Weyl Orbit Characters and Schur Functions

Transactions of the American Mathematical Society, 2014

We establish an irreducibility property for the characters of finite dimensional, irreducible representations of simple Lie algebras (or simple algebraic groups) over the complex numbers, i.e., that the characters of irreducible representations are irreducible after dividing out by (generalized) Weyl denominator type factors. For SL(r) the irreducibility result is the following: let λ = (a 1 ≥ a 2 ≥ • • • a r−1 ≥ 0) be the highest weight of an irreducible rational representation V λ of SL(r). Assume that the integers a 1 + r − 1, a 2 + r − 2, • • • , a r−1 + 1 are relatively prime. Then the character χ λ of V λ is strongly irreducible in the following sense: for any natural number d, the function χ λ (g d), g ∈ SL(r, C) is irreducible in the ring of regular functions of SL(r, C).

2018

We prove a character formula for irreducible highest weight modules over a simple affine vertex algebra of level k, attached to a simple Lie algebra g, which are locally g-finite, in the cases when g is of type An andCn (n≥2) and k = −1. We also conjecture a character formula for types D4, E6, E7, E8 and levels k = −1, ..., −b, where b = 2, 3, 4, 6 respectively.

arXiv (Cornell University), 2000

Advances in Mathematics, 1981

of preserving elements. 3.5. Alternative descriptions of the indicial polynomials. 4. The simple modules over certain skew-polynomial rings. 4.1. The S-torsion simple A-modules. 4.2. Indicial polynomials and S-torsion simple A-modules. 4.3. Conditions for A/A n S-'Ab to he simple. 4.4. The theorem determining the S-torsionfree simple A-modules. 4.5. Remarks on the commutative case. 5. The simple modules over sI(2, K). 5.1. Reduction to the Casimir element c acting as a scalar. 5.2. The embeddmgs p1 and CA of Us,= Us/Us(c -y) into 23. 5.3. The K[e]-torsion simple s-modules. 5.4. Sufficient conditions for the module M(u, y) to be simple. 5.5. The theorem determining the K[e]torsionfree simple s-modules. 6. The simple modules over b. 6.1. The K[e]-torsion simple bmodules. 6.2. The theorem determining the K[e]-torsionfree simple b-modules. 6.3. The relation between simple m-modules and simple %-modules. 6.4. The relation between simple b-modules and simple s-modules. 7. Examples. 7.1. The socles over 9I and p1 US of the degree one modules. 7.2. Harish-Chandra modules. References.

2005

In this paper we study the category C q of finite-dimensional representations of a quantum loop algebra U. Our aim is to study and to put into a common representation theoretic framework, two kinds of characters which have been associated to an object of C q . One is the notion of q-characters defined in which is analogous in this context, to the usual notion of a character of a finite-dimensional representation of a simple Lie algebra. The other, is the notion of the elliptic character defined in which plays the role of the central character for representations of semi-simple Lie algebras. Both kinds of characters are needed in our situation, because the category C q is not semi-simple and hence the problem of determining the blocks in this category becomes important.

Journal of Physics A: Mathematical and Theoretical, 2009

The orbits of Weyl groups W (An) of simple An type Lie algebras are reduced to the union of orbits of the Weyl groups of maximal reductive subalgebras of An. Matrices transforming points of the orbits of W (An) into points of subalgebra orbits are listed for all cases n ≤ 8 and for the infinite series of algebra-subalgebra pairs An ⊃ A n−k−1 × A k × U 1 , A 2n ⊃ Bn, A 2n−1 ⊃ Cn, A 2n−1 ⊃ Dn. Numerous special cases and examples are shown.

arXiv: Quantum Algebra, 2017

In this paper we study the family of prime irreducible representations of quantum affine $\lie{sl}_{n+1}$ which arise from the work of D. Hernandez and B. Leclerc. These representations can also be described as follows: the highest weight is a product of distinct fundamental weights with parameters determined by requiring that the representation be minimal by parts. We show that such representations admit a BGG-type resolution where the role of the Verma module is played by the local Weyl module. This leads to a closed formula (the Weyl character formula) for the character of the irreducible representation as an alternating sum of characters of local Weyl modules. In the language of cluster algebras our Weyl character formula describes an arbitrary cluster variable in terms of the generators $x_1,\cdots,x_n,x_1',\cdots, x_n'$ of an appropriate cluster algebra. Our results also exhibit the character of a prime level two Demazure module as an alternating linear combination of ...

