Simple Finite-Dimensional Modules and Monomial Bases from the Gelfand-Testlin Patterns

One of the most important classes of Lie algebras is , which are the matrices with trace 0. The representation theory for has been an interesting research area for the past hundred years and in it the simple finite-dimensional modules have become very important. They were classified and Gelfand and Tsetlin actually gave an explicit construction of a basis for every simple finite-dimensional module. This paper extends their work by providing theorems and proofs, and constructs monomial bases of the simple module.


Introduction
Let be a Lie algebra of all matrices of order . In this paper, we work with finite-dimensional modules and hence finite-dimensional representation of . This means for , there exists a matrix of order defined in such a way that Choose integers , , , such that the inequality is satisfied. These partitions are quite important because they appear to be the core in constructing representations. These chosen integers are used to construct some index set (the explicit construction of this index set will be given in the next section). For a Lie algebra with order , we could construct at least possible number of such with entries from a given partition. An example will be given in the next section.
Let be a matrix of order which has at the intersection of the row and the column and zeros in all other places and let be the matrix of order from . Note that , under our consideration corresponds to elements . It is easy to see that each matrix forms a linear combination of ; that is ∑ for some . Therefore, the set distinctly defines some representation. One could find all such representations by explicitly describing all linear transformations . The quest for irreducible representations of special linear algebra was reformulated: one needs matrices of order satisfying the following bracket relations: For irreducibility, the system is required to have no invariant subspaces. The representation theory of has a unique nature in choosing a partition. For the classification of simple finite dimensional modules, one sets the last choice in the partition. This controls differences between subsequent choices in a partition.
A comprehensive theory of infinitesimal transformations was first given by a Norwegian mathematician, Sophus Lie (1842-1899). I. M. Gelfand and M. L. Tsetlin gave an explicit construction of a basis for every simple finitedimensional module of . In their work, they gave all the irreducible representations of general linear algebra ( ) but without theorems [1]. Recently, V. Futorny, D. Grantcharov and L. E. Ramirez provided a classification and explicit bases of tableaux of all irreducible generic Gelfand-Tsetlin modules for the Lie algebra [2]. In , V. Futorny, D. Grantcharov, and L. E. Ramirez initiated the systematic study of a large class of non-generic Gelfand-Tsetlin modules -the class of singular Gelfand-Tsetlin modules. An explicit tableaux realization and the action of on these modules was provided using a new construction which they call derivative tableaux. Their construction of singular modules provides a large family of new irreducible Gelfand-Tsetlin modules of , and is a part of the classification of all such irreducible modules for [3]. This paper will show that the Gelfand-Tsetlin constructions given in the year [1] forms all the irreducible representations of special linear algebra by providing proofs to results. It will also show that module is simple and also construct monomial basis from these modules. Section discusses some previous work and gives some notations and Section presents proofs to results and shows that module is simple. Then a conclusion is drawn in Section .

Notations and Preliminaries
Definition 1 (Upper Triangular Matrix). This is a matrix with entries where are zeros. From now on, we will denote an upper triangular matrix by such that entry has a and all others are zeros. Let be the set of all upper triangular matrices. If , then . Therefore, is a Lie algebra and , is a basis of . So acts by zero. Hence are generators of . We will denote a sequence of upper triangular matrices by and a sequence of upper triangular matrices in relation to by . Definition 2 (Lower Triangular Matrix). This is a matrix with entries where are zeros. Similarly, from now on, we will denote a lower triangular matrix by such that entry has a and all others are zeros. Let be the set of all lower triangular matrices. If , then . Therefore, is a Lie algebra and is a basis of , with acting by zero, so are generators of . Similarly, we will denote a sequence of lower triangular matrices by and a sequence of lower triangular matrices in relation to by . Definition 3 (Diagonal Matrix). This is a matrix with some non-zero entries on its diagonal while all other entries away from the diagonal are zero.
It is well known that the entries of the diagonals of such a square matrix are the eigenvalues. Let be the set of all diagonal matrices with trace zero. For , . So is a basis of . Suppose . The map is defined by giving the image of , for all . By definition, Since generates all of , then is a basis of . Definition 4 (Representation [4]). Suppose is a Lie algebra and let . The operation is a Lie algebra representation. The vector space is the representation space. The bracket is bilinear and also an endomorphism. That means It is easy to see that the Lie algebra . If is a finite dimensional module, then (where ) acts on such that where runs over (a dual) and The weight spaces are infinitely many and different from zero when is infinite dimensional. is called a weight space, a weight vector and we called a weight of . A highest weight vector (maximal vectors) in module is a non-zero weight vector in weight space annihilated by the action of all upper triangular matrices. We will prove in this paper that a highest weight vector is indeed maximal and hence a generator.
The index set, is an interesting construction and we will show how it is built. is a vector space with bases [1]. These bases depend on the choice of integer partition

