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Principal symmetric ideals were recently introduced by Harada et al. in [The minimal free resolution of a general principal symmetric ideal, preprint (2023), arXiv:2308.03141], where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper, we determine when a product of two principal symmetric ideals is principal symmetric and when the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case. 

Abstract Background: In bioinformatics, network alignment algorithms have been applied to protein-protein interaction (PPI) networks to discover evolutionary conserved substructures at the system level. However, most previous methods aim to maximize the similarity of aligned proteins in pairwise networks, while concerning little about the feature of connectivity in these substructures, such as the protein complexes. Results: In this paper, we identify the problem of finding conserved protein complexes, which requires the aligned proteins in a PPI network to form a connected subnetwork. By taking the feature of connectivity into consideration, we propose ConnectedAlign, an efficient method to find conserved protein complexes from multiple PPI networks. The proposed method improves the coverage significantly without compromising of the consistency in the aligned results. In this way, the knowledge of protein complexes in well-studied species can be extended to that of poor-studied species. Conclusions: We conducted extensive experiments on real PPI networks of four species, including human, yeast, fruit fly and worm. The experimental results demonstrate dominant benefits of the proposed method in finding protein complexes across multiple species.