Tensor products and restrictions in type \(A\) (Q2776257)

Let \(G\) be a reductive algebraic group over an algebraically closed field \(\mathbb{F}\) of characteristic \(p>0\), and let \(\mathcal C\) denote the category of all rational \(\mathbb{F} G\)-modules. Writing \(X\) (resp. \(X^+\)) for the set of integral weights (resp. dominant integral weights) corresponding to the root system of \(G\), for each \(\lambda\in X^+\) the modules \(L(\lambda)\) and \(\nabla(\lambda)\) are the irreducible and costandard tilting modules of highest weight \(\lambda\), respectively. Let \({\mathcal C}(\mu)\) denote the linkage class corresponding to \(\mu\). There is an exact projection functor \(\text{pr}_\mu\colon{\mathcal C}\to{\mathcal C}(\mu)\) given on objects by taking the largest submodule belonging to \({\mathcal C}(\mu)\). Then the translation functor \(T^\mu_\lambda\colon{\mathcal C}(\lambda)\to{\mathcal C}(\mu)\) is the functor \(\text{pr}_\mu\circ(-\otimes\nabla(\upsilon))\) where \(\upsilon\) is the unique dominant weight conjugate under the Weyl group to \((\mu-\lambda)\), and \(\lambda,\mu\in X\) lying in the closure of the same alcove.NEWLINENEWLINENEWLINEQuestion. What does \(T^\mu_\lambda L(\mu)\) look like in general, for \(\mu,\lambda\in X^+\) lying in the closure of the same alcove but neither lying in the closure of the facet containing the other? For example, when is it non-zero? When is it irreducible or indecomposable? Can one give a lower bound on its Loewy length?NEWLINENEWLINENEWLINEThe paper under review is a survey article. The authors review some answers to these questions in the special case of \(G=\text{GL}_n(\mathbb{F})\) based on one of their papers [Proc. Lond. Math. Soc., III. Ser. 80, No. 1, 75-106 (2000; see the preceding review Zbl 1011.20042)]. They also review some general results from other papers of themselves [J. Algebra 217, No. 1, 335-351 (1999; Zbl 0974.20031)] bringing the structure of tensor products \(M\otimes\nabla(\upsilon)\) in characteristic \(p\), for a \(G\)-module \(M\). Section 4 of this paper is based on another paper by the authors [Math. Z. 232, No. 2, 287-320 (1999; Zbl 0945.20027)]. They also outline the relations between the LLT algorithm and the ideal structure of the group algebra of the finitary symmetric group.NEWLINENEWLINEFor the entire collection see [Zbl 0977.00027].