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We show that the rank 10 hyperbolic Kac-Moody algebra E10 contains every simply laced hyperbolic Kac-Moody algebra as a Lie subalgebra. Our method is based on an extension of earlier work of Feingold and Nicolai.

A symmetric space is a Riemannian manifold that is “symmetric” about each of its points: for each p ∈M there is an isometry σp of M such that (σp)∗ = −I on TpM . Symmetric spaces and their local versions were studied and classified by E.Cartan in the 1920’s. In 1980 D.Ferus [F2] introduced the concept of symmetric submanifolds of Euclidean space: A… (More)

We introduce a generalization of the classical Hall–Littlewood and Kostka–Foulkes polynomials to all symmetrizable Kac–Moody algebras. We prove that these Kostka–Foulkes polynomials coincide with the natural generalization of Lusztig’s t-analog of weight multiplicities, thereby extending a theorem of Kato. For g an affine Kac–Moody algebra, we define… (More)

We consider a large class of series of symmetrizable Kac-Moody algebras (generically denoted Xn). This includes the classical series An as well as others like En whose members are of Indefinite type. The focus is to analyze the behavior of representations in the limit n → ∞. Motivated by the classical theory of An = sln+1C, we consider tensor product… (More)

- Sankaran Viswanath
- Electr. J. Comb.
- 2006

We give an elementary combinatorial proof of a special case of a result due to Bazlov and Ion concerning the Fourier coefficients of the Cherednik kernel. This can be used to give yet another proof of the classical fact that for a complex simple Lie algebra g, the partition formed by the exponents of g is dual to that formed by the numbers of positive roots… (More)

In this note, we identify a natural class of subsets of affine Weyl groups whose Poincaré series are rational functions. This class includes the sets of minimal coset representatives of reflection subgroups. As an application, we construct a generalization of the classical lengthdescent generating function, and prove its rationality.

We study t-analogs of string functions for integrable highest weight representations of the affine Kac-Moody algebra A (1) 1 . We obtain closed form formulas for certain t-string functions of levels 2 and 4. As corollaries, we obtain explicit identities for the corresponding affine Hall-Littlewood functions, as well as higher-level generalizations of… (More)

The principal objects studied in this note are Coxeter groups W that are neither finite nor affine. A well known result of de la Harpe asserts that such groups have exponential growth. We consider quotients of W by its parabolic subgroups and by a certain class of reflection subgroups. We show that these quotients have exponential growth as well. To achieve… (More)

In this article, we consider infinite, non-affine Coxeter groups. These are known to be of exponential growth. We consider the subsets of minimal length coset representatives of parabolic subgroups and show that these sets also have exponential growth. This is achieved by constructing a reflection subgroup of our Coxeter group which is isomorphic to the… (More)

We continue the study of stabilization phenomena for Dynkin diagram sequences initiated in the earlier work of Kleber and the present author. We consider a more general class of sequences than that of this earlier work, and isolate a condition on the weights that gives stabilization of tensor product and branching multiplicities. We show that all the… (More)