reduced wigner coefficients for lie superalgebra gl m n corresponding to unitary representations and beyond in this paper fundamental wigner coefficients are determined algebraically by considering the eigenvalues of certain generalized casimir invariants here this method is applied in the context of both type 1 and type 2 unitary representations of the lie superalgebra gl m n extensions to the non unitary case are investigated a symmetry relation between two classes of wigner coefficients is given in terms of a ratio of dimensions

the discriminant of a decomposable symplectic involution a formula is given for the discriminant of the tensor product of the canonical involution on a quaternion algebra and an orthogonal involution on a central simple algebra of degree divisible by 4 as an application an alternative proof of shapiro s pfister factor conjecture is given for tensor products of at most five quaternion algebras

the induced representations of brauer algebras and the clebsch gordan coefficients of induced representations of from with are discussed the induction coefficients idcs or the outer product reduction coefficients of with up to a normalization factor are derived by using the linear equation method weyl tableaux for the corresponding gel fand basis of so n are defined the assimilation method for obtaining clebsch gordan coefficients of so n in the gel fand basis for no modification rule involved couplings from idcs of brauer algebras is proposed some isoscalar factors of for the resulting irrep with are tabulated

generalized cocycles with values in one units of henselian valued division algebras abstract let z f be an inertial galois extension of henselian valued fields and let d be a z central division algebra let g be a finite group acting on z with fixed field f we show that every generalized cocycle of g with values in the one units of d is cohomologous to one of the form 1 or in other words the existence of such a cocycle implies that the group action of g on z extends to a group action on d we provide applications to lifting of group actions on the residue division algebra and to the existence of kummer subfields in d given suitable data in formula

matrix elements and wigner coefficients for uq gl n in this series of papers the fundamental wigner coefficients and the matrix elements of the generators for the quantum group uq gl n are derived the uq gl n uq gl n 1 reduced wigner coefficients and reduced matrix elements are determined algebraically as eigenvalues of certain uq gl n 1 invariants in the quantum group uq gl n the matrix elements of the elementary quantum group generators in the gel fand tsetlin basis are derived below while all fundamental wigner coefficients are derived in the second paper of the series

clifford algebras of hyperbolic involutions for as a central simple algebra of even degree with hyperbolic orthogonal involution we describe the canonically induced involution underline sigma on the even clifford algebra c 0 a sigma underline sigma of a sigma when deg a equiv 0 mod 8 a cong m 2 b and the interesting part of underline sigma is isomorphic to the canonical involution on an exterior power algebra of b as a corollary we get some properties of the involution on the exterior power algebra

tensor operators and projection techniques in infinite dimensional representations of semi simple lie algebras polynomial identities satisfied by the infinitesimal generators of a semi simple lie group are employed to construct projection operators which project a tensor product representation v lambda x v v lambda finite dimensional and irreducible v an infinite dimensional representation admitting an infinitesimal character onto a primary subrepresentation such projection operators project out the generalised shift components of a tensor operator acting on an infinite dimensional representation v in direct analogy with the finite dimensional analysis of bracken and green 1971 applications of these methods to evaluating the matrix elements of the generators of the locally compact groups u p q and o p q in unitary discrete irreducible representations are discussed

the henselization of a valued division algebra the main result of this paper is the following if f v is a valued field fh its henselization d a division algebra with center f and d f then v extends to d iff d ffh is a division algebra it is also shown that if p char f p 1if char f 0 then d f d f d f pb for some nonnegative integer b

wigner coefficients for a semisimple lie group and the matrix elements of the o n generators a purely algebraic infinitesimal method for obtaining multiplicity free wigner coefficients is presented the method is applied to obtain analytic expressions for the complete matrix elements of all o n generators moreover the structure of these matrix elements in terms of reduced matrix elements wigner coefficients and reduced wigner coefficients is made explicit by comparison with the wigner eckart theorem explicit analytic expressions are obtained for the fundamental wigner coefficients of o n finally the results are presented in a form which is directly analogous with the corresponding results for u n

coupling coefficient for the symmetric and unitary groups two recursion formulae are deduced for the construction of inner and outer product isoscalar factors respectively for the symmetric group by which the corresponding coupling coefficients can be easily obtained a formula for the evaluation of the isoscalar factor for the subgroup chain u n contains implies u n 1 is proposed based on the relation between the outer product coupling coefficient for the standard basis of the symmetric group and clebsch gordan coefficient for the canonical basis of the unitary group the calculation procedures are presented and the properties of those coefficients are investigated

