Acta Polytechnica doi:10.14311/AP.2013.53.0450 Acta Polytechnica 53(5):450–456, 2013 © Czech Technical University in Prague, 2013 available online at http://ojs.cvut.cz/ojs/index.php/ap NOTE ON VERMA BASES FOR REPRESENTATIONS OF SIMPLE LIE ALGEBRAS Severin Pošta∗, Miloslav Havlíček Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, CZ-120 00 Prague, Czech Republic ∗ corresponding author: severin@km1.fjfi.cvut.cz Abstract. We discuss the construction of the Verma basis of the enveloping algebra and of finite dimensional representations of the Lie algebra An. We give an alternate proof of so-called Verma inequalities to the one given in [1] by P. Littelmann. Keywords: Verma basis, enveloping algebra, Lie algebra. Submitted: 7 May 2013. Accepted: 2 July 2013. 1. Introduction The theory of simple Lie groups and their representa- tions (and corresponding representations of simple Lie algebras) has been at the center of interest of modern mathematics for a long time, because it has many relationships with other areas of mathematics and physics. The simple Lie algebras over the field of complex numbers were classified in the famous works of Killing and Cartan in the 1930s. Since then we have known that there are four infinite series An, Bn, Cn, Dn, which are called the classical Lie algebras, and five Lie algebras E6, E7, E8, F4 and G2, which we call exceptional Lie algebras. The structure of these Lie algebras is described in terms of special finite sets of elements in a Euclidean space, called roots, which generate a root system. Weyl’s theorem assures that each finite dimensional reducible representation of such a Lie algebra is completely reducible. Therefore in the theory of finite dimensional representations of the semisimple Lie algebras, which are direct sums of simple ones, it is sufficient to restrict to irreducible finite dimensional representations. The complete classification of these irreducible finite dimensional representations is known. Their sets are parametrized by vectors of nonnegative integers called highest weights. Moreover, the characters and dimensions of such irreducible finite dimensional representations are explicitly known because of the Weyl formula [2–5]. Practical use of the simple Lie groups and Lie algebras, serving as a fundamental tool for studying the symmetries of systems examined in physics, often involve constructing the bases of the spaces on which their finite dimensional representations act. The best known example of such constructions are two works of Gelfand and Tsetlin. In two famous papers (see [6] and [7]), they gave an explicit construc- tion of bases for a general linear Lie algebra gl(n,C) (resp. special linear Lie algebra An) and for orthogo- nal Lie algebras Bn and Dn (for detailed comments, see also [8]). These papers contain no comments and no methods for deriving the explicit formulas. These papers also do not contain any references (the hint that one has to verify the commutation relations by direct calculation is not very useful for the proof). It is therefore no wonder that their formulas were re-derived and verified by other authors. Verification and/or an independent derivation of these formulas was given in the papers by Baird and Biedenharn (see [9, 10]), and also by other authors [11–14]. After Gelfand and Tsetlin’s construction of rep- resentations, in the second half of the 20th century and later, a range of different approaches were devel- oped and many techniques were adopted to construct the bases of the representations of classical series of Lie algebras. We can mention here the Gould paper [15], which made use of polynomial identities satisfied by