EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6458
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Gelfand-Tsetlin modules for Lie algebras of rank 2

Milica Andelić,1,∗, Carlos M. da Fonseca2,3, Vyacheslav Futorny4, Andrew Tsylke5

1 Department of Mathematics, Kuwait University, Al-Shadadiyah, Kuwait
2 Kuwait College of Science and Technology, Doha District, Safat 13133, Kuwait
3 Faculty of Applied Mathematics and Informatics, Technical University of Sofia, Kliment
Ohridski Blvd. 8, 1000 Sofia, Bulgaria
4 Shenzhen International Center for Mathematics, Southern University of Science and
Technology, China
5 Kyiv Taras Shevchenko University, Kyiv, Ukraine

Abstract. We explicitly construct families of simple modules for all simple Lie algebras of rank 2 on
which a certain commutative subalgebra acts diagonally with a simple spectrum. In type A, these
modules are the well-known generic Gelfand-Tsetlin modules.

2020 Mathematics Subject Classifications: 17B10, 16G99
Key Words and Phrases: Gelfand-Tsetlin module, Gelfand-Tsetlin basis, Lie algebras

1. Introduction

Let g be a simple finite-dimensional simple Lie algebra over the complex numbers and
let h be a fixed Cartan subalgebra of g. A g-module M is weight (with respect to h) if h is
diagonalizable on M , that is

M =
⊕
λ∈h∗

Mλ ,

where hv = λ(h)v, for any v ∈ Mλ and h ∈ h. The subspace Mλ is called a weight subspace
of weight λ, if Mλ ̸= 0.
Simple weight modules were studied extensively in the last 50 years. Classical results of

Fernando [1] and Mathieu [2] provided a complete classification of simple weight modules
with finite-dimensional weight subspaces. On the other hand, the classification of simple
weight modules with infinite-dimensional weight subspaces is still an open problem. The
most progress has been achieved in the case of Lie algebras of type A, where simple Gelfand-
Tsetlin modules were classified (see [3–6] and references therein). These are weight modules
with diagonalizable action of a certain commutative subalgebra of the universal enveloping
algebra U(g), called the Gelfand-Tsetlin subalgebra. Generically, such simple Gelfand-Tsetlin
modules have infinite-dimensional weight subspaces. In particular, in the case of sl(2) we
obtain in this way all simple weight modules. They depend on two parameters and have

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6458

Email addresses: milica.andelic@ku.edu.kw (M. Andelić),
c.dafonseca@kcst.edu.kw, carlos.fonseca@tu-sofia.bg (C.M. da Fonseca),
vfutorny@gmail.com (V. Futorny), andrew4tsylke@gmail.com (A. Tsylke)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



M. Andelić et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6458 2 of 21

1-dimensional weight subspaces. In the case of g = sl(3) a complete description of simple
Gelfand-Tsetlin modules was given in [7].
The original approach to the study of weight modules was based on the reduction to the

study of simple modules over the centralizer U0(g) of the Cartan subalgebra h in the universal
enveloping algebra U(g): ifM is a simple weight g-module, thenMλ is a simple U0(g)-module,
for any weight λ of M . Hence, every simple weight g-module corresponds to a simple (not
unique) U0(g)-module and, in its turn, any simple U0(g)-module corresponds to a unique
simple weight g-module. This approach was successfully used in the case of g = sl(3) [8–15],
etc.
As the structure of U0(g) is rather complicated (there are two commuting generators for

sl(2), and there are six generators for sl(3) with three polynomial relations and five commuting
generators among them), there were essentially no attempts beyond the sl(3) case.
The goal of the paper is to revise the centralizer approach and to construct new simple

weight modules with infinite-dimensional weight subspaces for all simple Lie algebras of rank
2. We explicitly construct simple generic modules in the category of Γ-pointed modules for
a commutative subalgebra Γ of the centralizer U0(g). In type A, Γ-pointed modules are the
celebrated Gelfand-Tsetlin modules, and similar constructions can be viewed analogously in
other types.
The structure of the paper is the following. In Section 2 we discuss the structure of the

centralizer U0(g) of the Cartan subalgebra h in the universal enveloping algebra U(g), prove
that U0(g) is finitely generated and finitely presented. We give a generating set of elements
and describe an algorithm for computing all relations between them. In Section 4 we consider
the Lie algebra of type A2. Our approach is a suitable modification of [9] and [14], and we
recover a construction of generic torsion free A2-modules with infinite-dimensional weight
spaces obtained in [14] and [7]. They are tame Gelfand-Tsetlin modules with diagonalizable
action of the Gelfand-Tsetlin subalgebra. In Section 5 we consider the Lie algebra g of type
C2 and give the generators and the defining relations of the centralizer U0(g). We construct
two 4-parameter families of simple torsion free C2-modules with infinite-dimensional weight
spaces. These modules are Γ-pointed, where Γ is the 4-generated Gelfand-Tsetlin subalgebra
of U0(g) which has a simple spectrum on such representations. Finally, in Section 6 we
construct a 3-parameter family of simple torsion free G2-modules with infinite-dimensional
weight spaces. These modules are Γ-pointed with respect to a 4-generated Gelfand-Tsetlin
subalgebra Γ of U0(g) which has a simple spectrum on such representations.
We hope to use the defined representations to construct new simple modules for all simple

finite-dimensional and Affine Lie algebras via the parabolic induction.

2. Cartan centralizers

Let ∆ = {α1, . . . , αk1} be the root system of (g, h), and let π = {β1, . . . , βk0} be a
basis of ∆. With respect to the basis π, we have the decomposition of ∆ into positive
and negative roots: ∆ = ∆+ ∪ ∆−. Let W be the Weyl group of the root system ∆.
Choose a basis G = G0 ∪ G1 of the Lie algebra g, where G0 = {hβ ∈ h |β ∈ π} and
G1 = {eα ∈ gα \ {0} |α ∈ ∆}. Set fα = e−α.
Denote by U0(g) the centralizer of the Cartan subalgebra h in the universal enveloping

algebra U(g). For each i ∈ N denote by U (i)(g) the vector subspace of U(g) spanned by the
monomials x1x2 · · ·xj , where x1, . . . , xj ∈ G and j ≤ i. Then we get an increasing sequence



M. Andelić et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6458 3 of 21

of subspaces
U (1)(g) ⊂ U (2)(g) ⊂ · · · ⊂ U (i)(g) ⊂ · · · ,

which defines a canonical filtration of U(g). The canonical filtration of U0(g) is the sequence
of subspaces

U
(1)
0 (g) ⊂ U

(2)
0 (g) ⊂ · · · ⊂ U

(i)
0 (g) ⊂ · · · ,

where U (i)
0 (g) = U (i)(g) ∩ U0(g).

For any monomial X = x1x2 · · ·xj denote by T0(X) the set of monomials obtained from
X by permuting the variables xi. We will treat each monomial as an element of the algebra
U(g), considering it as a monomial with coefficient 1. Define the degree function deg(y) as
follows: deg(y) = i if y ∈ U (i)(g), but y /∈ U (i−1)(g).
Now, fix some order on the set G:

x1 ≤ x2 ≤ · · · ≤ xk, (2.1)

where k is the dimension of g. Define standard monomials of U(g) with respect to this order
as follows:

XS = xs11 x
s2
2 · · ·xskk .

where S = (s1, . . . , sk) is a k-tuple of nonnegative integers, with at least one si ̸= 0.
For every monomial X define a lexicographical order on the set T0(X) with respect to

the order (2.1): if X1 = xi1xi2 · · ·xin and X2 = xj1xj2 · · ·xjn , with X1, X2 ∈ T0(X), then
X1 < X2 if either xi1 < xj1 or there exists an index 1 < s ≤ n such that xis < xjs and
xit = xjt , for 1 ≤ t < s.
Denote by P the set of all standard monomials and by P (i) the set of all standard monomi-

als of degree i. Set P0 = P∩U0(g) and P
(i)
0 = P (i)∩U0(g). In particular, T0(P ) =

⋃
X∈P T0(X)

will denote the set of all monomials in the algebra U(g).
The following lemma is an immediate consequence of the PBW theorem.

Lemma 2.1.

1. The set of the following standard monomials P̄ (i) = P (1)∪P (2)∪ · · ·∪P (i) forms a basis
for the vector space U (i)(g).

2. The set of the standard monomials P̄ (i)
0 = P

(1)
0 ∪ P (2)

0 ∪ · · · ∪ P (i)
0 forms a basis for the

vector space U (i)
0 (g).

3. If a ∈ U (i)(g) and b ∈ U (j)(g), then ab ∈ U (i+j)(g) and ab− ba ∈ U (i+j−1)(g).

4. If X ∈ P
(i)
0 and X1, X2 ∈ T0(X), then X1 −X2 ∈ U

(i−1)
0 (g).

5. For every monomial X ∈ T0(P ), the minimal element of the set T0(X) with respect to
the above lexicographical order is a standard monomial.

6. Let P ′ be a set of monomials of degree ≤ i, such that for every monomial X ∈ P̄ (i),
there exist a unique X ′ ∈ P ′ satisfying T0(X) = T0(X

′). Then P ′ is a basis of U (i)(g)

and P ′
0 = P ′ ∩ U (i)

0 (g) is a basis of U (i)
0 (g).



