Balanced notation for $\tau$-rigid pairs

Kiyoshi Igusa; Gordana Todorov

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arXiv · 2607.11456

Balanced notation for $\tau$-rigid pairs

Kiyoshi Igusa·Gordana Todorov

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Abstract

We introduce a new notation for $\tau$-rigid pairs called ``balanced pairs''. The notation $\frac AB$ allows for simplification of some formulas, for example the Jasso category and its dual. Using this balanced notation we show that the $g$-vector $g(\frac AB)$ has nice geometric and algebraic implications: It defines ``lower'' and ``upper'' chambers and we prove that this corresponds to generalized Bongartz and co-Bongartz completions of balanced pairs. We also show that the balanced notation agrees the wall labels for the semi-invariant picture in the hereditary case and, in the general case, we describe the relation between balanced notation and the wall labels.

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BibTeX RIS

Kiyoshi Igusa, Gordana Todorov. 2026-07-13. Balanced notation for $\tau$-rigid pairs. https://arxiv.org/abs/2607.11456

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