the generators of the corresponding Lie group, an approach which was then generalized then to Kac-Moody algebras [16], and the approach of Asherova, Smirnov and Tolstoy involving projection operators [17, 18], which prove their usefulness also in the field of Lie superalgebras and quantum algebras [19]. The results of Tarasov and Nazarov [20] also belong to this group. Another approach, based on using Weyl realization [21] of the representations of corresponding groups in tensor spaces, was developed in many papers [22–26]. So-called special bases were constructed by de Concini and Kazhdan [27], and their q-analogs by Xi [28]. Proper bases were constructed by Gelfand and Zelevinsky [29], Retakh and Zelevinsky [30], and similar good bases were constructed by Mathieu [31]. Another well known group of bases are crystal bases. They were constructed by Lusztig [32, 33], Kashiwara [34], Du [35, 36], Kang [37], and others [38–41]. 450 http://dx.doi.org/10.14311/AP.2013.53.0450 http://ojs.cvut.cz/ojs/index.php/ap vol. 53 no. 5/2013 Note on Verma Bases for Representations of Simple Lie Algebras 2. Verma bases Besides these approaches, an important role is played by bases which have special properties as bases in the universal enveloping algebra of a given simple Lie algebra, and which can then be restricted by taking a suitable subset to the basis of a given representation of that Lie algebra. Such bases are called monomial bases, and were constructed in the standard mono- mial theories developed by Lakshmibai, Musili and Seshadri [42], by Littelmann [1, 43], its q-analogs by Chari and Xi [44] and others. One of the advantages of these bases is that the basis vectors are eigenvec- tors of Cartan subalgebra, and therefore such a basis is suitable for various modifications. On the other hand, there is no explicit form of matrix elements of operators expressed in these bases. Verma bases as introduced in [45] are of this type. These bases were constructed for the Lie algebras An in [46] and for some concrete examples of other Lie algebras of low rank (see also [47]). In [48, 49] proof of the so-called Verma conjecture for the Lie algebra An was given by Raghavan and Sankaran. Note that the basis in enveloping algebra from which we can obtain the corresponding Verma basis by restriction was given in [1]. The basis vectors of the Verma basis are con- structed from the highest weight vector (vacuum state) in a way consisting of the action of some speci- fied sequence of the elements corresponding to simple roots. Each set of basis vectors is constructed using sequences given by a certain set of inequalities (called Verma inequalities). Let us briefly describe the main result for the Lie algebra An. Let An = sl(n + 1,C) = n+ ⊕h⊕n− be the decom- position of the Lie algebra sl(n + 1,C) into strictly upper triangular, diagonal and strictly lower trian- gular matrices. Denote U(An), U(A+n ) and U(A−n ) corresponding enveloping algebras of An, n+ and n−. Let Φ be the root system of An and fixed h such that Φ = Φ+ ∪ Φ−, where n+ = ⊕ β∈Φ+ gβ. For the positive root β ∈ Φ+ denote by fβ ∈ gβ and eβ ∈ g−β fixed elements of the Chevalley basis of An. For a fixed ordering of simple roots {β1, . . . ,βn} de- note fβj by fj and the corresponding eβj by ej. Put hj = [ej,fj]. Then the set of the following monomials (so-called Verma monomials), f a11 1 ( f a22 2 f a21 1 )( f a33 3 f a32 2 f a31 1 ) · · · ( f ann n f ann−1 n−1 · · ·f an2 2 f an1 1 ) , where ack ≤ a c k+1 (1) is a linear basis of U(A−n ). A similar basis consisting of vectors generated by appropriate sequences of ej’s spans U(A+n ). Together with the enveloping algebra of h one can