Residue Ext Table and Block Decomposition

The residue algebra is generated by

E = T_3,   N = T_11,   E^2 = E,   N^2 = EN = NE = 0.

A ≅ F_3 × F_3[ε]/(ε^2).

There are exactly two simple characters:

chi_v7: E=1, N=0
chi_pair: E=0, N=0

Regular module

In the cyclic basis [w, T_3w, T_11w], the residue regular module splits as

A_reg ≅ chi_v7 ⊕ P_pair

with - chi_v7 line spanned by [0, 1, 0] - P_pair plane spanned by [[1, 2, 0], [0, 0, 1]]

So the regular module is not an indecomposable length-3 bridge between the two characters. It is a direct sum of the split branch and the unique non-split self-extension of chi_pair.

Ext^1 classification

Ext^1(top, bottom)	nonzero Yoneda elements	non-split isomorphism classes	verdict
`Ext^1(chi_pair, chi_pair)`	`2`	`1`	`one-dimensional, unique non-split module`
`Ext^1(chi_pair, chi_v7)`	`0`	`0`	`split only`
`Ext^1(chi_v7, chi_pair)`	`0`	`0`	`split only`
`Ext^1(chi_v7, chi_v7)`	`0`	`0`	`split only`

Consequence

The only nontrivial residue extension is

Ext^1(chi_pair, chi_pair) ≅ F_3,

while the cross-block groups vanish:

Ext^1(chi_v7, chi_pair) = Ext^1(chi_pair, chi_v7) = 0.

So at residue level there is no genuine indecomposable bridge between the split V7 branch and the glued V1/V5 branch. The node is block-diagonal at the level of simples, and the only non-semisimplicity is the self-extension of the glued pair block.