Matrix Coefficients and Linearization Formulas for SL(2)
Matrix Coefficients and Linearization Formulas for SL(2) Robert W. Donley, Jr. (Queensborough Community College) November 17, 2017 Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 1 / 43 Goals of Talk 1 Review of last talk 2 Special Functions 3 Matrix Coefficients 4 Physics Background 5 Matrix calculator for cm;n;k (i; j) (Vanishing of cm;n;k (i; j) at certain parameters) Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 2 / 43 References 1 Andrews, Askey, and Roy, Special Functions (big red book) 2 Vilenkin, Special Functions and the Theory of Group Representations (big purple book) 3 Beiser, Concepts of Modern Physics, 4th edition 4 Donley and Kim, "A rational theory of Clebsch-Gordan coefficients,” preprint. Available on arXiv Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 3 / 43 Review of Last Talk X = SL(2; C)=T n ≥ 0 : V (2n) highest weight space for highest weight 2n, dim(V (2n)) = 2n + 1 ∼ X C[SL(2; C)=T ] = V (2n) n2N T X C[SL(2; C)=T ] = C f2n n2N f2n is called a zonal spherical function of type 2n: That is, T · f2n = f2n: Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 4 / 43 Linearization Formula 1) Weight 0 : t · (f2m f2n) = (t · f2m)(t · f2n) = f2m f2n min(m;n) ∼ P 2) f2m f2n 2 V (2m) ⊗ V (2n) = V (2m + 2n − 2k) k=0 (Clebsch-Gordan decomposition) That is, f2m f2n is also spherical and a finite sum of zonal spherical functions.
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