obtain a basis of the whole U(An). If we now restrict to the elements generated by sequences fulfilling Verma inequalities 0 ≤ ack ≤ min{a c k−1 + λn−c+k,a c+1 k+1}, where an+1k = +∞ and a k 0 = 0 for all k, acting on the highest weight vector |0〉 (vacuum state) with the highest weight (λ1, . . . ,λn), where λj are nonnegative integers, we obtain a basis of the representation space of the corresponding finite dimensional representa- tion. 3. Verma monomials inequalities As a contribution to the above discussion, we give an alternate proof of (1) to the proof given in [1]. Lemma 3.1. For any n,m ≥ 1 and k ≥ 0 we have fni f k i−1f m i ∈ span{f k i−1f n+m i , fk−1i−1 f n+m i fi−1, . . . ,f n+m i f k i−1}, (2) fn+mi−1 f n i ∈ span{f n i−1f n i f m i−1, fn−1i−1 f n i f m+1 i−1 , . . . ,f n i f m+n i−1 }, (3) and (2), (3) in which fi and fi−1 are interchanged. Proof. To show (2) we first prove the following iden- tity: for any m ≥ 1 and i = 2, 3, . . . ,n we have fifi−1f m i = 1 m + 1 ( fm+1i fi−1 + mfi−1f m+1 i ) . (4) For m = 1, (4) follows from the fact that[ fi, [fi,fi−1] ] = 0. Now let us assume validity for m and calculate the equation fifi−1f m+1 i = 1 2 (f2i fi−1 + fi−1f 2 i )f m i = 1 2(m + 1) fi(fm+1i fi−1 + mfi−1f m+1 i ) + 1 2 fi−1f m+2 i = 1 2(m + 1) fm+2i fi−1 + m 2(m + 1) fifi−1f m+1 i + 1 2 fi−1f m+2 i , from which, isolating the term fifi−1fm+1i we get the desired result. We now generalize formula (4) to the form fif k i−1f m i = 1 m + 1 ( kfk−1i−1 f m+1 i fi−1 + (m−k + 1)fki−1f m+1 i ) . (5) This formula is proved similarly by induction on k. Multiplying both sides of (5) by fi−1, we obtain fi−1fif k i−1f m i = 1 m + 1 ( kfki−1f m+1 i fi−1 + (m−k + 1)fk+1i−1 f m+1 i ) , fi−1fif k i−1f m i = 1 2 ( f2i−1fifif 2 i−1 ) fk−1i−1 f m i = 1 2 f2i−1 1 m + 1 ( (k − 1)fk−2i−1 f m+1 i fi−1 + (m−k + 2)fk−1i−1 f m+1 i ) + 1 2 fif k+1 i−1 f m i . 451 S. Pošta, M. Havlíček Acta Polytechnica Extracting terms fifk+1i−1 f m i we obtain (5) for k + 1. The last step is to prove the following identity: for any n,k,m ≥ 0 we have fni f k i−1f m i = 1( m+n n ) n∑ l=0 ( k l )( m−k + n n− l ) ×fk−li−1 f m+n i f l i−1. (6) This can be shown by induction on n. To show (3) we apply Dixmier’s antiisomorphism (see [3], 2.2.18., p. 73) to (6) to obtain the relation fmi f k i−1fi = 1 m + 1 ( kfi−1f m+1 i f k−1 i−1 + (m−k + 1)fm+1i f k i−1 ) , which allows inductively to prove fn+1i−1 f n i = n+1∑ l=1 (−1)l+1 ( n + 1 l ) fn+1−li−1 f n i f l i−1. (7) Multiplying (7) by fi−1 and repeatedly applying (7) to the right-hand side we finally obtain (3). Replacing fi and fi−1 and using a similar approach, we subsequently obtain the following formulas: fi−1fif m i−1 = 1 m + 1 ( fm+1i−1 fi + mfif m+1 i−1 ) , fi−1f k i f m i−1 = 1 m + 1 ( kfk−1i f m+1 i−1 fi + (m−k + 1)fki f m+1 i−1 ) , fni−1f k i f m i−1 = 1( m+n n ) n∑ l=0 ( k l )( m−k + n n− l ) ×fk−li f m+n i−1 f l i, fn+1i f n i−1 = n+1∑ l=1 (−1)l+1 ( n + 1 l ) fn+1−li f n i−1f l i. Due to the Poincaré-Birkhoff-Witt theorem, or- dered monomials es1212 e s13 13 · · ·e s1,n+1 1,n+1 e s23 23 e s24 24 · · ·e s2,n+1 2,n+1 · · ·esn−1,n+1n−1,n+1e sn,n+1 n,n+1 , sij ≥ 0, (8) where ei,i+1 = fi and eik = [ei,k−1,ek−1,k], i + 1 < k, (9) form a basis of U(A−n ). Let us consider any such monomial and denote it by v. The relations (9) express any generator eik, i < k as a commutator of simple ones f1, . . . , fn. Therefore v is a linear combination of (unordered) monomials from these simple generators and such monomials can be written in the form v′ = v1fr1n v2f r2 n · · ·vmf rm n vm+1, (10) where vi ∈ U(A−n−1) ⊂ U(A − n ) and the mono- mials v2,v3, . . . ,vm 6∈ U(A−n−2) (i. e. they con- tain generator fn−1, otherwise