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They have been heavily studied since their introduction and complete classification in the 1950s by Shephard and Todd, due to their many applications to combinatorics, representation theory, knot theory, and mathematical physics, to name a few examples. For each given complex reflection group G, we explain a new recipe for producing an integrable system of linear differential equations whose differential Galois group is precisely G. We exhibit these systems explicitly for many (low-rank) irreducible complex reflection groups in the Shephard-Todd classification.<\/jats:p>","DOI":"10.1145\/3717582.3717584","type":"journal-article","created":{"date-parts":[[2025,2,11]],"date-time":"2025-02-11T20:41:51Z","timestamp":1739306511000},"page":"57-61","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Complex Reflection Groups as Differential Galois Groups"],"prefix":"10.1145","volume":"58","author":[{"given":"Carlos E.","family":"Arreche","sequence":"first","affiliation":[{"name":"The University of Texas at Dallas"}]},{"given":"Avery","family":"Bainbridge","sequence":"additional","affiliation":[{"name":"The University of Texas at Dallas"}]},{"given":"Ben","family":"Obert","sequence":"additional","affiliation":[{"name":"The University of Texas at Dallas"}]},{"given":"Alavi","family":"Ullah","sequence":"additional","affiliation":[{"name":"The University of Texas at Dallas"}]}],"member":"320","published-online":{"date-parts":[[2025,2,11]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01393900"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.2307\/2372597"},{"key":"e_1_2_1_3_1","volume-title":"Differential Algebra and Algebraic Groups","author":"Kolchin E.R.","year":"1976","unstructured":"[Kol76] E.R. Kolchin. Differential Algebra and Algebraic Groups. AcademicPress, New York, (1976)."},{"key":"e_1_2_1_4_1","volume-title":"Lang Algebra","author":"S.","year":"2002","unstructured":"[Lan02] S. Lang Algebra, 3rd ed., Graduate Texts in Mathematics 211, Springer, (2002).","edition":"3"},{"key":"e_1_2_1_5_1","volume-title":"Unitary Reflection Groups","author":"Lehrer G.I.","year":"2009","unstructured":"[LT09] G.I. Lehrer and D.E. Taylor, Unitary Reflection Groups, Australian Mathematical Society Lecture Series 20, Cambridge University Press, (2009)."},{"key":"e_1_2_1_6_1","volume-title":"Grundlehren der Mathematischen Wissenschaften 328","author":"van der Put M.","year":"2002","unstructured":"[vdPS02] M. van der Put and M. F. Singer. Galois Theory of Linear Differential Equations, Grundlehren der Mathematischen Wissenschaften 328. Springer Science & Business Media, (2002)."},{"key":"e_1_2_1_7_1","volume-title":"Topics in Galois Theory","author":"Serre J.-P.","year":"2008","unstructured":"[Ser08] J.-P. Serre, Topics in Galois Theory, 2nd ed., notes written by Henri Darmon, Research Notes in Mathematics 1, A K Peters Ltd., (2008).","edition":"2"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1954-028-3"},{"key":"e_1_2_1_9_1","unstructured":"[WR24] Wolfram Research Inc. Mathematica Version 14.0 (2024)."}],"container-title":["ACM Communications in Computer Algebra"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3717582.3717584","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3717582.3717584","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T01:17:15Z","timestamp":1750295835000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3717582.3717584"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9]]},"references-count":9,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2024,9]]}},"alternative-id":["10.1145\/3717582.3717584"],"URL":"https:\/\/doi.org\/10.1145\/3717582.3717584","relation":{},"ISSN":["1932-2232","1932-2240"],"issn-type":[{"type":"print","value":"1932-2232"},{"type":"electronic","value":"1932-2240"}],"subject":[],"published":{"date-parts":[[2024,9]]},"assertion":[{"value":"2025-02-11","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}