标题： 权图的$gl(m|n)$模块 显示英文标题

标题： Weight Diagrams of $gl(m|n)$ Modules

摘要： 许多关于 gl(m|n) 的单有限维模的性质可以通过为相对于给定简单根基的最高权赋予权图来更好地理解。 本文我们考虑与标准 Borel 子代数在$gl(m|n)_0 = gl(m) \times gl(n)$中兼容的基；即通过奇反射序列从区分基$\Sigma^{dist}$的简单根得到的基。 我们研究了相对于此类基的单模$L(\lambda)$的最高权对应的权图。 此外，我们提供了组合工具，用以描述当仅提供区分最高权$\lambda$对应的$L(\lambda)$的权图时，$L(\lambda)$的所有最高权的权图。 Finally, we study the maximal cardinality of incomparable sets of positive odd roots with respect to $\Sigma^{dist}$ which are orthogonal to some highest weight of $L(\lambda)$ 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 M. Gorelik and Th. Heidersdorf, providing a counterexample to their Tail Conjecture. 显示英文摘要

摘要： Many properties of simple finite dimensional gl(m|n)-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 $gl(m|n)_0 = gl(m) \times gl(n)$; namely, the bases that differ from the distinguished base $\Sigma^{dist}$ of simple roots by a sequence of odd reflections. We examine the weight diagrams that arise from the highest weights of a simple module $L(\lambda)$ with respect to such bases. Further, we provide combinatorial tools to describe all the weight diagrams of highest weights of $L(\lambda)$ provided only with the the weight diagram of $L(\lambda)$ with respect to distinguished highest weight $\lambda$. Finally, we study the maximal cardinality of incomparable sets of positive odd roots with respect to $\Sigma^{dist}$ which are orthogonal to some highest weight of $L(\lambda)$ 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 M. Gorelik and Th. Heidersdorf, providing a counterexample to their Tail Conjecture.