M. Andelić et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6458 4 of 21

Denote by P̂ (i)
0 ⊂ P

(i)
0 the subset of monomials of degree i generated by G1, and set

P̂0 =
⋃

i P̂
(i)
0 .

LetX ∈ T0(P ). ThenX = hβ1 · · ·hβmeα1 · · · eαn , for some α1, . . . , αn ∈ ∆, β1, . . . , βm ∈ π.
Define two lists of roots associated with X as follows: L0(X) = (β1, . . . , βm) and L1(X) =
(α1, . . . , αm). Extend this definition to the set of all monomials as follows: if X ′ ∈ T0(X),
then L0(X

′) = L0(X), L1(X
′) = L1(X).

For any monomial X ∈ T0(P ), the sum of all roots in the list L1(X) is called the weight
of the list L1(X). Clearly, the weight of the list L1(X) is zero if and only if X ∈ T0(P0).
Denote by B(∆) the set of all zero-weight lists of roots. Furthermore, if L0(X) = L0(Y ) and
L1(X) = L1(Y ), for some monomials X,Y , then T0(X) = T0(Y ).
A list L1(X) for X ∈ P̄0 is called decomposable if there exist X1, X2 ∈ P̄0 such that

L1(X) = L1(X1) ⊔ L1(X2) (disjoint union of two lists). In this case, it follows that T0(X) =
T0(X1X2). Conversely, if no such decomposition exists, the list is called indecompossable.
Note that decompositions are not unique for certain monomials. For example, in the algebra
A2 (see Section 4.1), (α1, . . . , α6) = (α1, α6) ⊔ (α2, α5) ⊔ (α3, α4) = (α1, α2, α4) ⊔ (α3, α5, α6).
Denote by B1(∆) ⊂ B(∆) the set of all zero-weight indecomposable lists. Define the

action of the Weyl group W on the list of roots a = (α1, α2, . . . , αn) as follows:

w(a) = (wα1, wα2, . . . , wαn) , with ∈W .

Clearly, if a is indecomposable, then w(a) is also indecomposable. Moreover, if a has a
zero weight, then w(a) also has a zero weight. The simplest examples of indecomposable lists
are those containing only one positive or only one negative root.
Define the set of primitive lists B2(∆) ⊆ B1(∆) as follows: r ∈ B2(∆) if there exists

w ∈W such that the list w(r) contains only one negative or only one positive root.
Let Mk = {(n1, n2, . . . , nk) |ni ∈ N} be the set of vectors with nonnegative integer coor-

dinates. Define a partial order on the set Mk as follows:

(n1, n2, . . . , nk) ≤ (m1,m2, . . . ,mk) if ni ≤ mi , for all i = 1, . . . , k.

We will be using the following result [16, Lemma 2.6.2].

Lemma 2.2. If Sk ⊂ Mk is any infinite subset, then there exist two elements r1, r2 ∈ Sk
such that r1 ≤ r2.

We have the following properties of indecomposable lists of roots.

Lemma 2.3. Let g be a simple finite-dimensional Lie algebra with root system ∆. Then

1. The set of indecomposable lists B1(∆) is finite.

2. If g ∈ {A2, C2, G2}, then all indecomposable lists are primitive and hence B1(∆) =
B2(∆).

Proof. Define the function σ on the setB(∆) as follows: σ(α1, α2, . . . , αm) = (n1, n2, . . . , n|∆|),
where ni is the number of occurrences of the i-th root of ∆ (with respect to the ordering in
(2.1) in the list (α1, α2, . . . , αm). It is clear that a list r1 is a sublist of a list r2 if and only if
σ(r1) ≤ σ(r2). Now, the proof of the first statement follows from the previous lemma.
The second statement for the case A2 is obvious. We will now prove the second state-

ment for the case C2. Let π = {β1, β2} be a basis of the root system. For convenience,



M. Andelić et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6458 5 of 21

we will represent the roots of ∆ as vectors in this basis: (i, j) = iβ1 + jβ2. Then ∆ =
{(1, 0), (0, 1), (1, 1), (2, 1), (−1, 0), (0,−1), (−1,−1), (−2,−1)}. Let v = (i, j) ∈ ∆ be any root.
The action of the Weyl group is given by w1(v) = (2j − i, j) and w2(v) = (i, i− j), where w1

and w2 are simple reflections.
We will prove that non-primitive indecomposable lists do not exist for C2. By the definition

of a primitive list, any list consisting of two or three roots is primitive. Suppose there exists
a non-primitive indecomposable list r with more than three roots. First, observe that if
(2, 1), (−2,−1) /∈ r, then r cannot contain more than three roots.
Consider the case (2, 1) ∈ r. (The case (−2,−1) ∈ r is miror of first case.) Then necessarily

(−2,−1) /∈ r. Consider the list w1(r). Then w1(2, 1) = (0, 1) ∈ w1(r). First, suppose
(2, 1), (−2,−1) /∈ w1(r). In this case, w1(r) is primitive, and consequently, so is r. Second, if
(2, 1) ∈ w1(r), then both (2, 1) and (0, 1) belong to w1(r). Finally, if (−2,−1) ∈ w1(r), then
we obtain (2, 1), (0, 1) ∈ w1w2(r).
Now, suppose there exists w ∈W such that (2, 1), (0, 1) ∈ r′ = w(r). The possible negative

roots in the list r′ are (−1,−1) and (−1, 0). Since the second coordinate of the sum of all
positive roots is at least two, r′ must contain at least two occurrences of (−1,−1). Thus we
have r′ = {(2, 1), (0, 1), (−1,−1), (−1,−1)}. Now, applying w2, we obtain w2(r

′) =
= {(2, 1), (0,−1), (−1, 0), (−1, 0)}, which is primitive. Consequently r is also primitive.
The proof of the second statement for G2 is similar to the case of C2. For the sets of all

primitive lists in all three cases, see Sections 4-6. □

Define the following order on the set of roots ∆, derived from (2.1). For α1, α2 ∈ ∆

α1 ≤ α2 if and only if eα1 ≤ eα2 . (2.2)

A monomial X is called perfect if either X = hβ ∈ G0, or X = eα1 · eα2 · · · · · eαn and
the associated list L1(X) = (α1, α2, . . . , αn) is both indecomposable and ordered according
to (2.2), i.e., α1 ≤ α2 ≤ · · · ≤ αn.
Since the set of all indecomposable lists is finite, the set of all perfect monomials is also

finite. Let I(g) = {p1, . . . , pq} denote the set of all perfect monomials with degree greater
than one. Then I(g) ∪G0 is a generating set of U0(g) consisting of all perfect monomials.
Denote by d0 the maximal degree of perfect monomials in I(g).
Let J(g) = {c1, . . . , cq} be the set of new indeterminates, ri = ci − pi, i = 1, . . . , q and

D = ⟨r1, . . . , rq⟩ the 2-sided ideal of U0(g)[c1, . . . , cq] generated by these elements. Define

Û0(g) = U0(g)[c1, . . . , cq]/D.

Clearly, Û0(g) ∼= U0(g) and Û0(g) is generated by J(g) ∪ G0 with the ideal of relations K
inherited from U0(g). Define the degrees of new indeterminates ci by: deg(ci) = deg(pi).

Denote by Q0(g) the set of all monomials generated by J(g) ∪ G0 and set Q
(i)
0 (g) =

Q0(g) ∩ U (i)
0 (g). Define the function from Q0(g) to I(g) ∪G0 as follows:

ω : Q0(g) → I(g) ∪G0, ω(ci) = pi, ω(h) = h, h ∈ G0.

Let us fix some order on the set J(g) ∪G0:

x1 ≤ x2 ≤ · · · ≤ xq+k0 , xi ∈ J(g) ∪G0, i = 1, . . . , q + k0 . (2.3)



M. Andelić et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6458 6 of 21

For any monomial Y ∈ Q0(g), denote by T1(Y ) the set of all monomials X ∈ Q0(g) such
that T0(ω(Y )) = T0(ω(X)). Note that all monomials in the sets T0(ω(Y )) and T1(Y ) have the
same degree. Extend function L1 on the set J(g) as follows: L1(x) = L1(ω(x)), x ∈ Q0(g).
Define a lexicographical order on the set T1(Y ) with respect to the order given in (2.3) as

follows: if Y1 = a1a2 · · · an, Y2 = b1b2 · · · bm, for ai, bj ∈ J0(g) ∪G0 and Y1, Y2 ∈ T1(Y ), then
Y1 < Y2 if there exists an index 1 ≤ s ≤ min(n,m), such that as < bs and at = bt, for all
0 < t < s. Note that neither Y1 nor Y2 can be a prefix of the other, which ensure that our
definition is well-defined and that any two distinct elements are comparable.
A monomial Y ∈ Q0(g) is called semi-perfect if it is the minimal element of the set T1(Y )

under this lexicographical order. Denote by S(g) the set of all semi-perfect monomials, and

set S(i)(g) = S(g) ∩ Û (i)
0 (g).

Lemma 2.4.

1. For any i ≥ 1 and X ∈ P
(i)
0 there exists a unique semi-perfect monomial Y ∈ S(i)(g)

such that T0(X) = T0(ω(Y )).

2. Let X = a1a2 · · · am with ai ∈ J0(g) ∪ G0, be a semi-perfect monomial. Then, for any
s ∈ {1, . . . ,m}, the monomial X ′ = a1 · · · as−1as+1 · · · am is also semi-perfect.

