Permutation Weights for Affine Lie Algebras

The Weyl-Kac character formula gives a beautiful closed-form expression for the characters of integrable highest-weight modules of Kac-Moody algebras. It is not, however, a formula that is combinatorial in nature, obscuring positivity. In this paper we show that the theory of Hall-Littlewood polynomials may be employed to prove Littlewood-type combinatorial formulas for the characters of certain highest weight modules of the affine Lie algebras C_n^{(1)}, A_{2n}^{(2)} and D_{...

We introduce an affine Schur algebra via the affine Hecke algebra associated to Weyl group of affine type C. We establish multiplication formulas on the affine Hecke algebra and affine Schur algebra. Then we construct monomial bases and canonical bases for the affine Schur algebra. The multiplication formula allows us to establish a stabilization property of the family of affine Schur algebras that leads to the modified version of an algebra ${\mathbf K}^{\mathfrak c}_n$. We ...

We present a new combinatorial formula for Hall-Littlewood functions associated with the affine root system of type $\tilde A_{n-1}$, i.e. corresponding to the affine Lie algebra $\hat{\mathfrak{sl}}_n$. Our formula has the form of a sum over the elements of a basis constructed by Feigin, Jimbo, Loktev, Miwa and Mukhin in the corresponding irreducible representation. Our formula can be viewed as a weighted sum of exponentials of integer points in a certain infinite-dimensio...

Kostant's weight $q$-multiplicity formula is an alternating sum over a finite group known as the Weyl group, whose terms involve the $q$-analog of Kostant's partition function. The $q$-analog of the partition function is a polynomial-valued function defined by $\wp_q(\xi)=\sum_{i=0}^k c_i q^i$, where $c_i$ is the number of ways the weight $\xi$ can be written as a sum of exactly $i$ positive roots of a Lie algebra $\mathfrak{g}$. The evaluation of the $q$-multiplicity formula...

Weyl groups are ubiquitous, and efficient algorithms for them -- especially for the exceptional algebras -- are clearly desirable. In this paper we provide several of these, addressing practical concerns arising naturally for instance in computational aspects of the study of affine algebras or Wess-Zumino-Witten conformal field theories. We also discuss the efficiency and numerical accuracy of these algorithms.

We study connections between the ring of symmetric functions and the characters of irreducible finite-dimensional representations of quantum affine algebras. We study two families of representations of the symplectic and orthogonal Lie algebras. One is defined via combinatorial properties and is easy to calculate; the other is closely related to the $q=1$ limit of the ``minimal affinization'' representations of quantum affine algebras. We conjecture that the two families are ...

The behavior of objects associated with general extended affine Lie algebras is typically distinct from their counterparts in affine Lie algebras. Our research focuses on studying characters and Cartan automorphisms, which appear in the study of Chevalley involutions and Chevalley bases for extended affine Lie algebras. We show that for almost all extended affine Lie algebras, any finite order Cartan automorphism is diagonal, and its corresponding combinatorial map is a chara...

Let $\mathcal{A}$ be a Weyl arrangement in an $\ell$-dimensional Euclidean space. The freeness of restrictions of $\mathcal{A}$ was first settled by a case-by-case method by Orlik and the second author (1993), and later by a uniform argument by Douglass (1999). Prior to this, Orlik and Solomon (1983) had completely determined the exponents of these arrangements by exhaustion. A classical result due to Orlik, Solomon and the second author (1986), asserts that the exponents of ...

We give an explicit combinatorial Chevalley-type formula for the equivariant K-theory of generalized flag varieties G/P which is a direct generalization of the classical Chevalley formula. Our formula implies a simple combinatorial model for the characters of the irreducible representations of G and, more generally, for the Demazure characters. This model can be viewed as a discrete counterpart of the Littelmann path model, and has several advantages. Our construction is give...

The characters $\chi_\mu$ of nontwisted affine algebras at fixed level define in a natural way a representation $R$ of the modular group $SL_2(Z)$. The matrices in the image $R(SL_2(Z))$ are called the Kac-Peterson modular matrices, and describe the modular behaviour of the characters. In this paper we consider all levels of $(A_{r_1}\oplus\cdots\oplus A_{r_s})^{(1)}$, and for each of these find all permutations of the highest weights which commute with the corresponding Kac-...