Let
, and . All possible bases from this partition, as given by the construction of Figure 1, are Here, we discuss the module structure on . Our representation space is a module. Although this is true, we will not prove it. It is a module via actions of upper triangular matrices, lower triangular matrices and the diagonal matrices on [1].
In Gelfand [1], a comprehensive construction was presented for the action of upper triangular, lower triangular and diagonal matrices on basis vector . For upper triangular matrices in general, suppose is the pattern obtained from by replacing with , the upper triangular matrix acts on as ∑ For a matrix, the action of on raises the row in the basis by on every entry in that row accordingly. For the case , the formulas for computing the action can be found in Gelfand [1]. In general, is generated by . The action of on reduces the entries of the row in by accordingly. This is done in such a way that rules governing the size of entries are observed. Suppose is the pattern obtained from by replacing with . The lower triangular matrix acts on as ∑ The formulas for can be found in Gelfand [1]. In general, generates all and other actions can be computed using the Lie bracket operation.
The diagonal matrices can also be generated by and . Some coefficients from the action of can be zero but not all coefficients. In general is the coefficient of . The formulas for computing the action of diagonal matrices ( ) when can be found in Gelfand [1].
Theorem 5 (sln_module). The representation space is a module. A highest weight vector is the weight vector that is annihilated by every upper triangular matrix (that is with ). We fixed as our basis vector in , the representation space where is any integer depending on some conditions [1]. The nature of each basis vector depends on the dimension of operator acting on it and the partition. For , we choose some integers , ( ) such that the condition is satisfied. When , we choose three integers , , ( ). The bases vectors in the representation space are now numbered by triples, , , . The representation is given by We have our bases vectors of the form

( )
Every weight vector has a corresponding weight. The bases vectors are the weight vectors. Constructing these bases depends on the choices of as defined above. Suppose is a square diagonal matrix. The action , where is the eigenvalue of corresponding weight vector . There is a map such that for , such that . The map is the weight. Now, for arbitrary partition , has weight where is the weight for , the weight of and so on. Since is trace free, In general,

(∑ ∑ )
Suppose we let , and . Then where is a highest weight vector and is the maximum times each operator can act on while all conditions are observed to either raise the first row or the second row of . Due to the nature of transitions as a consequence of the action of the sequence, the result is unique (proved later).
The representation space is simple if for all , there exist upper triangular square matrices such that , a highest weight vector. The weight vectors could be of the form ∑ where is a weight vector. We will show that there exists a sequence of upper triangular matrices such that its action on any sum of weight vectors annihilates all but one. That resulting weight vector is a highest weight vector.

