Graph Convolution Networks (GCNs) have been widely used in skeleton-based action recognition. Although there are significant progress, the inherent limitation still lies in the restricted receptive field of GCN, hindering its ability to extract global dependencies effectively. And the joints that are structurally separated can also have strong correlation. Previous works rarely explore local and global correlations of joints, leading to insufficiently model the complex dynamics of skeleton sequences. To address this issue, we propose a GCN and Transformer complementary network (GTC-Net) that allows parallel communications between GCN and Transformer domains. Specifically, we introduce a graph convolution and self-attention combined module (GAM), which can effectively leverage the complementarity of GCN and self-attention to perceive local and global dependencies of joints for the human body. Furthermore, in order to address the problems of long-term sequence ordering and position detection, we design a position-aware module (PAM), which can explicitly capture the ordering information and unique identity information for body joints of skeleton sequence. Extensive experiments on NTU RGB+D 60 and NTU RGB+D 120 datasets are conducted to evaluate our proposed method. The results demonstrate that our method can achieve competitive results on both datasets.