Classification of thick tensor ideals of tilting $GL_m$-modules via nilpotent orbits

Classify the thick tensor ideals in $Tilt GL_m$ over fields of characteristic greater than $m$ in terms of orbits in the nilpotent cone and iteratively defined centralizers of the corresponding $SL_2$-homomorphisms.

Background

The paper constructs an embedding SL2×ℓ↪GLmSL_2^{\times\ell}\hookrightarrow GL_m in characteristic two as a replacement for a principal SL2SL_2-homomorphism, which does not exist for m>2m>2 in that characteristic. The construction is motivated by a conjectural description of thick tensor ideals in tilting-module categories in larger characteristic.

The cited classification is expected to use geometric data from the nilpotent cone, together with centralizers and an iterative procedure. The paper uses this perspective heuristically and confirms aspects of it in the characteristic-two setting, but does not establish the general classification for characteristic greater than mm.

References

We will use $\phi_{m}$ to pull back thick tensor ideals of $TiltSL_2{\times\ell}$ to $GL_m$. This is in line with a conjectural classification of thick tensor ideals in $TiltGL_m$ over fields of characteristic $p>m$ in cite{AHR2} (in which case $TiltGL_m=Tilt{\bullet}GL_m$). Indeed, the latter proposes to classify thick tensor ideals in terms of orbits in the nilpotent cone $\mathcal{N}$, and the centralisers of the $SL_2$-homomorphisms corresponding to these nilpotent elements (performed in an iterative manner, where one subsequently considers nilpotent orbits in these centralisers).

— Maximal ideals in the finitary symmetric group algebra in characteristic two  (2608.18782 - Coulembier, 19 Aug 2026) in Section 3, subsection “The replacement of the principal $SL_2$,” Remark following the examples