Triangulated Invariants of the Hecke Node

The singularity category of the pair factor has exactly two indecomposable objects:

B0,   B3,

with

Sigma(B0) = B3,
Sigma(B3) = B0,
Sigma^2 = Id.

Everything else follows formally.

Thick subcategories

Any nonzero thick subcategory containing B0 also contains Sigma(B0)=B3, hence the whole category. Likewise for B3.

Therefore the only thick subcategories are

0,   D_sg(R).

So the node is triangulated-irreducible.

Rouquier dimension

Because B3 = Sigma(B0), the closure of B0 under finite sums, summands, and shifts already contains every indecomposable object. Thus

dim D_sg(R) = 0.

Autoequivalences on isoclasses

Any exact autoequivalence must permute the two indecomposable isoclasses and commute with shift. Hence, on isomorphism classes, there are only two possibilities:

id,   Sigma.

So the autoequivalence group on isoclasses is

C2,

generated by the shift.

Grothendieck group and decomposability

The Grothendieck group is

K0(D_sg(R)) ≅ Z/2Z,

and there is no nontrivial semiorthogonal decomposition, since such a decomposition would induce a nontrivial thick subcategory.

Full singular Hecke order

For

O_sing ≅ Z_3(V7) × R,

the smooth V7 branch vanishes in the singularity category. So all these invariants carry over unchanged to the full Hecke node.

This is the sharpest compressed answer so far: after modding out perfect objects, the Hecke node is a 2-object, 2-periodic, triangulated-irreducible category of Rouquier dimension 0 and Grothendieck group Z/2.