
t1, t2, t3 ∈ Z≥0;
t−
2 ∈ Z<0
m ≤ t1;
t−
2 < n ≤ t2;
k ≤ m + t3
0 if α > t1 + t3
0 if α − β > t3 − t−
2 − 1
0 if β > t2 + t1
β − α + t3 − t−
2 , if β − α ≤ t2 − t3∧
β ≤ t−
2 + t1 + 1
t1 + t2 − β + 1, if β ≥ t−
2 + t1 + 1∧
β − α ≥ t2 − t3
t1 + t3 − α + 1, if β ≥ t−
2 + t1 + 1∧
β − α ≤ t2 − t3
t := t2 − t−
2 , if β ≤ t−
2 + t1 + 1∧
β − α ≥ t2 − t3
{
t1, t2 ∈ Z≥0;
t−
2 , t−
3 ∈ Z<0
m ≤ t1;
t−
2 < n ≤ t2;
k ≤ m + t3
0 if α − β < t−
3 − t1 + 1
0 if β > t2 + t1
0 if α < t−
2 + t−
3 + 2
t1 + t2 − β + 1, if β ≥ t−
2 + t1 + 1∧
α ≥ t2 + t−
3 + 1
α − t−
2 − t−
3 − 1, if α ≤ t2 + t−
3 + 1∧
β ≤ t1 + t−
2 + 1
α − β + t1 − t−
3 , if α ≤ t2 + t−
3 + 1∧
β ≥ t1 + t−
2 + 1
t := t1 − t−
1 , if β ≤ t−
2 + t1 + 1∧
α ≥ t2 + t−
3 + 1
{
t1, t2, t3 ∈ Z≥0;
t−
1 , t−
3 ∈ Z<0
n ≤ t2;
n + t−
3 < k ≤ m + t3;
t−
1 < m ≤ t1
0 if α > t1 + t3
0 if β − α ≥ t1 − t−
3
0 if β > t1 + t2
t := t1 − t−
1 if α ≤ t3 + t−
1 + 1∧
β − α ≤ t−
1 − t−
3 ∧
β ≤ t−
1 + t2 + 1
α − β − t−
3 + t1 if 2α − β ≤ t−
3 + t3 + 1∧
β − α ≤ t−
1 − t−
3 ∧
α ≥ t−
3 + t2 + 1
t1 + t3 − α − 1 if 2α − β ≥ t−
3 + t3 + 1∧
α ≥ t3 + t−
1 + 1∧
β − α ≤ t2 − t3
t1 + t2 − β − 1 if α − β ≤ t3 − t2∧
α ≥ t2 + t−
3 + 1∧
β ≥ t−
1 + t2 + 1
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And finally we have a description of the set of tableaux that define
finite dimensional sl(3)-modules. They have to satisfies the following
conditions:
• Conditions: t1, t2, t3 ∈ Z
≥0; t−
1 , t−
2 , t−
3 ∈ Z
<0
• B[L]: {[L](m,n,k) : t−
2 < n ≤ t2; n + t−
3 < k ≤ m + t3; t−
1 < m ≤ t1}
• Weight Multiplicities:
0 if α < t−
3 + t−
2 ∧ β − α ≥ t1 − t−
3
0 if β − α < t−
2 − t3 − 1 ∧ α > t1 + t3
0 if β ≤ t−
1 + t−
2 + 1
t1 − t−
1 if β − α ≤ t−
1 − t−
3 ∧ α ≤ t−
1 + t3 + 1
t2 − t−
2 if β − α ≥ t2 − t3 ∧ α ≥ t2 + t−
3 + 1
α − β + t1 − t−
3 if 2α − β ≤ t−
3 + t3 + 1 ∧ β − α ≥ t−
1 − t−
3
∧ α ≤ t2 + t−
3 + 1 ∧ β ≥ t1 + t−
2 + 1
t1 + t3 − α + 1 if 2α − β ≥ t−
3 + t3 + 1 ∧ α ≥ t3 + t−
1 + 1
∧ α − β ≥ t3 − t2 ∧ β ≤ t−
1 + t2 + 1
t1 + t2 − β + 1 if α ≥ t−
3 + t2 + 1 ∧ β − α ≥ t2 − t3
∧ β ≥ t2 + t−
1 + 1 ∧ β ≥ t1 + t−
2 + 1
α − t−
3 − t−
2 − 1 if 2α − β ≤ t−
3 + t3 + 1 ∧ α ≤ t2 + t−
3 + 1
∧ β − α ≥ t−
1 − t−
3 ∧ β ≤ t1 + t−
2 + 1
β − α + t3 − t−
2 if 2α − β ≥ t3 + t−
3 + 1 ∧ α ≥ t−
1 + t3 + 1
∧ α − β ≥ t3 − t2 ∧ β ≤ t1 + t−
2 + 1
β − t−
1 − t−
2 − 1 if β − α ≤ t−
1 − t−
3 ∧ α ≤ t−
1 + t3 + 1
∧ β ≥ t2 + t−
1 + 1 ∧ β ≤ t1 + t−
2 + 1
As an immediate consequence of the above description we can charac-
terize irreducible modules in GTT with 1-dimensional weight spaces and
those with bounded multiplicities.