3. The set of semi-perfect monomials S(i)(g) is a basis of the vector space Û (i)
0 (g).

Proof. The one-to-one correspondence between standard monomials in U (i)
0 (g) and semi-

perfect monomials in Û (i)
0 (g) follows from Lemma 2.1 and the definition of semi-perfect mono-

mials. Since Û (i)
0 (g) is a finite-dimensional vector space and the set of semi-perfect monomials

S(i)(g) satisfy the condition for the set P ′ in the part 6 of the Lemma 2.1, the lemma follows.
□

Since the set of semi-perfect monomials S(i)(g) forms a basis of the algebra U (i)
0 (g), any

element x ∈ Û0(g) can be expressed as a linear combination of semi-perfect elements. We
need to determine how to multiply any two semi-perfect monomials and express their product
as a linear combination of elements of S(g).
If an element X ∈ Û0(g) is expressed as a linear combination of semi-perfect monomials,

we say that X is in a normal form, denoted by N (X). The process of converting a given
element into its normal form is called normalization.
Define the set of relations as follows:

K ′ = {ci1ci2 · · · cim −N (ci1ci2 · · · cim) |m ≤ d0, ci ∈ J(g)}. (2.4)

Additionally, define the length function on K ′ ⊆ K as: Len(ci1 · · · cim −N (ci1 · · · cim)) = m.
Thus, K ′ is the set of all relations with length less or equal to d0. We are now ready to

state the main result regarding the Cartan centralizers of the universal enveloping algebras.

Theorem 2.5. The ideal of relations K is generated by the set K ′.

Proof. We prove that the product of semi-perfect elements can be expressed as a linear
combination of semi-perfect elements using only the relations K ′. This follows from the
reduction algorithm described below.
We will show that any monomial, written as product of perfect monomials in arbitrary

order, can be transformed into a linear combination of semi-perfect monomial.



M. Andelić et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6458 7 of 21

Let X = a1a2 · · · am ∈ Q
(i)
0 (g) be a monomial of degree i that is a product of perfect

elements. By Lemmas 2.1 and 2.4, there exists a semi-perfect element Y = b1b2 · · · bn ∈
Q

(i)
0 (g), such that T1(X) = T1(Y ) and X − Y ∈ Û

(i−1)
0 (g).

The proof proceeds by induction on the degree of the monomials. The statement is obvious
for the case i = 1. Assume that the statement holds for all elements in Û (i−1)

0 (g), that is, any
such element can be transformed into the normal form.
We will show that, using only relations K ′, X can be transformed into a sum X = Y +Y1,

where Y is above semi-perfect monomial and Y1 ∈ Û
(i−1)
0 (g). Denote Y ′ = b2 · · · bn. There

are two cases to consider.

1. Let b1 ∈ G0. Since L0(X) = L0(Y ), by Lemma 2.4, it follows that b1 = as for
some 1 ≤ s ≤ m. As b1 belongs to the center of Û0(g), we can rewrite X as X =
b1a1a2 · · · as−1as+1 · · · am = b1X

′. Thus, we obtain

X = Y + b1(X
′ − Y ′) .

It is easy to see that T1(X ′) = T1(Y
′), and Y ′ is semi-perfect. By the induction

hypothesis, we can apply our algorithm to the pair X ′, Y ′. This gives (X ′−Y ′) = X ′′ ∈
Û

(i−2)
0 , implying that b1X ′′ ∈ Û

(i−1)
0 . Therefore, the statement follows by the induction

hypothesis.

2. Let b1 ∈ J0(g). Since L1(X) = L1(Y ), it follows that L1(b1) is a sublist of L1(X).
Let {z1, z2, . . . , zt} ⊂ {a1, a2, . . . , am} be a minimal set of perfect monomials such that
L1(b1) ⊏ L1(z1z2 · · · zt). Set Z = z1z2 · · · zt. Since the cardinality of any indecomposable
list is bounded by d0, we have t ≤ d0.

Using the commutation relations for two perfect elements we can transform X into the
following form: X = ZX1 +X2, where T1(X) = T1(ZX1) and X2 ∈ Û

(i−1)
0 (g).

Denote n = deg(Z), then deg(X1) = i − n. There exists a semi-perfect monomial Z1

such that T1(Z) = T1(Z1). Since t ≤ d0, we can apply the normalization process to the

monomial Z. We obtain Z = Z1 + Z2, where Z2 ∈ Û
(n−1)
0 (g).

Since Z1 is a semi-perfect monomial, the first perfect element z′1 in its decomposition
Z1 = z′1z

′
2 · · · z′s = z′1Z

′
1 is the minimal perfect element with respect to the order (2.3).

This means that for every perfect monomial x such that L1(x) ⊑ L1(Z1), we have
z′1 ≤ x. Since L1(b1), L1(z1) ⊑ L1(Z) ⊑ L1(Y ), it follows that z′1 = b1.

Now we have

X = ZX1 +X2 = (Z1 + Z2)X1 +X2 = (b1Z
′
1 + Z2)X1 +X2

= Y − b1Y
′ + (b1Z

′
1 + Z2)X1 +X2 = Y + b1(Z

′
1X1 − Y ′) + Z2X1 +X2 .

It is easy to see that the pair of monomials X3 = Z ′
1X1 and Y ′ satisfies the initial

conditions of our lemma, with degree of the monomials less than i.

Since deg(Y ′) = i − k, T1(X3) = T1(Y
′) and Y ′ is semi-perfect, we can apply, by the

induction hypothesis, our algorithm to the pair X3, Y
′

This gives X3 = Y ′ +X4, where X4 ∈ Û
(i−k−1)
0 (g). Finally, we have X = Y +Y1, where

Y1 = b1X4 + Z2X1 +X2 ∈ Û
(i−1)
0 (g).



M. Andelić et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6458 8 of 21

The statement follows by the induction hypothesis. Note that we used relations K ′

twice: for commuting perfect elements and for normalizing Z.

□

Let A be an associative algebra defined by a set of generators S and a set of relation R
(as polynomials in the generators of S). The set of generators S can be partitioned into three
subsets, S = S1∪S2∪S3, where S1 generates the center of the algebra A and S1∪S2 generates
a commutative subalgebra of A.
The width of a monomialX, denoted by width(X), is defined as the number of occurrences

of variables from the set S3 in the monomial. The width of a polynomial P is then defined
as the maximal width among all monomials that make up the polynomial.
We can decompose the set of relation R into a union of subsets as follows: R = R0∪R1∪· · · ,

where Ri = {Y ∈ R | width(Y ) = i}, for i = 0, 1, 2, . . ..
Note that this decomposition is not unique and may vary depending on the choice of

generators S and the subset S2. Our goal will be to identify the ”best” set of generators S
and ”best” decomposition, such that the cardinality of the set S3 is minimal.

3. Category of Γ-pointed modules

Let Γ be a commutative subalgebra of U0(g) such that h ⊆ Γ ⊂ U0(g). By Hom(Γ,C) we
denote the set of all characters of Γ, that is the set of all C-algebra homomorphisms from Γ
to C.
Let M be a Γ-module. For each χ ∈ Hom(Γ,C) we set

Mχ = {v ∈M ; av = χ(a)v , ∀a ∈ Γ},

and call it the Γ-weight space of M with weight χ. When Mχ ̸= {0}, we say that χ is a
Γ-weight of M and the elements of Mχ are called Γ-weight vectors of weight χ. If a Γ-module
M satisfies

M =
⊕

χ∈Hom(Γ,C)

Mχ ,

then we call M a Γ-weight module. The dimension of the vector space Mχ ̸= 0 will be called
the Γ-multiplicity of χ in M . Module is called Γ-pointed if Γ-multiplicity of any character
χ equals 1, that is Γ separates the basis elements of M . In particular, M is a tame module
with diagonalizable action of Γ. A weight module M is torsion free provided all root vectors
of g act injectively on M .
In particular, if Γ = U(h), then Γ-weight module is a classical weight module.
Suppose g is of type A and Γ is a Gelfand-Tsetlin subalgebra of g [17]. Then Γ ⊂ U0(g)

and every generic Gelfand-Tsetlin g-module is Γ-pointed. We refer to [17] for details. Clearly,
every finite-dimensional g-module is also Γ-pointed. A family of simple Γ-pointed modules in
type A was studied in [18]. On the other hand, there exist Gelfand-Tsetlin modules which
are not pointed.

4. Construction of simple weight A2-modules

In this section we consider the Lie algebra g = sl(3). Even though this case is well
understood we give some details to illustrate our approach.



M. Andelić et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6458 9 of 21

4.1. Centralizer of the Cartan subalgebra of A2

Let ∆ = {α1, α2, α3 = α1+α2, α4 = −α1−α2, α5 = −α2, α6 = −α1} be the root system of
g. Fix a Chevalley basis G: e10 = E12, f10 = E21, e01 = E23, f01 = E32, e11 = E13, f11 = E31,
h10 = E11 − E22, h01 = E22 − E33.
Let us define the following order on the elements of G:

h01 < h10 < f01 < f10 < f11 < e11 < e10 < e01.

We have the following set of indecomposable lists of roots: {α1, α6}, {α2, α5}, {α3, α4},
{α1, α2, α4}, {α3, α5, α6}, and the following set of perfect monomials:

h1 = h01, h2 = h10, c1 = f01e01, c2 = f10e10, c3 = f11e11, c4 = f11e10e01, c5 = f01f10e11.

