The Swiss Cheese Theorem for Linear Operators with Two Invariant Subspaces

We study systems $(V,T,U_1,U_2)$ consisting of a finite dimensional vector space $V$, a nilpotent $k$-linear operator $T:V\to V$ and two $T$-invariant subspaces $U_1\subset U_2\subset V$. Let $\mathcal S(n)$ be the category of such systems where the operator $T$ acts with nilpotency index at most $n$. We determine the dimension types $(\dim U_1, \dim U_2/U_1, \dim V/U_2)$ of indecomposable systems in $\mathcal S(n)$ for $n\leq 4$. It turns out that in the case where $n=4$ there are infinitely many such triples $(x,y,z)$, they all lie in the cylinder given by $|x-y|,|y-z|,|z-x|\leq 4$. But not each dimension type in the cylinder can be realized by an indecomposable system. In particular, there are holes in the cylinder. Namely, no triple in $(x,y,z)\in (3,1,3)+\mathbb N(2,2,2)$ can be realized, while each neighbor $(x\pm1,y,z), (x,y\pm1,z),(x,y,z\pm1)$ can. Compare this with Bongartz' No-Gap Theorem, which states that for an associative algebra $A$ over an algebraically closed field, there is no gap in the lengths of the indecomposable $A$-modules of finite dimension

On saturated fusion systems and Brauer indecomposability of Scott modules

Let $p$ be a prime number, $G$ a finite group, $P$ a $p$-subgroup of $G$ and $k$ an algebraically closed field of characteristic $p$. We study the relationship between the category \Ff_P(G) and the behavior of $p$-permutation $kG$-modules with vertex $P$ under the Brauer construction. We give a sufficient condition for \Ff_P(G) to be a saturated fusion system. We prove that for Scott modules with abelian vertex, our condition is also necessary. In order to obtain our results, we prove a criterion for the categories arising from the data of $(b, G)$-Brauer pairs in the sense of Alperin-Brou\'e and Brou\'e-Puig to be saturated fusion systems on the underlying $p$-group

Multiple Flag Varieties of Finite Type

We classify all products of flag varieties with finitely many orbits under the diagonal action of the general linear group. We also classify the orbits in each case and construct explicit representatives. This generalizes the classical Schubert decompostion, which states that the GL(n)-orbits on a product of two flag varieties correspond to permutations. Our main tool is the theory of quiver representations.Comment: 18pp. to appear in Adv. Mat

Brauer-friendly modules and slash functors

This paper introduces the notion of Brauer-friendly modules, a generalisation of endo-p-permutation modules. A module over a block algebra OGe is said to be Brauer-friendly if it is a direct sum of indecomposable modules with compatible fusion-stable endopermutation sources. We obtain, for these modules, a functorial version of Dade's slash construction, also known as deflation-restriction. We prove that our slash functors, defined over Brauer-friendly categories, share most of the very useful properties that are satisfied by the Brauer functor over the category of p-permutation OGe-modules. In particular, we give a parametrisation of indecomposable Brauer-friendly modules, which opens the way to a complete classification whenever the fusion-stable sources are classified. Those tools have been used to prove the existence of a stable equivalence between non-principal blocks in the context of a minimal counter-example to the odd Z*p-theorem.Comment: 20 page