Osaka Journal of Mathematics

Simplical complexes associated to quivers arising from finite groups

Nobuo Iiyori and Masato Sawabe

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In this paper, we will introduce a simplicial complex $\mathrm{T}_{Q}(\mathcal{H})$ defined by a quiver $Q$ and a family $\mathcal{H}$ of paths in $Q$. We call $\mathrm{T}_{Q}(\mathcal{H})$ a path complex of $\mathcal{H}$ in $Q$. Let $G$ be a finite group, and denote by $\mathrm{Sgp}(G)$ and $\mathrm{Coset}(G)$ respectively the totality of subgroups of $G$, and that of left cosets $gL \in G/L$ of subgroups $L$ of $G$. We will particularly focus on quivers $Q_{G}$ and $Q_{\mathit{CG}}$ obtained naturally from posets $\mathrm{Sgp}(G)$ and $\mathrm{Coset}(G)$ ordered by the inclusion-relation. Then various properties of path complexes associated to $Q_{G}$ and $Q_{\mathit{CG}}$ will be studied.

Article information

Osaka J. Math., Volume 52, Number 1 (2015), 161-205.

First available in Project Euclid: 24 March 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20E15: Chains and lattices of subgroups, subnormal subgroups [See also 20F22]
Secondary: 55U10: Simplicial sets and complexes 06A11: Algebraic aspects of posets


Iiyori, Nobuo; Sawabe, Masato. Simplical complexes associated to quivers arising from finite groups. Osaka J. Math. 52 (2015), no. 1, 161--205.

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  • H. Cartan and S. Eilenberg: Homological Algebra, reprint of the 1956 original, Princeton Landmarks in Mathematics, Princeton Univ. Press, Princeton, NJ, 1999.
  • N. Iiyori and M. Sawabe: Representations of path algebras with applications to subgroup lattices and group characters, Tokyo J. Math. 37 (2014), 37–59.
  • D. Quillen: Homotopy properties of the poset of nontrivial $p$-subgroups of a group, Adv. in Math. 28 (1978), 101–128.
  • J.J. Rotman: An Introduction to Algebraic Topology, Graduate Texts in Mathematics 119, Springer, New York, 1988.
  • I.M. Singer and J.A. Thorpe: Lecture Notes on Elementary Topology and Geometry, reprint of the 1967 edition, Springer, New York, 1976.
  • J.W. Walker: Homotopy type and Euler characteristic of partially ordered sets, European J. Combin. 2 (1981), 373–384.