quantum line bundles on a noncommutative sphere a noncommutative nc sphere is introduced as a quotient of the enveloping algebra of the lie algebra su 2 following gurevich and saponov 2001 j phys a math gen 34 4533 69 and using the cayley hamilton identities we introduce projective modules which are analogues of line bundles on the usual sphere we call them quantum line bundles and define a multiplicative structure in their family also we compute a pairing between quantum line bundles and finite dimensional representations of the nc sphere in the spirit of the nc index theorem moreover we introduce projective modules being nc counterparts of tangent vector bundles and those of differential forms and define an analogue of the de rham complex making use of these modules

representation theory of the symplectic groups i certain aspects of symplectic group representation theory are investigated in particular it is shown that every irreducible representation of sp n admits a relatively large class of states referred to herein as canonical states which possess properties analogous to the gelfand tsetlin states appearing in the theory of the orthogonal and unitary groups the properties of canonical states are investigated and some matrix element formulas are derived

extensions of the conformal representations for orthogonal lie algebras abstract the conformal transformations with respect to the metric defining o n c give rise to an inhomogeneous polynomial representation of o n 2 c using shen s technique of mixed product we generalize the above representation to an inhomogeneous representation of o n 2 c on the tensor space of any finite dimensional irreducible o n c module with the polynomial space where a hidden central transformation is involved moreover we find a condition on the constant value taken by the central transformation such that the generalized representation is irreducible in our approach pieri s formulas invariant operators and the idea of kostant s characteristic identities play key roles the result could be useful in understanding higher dimensional conformal field theory with the constant value taken by the central transformation as the central charge our representations virtually provide natural extensions of the conformal transformations on a riemannian manifold to its vector bundles

matrix elements for type 1 unitary irreducible representations of the lie superalgebra gl m n using our recent results on eigenvalues of invariants associated to the lie superalgebra gl m n we use characteristic identities to derive explicit matrix element formulae for all gl m n generators particularly non elementary generators on finite dimensional type 1 unitary irreducible representations we compare our results with existing works that deal with only subsets of the class of type 1 unitary representations all of which only present explicit matrix elements for elementary generators our work therefore provides an important extension to existing methods and thus highlights the strength of our techniques which exploit the characteristic identities

poincare birkhoff witt theorems and generalized casimir invariants for some infinite dimensional lie groups ii analogues of the poincare birkhoff witt theorem for several pairs of lie groups acting on complex manifolds are studied these results lead to algebraic analogues of a theorem of i segal on the two sided regular representation of a unimodular locally compact group for dual representations of some infinite dimensional lie groups on generalized bargmann segal fock spaces generalized casimir invariants for these dual representations are also developed

the dual theory of choice under risk in this essay a new theory of choice under risk is being proposed it is a theory which in a sense that will become clear is dual to expected utility theory hence the title dual theory risky prospects are evaluated in this theory by a cardinal numerical scale which resembles an expected utility except that the roles of payments and probabilities are reversed this theme the reversal of the roles of probabilities and payments will recur throughout the paper i should emphasize that playing games with probabilities masquerading as payments and payments masquerading as probabilities is not my object rather i hope to convince the reader that the dual theory has intrinsic economic significance and that in some areas its predictions are superior to those of expected utility theory while in other areas the reverse will be the case two reasons have prompted me to look for an alternative to expected utility theory the first reason is methodological in expected utility theory the agent s attitude towards risk and the agent s attitude towards wealth are forever bonded together at the level of fundamental principles risk aversion and diminishing marginal utility of wealth which are synonymous under expected utility theory are horses of different colors the former expresses an attitute towards risk increased uncertainty hurts while the latter expresses an attitude towards wealth the loss of a sheep hurts more when the agent is poor than when the agent is rich a question arises therefore as to whether these two notions can be kept separate from each other in a full fledged theory of cardinal utility the dual theory will have this property the second reason that leads me to look for an alternative to expected utility theory is empirical behavior patterns which are systematic yet inconsistent with expected utility theory have often been observed two prominent references among many others are allais 1953 and kahneman tversky 1979 so deeply

