{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:18:59Z","timestamp":1760231939338,"version":"build-2065373602"},"reference-count":10,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,12]],"date-time":"2022-10-12T00:00:00Z","timestamp":1665532800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We define a Euler characteristic \u03c7(X,G) for a finite cell complex X with a finite group G acting cellularly on it. Then, each Ki(X) (a complex vector space with basis the i-cells of X) is a representation of G, and we define \u03c7(X,G) to be the alternating sum of the representations Ki(X), as elements of the representation ring R(G) of G. By adapting the ordinary proof that the alternating sum of the dimensions of the chain complexes is equal to the alternating sum of the dimensions of the homology groups, we prove that there is another definition of \u03c7(X,G) with the alternating sum of the representations Hi(X), again as elements of the representation ring R(G). We also show that the character of this virtual representation \u03c7(X,G), with respect to a given element g, is just the ordinary Euler characteristic of the fixed-point set by this element. Finally, we give a topological proof of a version of Artin\u2019s induction theorem. More precisely, we show that, if G is a group with an irreducible representation of dimension greater than 1, then each character of G is a linear combination with rational coefficients of characters induced up from characters of proper subgroups of G.<\/jats:p>","DOI":"10.3390\/sym14102121","type":"journal-article","created":{"date-parts":[[2022,10,12]],"date-time":"2022-10-12T22:45:29Z","timestamp":1665614729000},"page":"2121","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Topological Proof for a Version of Artin\u2019s Induction Theorem"],"prefix":"10.3390","volume":"14","author":[{"given":"M\u00fcge","family":"Saadeto\u011flu","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, North Cyprus, Via Mersin 10, 99628 Famagusta, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,12]]},"reference":[{"key":"ref_1","first-page":"361","article-title":"The Origin of Representation Theory","volume":"44","author":"Conrad","year":"1998","journal-title":"Enseign. Math."},{"key":"ref_2","first-page":"361","article-title":"Representations of Finite Groups: A Hundred Years. Part I, Part II","volume":"45","author":"Lam","year":"1998","journal-title":"Not. AMS"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1655","DOI":"10.3390\/sym7031655","article-title":"An Elementary Derivation of the Matrix Elements of Real Irreducible Representations of so(3)","volume":"7","author":"Stursberg","year":"2015","journal-title":"Symmetry"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Dally, M.M., and Abdulrahim, M.N. (2019). On the Unitary Representations of the Braid Group B6. Mathematics, 7.","DOI":"10.3390\/math7111080"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Chen, J.Q., Ping, J., and Wang, F. (2002). Group Representation Theory for Physicists, World Scientific Co.. [2nd ed.].","DOI":"10.1142\/5019"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Armstrong, M.A. (1983). Basic Topology, Springer.","DOI":"10.1007\/978-1-4757-1793-8"},{"key":"ref_7","unstructured":"Kao, D. (2010, October 06). Representations of the Symmetric Group. Available online: http:\/\/math.uchicago.edu\/~may\/VIGRE\/VIGRE2010\/REUPapers\/."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Serre, J.P. (1977). Linear Representation of Finite Groups, Graduate Texts in Mathematics 42; Springer.","DOI":"10.1007\/978-1-4684-9458-7"},{"key":"ref_9","unstructured":"Munkres, J.R. (1984). Elements of Algebraic Topology, Addison-Wesley."},{"key":"ref_10","unstructured":"Hatcher, A. (2002). Algebraic Topology, Cambridge University Press."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2121\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:50:28Z","timestamp":1760143828000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2121"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,12]]},"references-count":10,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["sym14102121"],"URL":"https:\/\/doi.org\/10.3390\/sym14102121","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,10,12]]}}}