Define the order on the set of perfect monomials: h1 < h2 < c1 < c2 < c3 < c4 < c5.
Note that any monomial containing both variables c4 and c5 is not semi-perfect since

m′ = c1c2c3 has the same associated list of roots as m = c4c5, but c1 < c4. Applying the
relations between the generators we easily obtain the following statement.

Proposition 4.1.

• The following set of monomials is a basis of Û0(g):

P = {hs11 h
s2
2 c

s3
1 c

s4
2 c

s5
3 c

s6
4 | s1, . . . , s6 ∈ N} ∪ {hs11 h

s2
2 c

s3
1 c

s4
2 c

s5
3 c

s6
5 | s1, . . . , s6 ∈ N}.

• The set K̃ = {X |X ∈ K ′,Len(X) = 2} is a generating set of the ideal of relations K.

The following K̃ is the list of all relations of length two:

cjhi = hicj , i = 1, 2, j = 1, . . . , 5 (4.1)

c2c1 = −c5 + c4 + c1c2, (4.2)

c3c1 = c5 − c4 + c1c3, (4.3)

c4c1 = −2c5 + (2 + h1)c4 + c1c4 − c1c3 + c1c2, (4.4)

c5c1 = −h1c5 + c1c5 + c1c3 − c1c2, (4.5)

c3c2 = −c5 + c4 + c2c3, (4.6)

c4c2 = h2c3 + h2c4 + c2c4 + c2c3 − c1c2, (4.7)

c5c2 = −h2c3 − h2c5 + c2c5 − c2c3 + c1c2, (4.8)

c4c3 = 2c5 − (h2 + h1 + 2)c4 + c3c4 − h2c3 − c2c3 + c1c3, (4.9)

c5c3 = (h2 + h1)c5 + c3c5 + h2c3 + c2c3 − c1c3, (4.10)

c5c4 = −(2h2 + h1)c5 − c3c5 − 2h2c3 + c2c5 − 2c2c3 − c1c5 + c1c2c3 + (h2 + h1 + 2)c1c2,
(4.11)

c4c5 = −h1c5 − c3c5 + h1h2c3 + c2c5 + h1c2c3 − c1c5 + h2c1c3 + c1c2c3. (4.12)

The following Casimir elements generate the center of the universal enveloping algebra
[14]:

z1 = c3 + c2 + c1 +
1

3
(h22 + 3h2 + h21 + 3h1 + h2h1) (4.13)



M. Andelić et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6458 10 of 21

z2 = c5 + c4 +
1

3
(h1 − h2)c3 −

1

3
(6 + 2h1 + h2)c2 +

1

3
(h1 + 2h2)c1

+
1

27
(−h2 − 3 + h1)(6 + 2h1 + h2)(h1 + 2h2) (4.14)

Using the relations (4.1)-(4.14), we can reduce the number of generators of U0(g): applying
(4.2), (4.13), and (4.14) we can exclude c3, c4, c5 from all other relations. As a result we will
get a generating set S = {h1, h2, z1, z2, c1, c2} of U0(g) with the following decomposition:
S1 = {h1, h2, z1, z2}, S2 = {c1}, S3 = {c2}. For the set of relations we have R = R1 ∪ R2,
where R1 consist of the relations obtained from (4.1)-(4.5), and R2 consists of the relations
obtained from (4.6)-(4.12).

4.2. Generic torsion free A2-modules

Let Γ be the commutative subalgebra of U0(g) generated by the elements h1, h2, z1, z2, c1.
This is a Gelfand-Tsetlin subalgebra of sl(3). We give a construction of a family of Γ-pointed
modules V (a1, a2, a3, ξ, µ) which depend on five complex parameters with the restriction a3 /∈
Z. These are essentially the universal generic Gelfand-Tsetlin modules initially constructed
in [17] and [7] using a Gelfand-Tsetlin basis [19].
Fix arbitrary a1, a2, a3, ξ, µ ∈ C such that a3 /∈ Z. To simplify formulas we define the

following set of indexed variables:

h
(1)
ij = a1 + 2i− j, h

(2)
ij = a2 − i+ 2j, sj,k = a3 − j + 2k − 1,

S+
ijk =

1

2
(sj,k + h

(1)
i,j ) =

1

2
(a1 + a3 − 1) + i− j + k,

S−
ik =

1

2
(s0,k − h

(1)
i,0 ) =

1

2
(−a1 + a3 − 1)− i+ k,

T+
k =

1

2
(s0,k +

1

3
(h

(1)
0,0 + 2h

(2)
0,0)) =

1

6
(a1 + 2a2 + 3a3) + k − 1

2
,

T−
jk =

1

2
(sj,k −

1

3
(h

(1)
0,j + 2h

(2)
0,j )) =

1

6
(−a1 − 2a2 + 3a3)− j + k − 1

2
,

Q−
jk = −µ+ T−

j,k−2ξ − T−
j,kT

−
j,k−1T

−
j,k−2 = −(T−

j,k−1 − t1)(T
−
j,k−1 − t2)(T

−
j,k−1 − t3),

Q+
k = µ+ T+

k ξ − T+
k T

+
k−1T

+
k−2 = (T+

k−1 + t1)(T
+
k−1 + t2)(T

+
k−1 + t3).

where t1, t2, t3 are three roots of the equation t
3 − t(ξ + 1) + µ+ ξ = 0

(4.15)

Define an action of the Lie algebra g on the vector space V (a1, a2, a3, ξ, µ) = spanC{vijk | i, j, k ∈
Z} as follows:

z1(vijk) = ξvijk, z2(vijk) = µvijk, h01(vijk) = h
(1)
ij vijk, h10(vijk) = h

(2)
ij vijk,

e01(vijk) = S+
ijkvi+1,j,k, f01(vijk) = S−

ikvi−1,j,k,

e10(vijk) = Q−
jkvi,j+1,k +

S−
i,k

sj+1,ksjk
vi,j+1,k+1,

f10(vijk) = Q+
k vi,j−1,k−1 +

S+
i,j,k

sj+1,ksjk
vi,j−1,k.

(4.16)

We have the following statement (cf. [7, 14]).

Theorem 4.2. Let a1, a2, a3, ξ, µ ∈ C and a3 /∈ Z. Then



M. Andelić et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6458 11 of 21

1. The space V (a1, a2, a3, ξ, µ) is a torsion free Γ-pointed g-module if and only if S+
ijk and

S−
ik are non-zero, for all i, j, k ∈ Z.

2. The module V (a1, a2, a3, ξ, µ) is simple if and only if S
+
ijk, S

−
ik, Q

−
jk, Q

+
k are non-zero,

for all i, j, k ∈ Z.

Proof. The fact that V (a1, a2, a3, ξ, µ) is a g-module was shown in [14] and [7]. It is Γ-
pointed as Γ separates the basis elements. It follows from formulas above that the module
V (a1, a2, a3, ξ, µ) is torsion free if and only if S

+
ijk and S

−
ik are different from zero for all

i, j, k ∈ Z. Note that for every h-weight λ of V (a1, a2, a3, ξ, µ), the λ-weight subspace is
infinite-dimensional with basis vijk, where i, j are determined by λ and k runs through Z. In
this basis the operator c1 is presented by an infinite diagonal matrix and the operator c2 is
presented by a 3-diagonal matrix (bst), with bst = 0, for all s, t such that |s− t| > 1. This is
a consequence of relations above. The simplicity of V (a1, a2, a3, ξ, µ) results in the simplicity
of each λ-weight subspace as U0(g)-module. This implies the required conditions in item 2.
□

Simple subquotients of the module V (a1, a2, a3, ξ, µ) give all simple generic Gelfand-Tsetlin
g-modules with finite or infinite weight multiplicities [20]. Moreover, since any simple sub-
quotient V ′ of V (a1, a2, a3, ξ, µ) is torsion free and we can choose any weight from the weight
lattice as part of the parameter set, we have

Corollary 4.3. If V ′ is a simple generic Gelfand-Tsetlin g-module, then it is isomorphic to
a subquotient of V (a1, a2, a3, ξ, µ) for some suitable parameters such that 0 ≤ Rea1 < 1,
0 ≤ Rea2 < 3, 0 < Rea3 < 2, a3 ̸= 1, where Rea stands for the real part of a.

4.3. Subquotients of V (a1, a2, a3, ξ, µ)

As we saw in the previous section, the Γ-pointed module V (a1, a2, a3, ξ, µ) is simple when
S+
ijk, S

−
ik, Q

−
jk, Q

+
k are non-zero, for all i, j, k ∈ Z. If one or more these coefficients are

equal to zero, then a system of sub modules arises. Below, we present two examples of such
sub modules. Let I = Z3 ⊂ R3 be the weight lattice (the lattice of indices) of the module
V (a1, a2, a3, ξ, µ).
Case 1. Assume that S+

i0,j0,k0
= 0 for some i0, j0, k0 ∈ Z. Then, it follows that a1 + a3 ∈

2Z+1. Consider the function F (i, j, k) = S+
ijk defined on the set I. The subset of zero points

P (F ) = {(i, j, k) |F (i, j, k) = 0} is called the splitting hyperplane of the function F . The
splitting hyperplane F (i, j, k) divides the set I into two subsets, called the positive and the
negative components, given by

I+F = {(i, j, k) |F (i, j, k) > 0} and I−F = {(i, j, k) |F (i, j, k) ≤ 0}.