a projection based approach to the clebsch gordan multiplicity problem for compact semisimple lie groups iii the classical limit for pt ii see ibid vol 19 p 1531 1986 this paper is the third in a series directed towards a projection based solution to the clebsch gordan multiplicity problem for semisimple compact lie groups the authors show that the project states for the clebsch gordan problem approach orthogonality in the classical limit of large quantum numbers in a manner analogous to that of elliot s 1962 well known solution to the u 3 contains implies o 3 state labelling problem

division algebras over henselian fields in this chapter we focus on the tame division algebras d with center a field f with henselian valuation v as usual we approach this by first obtaining results for graded division algebras then lifting back from operatorname mathsf gr d to d this is facilitated by results in 8 1 on existence and uniqueness of lifts of tame subalgebras from operatorname mathsf gr d to d in 8 2 we describe four fundamental canonical up to conjugacy subalgebras of d that reflect its valuative structure the rest of the chapter is devoted to brauer group factorizations of d corresponding to the noncanonical direct product decomposition of operatorname mathit br t f given in cor 7 85 the factor operatorname mathit hom c operatorname mathcal g overline f mathbb t gamma f is represented by a type of division algebra n called decomposably semiramified defined in 8 3 and characterized by the property that n contains a maximal subfield inertial over f and another totally ramified over f we show in 8 4 that every tame division algebra d is brauer equivalent to some s f t where s is inertially split and t is tame and totally ramified over f we show further that every inertially split division algebra s is brauer equivalent to some i f n where i is inertial over f and n is decomposably semiramified the classes overline i for the i appearing in the i f n decompositions of s are shown to range over a single coset of mathit dec z overline s overline f in operatorname mathit br overline f called the specialization coset of s in the final subsection 8 4 6 we summarize what happens in the special case that v is discrete of rank 1 where substantial simplifications occur

the theory of angular momenta and higher su n symmetries the authors discuss interrelations between generating invariants gi for the reduction pi mi 1 d pi to d on 1 of the m fold tensor product of su n irreducible representations irreps d pi and polynominal bases of the su m irreps d on 1j om n 1 ok 0 0 0 k times a realisation of the su m irrep d oj om 3 bases is given in terms of gi for su 2 coupling wigner coefficients as a byproduct an expression is obtained for su 2 6 j symbols in terms of only two wigner coefficients the authors also discuss some possibilities of the analysis involved in solving the wigner biedenharn problem construction of orthonormal sets of the wigner coefficients for su n groups n or 3

multiplicity free wigner coefficients for semisimple lie groups i the u n pattern calculus this is the first paper in a series of two dedicated to a new algebraic determination of the multiplicity free reduced wigner coefficients for the lie groups u n and o n the approach employed enables a direct nonrecursive derivation of reduced wigner coefficients the absolute value squared of the reduced wigner coefficients is expressed as a rational polynomial function numerator polynomial divided by denominator polynomial whose weyl group symmetries are seen to fall out naturally in our approach from the transformation properties of polynomial functions determined by casimir invariants a unified treatment of the composition laws of reduced wigner coefficients and the evaluation of their associated denominator polynomials is presented which applies to both u n and o n an explicit formula for the numerator polynomials of u n is also derived the numerator polynomials for the orthogonal groups will be given in the second paper of the series

weight bases of gelfand tsetlin type for representations of classical lie algebras this paper completes a series devoted to explicit constructions of finite dimensional irreducible representations of the classical lie algebras here the case of odd orthogonal lie algebras of type b is considered two previous papers dealt with c and d types a weight basis for each representation of the lie algebra 2n 1 is constructed the basis vectors are parametrized by gelfand tsetlin type patterns explicit formulae for the matrix elements of generators of 2n 1 in this basis are given the construction is based on the representation theory of the yangians

generalized projective representations for sl n 1 it is well known that n dimensional projective group gives rise to a non homogeneous representation of the lie algebra sl n 1 on the polynomial functions of the projective space using shen s mixed product for witt algebras also known as larsson functor we generalize the above representation of sl n 1 to a non homogeneous representation on the tensor space of any finite dimensional irreducible gl n module with the polynomial space moreover the structure of such a representation is completely determined by employing projection operator techniques and well known kostant s characteristic identities for certain matrices with entries in the universal enveloping algebra in particular we obtain a new one parameter family of infinite dimensional irreducible sl n 1 modules which are in general not highest weight type for any given finite dimensional irreducible sl n module the results could also be used to study the quantum field theory with the projective group as the symmetry

