LieART 2.0 -- A Mathematica Application for Lie Algebras and Representation Theory

Abstract

We present LieART 2.0 which contains substantial extensions to the Mathematica application LieART (Lie Algebras and Representation Theory) for computations frequently encountered in Lie algebras and representation theory, such as tensor product decomposition and subalgebra branching of irreducible representations. The basic procedure is unchanged-it computes root systems of Lie algebras, weight systems and several other properties of irreducible representations, but new features and procedures have been included to allow the extensions to be seamless. The new version of LieART continues to be user friendly. New extended tables of properties, tensor products and branching rules of irreducible representations are included in the appendix for use without Mathematica software. LieART 2.0 now includes the branching rules to special subalgebras for all classical and exceptional Lie algebras up to and including rank 15.

Spin chains with boundary inhomogeneities

“…, m n ) for an eigenvalue Λ (m 1 ,...,mn) (u), Eqs. (4.17)-(4.18) determine the Dynkin labels of the corresponding "left" and "right" representations, from which one can deduce (e.g., using LieART [36]) their dimensions, and therefore the eigenvalue's degeneracy.…”

Machine Learning Lie Structures & Applications to Physics

“…For the SU (5) branching, we use 1,000 irreps below 70,000 dimension as input vectors, with binary YES(=1)/NO(=0) output depending on presence/absence of a bi-fundamental rep of SU (3) × SU (2). Training is done by choosing 20% of the data, and validation on the complementary data.…”

“…Traditionally, an indispensable tool to the high energy physicist is the extensive tables of [1]. More contemporary usage, with the advent of computing power of the ordinary laptop, have relied on the likes of highly convenient software such as "LieART" [2]. Such computer algebra methods, especially in conjunction with the familiarity of the Wolfram programming language to the theoretical physicists, are clearly destined to play a helpful rôle.…”

GroupMath: A Mathematica package for group theory calculations

“…Less stringent conditions apply to a semi-standard Young tableau. 11 It is filled with the natural numbers up to some value c which does not need to be related to the shape λ, and repetitions are allowed. Numbers must increase along each column and they cannot decrease along rows.…”

“…This is by no means the only available tool for performing group theory calculations. A likely incomplete list of other programs includes LiE [7], GAP [8], Affine.m [9] and LieART [10,11], the latter two being Mathematica packages just like GroupMath. Furthermore, the reader also has at its disposal several references containing a large amount of pre-computed Lie algebra data [12,13].…”