Definition 8. A weight g-module V is called pointed if dim(Vλ) = 1
for all weight λ such that dim(Vλ) 6= 0.
Corollary 11. The irreducible sl(3)-module generated by [L] is a pointed
module if and only if [L] satisfies the following conditions:
t1 = 0, t−
1 = −1 or t2 = 0, t−
2 = −1.
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292 Combinatorics of irreducible sl(3)-modules
Definition 9. A weight module V is bounded if there exist N ∈ N such
that dim(Vλ) ≤ N for all weight λ.
Corollary 12. The irreducible sl(3)-module generated by [L] is bounded
if and only if [L] satisfies the following conditions:
t1 ∈ Z
≥0; t−
1 ∈ Z
<0 or t2 ∈ Z
≥0, t−
2 ∈ Z
<0
Example 2. Let be c = −3 − π −
√
2, the following tableau satisfies t1 =
0, t2 = 0, t3 = 0; t−
1 , t−
2 , t−
3 /∈ Z. Hence we are in the case t1, t2, t3 ∈ Z
≥0.
π
√
2 c
[L]:= π
√
2
π
1) Basis: {L(m,n,k) : k ≤ m ≤ 0; n ≤ 0}
2) Weights Multiplicities:
b b b b b b
b b b b b b
b b b b b b
b b b b b b
b b b b b b
b b b b b b
1 1 1 1 1 1
2 2 2 2 2 1
3 3 3 3 2 1
4 4 4 3 2 1
5 5 4 3 2 1
6 5 4 3 2 1
E23
E32
E12E21
4. On tableaux realizations of highest weight sl(3)-modules
In this section we will discuss the tableaux realizations of highest weight
sl(3)-modules with respect to different choices of GT-subalgebras [6].
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i) Let Γ1 := Γ the standard GT-subalgebra obtained by the inclusions
with respect to the left upper corner. The formulas in this case are
given by:
h1([L]) = (2z − (x + y + 1))[L]
h2([L]) = (2(x + y + 1) − z)[L]
E12([L]) = −(x − z)(y − z)[L + δ11]
E23([L]) = (a−x)(b−x)(c−x)
(x−y) [L + δ21] − (a−y)(b−y)(c−y)
(x−y) [L + δ22]
Then, looking at the formulas, the only possible tableau that can
represent a highest weight vector is:
x y c
[T ]:= x y
x
where c = −3 − x − y and the highest weight is λ = (x − y − 1, x +
2y + 2). But in this case we can not represent highest weights with
tableau where t0([T ]) = 0 (i.e. λ = (−1, 3x + 2), x ∈ C). then we
obtain highest weight tableau for λ 6= (−1, h2) with h2 ∈ C.
ii) Let Γ2 the GT-subalgebra induced by the inclusions with respect to
the lower right corner. The GT formulas in this case are given by:
h2([L]) = (2z − (x + y + 1))[L]
h1([L]) = (2(x + y + 1) − z)[L]
E23([L]) = −(x − z)(y − z)[L + δ11]
E12([L]) = (a−x)(b−x)(c−x)
(x−y) [L + δ21] − (a−y)(b−y)(c−y)
(x−y) [L + δ22]
Then, if c := −3 − x − y; the possible highest weights vectors are
represented by the following tableau:
x y c
[T ]:= x y
x
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294 Combinatorics of irreducible sl(3)-modules
with highest weight λ = (x + 2y + 2, x − y − 1); (as in the case of Γ1
we have the restriction x 6= y; i.e. λ 6= (3x + 2, −1); x ∈ C); then
we obtain highest weight tableaux realization for λ 6= (h1, −1) with
h1 ∈ C.