Using (4.16), we observe that, for any (i, j, k) ∈ I−F , the action of the elements e01, f01, e10, f10
on vijk results in elements that remain in the subspace V ′ = spanC{vijk | (i, j, k) ∈ I−F } ⊂ V .
Thus V ′ forms a submodule of V .
It is simple if none of S−

ik, Q
+
jk or Q

−
jk vanish for any choice of indices.

Case 2. Assume that Q+
k = 0 for some k ∈ Z. As we can see, Q+

k is the product of three
first degree polynomials Fm(i, j, k) = (T+

k + tm), with m = 1, 2, 3, in the variable k. Let us



M. Andelić et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6458 12 of 21

consider the case when Q+
k = 0 has three distinct roots. We need to find possible values of

the parameters a1, a2, a3, t1, t2, t3 such that the system of three equations: Fm(i, j, km) = 0,
m = 1, 2, 3 has integer solutions for the variables km. We find that Q

+
k = 0, for k =

k1, k2, k3 ∈ Z if the following conditions hold: tm = 1
3(k1 + k2 + k3) − km, m = 1, 2, 3 and

a3 = 3 − 1
3(2k1 + 2k2 + 2k3 + a1 + 2a2). Suppose that k1 > k2 > k3. Then we have three

splitting hyperplanes Pm = {(i, j, k) |Fm(i, j, k) = 0} and I = I+Fm
∪ I−Fm

, where

I+Fm
= {(i, j, k) |Fm(i, j, k) ≥ 0} , I−Fm

= {(i, j, k) |Fm(i, j, k) < 0} , for m = 1, 2, 3 .

Again, using (4.16) we immediately obtain

Proposition 4.4. The subspaces V ′
m = spanC{vijk | (i, j, k) ∈ I+Fm

} ⊂ V (a1, a2, a3, ξ, µ) are
submodules of V (a1, a2, a3, ξ, µ) satisfying the inclusion V ′

1 ⊂ V ′
2 ⊂ V ′

3 .

If none of S+
ijk or S

−
ik or Q

−
jk are zero for all choices of indices, then V

′
1 and V

′
2/V

′
1 and

V ′
3/V

′
2 and V (a1, a2, a3, ξ, µ)/V

′
3 are simple, torsion-free modules. All non-zero weight spaces

of these modules have dimensions respectively: ∞ for V ′
1 and V (a1, a2, a3, ξ, µ)/V

′
3 , k1 − k2

for V ′
2/V

′
1 , and k2 − k3 for V ′

3/V
′
2 .

A similar construction of a family of torsion-free submodules can be obtained in the case
when Q−

j,k = 0. Combining the conditions under which one or more coefficients are equal
to zero we can obtain various distinct modules. The full classification of sl(3) modules was
obtained in [7] using the Gelfand-Tsetlin tableaux technique.

5. Construction of simple weight C2-modules

In this section we use the technique developed in the previous section to construct simple
Gelfand-Tsetlin modules for the Lie algebra of type C2.

5.1. The centralizer for C2

Consider the root system of C2:

∆ = {α1, α2, α3 = α1 + α2, α4 = 2α1 + α2, α5 = −α4, α6 = −α3, α7 = −α2, α8 = −α1}

and the Chevalley basis: e10 = E12 − E43, f10 = E21 − E34, e01 = E31, f01 = E13, h10 =
(E11−E22−E33+E44), h01 = −E11+E33, e11 = −E32−E41, e21 = 2E42, f11 = −E14−E23,
f21 = 2E24 with the order h10 < h01 < f10 < f01 < f11 < f21 < e21 < e11 < e01 < e10.
The following is a complete set of indecomposable lists of roots: {α1, α8}, {α2, α7},

{α3, α6}, {α4, α5}, {α1, α2, α6}, {α3, α7, α8}, {α1, α3, α5}, {α4, α6, α8}, {α1, α1, α2, α5}, {α4, α7, α8, α8},
{α3, α3, α5, α7}, {α2, α4, α6, α6}.
Let us choose the following generators of U0 = U0(C2):

h1 = h01, h2 = h10, c1 = f01e01, c2 = f21e21, c3 = f10e10, c4 = f11e11,

c5 = f11e01e10, c6 = f10f01e11, c7 = f21e11e10, c8 = f10f11e21,

c9 = f21e01e10e10, c10 = f10f10f01e21, c11 = f01f21e11e11, c12 = f11f11e21e01.

Order the set of perfect monomials as follows:

h1 < h2 < c1 < c2 < c3 < c4 < c5 < c6 < c7 < c8 < c9 < c10 < c11 < c12. (5.1)

As in type A2 we have

Proposition 5.1.



M. Andelić et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6458 13 of 21

1) The following set of monomials is a basis of U0:

P ={hs11 h
s2
2 c

s3
1 c

s4
2 c

s5
3 c

s6
4 c

sm
m csnn | s1, s2, . . . , s6, sm, sn ∈ N and

(m,n) ∈ {(5, 9), (5, 12), (6, 10), (6, 11), (7, 9), (7, 11), (8, 10), (8, 12)}}. (5.2)

2) The set K̃ = {X|X ∈ K ′,Len(X) = 2} is a generating set of the ideal of relations K.

Proof. Let X = cmcnck ∈ Q0(C2), where 4 < m < n < k ≤ 12 be a monomial. Manual
calculation shows that for any triple m,n, k satisfying above condition there is 1 ≤ p ≤ 4 such
that L1(cp) is sublist of L1(X). For example, if (m,n, k) = (5, 6, 7), then L1(c1) ⊏ L1(X).
This means that no semi-perfect element can contain above three perfect monomials.
Now, let X = cmcn ∈ Q0(C2), where 4 < m < n ≤ 12 be a monomial. One can easily see

that for any pair m,n which is not in the (5.2), there exists 1 ≤ p ≤ 4 such that L1(cp) is a
sublist of L1(X). For example, if (m,n) = (9, 12), then L1(c2) ⊏ L1(X). It follows that the
only possible perfect monomials in semi-perfect monomials are the ones given above, proving
1. In particular this means that we can take d0 = 2 in the definition of K ′ (cf. (2.4)) and in
Theorem 2.5 with sort order defined in (5.1).

□

Denote h3 = h1 + h2 and let U ′
0 = U0[h

−1
1 , h−1

3 ]. It will be convenient for us to work with
U ′
0.
The list of all relations is rather big, so we will give only those of them that are used in

our calculations. The following list of relations is used to express c12, . . . , c5 via c4, c3, c2, c1,
h1, h3, z1 in U ′

0:

c12 =− c10 − c2 − 2c8 + [c1, c8], (5.3)

c11 =− c9 − 2c7 + c2 − [c1, c7], (5.4)

c10 =c9 − c8 + c7 +
1

2
[c5, c2] + [c1, c8] (5.5)

c9 =
1

2h1
([c1, [c1, c7]]− 2c1c2 − 2c7c1 − 2c1c7) +

1

2
[c7, c1]− 2c7 − c2, (5.6)

c8 =c7 +
1

2
[c2, c3] (5.7)

c7 =
1

16h3
(−[c2, [c2, c3]] + 8c3c2 − 8c2c4 − 8h1c2) +

1

4
[c3, c2]−

1

2
c2, (5.8)

c6 =c5 + [c3, c1] (5.9)

c5 =− c4 + c1c3 +
1

2h1
(−[c1, c1, c3]− c3c1(h1 − 2)− 2c1c4). (5.10)

The following relations are used to find relations between the elements c4, c3, c2, c1, h1,
h3, z1.

c5c1 = −c6 + (h1 + 1)c5 + h1c4 + c1c5 + c1c4 − c1c3 (5.11)

c7c2 = −4h3c7 + c2c7 − 2c2c4 + 2c2c3 + (−2h3 − 2h1)c2 (5.12)

c5c3 = c9 + c8 − 2c6 + (h3 − h1)c5 + c3c5 − c3c4 + 2c1c3 (5.13)



M. Andelić et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6458 14 of 21

c7c3 = −2c9 − 2c8 + (h3 − h1)c7 + 4c6 + c3c7 + 2c3c4 − c2c3 (5.14)

c6c5 = c10 + 2c8 − c7 + (h3 − h1 − 2)c6 − c4c5

− c3c6 − 2c3c4 − c1c8 + 2c1c6 + c1c3c4 + (h3 + h1 + 2)c1c3. (5.15)

The center of the universal enveloping algebra of g is generated by the following Casimir
elements:

z1 =4c1 + c2 + 2c3 + 2c4 + 2h21 + 2h23 + 2h1 + 4h3, (5.16)

z2 =2c12 + 2c11 − 2c10 − 2c9 + (2h1 + 1)c8 + (2h1 − 1)c7

+ (4h3 + 6)c6 + (4h3 + 10)c5 − c24 + (−2h1h3 − 4h1 + 2h3 + 6)c4 − 2c3c4

− c23 + (2h1h3 + 4h1 + 2h3 + 6)c3 − (h1 − 1)(h1 + 1)c2 − 4c1c2

− 4(h2 + 3)(h2 + 1)c1 − h1(h3 + 3)(h3 + 1)(h1 + 2).

As in type A, our goal is to find the ”best” choice of a generating set S of U0, such that
the cardinality of the set S3 is minimal.

