The Benson-Symonds Invariant for Permutation Modules

The Benson -- Symonds Invariant for Ordinary and Signed Permutation Modules

The signed permutation modules are a simultaneous generalization of the ordinary permutation modules and the twisted permutation modules of the symmetric group. In a recent paper Dave Benson and Peter Symonds defined a new invariant $γ_G(M)$ for a finite dimensional module $M$ of a finite group $G$ which attempts to quantify how close a module is to being projective. In this paper, we determine this invariant for all the signed permutation modules of the symmetric group using tools from representation theory and combinatorics.

On the Castelnuovo-Mumford regularity of the cohomology of fusion systems and of the Hochschild cohomology of block algebras

Symonds' proof of Benson's regularity conjecture implies that the regularity of the cohomology of a fusion system and that of the Hochschild cohomology of a p-block of a finite group is at most zero. Using results of Benson, Greenlees, and Symonds, we show that in both cases the regularity is equal to zero.

张量积与对称群代数的稳定绿色环

Tensor Product and the Stable Green Ring of the Symmetric Group Algebra $F\mathfrak{S}_p$

研究对称群代数F S_p的不可约非投射模的张量积分解，证明其模投射模为半单，并计算Benson-Symonds不变量。

Recursive sequences attached to modular representations of finite groups

The core of a finite-dimensional modular representation $M$ of a finite group $G$ is its largest non-projective summand. We prove that the dimensions of the cores of $M^{\otimes n}$ have algebraic Hilbert series when $M$ is Omega-algebraic, in the sense that the non-projective summands of $M^{\otimes n}$ fall into finitely many orbits under the action of the syzygy operator $Ω$. Similarly, we prove that these dimension sequences are eventually linearly recursive when $M$ is what we term $Ω^{+}$-algebraic. This partially answers a conjecture by Benson and Symonds. Along the way, we also prove a…

离散群的下降与有限置换分解

Descent and finite permutation resolutions for discrete groups

对于具有有限维proper作用模型空间的离散群和有限整体维数的交换Noether环，证明了投射kG-模的同伦范畴、Mazza-Symonds定义的kG-模的稳定模范畴以及具有有限迷向的置换kG-模的导出范畴可以下降至有限子群，并应用该结果推广了Balmer-Gallauer的定理。

模 $p$-群上模表示的渐进行为

Asymptotic behavior of modular representations over abelian $p$-groups

研究模表示论中 $p$-群上表示环的渐进行为，推导了张量幂核心维度的渐近阶，反驳了 Benson 和 Symonds 的问题，并系统计算了 $Ω$-代数模的核心级数。