Main Results
Theorem 6. The representation space is a simple module. This theorem requires a proof for many parts so we break it down into two propositions and two lemmas. Proposition 7. For every given partition there is a highest weight vector, . Proof. Suppose for integers , , , that ( ) Suppose there exists some such that (we have a total ordering) and and and so on. Also, suppose that entries in both basis vectors , are equal at the bottom, except for a certain row such that in that row, the sum of the entries (for , denote the first entry of the row and the second entry and so on) and that We can write , where is some weight vector and ∑ , is a complex number. The action ∑ where is the set of all resulting weight vectors the sum of whose entries in the row are greater than that of . Now, with a sequence of upper triangular matrices which raises the entries of ,

∑
The sequence is actually raising the weight vectors by the series of actions and the supposedly the smallest basis vector becomes a highest weight vector as a consequence. So

∑ ∑
Therefore, is a highest weight vector.
The weight for is such that So has weight , which is a highest weight. Q.E.D. Proposition 8. For any basis vector , there exists a set of upper triangular matrices, , such that where Proof. From the order introduced in Proposition 7, we see that is smaller than all other basis vectors. The action

( ∑ )
But ∑ will be annihilated by the action since its elements are bigger and will be the sequence that raises to , which is a highest weight. Q.E.D.
Lemma 9. Suppose is a non-zero element in ,

∑
Then there exists a sequence of upper triangular matrices such that Proof. From Proposition 7, for , we established that Then, for all , Since is the smallest basis, the action will be ∑ Therefore, . Q.E.D. This implies Corollary 10. If is a non-zero submodule, then . We proved from Proposition 7 that there is a highest weight vector . So if is a non-zero submodule, then . Let be a simple finite-dimensional module and be a highest weight vector, the following result claims that is generated by through applying iterative lower triangular matrices on . We can view this iterated applying as being a product in some algebra (namely the universal enveloping algebra).
Definition 11 (Monomial Basis). For a finite-dimensional module and a highest weight vector, consider the fixed basis and the monomials in these only. A given set of monomials is called a monomial basis of if is a basis of , where is a product (sequence) of some lower triangular matrices. Lemma 12. Let Then is a basis. Proof. Now, we want to show that is generated by . From the ordering in Proposition 7, we see that at least (for the lower triangular matrix ) Also, for Therefore, we can write ∑ where and ∑ . Suppose ∑ where is the size of the basis , and all . Then

∑ ( ∑ )
We can fix such that . So We know is the smallest and are linearly independent for , then . Therefore, the set is linearly independent.
We are given that is a basis of implying ( in particular) is a basis element. The cardinality of is (that is , in other words the number of basis vectors one can make from a given partition). Since has linearly independent elements, then . So spans and is a basis in . Since spans and all its elements are linearly independent, then it is all of . Therefore, the weight vector generates all of . Q.E.D.
From the above proofs, we can make out that if is a highest weight vector, a submodule of (i.e and is all of ) implies generates all of . Therefore, there is no invariant subspace of .
Corollary 13. The representation space is generated by , and moreover if is a non-zero submodule, then . This completes the proof for Theorem 6. So, the representation space is a simple module. Already, a monomial basis is constructed in Lemma 12.

Conclusion
In this paper, our representation is actually where . The map is linear and also the identity. Suppose and where are basis vectors and are non-zero coefficients. Let and . Then and are both well defined operations in our representation. Now, let and . Then and again are both well defined operations in our representation. The diagonal matrices act by a scalar; that is . In all the actions above, the results are all accounted for in formulas of Equations (2), (3) and (4). If , then is all of . So, has no invariant subspace. Therefore, is an irreducible representation of the special linear algebra, . For any partition, we can construct all possible basis vectors and modules as discussed above. We apply total ordering on basis vectors to identify the smallest basis vector. A sequence of upper triangular matrices that acts maximally on the smallest bases vector will eventually act on a set of bases vectors resulting in a total annihilation of all bases vectors but raising the smallest basis vector maximally, to a highest weight vector which has weight . We also proved that every basis vector has a sequence of upper triangular matrices that acts on it maximally to yield a highest weight vector. We proved the existence of monomial basis and gave a construction. Each of these results contributes in proving our main result, that module is simple, and has monomial basis.