we use the relation frnvjf s n = fr+sn vj and the product can be truncated). Theorem 3.2. Let us denote VR,m = span { v1f r1 n v2f r2 n · · ·vmf rm n vm+1 ∣∣ vi ∈ U(A−n−1), r1 + r2 + · · · + rm = R } . (11) Then we have v′ ∈ VR,1. Proof. By induction on n. If n = 2, we have v′ = fs11 f r1 2 f s2 1 f r2 2 · · ·f sm 1 f rm 2 f sm+1 1 . (12) If m ≥ 2, we use formula (2) from lemma 3.1 applied to the product fr12 f s2 1 f r2 2 and we obtain v ′ ∈ VR,m−1. In the general case we write vi = wifsin w ′ i, w,w ′ ∈ U(A−n−1), and, therefore, [w,fn+1] = [w′,fn+1] = 0. For monomial v′ we can write v′ = v1fr1n+1w2f s2 n w ′ 2f r2 n+1 · · · = v1w2fr1n+1f s2 n f r2 n+1w ′ 2 · · · and we use the same argument as in the case n = 2. It follows from the above theorem that the set U(A−n ) is spanned by monomials of a special type. When n = 2 these monomials are of the form {fs11 f r1 2 f s2 1 |s1,s2,r1 ≥ 0}. Due to formula (3) from lemma 3.1 the monomials fs11 f r1 2 f s2 1 , where s1 > r1 are linearly dependent on those having s1 ≤ r1, therefore we can restrict to the set {fs11 f r1 2 f s2 1 |s1,s2,r1 ≥ 0, s1 ≤ r1}. (13) We can generalize this assertion for n > 2 as follows. Theorem 3.3. U(A−n ) = span { fk1n1 f k2n 2 · · ·f knn n f k1,n−1 1 f k2,n−1 2 · · · f kn−1,n−1 n−1 . . .f k12 1 f k22 2 f k11 1 ∣∣kij ≤ ki+1,j, i = 1, . . . ,n− 1, j = 1, . . . ,n } . (14) (We call Verma monomials those monomials appear- ing on the right hand side of this equality.) Proof. We proceed by induction on n. For n = 2 the assertion is true, now assume validity for n and we prove for n + 1. It follows from the preceeding lemma that it is sufficient to consider v ∈ U(A−n+1) such that 452 vol. 53 no. 5/2013 Note on Verma Bases for Representations of Simple Lie Algebras it can be written as v1fRn+1v2, where v1,v2 ∈ U(A−n ) and v1 is a Verma monomial of the form v1 = fk1n1 f k2n 2 · · ·f knn n f k1,n−1 1 f k2,n−1 2 . . . ×fkn−1,n−1n−1 . . .f k12 1 f k22 2 f k11 1 . Therefore v1f R n+1v2 = f k1n 1 f k2n 2 · · ·f knn n f R n+1 ×fk1,n−11 f k2,n−1 2 · · ·f kn−1,n−1 n−1 . . .f k12 1 f k22 2 f k11 1 v2︸ ︷︷ ︸ v′ and, due to the induction hypothesis, v′ is a linear combination of Verma monomials. Now if R ≥ knn, then the product fk1n1 f k2n 2 · · ·f knn n f R n+1 is already a Verma monomial and the proof is finished. When R < knn, we rewrite the product fknnn fRn+1 using (2) from lemma 3.1 to the form fknnn f R n+1 = a0f R n f R n+1f knn−R n + a1fR−1n f R n+1f knn−R+1 n + · · · + aRf R n+1f knn n , ai being suitable complex constants. From this we conclude that fk1n1 f k2n 2 · · ·f knn n f R n+1 ∈ span{u1fRn+1w1,u2f R n+1w2, . . . ,unf R n+1wn}, (15) where ui,wi are Verma monomials from U(A−n ) and the highest degree of the simple root fn in vi is less or equal to R. Therefore the product (15) is a linear combination of Verma monomials from U(A−n+1), as desired. Linear independence of Verma monomials can be shown as follows. We make use of commuting opera- tors ad hi : U(A−n ) → U(A−n ) defined by ad hiv = [hi,v], i = 1, . . . ,n. (16) The algebra U(A−n ) decomposes into the direct sum U(A−n ) = ⊕ z1,...,zn V (z1, . . . ,zn) of common eigenspaces of operators ad hi V (z1, . . . ,zn) = { v ∈ U(A−n ) ∣∣ ad hiv = ziv, i = 1, . . . ,n } . Two vectors belonging to different subspaces are linearly independent. Lemma 3.4. (1.) PBW monomials (8) of U(A−n ) are eigenvectors of all ad hi. (2.) v ∈ V (z1, . . . ,zn) iff s12 + s13 + · · · + s1,n+1 = m1, s23 + · · · + s2,n+1 = m2 + s12, ... sn,n+1 = mn + s1n + s2n + · · · + sn−1,n, (17) where m = − 1 n + 1 C1z, (18) and m =   m1 m2 m3 ... mn−1 mn   , C1 =   −n −(n− 1) −(n− 2) · · · −2 −1 1 −(n− 1) −(n− 2) · · · −2 −1 1 2 −(n− 2) · · · −2 −1 ... ... ... ... ... 