We present a sum-product formula for the classical Hall polynomial which is based on tableaux that have been introduced by T. Klein in 1969. In the formula, each summand corresponds to a Klein tableau, while the product is taken over the cardinalities of automorphism groups of short exact sequences which are derived from the tableau. For each such sequence, one can read off from the tableau the summands in an indecomposable decomposition, and the size of their homomorphism and automorphism groups. Klein tableaux are refinements of Littlewood–Richardson tableaux in the sense that each entry ℓ ≥ 2 carries a subscript r. We describe module theoretic and categorical properties shared by short exact sequences which have the same symbol ℓ r in a given row in their Klein tableau. Moreover, we determine the interval in the Auslander–Reiten quiver in which the indecomposable sequences of p n -bounded groups which carry such a symbol occur.
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机译：我们提出了基于经典的Hall多项式的和积公式，该公式基于T. Klein于1969年引入的tableaux。在该公式中，每个被加数都对应于Klein tableau，而乘积则取自自同构的基数从画面派生的一组短精确序列。对于每个这样的序列，可以从表格中读取不可分解的分解中的求和数，以及它们的同构和同构群的大小。 Klein tableaux是Littlewood–Richardson tableaux的改进，从意义上说，每个条目≥2都带有一个下标r。我们将描述由短精确序列共享的模块理论和分类属性，这些精确序列在其Klein表的给定行中具有相同的符号ℓ r 。此外，我们确定在Auslander–Reiten颤动中的间隔，在该间隔中出现带有该符号的p n 界的不可分解序列。
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