First cohomology for finite groups of Lie type: simple modules with small dominant weights

Let $k$ be an algebraically closed field of characteristic $p > 0$, and let $G$ be a simple, simply connected algebraic group defined over $\mathbb{F}_p$. Given $r \geq 1$, set $q=p^r$, and let $G(\mathbb{F}_q)$ be the corresponding finite Chevalley group. In this paper we investigate the structure of the first cohomology group $H^1(G(\mathbb{F}_q),L(\lambda))$ where $L(\lambda)$ is the simple $G$-module of highest weight $\lambda$. Under certain very mild conditions on $p$ and $q$, we are able to completely describe the first cohomology group when $\lambda$ is less than or equal to a fundamental dominant weight. In particular, in the cases we consider, we show that the first cohomology group has dimension at most one. Our calculations significantly extend, and provide new proofs for, earlier results of Cline, Parshall, Scott, and Jones, who considered the special case when $\lambda$ is a minimal nonzero dominant weight.Comment: 24 pages, 5 figures, 6 tables. Typos corrected and some proofs streamlined over previous versio

Second cohomology for finite groups of Lie type

Let $G$ be a simple, simply-connected algebraic group defined over $\mathbb{F}_p$. Given a power $q = p^r$ of $p$, let $G(\mathbb{F}_q) \subset G$ be the subgroup of $\mathbb{F}_q$-rational points. Let $L(\lambda)$ be the simple rational $G$-module of highest weight $\lambda$. In this paper we establish sufficient criteria for the restriction map in second cohomology $H^2(G,L(\lambda)) \rightarrow H^2(G(\mathbb{F}_q),L(\lambda))$ to be an isomorphism. In particular, the restriction map is an isomorphism under very mild conditions on $p$ and $q$ provided $\lambda$ is less than or equal to a fundamental dominant weight. Even when the restriction map is not an isomorphism, we are often able to describe $H^2(G(\mathbb{F}_q),L(\lambda))$ in terms of rational cohomology for $G$. We apply our techniques to compute $H^2(G(\mathbb{F}_q),L(\lambda))$ in a wide range of cases, and obtain new examples of nonzero second cohomology for finite groups of Lie type.Comment: 29 pages, GAP code included as an ancillary file. Rewritten to include the adjoint representation in types An, B2, and Cn. Corrections made to Theorem 3.1.3 and subsequent dependent results in Sections 3-4. Additional minor corrections and improvements also implemente

(0,2)-graphs and root systems

We construct (0, 2)-graphs from root systems with simply laced diagram and study their properties

(0,2)-graphs and root systems

We construct (0, 2)-graphs from root systems with simply laced diagram and study their properties

Protein tyrosine kinase 6 promotes ERBB2-induced mammary gland tumorigenesis in the mouse

Protein tyrosine kinase 6 (PTK6) expression, activation, and amplification of the PTK6 gene have been reported in ERBB2/HER2-positive mammary gland cancers. To explore contributions of PTK6 to mammary gland tumorigenesis promoted by activated ERBB2, we crossed Ptk6(−/−) mice with the mouse mammary tumor virus-ERBB2 transgenic mouse line expressing activated ERBB2 and characterized tumor development and progression. ERBB2-induced tumorigenesis was significantly delayed and diminished in mice lacking PTK6. PTK6 expression was induced in the mammary glands of ERBB2 transgenic mice before tumor development and correlated with activation of signal transducer and activator of transcription 3 (STAT3) and increased proliferation. Disruption of PTK6 impaired STAT3 activation and proliferation. Phosphorylation of the PTK6 substrates focal adhesion kinase (FAK) and breast cancer anti-estrogen resistance 1 (BCAR1; p130CAS) was decreased in Ptk6(−/−) mammary gland tumors. Reduced numbers of metastases were detected in the lungs of Ptk6(−/−) mice expressing activated ERBB2, compared with wild-type ERBB2 transgenic mice. PTK6 activation was detected at the edges of ERBB2-positive tumors. These data support roles for PTK6 in both ERBB2-induced mammary gland tumor initiation and metastasis, and identify STAT3, FAK, and BCAR1 as physiologically relevant PTK6 substrates in breast cancer. Including PTK6 inhibitors as part of a treatment regimen could have distinct benefits in ERBB2/HER2-positive breast cancers