iii) Let Γ3 the GT-subalgebra induced by the subalgebras inclusions:
〈E31〉 ⊂ 〈E11, E13, E31, E33〉 ⊂ gl(3)
The GT-formulas in this case are given by:
h1([L]) = [(2(x + y + 1) − z) + (2z − (x + y + 1))][L]
h2([L]) = −(2z − (x + y + 1))[L]
E12([L]) = [L − δ11]
E23([L]) =
(a − x)(b − x)(c − x)(y − z)
(x − y)
[L + δ21 + δ11]−
− (a − y)(b − y)(c − y)(x − z)
(x − y)
[L + δ22 + δ11]
Then, the possible highest weights vectors are represented by the
following tableau:
x z-1 c̃
[T1]:= x z-1
z
where c̃ = −2 − x − z and the highest weight is λ = (x + 2z, x − z).
Then we obtain highest weight tableaux for x 6= z − 1 that means
λ 6= (3z − 1, −1) with z ∈ C. Then, with Γ3 we obtain tableaux
realizations of highest weight modules such that the highest weight
satisfies λ 6= (h1, −1) with h1 ∈ C.
Proposition 13. If λ 6= (−1, −1); the irreducible highest weight sl(3)-
module with highest weight λ admits a tableaux realization with respect
to some GT-subalgebra.
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L. E. Ramirez 295
5. Harish Chandra sl(3)-modules in GTT
Let B a Chevalley basis for sl(3) given by:
Xα := E12 Yα := E21 Hα := E11 − H22 Xα+β := E13
Xβ := E23 Yβ := E32 Hβ := E22 − E33 Yα+β := E31
and set g̃ the Lie subalgebra 〈Xα, Yα, Hα〉 ∼= sl(2).
Definition 10. An sl(3)-module V is called left (respectively right)
Harish-Chandra module if can be expressed as a sum of lowest weight
(respectively highest weight) sl(2)-modules.
Definition 11. An sl(3)-module V is called Harish-Chandra mod-
ule if can be expressed as a sum of finite dimensional sl(2)-modules.
Equivalently; if the module is a left and right Harish-Chandra module.
Lemma 14. Let V be an irreducible sl(3)-module and 0 6= v ∈ V . If
there exists n ∈ Z
≥0 (respectively n ∈ Z
<0) such that Xn
αv = 0 then, for
all u ∈ V there exist r = r(u) ∈ Z
≥0 (respectively r ∈ Z
<0) such that
Xr
αu = 0.
Proof:. As V is irreducible, each u ∈ V can be expressed as u =
∑
k akv
where ak are elements of U(sl(3)). Then the statement of lemma is a
consequence of the fact that for all N ∈ Z we have:
XN
α Xβ = NXα+βXN−1
α + XβXN
α , XN
α Hα = HαXN
α − 2NXN
α
XN
α Yα+β = Yα+βXN
α − 2YβXN−1
α , XN
α Hβ = HβXN
α + NXN
α
XαYα = YαXN
α + NHαXN−1
α − 2NXN−1
α .
Corollary 15. An irreducible sl(3)-module V is a Harish-Chandra mod-
ule (with respect to g̃) if and only if there exist 0 6= v ∈ V and n ∈ N
such that X±n
α v = 0.
As a consequence of the description of bases for irreducible sl(3)-
modules in GTT we have the following corollaries:
Corollary 16. The irreducible sl(3)-module generated by [L] is a left
(respectively right) Harish-Chandra module (with respect to g̃) if and
only if
t3 ∈ Z
≥0 (respectively t−
3 ∈ Z
<0)
Corollary 17. The irreducible sl(3)-module generated by [L] is a Harish-
Chandra module (with respect to g̃) if and only if at least the conditions
holds:
t3 ∈ Z
≥0, t−
3 ∈ Z
<0
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296 Combinatorics of irreducible sl(3)-modules
Acknowledgments
I would like to thank Vyacheslav Futorny for stimulating discussions
and patience during the preparation of this paper. Also I would like thank
Volodymyr Mazorchuk for his attention and helpful suggestions.
References
[1] Drozd Yu.A., Ovsienko S.A., Futorny V.M.,Irreducible Weighted sl(3)-Modules.
