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| In mathematics, the '''Frobenius determinant theorem''' is a discovery made in 1896 by the mathematician [[Richard Dedekind]], who wrote a letter to [[Ferdinand Georg Frobenius|F. G. Frobenius]] about it (reproduced in {{harv|Dedekind|1968}}, with an English translation in {{harv|Curtis|2003|loc=p. 51}}).
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| If one takes the multiplication table of a [[Group (mathematics)|group]] ''G'' and replaces each entry ''g'' with the variable ''x''<sub>''g''</sub>, and subsequently takes the [[determinant]], then the determinant factors as a product of ''n'' irreducible polynomials, where ''n'' is the number of conjugacy classes. Moreover, each polynomial is raised to a power equal to its degree. Frobenius proved this surprising fact, and this theorem became known as the Frobenius determinant theorem.
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| ==Formal statement==
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| Let a [[finite group]] <math>G</math> have elements <math>g_1, g_2,\dots,g_n</math>, and let <math>x_{g_i}</math> be associated with each element of <math>G</math>. Define the matrix <math>X_G</math> with entries <math>a_{ij}=x_{g_i g_j}</math>. Then
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| : <math> \det X_G = \prod_{j=1}^r P_j(x_{g_1},x_{g_2},\dots,x_{g_n})^{\deg P_j}</math>
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| where ''r'' is the number of conjugacy classes of ''G''.
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| ==References==
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| *{{Citation | last1=Curtis | first1=Charles W. | authorlink = Charles W. Curtis | title=Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer | url=http://books.google.com/books?isbn=0821826778 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=History of Mathematics | isbn=978-0-8218-2677-5 |doi=10.1090/S0273-0979-00-00867-3 | mr=1715145 | year=2003}} [http://www.ams.org/journals/bull/2000-37-03/S0273-0979-00-00867-3/ Review]
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| *{{Citation | last1=Dedekind | first1=Richard | author1-link=Richard Dedekind | editor1-last=Fricke | editor1-first=Robert | editor2-last=Noether | editor2-first=Emmy | editor2-link=Emmy Noether | editor3-last=Ore | editor3-first=öystein | title=Gesammelte mathematische Werke. Bände I--III | origyear=1931 | publisher=Chelsea Publishing Co. | location=New York | mr=0237282 |jfm=56.0024.05 | year=1968}}
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| *Etingof, Pavel. ''[http://www-math.mit.edu/~etingof/cltrunc.pdf Lectures on Representation Theory]''.
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| *{{Citation | last1=Frobenius | first1=Ferdinand Georg | author1-link=Ferdinand Georg Frobenius | title=Gesammelte Abhandlungen. Bände I, II, III | publisher=[[Springer-Verlag]] | location=Berlin, New York | editor-first= J.-P.|editor-last= Serre | isbn=978-3-540-04120-7 | mr=0235974 | year=1968}}
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| [[Category:Theorems in algebra]]
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| [[Category:Determinants]]
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| [[Category:Group theory]]
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| [[Category:Matrix theory]]
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