generalized gel fand invariants and characteristic identities for quantum groups the generalized gel fand invariants for an arbitrary quantum group are explicitly constructed and their eigenvalues in any irreducible representation are computed these invariants enable one to develop characteristic identities for the quantum group and as a natural application these identities are used to construct projection operators for tensor products of representations to illustrate the general theory the quantum group uq gl m is studied in detail

the pattern calculus for tensor operators in quantum groups an explicit algebraic evaluation is given for all q tensor operators with unit norm belonging to the quantum group uq u n and having extremal operator shift patterns acting on arbitrary uq u n irreps these rather complicated results are shown to be easily comprehensible in terms of a diagrammatic calculus of patterns a more conceptual derivation of these results is discussed using fundamental properties of the q 6j coefficients

new perspective on the u n wigner racah calculus ii elementary reduced wigner coefficients for u n for pt i see ibid vol 20 no 9 p 2241 50 1987 exploiting the powerful formalism of the vector coherent state theory expounded in part i the group theoretical content of the complementarity principle and a novel interpretation of the operator pattern of biedenharn and louck the authors rederive in a simple fashion all u n u n 1 reduced wigner coefficients for the elementary u n tensors

decomposition of involutions on inertially split division algebras let f be a henselian valued field with mathrm char overline f neq 2 and let s be an inertially splitf central division algebra with involution sigma ast that is trivial on an inertial lift in s of the field z overline s we prove necessary and sufficient conditions for s to contain a sigma ast stable quaternion it f subalgebra and for s sigma ast to decompose into a tensor product of quaternion algebras these conditions are in terms of decomposability of an associated residue central simple algebra overline i that arises from a brauer group decomposition of s

casimir invariants and characteristic identities for gl a full set of higher order casimir invariants for the lie algebra gl is constructed and shown to be well defined in the category ofs generated by the highest weight unitarizable irreducible representations with only a finite number of nonzero weight components moreover the eigenvalues of these casimir invariants are determined explicitly in terms of the highest weight characteristic identities satisfied by certain infinite matrices with entries from gl are also determined and generalize those previously obtained for gl n by bracken and green a j bracken and h s green j math phys 12 2099 1971 h s green ibid 12 2106 1971

the state labeling problem a universal solution a new method is proposed for labeling a basis for any irreducible representation with top weight of a simple lie algebra la it has two advantages over gel fand zetlin patterns i being exactly the same for all la s both exceptional and classical and ii providing labels that are much more compact thus it may prove useful in discussing large representations such as those e8 e8 that occur in superstring theory the method makes essential use of the weyl group and is based on a theorem that associates to each weight a symmetric matrix with integer coefficients whose rank equals the multiplicity of the weight

discriminant and clifford algebras the centralizer of a square central skew symmetric unit in a central simple algebra with orthogonal involution carries a unitary involution the discriminant algebra of this unitary involution is shown to be an orthogonal summand in one of the components of the clifford algebra of the orthogonal involution as an application structure theorems for orthogonal involutions on central simple algebras of degree 8 are obtained

a question on the discriminants of involutions of central division algebras let k be a field of characteristic not equal to 2 and d a finite dimensional central simple algebra over k whose class is 2 torsion in the brauer group br k of k by classical results of albert a ch x th 19 d admits an involution which is identity on k an invariant with values in k modulo squares in k called

orthogonal involutions on central simple algebras and function fields of severi brauer varieties abstract an orthogonal involution on a central simple algebra a after scalar extension to the function field f a of the severi brauer variety of a is adjoint to a quadratic form q over f a which is uniquely defined up to a scalar factor some properties of the involution such as hyperbolicity and isotropy up to an odd degree extension of the base field are encoded in this quadratic form meaning that they hold for the involution if and only if they hold for q as opposed to this we prove that there exists non totally decomposable orthogonal involutions that become totally decomposable over f a so that the associated form q is a pfister form we also provide examples of nonisomorphic involutions on an index 2 algebra that yield similar quadratic forms thus proving that the form q does not determine the isomorphism class of even when the underlying algebra has index 2 as a consequence we show that the e 3 invariant for orthogonal involutions is not classifying in degree 12 and does not detect totally decomposable involutions in degree 16 as opposed to what happens for quadratic forms

