Support varieties of line bundle cohomology groups for

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• 作者：William D. Hardesty ^{hardesty@math.uga.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Support varieties ; Sheaf cohomology ; SL3SL3 ; A2A2 ; Character formulas ; Generic dimension ; Quantum dimension
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 February 2016
• 年：2016
• 卷：448
• 期：Complete
• 页码：127-173
• 全文大小：1403 K

文摘

Let 11057c1d6" title="Click to view the MathML source">G=SL₃(k)$G={\mathit{SL}}_3(k)$ where k   is a field of characteristic p>0$p>0$ and let λ∈X(T)$\lambda \in X(T)$ be any weight with corresponding line bundle L(λ)$\mathcal{L}(\lambda )$ on G/B$G/B$. In this paper we compute the support varieties for all modules of the form Hⁱ(λ):=Hⁱ(G/B,L(λ))$H^i(\lambda ):=H^i(G/B,\mathcal{L}(\lambda ))$ over the first Frobenius kernel G₁$G_1$. The calculation involves certain recursive character formulas given by Donkin (cf. [5]) 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=Hⁱ(λ)≠0$M=H^i(\lambda )\ne 0$ for some i, these formulas are used to show that any pth 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 (cf. [7]) to show that the module Hⁱ(λ)$H^i(\lambda )$ is not projective over G₁$G_1$ whenever it is nonzero and λ lies outside of the Steinberg block.