Lemma 5.2.

1. The set S = {h1, h2, z1, z2, c1, c2, c3} is a generating set of the centralizer U ′
0 with the

following decomposition: S1 = {h1, h2, z1, z2}, S2 = {c1, c2}, S3 = {c3}.

2. The decomposition of the set of relations is the following: R = R1 ∪ R2, where R1

consists of two relations obtained from (5.11) and (5.12), while R2 consists of three
relations obtained from (5.13), (5.14), and (5.15).

Proof. Using the (5.3)-(5.10) and (5.16) for the Casimir element z1 we can exclude the
generators c12, . . . , c4 from all other relations. As the result, we will get the generating set
S = {h1, h2, z1, c1, c2, c3}. The rest can be verified by direct computations. □

Remark. We can not claim that S is a generating set of U0. Nevertheless, any U0-module
M with a non-zero scalar action of h1 is a U ′

0-module.

5.2. Construction of torsion free C2-modules

Let Γ be the commutative subalgebra of U0(C2) generated by the elements h1, h2, z1, z2, c1.
We construct two families of Γ-pointed modules, each depending on four complex parameters.
Fix arbitrary complex number a1, a2, a3, a4, η, and define the following set of indexed

variables:

h
(1)
ij = a1 + 2i− j, h

(2)
ij = a2 − 2i+ 2j, sjk = a3 − j + 2k − 1, Q±

jk =
η

sjk
± 1,

S+
ijk =

1

2
(a1 + a3 + 2i− 2j + 2k − 1), S−

ik =
1

2
(−a1 + a3 − 2i+ 2k − 1),

T+
k =

1

2
(a1 + a2 + a4 + 2k − 1), T−

jk =
1

2
(−a1 − a2 + a4 − 2j + 2k − 1),

where i, j, k ∈ Z.

(5.17)



M. Andelić et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6458 15 of 21

Consider the vector space V (a1, a2, a3, a4, η) = spanC{vijk | i, j, k ∈ Z} and define the
following operators on V (a1, a2, a3, a4, η) (using the same letters as for the generators of g):

h1(vijk) = h
(1)
ij vijk, h2(vijk) = h

(2)
ij vijk,

e01(vijk) = S+
ijkvi+1,j,k, f01(vijk) = S−

ikvi−1,j,k,

e10(vijk) = S−
ikQ

+
j+1,kvi,j+1,k+1 + T−

j+1,kQ
−
j+1,kvi,j+1,k,

f10(vijk) = S+
ijkQ

+
j+1,kvi,j−1,k + T+

k−1Q
−
j+1,kvi,j−1,k−1,

e11(vijk) = −S+
ijkQ

+
j+1,kvi+1,j+1,k+1 + T−

j+1,kQ
−
j+1,kvi+1,j+1,k,

f11(vijk) = S−
i,kQ

+
j+1,kvi−1,j−1,k − T+

k−1Q
−
j+1,kvi−1,j−1,k−1,

e21(vijk) = 2T−
j+1,kvi+1,j+2,k+1, f21(vijk) = 2T+

k−1vi−1,j−2,k−1.

(5.18)

These formulas define a Γ-module structure on V (a1, a2, a3, a4, η), but at this point we
can not claim a g-module structure.

Lemma 5.3. The subalgebra Γ has a simple spectrum on V (a1, a2, a3, a4, η), and hence
separates the basis elements vijk, if and only if a3 /∈ Z.

Proof. We need to show that Γ acts with different characters on basis elements vi,j,k. It is
sufficient to consider vectors from the same weight space. Suppose

h1(vijk) = (a1 + 2i− j)vijk = λvijk, h2(vijk) = (a2 − 2i+ 2j)vijk = µvijk,

for some fixed λ and µ. Then i and j are uniquely determined: j = λ + µ − a1 − a2 and
i = 1

2(λ + j − a1). Hence, basis elements of this weight subspace differ by the third index.
Consider vijk1 and vijk2 for arbitrary integer i, j, k1, k2. We have

c1(vijk) = f01e01(vijk) = f01(S
+
i,j,kvi+1jk) = S−

i+1,kS
+
i,j,k(vijk) (5.19)

Suppose that c1 has the same value on vijk1 and vijk2 . Then we have the equation
0 = S−

i+1,k1
S+
i,j,k1

−S−
i+1,k2

S+
i,j,k2

which simplifies to 0 = 1
4(k1− k2)(a3− j+ k1+ k2− 2). Since

k1 ̸= k2, it follows that a3 = j − k1 − k2 + 2, which must be an integer. □

Denote V1(a1, a2, a3, η) = V (a1, a2, a3, a3, η) and V2(a1, a2, a3, a4) = V (a1, a2, a3, a4, 0) and
ξ = 2(η + 1)(η − 2).

Theorem 5.4. Let V = V (a1, a2, a3, a4, η). Then

1. V is a g-module if and only if V = V1(a1, a2, a3, η) with a3 /∈ Z, or V = V2(a1, a2, a3, a4).

2. The g-module V is a torsion free Γ-pointed g-module if and only if S+
ijkS

−
ikT

−
jkT

+
k ̸= 0

for all i, j, k ∈ Z.

3. The torsion free Γ-pointed g-module V is simple if and only if Q+
jkQ

−
jk ̸= 0 for all

j, k ∈ Z.

4. If V ′ is a simple torsion free Γ-pointed g-module with a basis parametrized by the lattice
Z3 and with separating action of Γ on basis elements, then V ′ ≃ V (a′1, a

′
2, a

′
3, a

′
4, η

′)
for some suitable choice of parameters such that 0 ≤ Rea′1 < 1, 0 ≤ Rea′2 < 2,
0 < Rea′3 < 2, where Rea denotes the real part of a.



M. Andelić et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6458 16 of 21

5. The action of the Casimir elements in the case a3 = a4:

z1(vijk) = ξvijk, z2(vijk) = −1

4
ξ(ξ + 4)vijk

and in the case a3 ̸= a4: z1(vijk) = ((a3 − a4)
2 − 4)vijk, z2(vijk) = 0.

Proof. The formulas (5.17) and (5.18) are well defined if and only if V = V1(a1, a2, a3, η)
with a3 /∈ Z, or V = V2(a1, a2, a3, a4). The statement that V is a g-module follows by
checking all defining relations of the Lie algebra. The relations 5.11-5.15 give only two al-
ternatives V1(a1, a2, a3, η) and V2(a1, a2, a3, a4). It follows immediately from the formulas of
the action of g that V (a1, a2, a3, a4, η) is a torsion free module if and only if S

+
i,j,k, S

−
i,k, T

−
jk

and T+
k−1 are different from zero, for all i, j, k ∈ Z. Note that Γ has a simple spectrum on

V (a1, a2, a3, a4, η) and hence, the action of Γ separates the basis elements by Lemma 5.3. In
particular, V (a1, a2, a3, a4, η) is Γ-pointed. This implies the second statement.
Suppose the module V (a1, a2, a3, a4, η) is torsion free. Similarly to case of A2, using

the relations in U0(g) we get that c1, c2 are presented by infinite diagonal matrices, while the
element c3 is presented by a 3-diagonal matrix on every weight subspace of V (a1, a2, a3, a4, η).
Using this fact and the separating action of Γ on basis elements, it is easy to see that conditions
Q+

jkQ
−
jk ≠ 0, for all i, j, k ∈ Z are necessary and sufficient to guarantee that any element of

V (a1, a2, a3, a4, η) generates the whole module, which is equivalent to the simplicity of the
module. This implies the third statement. Let V ′ be a simple torsion free Γ-pointed C2-
module with a basis {v′

ijk, i, j, k ∈ Z} such that Γ acts by different characters on the basis
elements v′

ijk. Fix one basis element v
′
ijk and apply the generators of the centralizer U0(g).

One can see directly from the action that U0(g)v
′
ijk will be equal to the whole weight space of

V ′, which v′
ijk belongs to. Moreover, the action of the generators of Γ will determine uniquely

the corresponding parameters a1, a2, a3, a4, η. Hence, we get a non-zero homomorphism θ of
U0(g)-modules U0(g)vijk and U0(g)v

′
ijk: θ(vijk) = v′

ijk. Moreover, U0(g)v
′
ijk is a simple U0(g)-

module as V ′ is simple. Hence, θ is surjective. It extends to a surjective homomorphism from
V (a1, a2, a3, a4, η) to V ′. Comparing the bases of both modules we conclude the isomorphism
V ′ ≃ V (a1, a2, a3, a4, η). Since V ′ is torsion free we can choose any weight of the weight lattice
as part of the parameter set. This proves the fourth statement.
The last statement follows by direct computation. □

Hence, Theorem 5.4 provides two 4-parameter families of simple torsion free Γ-pointed
g-modules. If V (a1, a2, a3, a4, η) is a torsion free Γ-pointed g-module which is not simple,
then all its simple subquotients are torsion free Γ-pointed modules. They exhaust all generic
simple torsion free Γ-pointed g-modules, which are analogs of generic simple Gelfand-Tsetlin
modules in type A.

6. Gelfand-Tsetlin modules for G2

In this section we extend the results of previous sections to the of the Lie algebra of type
G2.