Lettere Al Nuovo Cimento, 1977

ricevuto il 5 0ttobre 1976)

International Mathematics Research Notices, 2010

We show that the characters of all highest weight modules over an affine Lie algebra with the highest weight away from the critical hyperplane are meromorphic functions in the positive half of the Cartan subalgebra, their singularities being at most simple poles at zeros of real roots. We obtain some information about these singularities.

1996

We associate to outer automorphisms of generalized Kac-Moody algebras generalized character-valued indices, the twining characters. A character formula for twining characters is derived which shows that they coincide with the ordinary characters of some other generalized Kac-Moody algebra, the so-called orbit Lie algebra. Some applications to problems in conformal field theory, algebraic geometry and the theory of sporadic simple groups are sketched.

2020

We generalize the famous weight basis constructions of the finite-dimensional irreducible representations of sl(n,C) obtained by Gelfand and Tsetlin in 1950. Using combinatorial methods, we construct one such basis for each finite-dimensional representation of sl(n,C) associated to a given skew Schur function. Our constructions use diamond-colored distributive lattices of skew-shaped semistandard tableaux that generalize some classical Gelfand–Tsetlin (GT) lattices. Our constructions take place within the context of a certain programmatic study of poset models for semisimple Lie algebra representations and Weyl group symmetric functions undertaken by the first-named author and many collaborators. Key aspects of the methodology of that program are recapitulated here. This work is applied here to extend combinatorial results about classical GT lattices to our more general lattices; to obtain a new combinatorial proof of a Zelevinsky–Stembridge generalization of the Littlewood–Richards...

International Journal of Theoretical Physics, 2014

The characters of irreducible finite dimensional representations of compact simple Lie group G are invariant with respect to the action of the Weyl group W (G) of G. The defining property of the new character-like functions ('hybrid characters') is the fact that W (G) acts differently on the character term corresponding to the long roots than on those corresponding to the short roots. Therefore the hybrid characters are defined for the simple Lie groups with two different lengths of their roots. Dominant weight multiplicities for the hybrid characters are determined. The formulas for 'hybrid dimensions' are also found for all cases as the zero degree term in power expansion of the 'hybrid characters'.

Noncommutative Geometry and Physics 3, 2012

Nuclear Physics B, 1990

General field theoretic methods are developed which will allow a path integral derivation of the character formula for loop groups. The methods are introduced in the classical Weyl character case. The irreducible representations of a compact semi-simple Lie group G are realized as the ground States of a supersymmetric quantum mechanical system. The Hilbert space for the quantum mechanical system is the space of sections of a holomorphic line bundle L over the complex manifold G/T, where T is the maximal torus of G. The Weyl character formula is derived by an explicit path integral computation of the index of the Dolbeault operator 3L * This work was supported in part by the * We use the quantum mechanical convention of not distinguishing between the angular momentum operators in a specific representation and the abstract generators.

Given any simple Lie superalgebra g, we investigate the structure of an arbitrary simple weight g-module. We introduce two invariants of simple weight modules: the shadow and the small Weyl group. Generalizing results of Fernando and Futorny we show that any simple module is obtained by parabolic induction from a cuspidal module of a Levi subsuperalgebra. Then we classify the cuspidal Levi subsuperalgebras of all simple classical Lie superalgebras and of the Lie superalgebra W(n). Most of them are simply Levi subalgebras of g 0 , in which case the classification of all finite cuspidal representations has recently been carried out by one of us (Mathieu). Our results reduce the classification of the finite simple weight modules over all classical simple Lie superalgebras to classifying the finite cuspidal modules over certain Lie superalgebras which we list explicitly.