arXiv 1408.22732014-08math.RT

SL3（k）的线丛上同调群的支撑簇

Support varieties of line bundle cohomology groups for SL3 (k)

William D. Hardesty

备注：: 32 pages

摘要：

令$G= SL_3(k)$，其中$k$是特征$p > 0$的字段，并且令$λ\in X(T)$是具有$G/B$上的相应线丛$\mathscr{L}(λ)$的任何权重。在本文中，我们计算支持品种的形式$H^i(λ):= H^i(G/B, \mathscr{L}(λ))$的所有模块的第一Frobenius核$G_1$。该计算涉及到Donkin给出的某些递归特征标公式，这些公式可用于计算线丛上同调群的特征标。在$λ$是$p$ -正则权重且对于某些$i$是$M=H^i(λ)\neq 0$的情况下，这些公式用于表明单位$ζ$的任何$p^{th}$根不是$M$的类属维数的根。为了处理的情况下，$λ$是不$p$ -定期，我们采用类似的技术所使用的Drupieski，中野和Parshall表明，模块$H^i(λ)$是不投射在$G_1$，只要它是非零和$λ$在于外面的斯坦伯格块。

英文摘要：

Let $G= SL_3(k)$ where $k$ is a field of characteristic $p > 0$ and let $λ\in X(T)$ be any weight with corresponding line bundle $\mathscr{L}(λ)$ on $G/B$. In this paper we compute the support varieties for all modules of the form $H^i(λ):= H^i(G/B, \mathscr{L}(λ))$ over the first Frobenius kernel $G_1$. The calculation involves certain recursive character formulas given by Donkin which can be used to compute the characters of the line bundle cohomology groups. In the case where $λ$ is a $p$-regular weight and $M=H^i(λ)\neq 0$ for some $i$, these formulas are used to show that any $p^{th}$ root of unity $ζ$ is not a root of the generic dimension of $M$. To handle the case where $λ$ is not $p$-regular, we employ techniques similar to those used by Drupieski, Nakano and Parshall to show that the module $H^i(λ)$ is not projective over $G_1$ whenever it is nonzero and $λ$ lies outside of the Steinberg block.

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