1 2 3 · · · −2 −1 1 2 3 · · · (n− 1) −1   , z =   z1 z2 z3 ... zn−1 zn   . Proof. (1.) The assertion is direct consequence of the fact that generators ejk, j < k are eigenvectors of ad hi. (2.) System (17) was obtained using generators eii, eii = 1 n + 1 ( c1 − n+1−i∑ j=1 jhn+1−j+ n∑ j=n+2−i (n + 1 − j)hn+1−j ) , ad eiiv = 1 n + 1 ( − n+1−j∑ j=1 jzn+1−j+ n∑ j=n+2−i (n + 1 − j)zn+1−j ) v ≡ miv. c1 = e11 +· · ·+enn stands for the Casimir operator. Note that the matrix which appears in (18) is the inverse of the Cartan matrix of the algebra An. From equations (17) we see that mi are all nonnegative 453 S. Pošta, M. Havlíček Acta Polytechnica integers. The dimension of V (z1, . . . ,zn) is finite since for fixed right hand sides of equations (18) there is only a finite number of decompositions of m1 into the sum of s12, s13, . . . , s1,n+1, etc. For each of the possibilities (s12, . . . ,s1,n+1,s21, . . . ,s2,n+1, . . . , . . . ,sn,n+1) (19) we obtain a basis vector v ∈ V (z1, . . . ,zn). By ex- hausting all these possibilities we obtain the basis of V (z1, . . . ,zn). Lemma 3.5. Verma monomial v = ( fl1n1 · · ·f ln−1,n n−1 f kn n )( f l1,n−1 1 · · ·f ln−2,n−1 n−2 f kn−1 n−1 ) · · · ( fl121 f k2 2 )( fk11 ) ∈ V (z1, . . . ,zn) (20) iff   z1 z2 ... zn−1 zn   =   −2 −1 0 · · · 0 0 −1 −2 −1 · · · 0 0 ... ... ... ... ... 0 0 0 · · · −2 −1 0 0 0 · · · −1 −2   ×   l1n + l1,n−1 + · · · + l12 + k1 l2n + l2,n−1 + · · · + l2,3 + k2 ... ln−1,n + kn−1 kn   . (21) Proof. By direct calculation using relations [fi,hj] = cijfi, cij = 2δij − δi,j+1 −δi,j−1, cij = 0 for |i− j| > 1 and, consequently [fαi ,hj] = αcijf α i . Note that inverting the Cartan matrix and matrix from eq. (21) we can rewrite system (21) to the form l1n + l1,n−1 + · · · + l12 + k1 = m1, l2n + l2,n−1 + · · · + k2 = m1 + m2, ... ln−1,n + kn−1 = m1 + m2 + · · · + mn−1, kn = m1 + m2 + · · · + mn. (22) Theorem 3.6. If the numbers m1, . . . ,mn are fixed, then there is a bijective mapping between the set of all solutions (19) of (17) and the set of all solutions( l1n, l1,n−1, . . . , l12,k1, l2n, l2,n−1, . . . , l23,k2, . . . , . . . , ln−1,n,kn−1 ) of the system (22). Proof. The explicit form of this bijection is s12 = k1 s1t = l1,t−1, t = 3, . . . ,n + 1, sr,r+1 = kr − lr−1,r, srt = lr,t−1 − lr−1,t−1, t = r + 2, . . . ,n + 1, sn,n+1 = mn + kn−1. Bijectivity is the consequence of the fact that Verma monomials form the spanning set of U(A−n ). Corollary 3.7. All Verma monomials are linearly independent. 4. Conclusions Problems of unified construction of Verma bases for other series of simple Lie algebras (namely orthogonal and symplectic) and of an effective determination of matrix elements in these cases are still open. In [50, 51], Kang and Lee developed the notion of Gröbner- Shirshov pairs. In this way, the reduction problem in representation theory was solved and monomial bases of representations of various associative algebras could be constructed. The algebra An was among the first examples. Note that the bases obtained there are different from Verma bases. It is an interesting question whether Verma bases can be derived this way. Acknowledgements S. P. acknowledges support from grant no. P201/10/1509, a project of the Grant Agency of the Czech Republic. References [1] P. Littelmann. An algorithm to compute bases and representation matrices for SLn+1-representations. J Pure Appl Algebra 117/118:447–468, 1997. Algorithms for algebra (Eindhoven, 1996). [2] N. Bourbaki. Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines. Actualités Scientifiques et Industrielles, No. 1337. Hermann, Paris, 1968. [3] J. Dixmier. Algèbres enveloppantes. Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics]. Éditions Jacques Gabay, Paris, 1996. Reprint of the 1974 original. [4] J. E. Humphreys. Introduction to Lie algebras and representation theory, vol. 9 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1978. Second printing, revised. [5] K. Erdmann, et al. Introduction to Lie algebras. Springer Undergraduate Mathematics Series. Springer-Verlag London Ltd., London, 2006. [6] I. M. Gel′fand, et al. Finite-dimensional representations of the group of unimodular matrices. Doklady Akad Nauk SSSR (NS) 71:825–828, 1950. 454 vol. 53 no. 5/2013 Note on Verma Bases for Representations of Simple Lie Algebras [7] I. M. Gel′fand, et al. Finite-dimensional representations of groups of orthogonal matrices. Doklady Akad Nauk SSSR (NS) 71:1017–1020, 1950. [8] A. I. Molev. Gelfand-Tsetlin bases for classical Lie algebras. In Handbook of algebra. Vol. 4, vol. 4 of Handb. Algebr., pp. 109–170. Elsevier/North-Holland, Amsterdam, 2006. [9] L. C. Biedenharn. On the representations of the semisimple Lie groups. I. The explicit construction of invariants for the unimodular unitary group in N dimensions. J Math Phys 4:436–445, 1963. [10] G. E. Baird, et al. On the representations of the semisimple Lie groups. II. J Math Phys 4:1449–1466, 1963. [11] D. P. Želobenko. Compact Lie groups and their representations. American Mathematical Society, Providence, R.I., 1973. Translated from the Russian by the Israel Program for Scientific Translations, Translations of Mathematical Monographs, Vol. 40. [12] J. G. Nagel, et al. Operators that lower or raise the irreducible vector spaces of Un−1 contained in an irreducible vector space of Un. J Math Phys 6:682–694, 1965. [13] H. Pei-yu. Orthonormal bases and infinitesimal operators of the irreducible representations of group Un. Sci Sinica 15:763–772, 1966. [14] F. Lemire, et al. Formal analytic continuation of Gel′fand’s finite-dimensional representations of gl(n, C). J Math Phys 20(5):820–829, 1979. [15] M. D. Gould. The characteristic identities and reduced matrix elements of the unitary and orthogonal groups. J Austral Math Soc Ser B 20(4):401–433, 1977/78. [16] M. D. Gould, et al. Characteristic identities for Kac- Moody algebras. Lett Math Phys 22(2):91–100, 1991. [17] R. M. Ašerova, et al. Projection operators for simple Lie groups. II. General scheme for the construction of lowering operators. The case of the groups SU(n). Teoret Mat Fiz 15(1):107–119, 1973. [18] R. M. Ašerova, et al. Projection operators for simple Lie groups. Teoret Mat Fiz 8(2):255–271, 1971. [19] Y. F. Smirnov. Projection operators for Lie algebras, superalgebras, and quantum algebras. In Latin-American School of Physics XXX ELAF (Mexico City, 1995), vol. 365 of AIP Conf. Proc., pp. 99–116. Amer. Inst. Phys. Press, Woodbury, NY, 1996. [20] M. Nazarov, et al. Representations of Yangians with Gelfand-Zetlin bases. J Reine Angew Math 496:181–212, 1998. [21] H. Weyl. The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J., 1939. [22] R. C. King, et al. Standard Young tableaux and weight multiplicities of the classical Lie groups. J Phys A 16(14):3153–3177, 1983. [23] A. Berele. Construction of Sp-modules by tableaux. Linear and Multilinear Algebra 19(4):299–307, 1986. [24] R. C. King, et al. Construction of orthogonal group modules using tableaux. Linear and Multilinear Algebra 33(3-4):251–283, 1993. [25] K. Koike, et al. Young-diagrammatic methods for the representation theory of the classical groups of type Bn, Cn, Dn. J Algebra 107(2):466–511, 1987. [26] P. Exner, et al. Canonical realizations of classical Lie algebras. Czech J Phys 26B(11):1213–1228, 