6.1. Construction of Centralizer of G2

Define the root system for G2 (for convenience will use notation βi,j for α−i,−j):

∆ = {α01, α10, α11, α21, α31, α32, β32, β31, β21, β11, β10, β01},



M. Andelić et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6458 17 of 21

where αij = iα10 + jα01. Fix a Chevalley basis: e01 = E31 + E64, f01 = E13 + E46, e10 =
2E17 + E23 − E45 − E76, f10 = E32 − E54 − 2E67 + E71, e11 = −E21 + 2E37 − E65 + E74,
f11 = −E12 +2E47 −E56 +E73, e21 = E14 +2E27 +E36 +E75, f21 = E41 +2E57 +E63 +E72,
e31 = E15 + E26, f31 = E51 + E62, e32 = −E24 + E35, f32 = −E42 + E53, h01 = −E11 +
E33 − E44 + E66, h10 = 2E11 + E22 − E33 + E44 − E55 − 2E66, h31 = E11 + E22 − E55 − E66,
h21 = E11 + 2E22 + E33 − E44 − 2E55 − E66.
Here we are using a standard realization of G2 with matrix units Eij . All indecomposable

lists of roots of G2 can be obtained from primitive lists of roots by Lemma 2.3.2.
Then we obtain the following description of perfect monomials (we omit the details).

Lemma 6.1. The following is the set of all perfect monomials:

h1 = h01, h2 = h21, c1 = f01e01, c2 = f10e10, c3 = f11e11, c4 = f11e10e01, c5 = f01f10e11,

c6 = f21e21, c7 = f21e11e10, c8 = f10f11e21, c9 = f21e
2
10e01, c10 = f01f

2
10e21, c11 = f211e21e01,

c12 = f01f21e
2
11, c13 = f31e31, c14 = f31e21e10, c15 = f10f21e31, c16 = f31e11e

2
10, c17 = f210f11e31,

c18 = f31e
3
10e01, c19 = f01f

3
10e31, c20 = f32e32, c21 = f32e31e01, c22 = f32e21e11, c23 = f01f31e32,

c24 = f11f21e32, c25 = f32e21e10e01, c26 = f11f21e31e01, c27 = f32e
2
11e10, c28 = f01f31e21e11,

c29 = f01f10f21e32, c30 = f10f
2
11e32, c31 = f32e11e

2
10e01, c32 = f10f

2
11e31e01, c33 = f01f31e

2
11e10,

c34 = f01f
2
10f11e32, c35 = f32e

3
10e

2
01, c36 = f201f

3
10e32, c37 = f311e32e01, c38 = f01f32e

3
11,

c39 = f311e31e
2
01, c40 = f201f31e

3
11, c41 = f11f31e32e10, c42 = f221e32e10, c43 = f10f32e31e11,

c44 = f221e31e11, c45 = f10f32e
2
21, c46 = f11f31e

2
21, c47 = f221e31e10e01, c48 = f01f10f31e

2
21,

c49 = f11f32e
2
21e01, c50 = f01f

2
21e32e11, c51 = f21f31e32e

2
10, c52 = f210f32e31e21, c53 = f21f32e31e

2
11,

c54 = f211f31e32e21, c55 = f231e32e
3
10, c56 = f310f32e

2
31, c57 = f31f32e

3
21, c58 = f32,1e32e31,

c59 = f321e
2
31e01, c60 = f01f

2
31e

3
21, c61 = f232e

3
21e01, c62 = f232e31e

3
11, c63 = f311f31e

2
32, c64 = f01f

3
21e

2
32.

Define the order on the set of perfect monomials as follows: h1 < h2 < c1 < c2 < · · · <
c63 < c64. The following proposition can be easily checked.

Proposition 6.2. The centralizer subalgebra U0(G2) is generated by a finite set of monomi-
als {h1, h2, c1, . . . , c64} with a finite number of relations {r1, . . . , rm}, where ri is a polynomial
in h1, h2, c1, . . . , c64 of length ≤ 4.

We will use the following quadratic Casimir element of U0(g):

z1 = 3c1 + c2 + c3 + c6 + 3c13 + 3c20 + h201 + h01h10 + h210 + 4h01 + 5h10.

Tedious computations show that using the relations, the elements c64, . . . , c5 can be written
in terms of c4, c3, c2, c1, h1, h2, z1.
The Lie algebra G2 contains the subalgebra ĝ of type A2 generated by the following

elements h01, h31, e01, f01, e31, f31, e32, f32. We will use the results from the previous
sections just adding ”hat” to all generators and to all formulas. Consider a natural embedding
φ : ĝ → g:

φ(ĥ01) = h01, φ(ĥ10) = h31, φ(ê01) = e01, φ(f̂01) = f01, φ(ê10) = e31,

φ(f̂10) = f31, φ(ê11) = e32, φ(f̂11) = f32



M. Andelić et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6458 18 of 21

and extend it to the embedding of the universal enveloping algebras.
The images of the Casimir elements ẑ1, ẑ2 of U(ĝ) in U(g) are:

Z1 = φ(ẑ1) = c20 + c13 + c1 +
1

3
(h231 + 3h31 + h201 + 3h01 + h31h01)

Z2 = φ(ẑ2) = c23 + c21 +
1

3
(h01 − h31)c20 −

1

3
(6 + 2h01 + h31)c13 +

1

3
(h01 + 2h31)c1

+
1

27
(−h31 − 3 + h01)(6 + 2h01 + h31)(h01 + 2h31).

6.2. Torsion free G2-modules

Let Γ be commutative subalgebra of U0(G2) generated by elements h1, h2, z1, c1. This our
Gelfand-Tsetlin subalgebra for G2. We will give a construction of a 3-parameter family of
Γ-pointed modules with separating action of Γ on basis elements.
Fix a1, a2, a3 ∈ C such that a3 /∈ Z. Define the following set of indexed variables:

h01(i, j) = a1 + 2i− j, h10(i, j) =
1

2
(h21(i, j)− 3h01(i, j)) =

1

2
(a2 − 3a1)− 3i+ 2j,

h21(i, j) = a2 + j, h11(i, j) =
1

2
(h21(i, j) + 3h01(i, j)) =

1

2
(a2 + 3a1) + 3i− j,

h31(i, j) =
1

2
(h21(i, j)− h01(i, j)) =

1

2
(a2 − a1)− i+ j,

h32(i, j) =
1

2
(h21(i, j) + h01(i, j)) =

1

2
(a2 + a1) + i, sjk = a3 − j + 2k − 1,

S+
ijk =

1

2
(sjk + h01(i, j)) =

1

2
(a1 + a3 + 2i− 2j + 2k − 1),

S−
ik =

1

2
(s0k − h01(i, 0)) =

1

2
(−a1 + a3 − 2i+ 2k − 1),

T+
jk =

1

2
(sjk +

1

3
h2,1(0, j)) =

1

6
(a1 + 2a2 + 3a3 − 2j + 6k − 3),

T−
jk =

1

2
(sjk −

1

3
h2,1(0, j)) =

1

6
(−a1 − 2a2 + 3a3 − 4j + 6k − 3),

A+
jk =

T−
j−1,k−1T

+
jkT

+
j+1,k

9sjksj+1,k
, A−

jk =
T−
j−1,k−1T

−
jkT

+
j+1,k

9sjksj+1,k
,

B+
jk =

T+
j−1,kT

+
jkT

+
j+1,k

27sjksj+1,k
, B−

jk =
T−
j−1,k−1T

−
jkT

−
j+1,k+1

27sjksj+1,k
, i, j, k ∈ Z.

(6.1)

Define the action of the Lie algebra g on V (a1, a2, a3) = spanC{vijk | i, j, k ∈ Z} as follows:

h01(vijk) = h01(i, j)vijk, h10(vijk) = h10(i, j)vijk, h11(vijk) = h11(i, j)vijk, h21(vijk) = h21(i, j)vijk,

h31(vijk) = h31(i, j)vijk, h32(vijk) = h32(i, j)vijk, e01(vijk) = S+
ijkvi+1,j,k, f01(vijk) = S−

ikvi−1,j,k,

e21(vijk) = T+
j+1,kvi+1,j+2,k+1, f21(vijk) = T−

j−1,k−1vi−1,j−2,k−1, e10(vijk) = 3vi,j+1,k +A+
jkS

−
ikvi,j+1,k+1,

f10(vijk) = −3vi,j−1,k−1 −A−
jkS

+
ijkvi,j−1,k, e11(vijk) = −3vi+1,j+1,k +A+

jkS
+
i+1,j+1,kvi+1,j+1,k+1,

f11(vijk) = −3vi−1,j−1,k−1 +A−
jkS

−
i+1,k+1vi−1,j−1,k, e31(vijk) = vi+1,j+3,k+1 −B+

jkS
−
i+1,k+1vi+1,j+3,k+2,

f31(v,jk) = vi−1,j−3,k−2 −B−
jkS

+
i+1,j+1,kvi−1,j−3,k−1, e32(v,jk) = −vi+2,j+3,k+1 −B+

jkS
+
i+1,j+1,kvi+2,j+3,k+2,

f32(vijk) = vi−2,j−3,k−2 +B−
jkS

−
i+1,k+1vi−2,j−3,k−1, z1(vijk) =

14

3
vijk.



M. Andelić et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6458 19 of 21

Lemma 6.3. Subalgebra Γ has a simple spectrum on V (a1, a2, a3), and hence separates the
basis elements vijk, if and only if a3 /∈ Z.