a class of quadratic deformations of lie superalgebras we study certain graded finite dimensional polynomial algebras of degree 2 which are a special class of deformations of lie superalgebras which we call quadratic lie superalgebras starting from the formal definition we discuss the generalized jacobi relations in the context of the koszul property and give a proof of the poincare birkhoff witt basis theorem we give several concrete examples of quadratic lie superalgebras for low dimensional cases and discuss aspects of their structure constants for the type i class we derive the equivalent of the kac module construction for typical and atypical modules and a related direct construction of irreducible modules due to gould we investigate in detail one specific case the quadratic generalization gl2 n 1 of the lie superalgebra sl n 1 we formulate the general atypicality conditions at level 1 and present an analysis of zero and one step atypical modules for a certain family of kac modules

spinor group and its restrictions a realization of the spinor algebra of the rotation group so n n 2n or 2n 1 in the covering algebra of u 2n is exploited to obtain explicit representation matrices for the so n generators in the basis adapted to the subgroup chain so n u n u n 1 u 1 as a special case the computation of matrices of u n representations characterized by a k column young tableau is reduced to the evaluation of at most k box totally symmetric representations of u 2n

invariants and reduced matrix elements associated with the lie superalgebra gl m n we construct explicit formulae for the eigenvalues of certain invariants of the lie superalgebra gl m n using characteristic identities we discuss how such eigenvalues are related to reduced wigner coefficients and the reduced matrix elements of generators and thus provide a first step to a new algebraic derivation of matrix element formulae for all generators of the algebra

on the matrix elements of the u n generators a straightforward derivation of the matrix elements of the u n generators is presented using algebraic infinitesimal techniques an expression for the general fundamental wigner coefficients of the group is obtained as a polynomial in the group generators this enables generalized matrix elements to be defined without explicit reference to basis states such considerations are important for treating groups such as sp 2n whose basis states are not known

parastatistics and the clifford algebra unitary group approach to the many electron correlation problem it is shown that the clifford algebra unitary group approach which is based on the subgroup chain u 2n so 2n 1 so 2n u n may be described in terms of the para fermi algebra applications to the development of efficient algorithms for the evaluation of matrix elements of u n generators and of their products are also briefly discussed

calibration results for non expected utility theories rabin 2000 proved that a low level of risk aversion with respect to small gambles leads to a high and absurd level of risk aversion with respect to large gambles rabin s arguments strongly depend on expected utility theory but we show that similar arguments apply to general non expected utility theories

reduced wigner coefficients for uq gl n this is the second paper in a series concerned with the racah wigner calculus for the quantum group uq gl n in this approach all reduced wigner coefficients are determined algebraically by the eigenvalues of certain generalized casimir invariants here the method is applied to evaluate all fundamental wigner coefficients and reduced wigner coefficients for uq gl n

mixture symmetry and quadratic utility the independence axiom of expected utility theory has recently been weakened to the betweenness axiom in this paper an even weaker axiom called mixture symmetry is presented the corresponding functional structure is such that utility is a betweenness functional on part of its domain and quadratic in probabilities elsewhere the experimental evidence against betweenness provides one motivation for the more general theory presented here another advantage of the mixture symmetric class of utility functions is that it is sufficiently flexible to permit the disentangling of attitudes towards risk and towards randomization

multiplicity free wigner coefficients for semisimple lie groups ii a pattern calculus for o n in this paper a direct algebraic derivation of a large class of multiplicity free reduced wigner coefficients is presented for the orthogonal groups in particular all elementary reduced wigner coefficients including those for the fundamental spinor and tensor representations are obtained the results are presented in a form directly analogous to the corresponding results obtained for the unitary groups

on the symmetry properties of outer product reduction the symmetry properties of outer product coupling coefficients for the symmetric group and of clebsch gordan coefficients for the unitary group and of the corresponding isoscalar factors are discussed a simple rule is given for determination of a phase factor of outer reduction symmetry of sn and u n the rule is provided with a relationship between the phase factor of a young diagram with its subdiagrams

new perspective on the u n wigner racah calculus i vector coherent state theory and construction of gel fand bases using a vector coherent state theory it is shown that the construction of gel fand bases for the unitary group is particularly simple the very specific rules for construction of the states greatly facilitate the subsequent computation of matrix elements of the generators of the u n lie algebra

