Positivity Problems And Conjectures In Algebraic Combinatory

Positivity questions occupy a central place in algebraic combinatory: given a naturally occurring symmetric or quasisymmetric, or polynomial function, when does it expand with nonnegative coefficients in a preferred basis monomial, elementary, Schurz, etc. This survey/research-style paper organizes classical and recent positivity problems, summarizes principal techniques, records key breakthroughs, and lists open conjectures and directions. We emphasize (i) classical positivity phenomena Littlewoods–Richardson, Schurz- and e- positivity, (ii) structural conjectures such as the Stanley–Stem bridge and Macdonald positivity problems and their recent status, (iii) positivity for representation-theoretic multiplicities Kroenke, platysma, and (iv) modern tools that have proved or advanced these questions. We close with a curated bibliography of key references and suggested research directions.