Steinberg Tensor Products with Good Filtrations

Definitions and mathematical statement

The supplied record does not specify the algebraic group $G$, its base field or characteristic, or the allowed values of $r$. Thus its intended scope cannot be recovered completely. The following gives an explicit conditional interpretation in a conventional setting; these hypotheses are not supplied source facts.

Suppose that $k$ is an algebraically closed field of characteristic $p>0$, that $G$ is a connected, simply connected semisimple algebraic group over $k$, and that $r\geq 1$ is an integer. Fix a maximal torus $T\subseteq G$ and a system of positive roots. Let $X(T)=\operatorname{Hom}(T,\mathbb G_m)$ be the character lattice, let $X^+\subseteq X(T)$ consist of the weights whose pairing with every simple coroot is nonnegative, and let $\rho$ be half the sum of the positive roots. For $\lambda\in X^+$, let $L(\lambda)$ denote the finite-dimensional simple rational $G$-module of highest weight $\lambda$. In this setting, define the $r$-th Steinberg module by \[ \mathrm{St}_r=L\left((p^r-1)\rho\right). \] For $\mu\in X^+$, let $\nabla(\mu)$ denote the dual Weyl, or costandard, rational $G$-module of highest weight $\mu$. A finite-dimensional rational $G$-module $M$ has a good filtration if it admits a finite filtration by rational $G$-submodules \[ 0=M_0\subset M_1\subset\cdots\subset M_n=M \] such that, for every $1\leq i\leq n$, there is a weight $\mu_i\in X^+$ with $M_i/M_{i-1}\cong\nabla(\mu_i)$.

For fixed parameters in this conditional setting, the source question is to determine exactly the set \[ \left\{\lambda\in X^+ : \mathrm{St}_r\otimes_k L(\lambda)\text{ has a good filtration as a rational }G\text{-module}\right\}, \] where the tensor product has the diagonal $G$-action. The intended hypotheses on $G$, $k$, $p$, and $r$, and whether the classification is requested uniformly over any of these parameters, remain unresolved in the supplied record.

Short English statement

Determine which dominant weights $\lambda$ make the tensor product of the $r$-th Steinberg module with the simple module $L(\lambda)$ admit a filtration with dual Weyl module factors. The source omits the ambient hypotheses, so the displayed formulation is conditional and requires review against the original report.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002876$, title "Steinberg Tensor Products with Good Filtrations," Oberwolfach Reports Open Problems collection, exact source problem code catalogue entry. No individual proposer is supplied; the proposed year is recorded as 2015.
• Supplied report citation: Cohomology of Finite Groups: Interactions and Applications (2016), Oberwolfach Reports, pp. 1317--1384, DOI 10.4171/owr/2015/24. https://doi.org/10.4171/owr/2015/24. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The report citation has not been independently consulted here.
• Supplied literature reference: On tensoring with the Steinberg representation. https://arxiv.org/abs/1804.00613.
• Supplied literature reference: Counterexamples to the Tilting and (p,r)-Filtration Conjectures. https://arxiv.org/abs/1901.06687.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the Steinberg-tensor good-filtration question is settled only in broad special ranges and cases, while the general classification of $\lambda$ remains open. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that its mathematical content was preserved while notation, typography, or omitted definitions were repaired for publication. This is a supplied editorial assertion, not mathematical verification of this draft.