invariants and matrix elements of the quantum group u_q gl n mathbb c revisited in a previous paper the generator matrix elements and dual vector reduced wigner coefficients rwcs were evaluated via the polynomial identities satisfied by a certain matrix constructed from the r matrix r and its twisted counterpart r t t circle r here we provide an alternative evaluation utilising the r matrix r over tilde r t 1 this provides a new direct derivation of the vector rwcs obtained indirectly in earlier work via a symmetry relation this approach has the advantage that it generalises to the lie superalgebra case which will be investigated elsewhere

a projection based solution to the sp 2n state labeling problem a projection based solution to the symplectic group state labeling problem is presented the approach yields a nonorthogonal gel fand tsetlin basis for the irreducible representations of sp 2n a method for evaluating the corresponding overlap coefficients is discussed the action of the sp 2n generators in the basis obtained is determined and some matrix element formulas are derived the results obtained are comparable to the matrix element formulas for o n and u n

enveloping algebra annihilators and projection techniques for finite dimensional cyclic modules of a semisimple lie algebra some results on the structure of finite dimensional cyclic modules for a semisimple lie algebra are presented cyclic modules arise naturally in constructing symmetry adapted states of a system using projection projecting out states with definite symmetry from an arbitrary state is related to the properties of the cyclic module generated by an important example of a cyclic module is the tensor product of two irreducible modules v v which is cyclically generated by the vector v v where v resp v is the minimal resp maximal weight vector of v resp v for this particular case we determine the explicit form of the annihilator in the universal enveloping algebra of the cyclic vector v v it is hoped that this result may add new insight into the clebsch gordan multiplicity problem as an application of this result projection operators are constructed which project from an arbitrary vector of weight a maximal weight vector of weight

relationship between sn and u n isoscalar factors and higher order u n invariants general relationships expressing u n and sn coupling and transformation isoscalar factors in terms of u n racah and 9 coefficients are derived the absolute values of u n racah coefficients involving at most k column irreducible representations are shown to be identical with su k racah coefficients in particular an explicit relationship is established between the u n and su 2 approaches to the many electron correlation problem

the generalized even clifford algebra abstract for a central simple algebra a with an orthogonal involution over a field f of characteristic not equal to two we associate to it the generalized even clifford algebrac0 a we will present results on the structure of this algebra c0 a some of which are due to jacobson in j algebra1 1964 288 300 using a generic method we then give a formula for computing c0 a when a is decomposable as algebras with involution using these results a necessary condition for decomposability of an algebra with orthogonal involution compatible with the involution is then obtained

a projection based approach to the clebsch gordan multiplicity problem for compact semisimple lie groups i general formalism the authors describe a method analogous to the elliott method 1958 for o 3 contained in implied by u 3 for resolution of the multiplicity in the decomposition into irreducibles of the tensor product of two irreducible representations of a compact semisimple lie group g the method is based on a decomposition of highest weight vectors in the product representation into direct product states focusing on components of the form e x e mu with e a weight vector and e mu a highest weight vector the weight of the vector e corresponds to the shift weight in a tensor operator formulation of the problem a result from an earlier paper gould et al 1984 based on the fact that a tensor product representation can be generated cyclically from the product of a highest and a lowest weight vector is used to give an explicit characterisation of the space of shift weight vectors e that can appear in the decomposition of a highest weight state this characterisation is in terms of lowering operators in the complexified lie algebra l of g and closely parallels verma s well known enveloping algebra characterisation 1968 of the highest weight states of finite dimensional irreducible representations of complex semisimple lie algebra

matrix elements and duality for type 2 unitary representations of the lie superalgebra gl m n the characteristic identity formalism discussed in our recent articles is further utilized to derive matrix elements of type 2 unitary irreducible gl m n modules in particular we give matrix element formulae for all gl m n generators including the non elementary generators together with their phases on finite dimensional type 2 unitary irreducible representations which include the contravariant tensor representations and an additional class of essentially typical representations remarkably we find that the type 2 unitary matrix element equations coincide with the type 1 unitary matrix element equations for non vanishing matrix elements up to a phase

highest weight irreducible unitary representations of lie algebras of infinite matrices i the algebra gl two classes of irreducible highest weight modules of the general linear lie algebra gl corresponding to two different borel subalgebras are constructed both classes contain all unitary representations within each module a basis is introduced expressions for the transformation of the basis under the action of the algebra are written down