Hypermatrix factors for string and membrane junctions

The adjoint representations of the Lie algebras of the classical groups SU(n), SO(n), and Sp(n) are, respectively, tensor, antisymmetric, and symmetric products of two vector spaces, and hence are matrix representations. We consider the analogous products of three vector spaces and study when they appear as summands in Lie algebra decompositions. The Z3-grading of the exceptional Lie algebras provide such summands and provides representations of classical groups on hypermatrices. The main natural application is a formal study of three-junctions of strings and membranes. Generalizations are also considered.Comment: 25 pages, 4 figures, presentation improved, minor correction

XTREME-UP: A User-Centric Scarce-Data Benchmark for Under-Represented Languages

Data scarcity is a crucial issue for the development of highly multilingual NLP systems. Yet for many under-represented languages (ULs) -- languages for which NLP re-search is particularly far behind in meeting user needs -- it is feasible to annotate small amounts of data. Motivated by this, we propose XTREME-UP, a benchmark defined by: its focus on the scarce-data scenario rather than zero-shot; its focus on user-centric tasks -- tasks with broad adoption by speakers of high-resource languages; and its focus on under-represented languages where this scarce-data scenario tends to be most realistic. XTREME-UP evaluates the capabilities of language models across 88 under-represented languages over 9 key user-centric technologies including ASR, OCR, MT, and information access tasks that are of general utility. We create new datasets for OCR, autocomplete, semantic parsing, and transliteration, and build on and refine existing datasets for other tasks. XTREME-UP provides methodology for evaluating many modeling scenarios including text-only, multi-modal (vision, audio, and text),supervised parameter tuning, and in-context learning. We evaluate commonly used models on the benchmark. We release all code and scripts to train and evaluate model