1976. [27] C. De Concini, et al. Special bases for SN and GL(n). Israel J Math 40(3-4):275–290 (1982), 1981. [28] N. H. Xi. Special bases of irreducible modules of the quantized universal enveloping algebra Uv (gl(n)). J Algebra 154(2):377–386, 1993. [29] I. M. Gel′fand, et al. Multiplicities and proper bases for gln. In Group theoretical methods in physics, Vol. II (Yurmala, 1985), pp. 147–159. VNU Sci. Press, Utrecht, 1986. [30] A. V. Zelevinskĭı, et al. The fundamental affine space and canonical basis in irreducible representations of the group Sp4. Dokl Akad Nauk SSSR 300(1):31–35, 1988. [31] O. Mathieu. Good bases for G-modules. Geom Dedicata 36(1):51–66, 1990. [32] G. Lusztig. Canonical bases arising from quantized enveloping algebras. J Amer Math Soc 3(2):447–498, 1990. [33] G. Lusztig. Canonical bases arising from quantized enveloping algebras. II. Progr Theoret Phys Suppl (102):175–201 (1991), 1990. Common trends in mathematics and quantum field theories (Kyoto, 1990). [34] M. Kashiwara. Crystalizing the q-analogue of universal enveloping algebras. Commun Math Phys 133(2):249–260, 1990. [35] J. Du. Canonical bases for irreducible representations of quantum GLn. Bull London Math Soc 24(4):325–334, 1992. [36] J. Du. Canonical bases for irreducible representations of quantum GLn. II. J London Math Soc (2) 51(3):461–470, 1995. [37] S.-J. Kang. Representations of quantum groups and crystal base theory. In Algebra and Topology 1992 (Taejŏn), pp. 189–210. Korea Adv. Inst. Sci. Tech., Taejŏn, 1992. [38] K. Misra, et al. Crystal base for the basic representation of Uq (sl(n)). Commun Math Phys 134(1):79–88, 1990. [39] Y. M. Zou. Crystal bases for Uq (sl(2, 1)). Proc Amer Math Soc 127(8):2213–2223, 1999. [40] G. Cliff. Crystal bases and Young tableaux. J Algebra 202(1):10–35, 1998. [41] D. P. Zhelobenko. Crystal bases and the problem of reduction in classical and quantum modules. In Lie groups and Lie algebras: E. B. Dynkin’s Seminar, vol. 169 of Amer. Math. Soc. Transl. Ser. 2, pp. 183–202. Amer. Math. Soc., Providence, RI, 1995. [42] V. Lakshmibai, et al. Geometry of G/P . IV. Standard monomial theory for classical types. Proc Indian Acad Sci Sect A Math Sci 88(4):279–362, 1979. 455 S. Pošta, M. Havlíček Acta Polytechnica [43] P. Littelmann. Cones, crystals, and patterns. Transform Groups 3(2):145–179, 1998. [44] V. Chari, et al. Monomial bases of quantized enveloping algebras. In Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998), vol. 248 of Contemp. Math., pp. 69–81. Amer. Math. Soc., Providence, RI, 1999. [45] S. P. Li, et al. Verma bases for representations of classical simple Lie algebras. J Math Phys 27(3):668–677, 1986. [46] J. Patera. Verma bases for representations of rank two Lie and Kac-Moody algebras. In Proceedings of the 14th ICGTMP (Seoul, 1985), pp. 289–292. World Sci. Publishing, Singapore, 1986. [47] M. E. Hall. Verma bases of modules for simple Lie algebras. ProQuest LLC, Ann Arbor, MI, 1987. Thesis (Ph.D.)–The University of Wisconsin - Madison. [48] K. N. Raghavan, et al. On Verma bases for representations of sl(n, C. J Math Phys 40(4):2190–2195, 1999. [49] K. N. Raghavan, et al. Erratum: “On verma bases for representations of sl(n, C)” [J. Math. Phys. 40 (1999), no. 4, 2190–2195; MR1683149 (2000b:17011)]. J Math Phys 41(5):3302, 2000. [50] S.-J. Kang, et al. Gröbner-Shirshov bases for representation theory. J Korean Math Soc 37(1):55–72, 2000. [51] S.-J. Kang, et al. Gröbner-Shirshov bases for irreducible sln+1-modules. J Algebra 232(1):1–20, 2000. 456 Acta Polytechnica 53(5):450–456, 2013 1 Introduction 2 Verma bases 3 Verma monomials inequalities 4 Conclusions Acknowledgements References