The generating hypothesis for the stable module category of a $p$-group

A type B analogue of the category of finite sets with surjections

We define a type B analogue of the category of finite sets with surjections, and we study the representation theory of this category. We show that the opposite category is quasi-Grobner, which implies that submodules of finitely generated modules are again finitely generated. We prove that the generating functions of finitely generated modules have certain prescribed poles, and we obtain restrictions on the representations of type B Coxeter groups that can appear in such modules. Our main example is a module that categorifies the degree i Kazhdan-Lusztig coefficients of type B Coxeter arrangem…

量子自旋Brauer范畴

The quantum spin Brauer category

本文引入量子自旋Brauer范畴，构造其到对应量子群有限维1型模范畴的满函子，该范畴是自旋Brauer范畴的量子版本及Kauffman范畴的扩张，为相关研究提供新工具。

A canonical torsion theory for pro-p Iwahori-Hecke modules

Let $\mathfrak F$ be a locally compact nonarchimedean field with residue characteristic $p$ and $G$ the group of $\mathfrak{F}$-rational points of a connected split reductive group over $\mathfrak{F}$. We define a torsion pair in the category Mod$(H)$ of modules over the pro-$p$-Iwahori Hecke $k$-algebra $H$ of $G$, where $k$ is an arbitrary field. We prove that, under a certain hypothesis, the torsionfree class embeds fully faithfully into the category Mod${}^I(G)$ of smooth $k$-representations of $G$ generated by their pro-$p$-Iwahori fixed vectors. If the characteristic of $k$ is different …

Computations of generating lengths with GAP

In this paper, we discuss how to apply GAP to do computations in modular representation theory. Of particular interest is the generating number of a group algebra, which measures the failure of the generating hypothesis in the stable module category. We introduce a computational method to do this calculation and present it in pseudo-code. We have also implemented the algorithm in GAP and managed to do computations of examples that we were not able to do before. The computations lead to conjectures on the ghost numbers of the groups $Q_8$ and $A_4$.

Ghosts and Strong Ghosts in the Stable Module Category

Suppose that $G$ is a finite group and $k$ is a field of characteristic $p>0$. A ghost map is a map in the stable category of finitely generated $kG$-modules which induces the zero map in Tate cohomology in all degrees. In an earlier paper we showed that the thick subcategory generated by the trivial module has no nonzero ghost maps if and only if the Sylow $p$-subgroup of $G$ is cyclic of order 2 or 3. In this paper we introduce and study some variations of ghosts maps. In particular, we consider the behavior of ghost maps under restriction and induction functors. We find all groups satisfyin…

Triangulations of projective modules

We show that the category of projective modules over a graded commutative ring admits a triangulation with respect to module suspension if and only if the ring is a finite product of graded fields and exterior algebras on one generator over a graded field (with a unit in the appropriate degree). We also classify the ungraded commutative rings for which the category of projective modules admits a triangulation with respect to the identity suspension. Applications to two analogues of the generating hypothesis in algebraic topology are given, and we translate our results into the setting of modul…

Pullback diagrams, syzygy finite classes and Igusa-Todorov algebras

For an abelian category $\mathcal{A}$, we define the category PEx($\mathcal{A}$) of pullback diagrams of short exact sequences in $\mathcal{A}$, as a subcategory of the functor category Fun($Δ, \mathcal{A}$) for a fixed diagram category $Δ$. For any object $M$ in ${\rm PEx}(\mathcal{A}),$ we prove the existence of a short exact sequence $0 {\to} K {\to} P {\to} M {\to} 0$ of functors, where the objects are in PEx($\mathcal{A}$) and $P(i) \in {\rm Proj(\mathcal{A})}$ for any $i \in Δ$. As an application, we prove that if $(\mathcal{C}, \mathcal{D}, \mathcal{E})$ is a triple of syzygy finite cla…