Simple Lie Algebras

A new family of polyhedral models for tensor product multiplicities of all simply-connected simple Lie groups (or simple Lie algebras) were obtained in [2]. They improve Berenstein-Zelevinsky models [1] and are expected to be as nice as hive model of Knutson-Tao for GLn [3].

The new models have clean presentations:

{g | Hg ≥ 0, g^Tσ = (μ, ν, λ)}

The lattice points in the above polytope count the tensor product multiplicity c(μ, ν, λ) in the decomposition

L(μ) Ä L(ν) = ⊕ c(μ, ν, λ) L(λ),

where L(λ) is the irreducible representation of highest weight λ. We identify each λ by non-negative integral vectors via fundamental weights as the software LiE.

The matrix σ can be directly read off from the iARt quiver. (Here is a picture of an iARt quiver of E8, see [2, Appendix A] for more iARt quivers). The rows of H are given by dimension vectors of subrepresentations of 3n boundary representations of the quiver (see [2] for detail).

For type A,B,C,D, and G2, the matrix H can be easily written down. For type F4 and E6, it is still possible to be done by a human being. An algorithm is provided in [2, Appendix B.3] to compute H for all types. Here we offer the H-matrices for exceptional types for downloading. You can plug the H and σ matrices to your favorite lattice point counting programs (eg., Latte, Polymake, and Normaliz). It is expected to be much faster than LiE if (μ, ν, λ) is sufficiently large.

1. A. Berenstein, A. Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 (2001), 77-128.

2. J. Fei, Tensor product multiplicities via upper cluster algebras, to appear Annales de l'ENS (2021), arXiv:1603.02521.

3. A. Knutson, T. Tao, The honeycomb model of GLn(C) tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), no. 4, 1055-1090.