The Endpoint Adjugate Identity: A Universal PIM Relation for Brauer Tree Algebras and the Torus-Crossing Phenomenon for GL(2)

Abstract

We prove that for the e × e tridiagonal Cartan matrix C(e,m) of a Brauer tree algebra with e-edge line tree and exceptional multiplicity m ≥ 1 at one endpoint, the last row of the adjugate matrix adj(C(e,m)) is independent of m, with entries (−1)e−1−j(j + 1). This independence is proved via an explicit continuant-recurrence computation of the (j,e − 1)cofactors. As a corollary, for any cyclic defect block whose Brauer tree is an e-edge line with exceptional vertex at an endpoint—including unipotent cyclic-defect blocks of GL(n,Fq) in the line-tree endpoint-exceptional regime—the normalized PIM dimensions satisfy a universal linear relation: after normalization by |D|, the resulting linear combination depends only on the ordinary character degrees of the block, and is independent of m and the ℓ-part parameter encoded by m. For GL(2,Fq) with ℓ | q+1 (the case e = 2), the identity specializes to , recovering the order of a single factor of the split maximal torus from block data governed by the nonsplit torus. We interpret this torus-crossing phenomenon in the context of the Drinfeld curve cohomology, the Bonnaf´e–Rouquier derived equivalence, and the Hiraga–Ichino–Ikeda formal degree formula, and formulate conjectures connecting the identity’s coefficients to Artin conductor data under the local Langlands correspondence. Computational verification spans 224 distinct blocks for GL(2), 19 blocks for GL(3) with e = 3 (where the identity recovers the cuspidal degree (q − 1)2(q + 1)), adjugate computations for all e ≤ 8 and m ≤ 100, and 10 non-line tree topologies supporting a generalization conjecture.

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