Funksionalnyi Analiz i Ego Prilozheniya, 23 (1989), 57-58.
[2] Drozd Yu.A., Ovsienko S.A., Futorny V.M., Gelfand-Tsetlin Modules Over Lie
Algebra sl(3). Contemporary Mathematics, 131 (1992) 23-29.
[3] Drozd Yu.A., Ovsienko S.A., Futorny V.M., Harish-Chandra subalgebras and
Gelfand-Zetlin modules, Math. and Phys. Sci., 424 (1994), 72-89.
[4] S. Fernando; Lie Algebra Modules with finite dimensional weight spaces I. Trans.
Amer. Math. Soc. 322 (1990), 757-781.
[5] V. Futorny; A Generalization of Verma Modules, and Irreducible Representations
of the Lie Algebra sl(3). Ukrainskii Matematicheskii Zhurnal, Vol. 38, No. 4, pp.
492-497, July-August, 1986.
[6] V. Futorny, S. Ovsienko, M. Saorin; Gelfand-Tsetlin categories. Contemporary
Mathematics - American Mathematical Society (Print) 537 (2011), 193-203,
[7] I.M Gelfand, M.L. Tsetlin, Finite-dimensional representations of the group of
unimodular matrices, Doklady Akad. Nauk SSSR (N.s.), 71 (1950), 825-828.
[8] V. Mazorchuk; Lectures on sl(2)-modules. Imperial College Press, London, 2010.
[9] V. Mazorchuk; Tableaux Realization of Generalized Verma Modules. Can. J. Math.
Vol. 50(4) (1998), 816-828.
[10] V. Mazorchuk; On Categories Of Gelfand-Tsetlin Modules. Noncommutative
Structures in Mathematics and Physics, (2001), 299-307.
[11] A.I. Molev; Gelfand-Tsetlin Bases for Classical Lie Algebras. Handbook of Algebra",
Vol. 4, (M. Hazewinkel, Ed.), Elsevier, 2006, pp. 109-170.
Contact information
L. E. Ramirez Institute of Mathematics and Statistics,
University of Sao Paulo, Sao Paulo, Brazil
E-Mail: luisenrique317@gmail.com
Received by the editors: 14.02.2012
and in final form 14.02.2012.
|
| id | nasplib_isofts_kiev_ua-123456789-152244 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T16:29:56Z |
| publishDate | 2012 |
| publisher | Інститут прикладної математики і механіки НАН України |
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| spelling | Ramirez, L.E. 2019-06-09T06:11:38Z 2019-06-09T06:11:38Z 2012 Combinatorics of irreducible Gelfand-Tsetlin sl(3)-modules / L.E. Ramirez // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 276–296. — Бібліогр.: 11 назв. — англ. 1726-3255 2010 MSC:17B35, 17B37, 17B67, 16D60, 16D90, 16D70, 81R10. https://nasplib.isofts.kiev.ua/handle/123456789/152244 In this paper we present an explicit description of all irreducible sl(3)-modules which admit a Gelfand-Tsetlin tableaux realization with respect to the standard Gelfand-Tsetlin subalgebra. The author is supported by the CNPq grant (processo 142407/2009-7). I would like to thank Vyacheslav Futorny for stimulating discussions and patience during the preparation of this paper. Also I would like thank Volodymyr Mazorchuk for his attention and helpful suggestions. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Combinatorics of irreducible Gelfand-Tsetlin sl(3)-modules Article published earlier |
| spellingShingle | Combinatorics of irreducible Gelfand-Tsetlin sl(3)-modules Ramirez, L.E. |
| title | Combinatorics of irreducible Gelfand-Tsetlin sl(3)-modules |
| title_full | Combinatorics of irreducible Gelfand-Tsetlin sl(3)-modules |
| title_fullStr | Combinatorics of irreducible Gelfand-Tsetlin sl(3)-modules |
| title_full_unstemmed | Combinatorics of irreducible Gelfand-Tsetlin sl(3)-modules |
| title_short | Combinatorics of irreducible Gelfand-Tsetlin sl(3)-modules |
| title_sort | combinatorics of irreducible gelfand-tsetlin sl(3)-modules |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/152244 |
| work_keys_str_mv | AT ramirezle combinatoricsofirreduciblegelfandtsetlinsl3modules |