Hall polynomials via automorphisms of short exact sequences

TL;DR

This paper introduces a new sum-product formula for Hall polynomials using Klein tableaux, linking combinatorial tableau data with automorphism groups of short exact sequences in module theory.

Contribution

It provides a novel combinatorial and categorical framework connecting Klein tableaux with automorphism groups and indecomposable sequences in module categories.

Findings

Sum-product formula for Hall polynomials based on Klein tableaux

Identification of automorphism group sizes for short exact sequences

Characterization of indecomposable sequences in the Auslander-Reiten quiver

Abstract

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 $\ell \geq 2$ carries a subscript $r$. We describe module theoretic and categorical properties shared by short exact sequences which have the same symbol $\ell_r$ in a given row in their Klein tableau. Moreover, we determine the interval in the Auslander-Reiten quiver in which indecomposable sequences of $p^n$-bounded groups which carry such a symbol occur.