Weight diagrams of finite-dimensional highest weight 𝔤𝔩(m|n)-modules

Many properties of simple finite-dimensional [Formula: see text]-modules may be better understood by assigning weight diagrams to the highest weights with respect to a given base of simple roots. In this paper, we consider bases that are compatible with the standard Borel subalgebra in [Formula: see text]; namely, the bases that differ from the distinguished base [Formula: see text] of simple roots by a sequence of odd reflections. We examine the weight diagrams that arise from the highest weights of a simple module [Formula: see text] with respect to such bases. Further, we provide combinatorial tools to describe all the weight diagrams of highest weights of [Formula: see text] provided only with the weight diagram of [Formula: see text] with respect to distinguished highest weight [Formula: see text]. Finally, we study the maximal cardinality of incomparable sets of positive odd roots with respect to [Formula: see text] which are orthogonal to some highest weight of [Formula: see text] with respect to a base as above. We provide explicit formulas for this value, connecting it to the combinatorics of the weight diagrams. Based on this study, we respond to the work of Gorelik and Heidersdorf by providing a counterexample to the Tail Conjecture appearing in Gorelik and Heidersdorf, Gruson–Serganova character formulas and the Duflo–Serganova cohomology functor, J. Reine Angew. Math. (Crelles J.) 2023(798) (2023) 1–54.