Proof. It is sufficient to consider vectors from the same weight space. Suppose

h1(vijk) = (a1 + 2i− j)vijk = λvijk, h2(vijk) = (a2 + j)vijk = µvijk,

for some λ and µ. Then j = µ− a2 and i = 1
2(λ+ µ− a1 − a2). Hence, basis elements of this

weight subspace differ by the third index.
Consider vijk1 and vijk2 for arbitrary integers i, j, k1, k2. We have

c1(vijk) = f01e01(vijk) = f01(S
+
i,j,kvi+1jk) = S−

i+1,kS
+
i,j,k(vijk).

Since the formulas for S−
i,k and S

+
i,j,k are the same for C2 (5.17) and G2 (6.1), we obtain

similar results as in (5.19) and conclude that c1 separates the basis elements vijk1 and vijk2 .
This implies the statement. □

Theorem 6.4. For any complex a1, a2, a3 such that a3 /∈ Z, the above formulas define the
G2-module structure on the space V (a1, a2, a3).

Proof. Follows by checking that the defining relations of the Lie algebra G2 are satisfied on
V (a1, a2, a3). We omit the details. □

Theorem 6.5. Let a1, a2, a3 ∈ C and a3 /∈ Z. Then

1. V (a1, a2, a3) is a torsion free simple Γ-pointed G2-module if and only if S
+
ijkS

−
ikT

+
jkT

−
jk ̸=

0, for all i, j, k ∈ Z.

2. If V ′ is a simple torsion free Γ-pointed G2-module with a basis parametrized by the
lattice Z3 and with separating action of Γ on basis elements, then it is isomorphic to
V (a1, a2, a3) for suitable parameters a1, a2, a3 such that 0 ≤ Rea1 < 1, 0 ≤ Rea2 < 2,
0 < Rea3 < 2 and a3 ̸= 1.

Proof. It follows immediately from the formulas of the action of G2 that V (a1, a2, a3) is a
torsion free module if and only if S+

ijk ̸= 0, S−
ik ̸= 0, T+

jk ≠ 0 and T−
jk ̸= 0, for all i, j, k ∈ Z.

Note that Γ has a simple spectrum on V (a1, a2, a3) and hence, the action of Γ separates the
basis elements by Lemma 6.3. In particular, V (a1, a2, a3) is Γ-pointed. Using this fact, it is
easy to see that conditions S+

ijkS
−
ikT

+
jkT

−
jk ≠ 0, for all i, j, k ∈ Z are necessary and sufficient

to guarantee that any element of V (a1, a2, a3) generates the whole module, and hence the
simplicity of the module.
Suppose that V ′ is a simple torsion free Γ-pointedG2-module with a basis {v′ijk, i, j, k ∈ Z}

such that Γ acts by different characters on the basis elements v′ijk. Then the same argument
as in the proof of Theorem 5.4 shows that V ′ ≃ V (a1, a2, a3) for some choice of complex
parameters a1, a2, a3 with a3 /∈ Z. Since module V ′ is torsion free, we can choose any weight
of the weight lattice and hence the parameters can be chosen in such a way that their real
parts satisfy the inequalities 0 ≤ Rea1 < 1, 0 ≤ Rea2 < 3, 0 < Rea3 < 2. This completes
the proof.

□

Consider the following standard embedding θ : A2 → G2: θ(ê01) = e01, θ(f̂01) = f01,
θ(ê10) = e31, θ(f̂10) = f31, θ(ê11) = e32, θ(f̂11) = f32, θ(ĥ01) = h01, θ(ĥ10) = h31, and denote



M. Andelić et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6458 20 of 21

by ĝ its image. Consider the restriction V̂ (a1, a2, a3) of V (a1, a2, a3) on ĝ. The images Z1

(respectively Z2) of the Casimir elements of A2 act on V̂ (a1, a2, a3) by −8
9 (respectively

8
9).

We have the following decomposition of V̂ (a1, a2, a3).

Theorem 6.6. Suppose that V (a1, a2, a3) is torsion free. The A2-module V̂ (a1, a2, a3) decom-
poses into a direct sum of three torsion free submodules V̂ (a1, a2, a3) = V̂ (1)⊕ V̂ (2)⊕ V̂ (3),
where

V̂ (m) = spanC{vi,3j+m,k | i, j, k ∈ Z} ≃W = V (a
(m)
1 , a

(m)
2 , a3,

−8

9
,
8

9
),

a
(m)
1 = a1 −m, a(m)

2 = 1
2(a2 − a1) +m, for m = 0, 1, 2.

Proof. Consider the subspace V̂ (m) = spanC{vi,3j+m,k | i, j, k ∈ Z} of V̂ (a1, a2, a3), for m =

0, 1, 2. Clearly, V̂ (m) is a ĝ-submodule of V̂ (a1, a2, a3). The ĝ-moduleW = spanC{wijk | i, j, k ∈
Z} is defined by the formulas (4.15), (4.16), and we have a homomorphism of ĝ-modules
ψm :W → V̂ (m), such that ψm : wijk 7→ vi+j,3j+m,k+j , and for any x ∈ ĝ holds ψm(x(wijk)) =
θ(x)(ψm(wijk)) = θ(x)(vi+j,3j+m,k+j), for i, j, k ∈ Z and m = 0, 1, 2. Since ψm is a linear
isomorphism, the statement follows. □

7. Conclusion

For all simple Lie algebras of rank 2 we constructed families of simple modules with
infinite-dimensional weight spaces. These modules admit a diagonalizable action of a certain
commutative subalgebra with a simple spectrum. In type A2, this commutative subalgebra
is a famous Gelfand-Tsetlin subalgebra and the corresponding modules are generic Gelfand-
Tsetlin modules. In type C2 we construct two 4-parameter families of simple modules which
are analogs of generic Gelfand-Tsetlin modules, while in type G2 we construct a 3-parameter
family of such simple modules.
Detailed computations are available at

https://icm.sustech.edu.cn/en/people/faculty/vyacheslav-futorny-en .

Acknowledgements

This work was supported by Kuwait University, Research Grant SM02/22.

References

[1] S Fernando. Lie algebra modules with finite dimensional weight spaces i. Transactions
of the American Mathematical Society, 322:757–781, 1990.

[2] O Mathieu. Classification of irreducible weight modules. Annales de l’Institut Fourier,
50:537–592, 2000.

[3] V Futorny, D Grantcharov, and L E Ramírez. Drinfeld category and the classification of
singular gelfand-tsetlin gln-modules. International Mathematics Research Notices, 2017.

[4] L E Ramírez and P Zadunaisky. Gelfand-tsetlin modules over gl(n) with arbitrary char-
acters. Journal of Algebra, 502:328–346, 2018.

[5] E Vishnyakova. Geometric approach to p-singular gelfand-tsetlin gln-modules. Differen-
tial Geometry and Applications, 56:155–160, 2018.

[6] B Webster. Gelfand-tsetlin modules in the coulomb context, 2019. arXiv:1904.05415.



M. Andelić et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6458 21 of 21

[7] V Futorny, D Grantcharov, and L E Ramírez. Classification of simple gelfand-tsetlin
modules of sl(3). Bulletin of Mathematical Sciences, pages 1–109, 2021.

[8] D Britten and F Lemire. A classification of pointed an-modules. In Lecture Notes in
Mathematics, volume 933, pages 63–70. 1982.

[9] D Britten and F Lemire. Irreducible representations of an with one-dimensional weight
space. Transactions of the American Mathematical Society, 273:509–540, 1982.

[10] D Britten and F Lemire. A classification of simple lie modules having a one-dimensional
weight space. Transactions of the American Mathematical Society, 299:111–121, 1987.

[11] D J Britten, V Futorny, and F W Lemire. Simple a2-modules with a finite dimensional
weight space. Communications in Algebra, 23(2):467–510, 1995.

[12] Y Drozd, V Futorny, and S Ovsienko. Irreducible weighted sl(3)-modules. Funksionalnyi
Analiz i Ego Prilozheniya, 23:57–58, 1989.

[13] Y Drozd, V Futorny, and S Ovsienko. Gelfand-tsetlin modules over lie algebra sl(3). In
Contemporary Mathematics, volume 131, pages 23–29. 1992.

[14] V Futorny. Irreducible sl(3)-modules with infinite-dimensional weight subspaces. Ukrain-
skii Matematicheskii Zhurnal, 41:856–859, 1989.

[15] V Futorny. Weight sl(3)-modules generated by semiprimitive elements. Ukrainskii
Matematicheskii Zhurnal, 43:281–285, 1991.

[16] J Dixmier. Enveloping algebras, volume 11 of Graduate Studies in Mathematics. Ameri-
can Mathematical Society, Providence, RI, 1996.

[17] Y Drozd, V Futorny, and S Ovsienko. Harish-chandra subalgebras and gelfand-zetlin
modules. Mathematical and Physical Sciences, 424:72–89, 1994.

[18] V Futorny, L E. Ramírez, and J Zhang. Combinatorial construction of gelfand-tsetlin
modules for gln. Advances in Mathematics, 343:681–711, 2019.

[19] I Gelfand and M Tsetlin. Finite-dimensional representations of the group of unimodular
matrices. Doklady Akademii Nauk SSSR (N.S.), 71:825–828, 1950.

[20] V Futorny, D Grantcharov, and L E. Ramírez. Irreducible generic gelfand-tsetlin modules
of gl(n). SIGMA, 11:018, 13, 2015.