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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. m+n X f2m · f2n = c(2m; 2n; 2p) f2p p=jm−nj Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 5 / 43 Gap 4 1 For what m; n; p is c(2m; 2n; 2p) nonzero? 2 Explicit calculation of c(2m; 2n; 2p). Main result (Gap 4) min(m;n) X f2m · f2n = c(2m; 2n; 2m + 2n − 4k) f2m+2n−4k k=0 Compare with Clebsch-Gordan Decomposition: V (2m) ⊗ V (2n) =∼ V (2m + 2n) ⊕ V (2m + 2n − 2) ⊕ · · · ⊕ V (j2m − 2nj) f2m f2n 2 V (2m + 2n) ⊕ V (2m + 2n − 4) ⊕ · · · ⊕ V (j2m − 2nj) Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 6 / 43 Special Functions Background Orthogonal Polynomials 1 Basis fpn(x)gn=0 of C[x] such that 1 p0(x) = 1 2 deg(pn(x)) = n; and b 3 R a pm(x)pn(x)dµ(x) = hnδm;n for some positive measure dµ(x) on [a; b]: Recursion: xpn(x) = anpn+1(x) + bnpn(x) + cnpn−1(x) Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 7 / 43 (α,β) Jacobi Polynomials Pn (x) \Rodrigues" Formula: (−1)n dn (1 − x)α(1 + x)β P(α,β)(x) = [(1 − x)n+α(1 + x)n+β] n 2nn! dxn Positive measure: dµ = (1 − x)α(1 + x)βdx on [−1; 1] Formula for hn Z 1 α+β+1 (α,β) (α,β) 2 Γ(n + α + 1) Γ(n + β + 1) Pm (x)Pn (x) dµ = δm;n −1 (2n + α + β + 1) Γ(n + α + β + 1) n! Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 8 / 43 (0;0) Legendre Polynomials Pn(x) = Pn (x) Rodrigues Formula (1816): (−1)n dn P (x) = [(1 − x2)n] n 2nn! dxn Positive measure: dµ = dx on [−1; 1] Formula for hn Z 1 2 Pm(x) Pn(x) dµ = δm;n −1 (2n + 1) Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 9 / 43 Linearization Formula for Pn(x) Ferrers (1877), Adams (1877), independently: min(m;n) X Pm(x) · Pn(x) = am;n;k Pm+n−2k (x) k=0 with 2m + 2n + 1 − 4k (1=2)k (1=2)m−k (1=2)n−k (m + n − k)! am;n;k = · 2m + 2n + 1 − 2k k!(m − k)! (n − k)! (1=2)m+n−k Shifted factorial: (x)k = x(x + 1)(x + 2) ::: (x + k − 1) Gaps: Pn(x) has same parity as n (even/odd as functions) Note: am;n;k > 0 Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 10 / 43 Representations to Functions Compact picture: SU(2) has same representation theory as SL(2; C): Recall: (π2n; V (2n)) can be realized as (L; C[x; y]2n) with SU(2)-invariant inner product h·; ·i2n: Since π2n(−I ) = id for −I 2 SU(2) ⊂ SL(2; C), ∼ (π2n; V (2n)) admits an action of SO(3) since SO(3) = SU(2)=±I Implement by Adjoint map: k 2 SU(2) acts on traceless skew-Hermitian matrices of size 2 by Ad : SU(2) ! SO(sSHerm) =∼ SO(3); Ad(k): sSHerm ! sSHerm; Ad(k)X = kXk−1 3 3 Geometry: S ! RP Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 11 / 43 Fourier Analysis Features of L2(SO(3)): 1 −1 joint action by SO(3) × SO(3): (π(k1; k2)f )(g) = f (k1 gk2) 2 R bi-invariant Haar measure dk: SO(3) dk = 1; Z Z f (k1kk2) dk = f (k) dk; SO(3) SO(3) 3 (Peter-Weyl) L2(SO(3)) =∼ P V (2n) ⊗ V (2n)∗ Compare with K = S1 acting on 2 1 ∼ X inθ L (S ) = Ce n2Z Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 12 / 43 Matrix coefficients To pass from representations to functions in L2(SO(3)), Definition: Choose u; v in V (2n). Define the matrix coefficient φu;v : SO(3) ! C by φu;v (k) = hπ2n(k)u; vi: This includes X V (2n) ⊗ V (2n)∗ ! L2(SO(3)) =∼ V (2m) ⊗ V (2m)∗ as an irreducible SO(3) × SO(3) representation. Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 13 / 43 Schur Orthogonality Key formula (Schur orthogonality): If u; v; u0; v 0 in V (2n), then Z 1 0 0 φu;v (k) φu0;v 0 (k) dk = hu; u ihv; v i: SO(3) 2n + 1 If u; v in V (2m) and u0; v 0 in V (2n) with m 6= n then Z φu;v (k) φu0;v 0 (k) dk = 0: SO(3) ONB for each V (2n) ! ONB of matrix coefficients for L2(SO(3)) That is, X p f (k) = c2n;i;j 2n + 1 φui ;uj (k) Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 14 / 43 Summands in L2(SO(3)=T ) For the summand of \V (2n)" in X L2(SO(3)=T ) =∼ \V (2m)"; fix any abstract realization of (π2n; V (2n)); fix a nonzero spherical vector φ0 in V (2n) and any v in V (2n). Then \V (2n)" is spanned by the matrix coefficients φv,φ0 (k) = hv; π2n(k)φ0i: with group action L(g)φ(k) = φ(g −1k) Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 15 / 43 Tensors Consider V (2m) ⊗ V (2n): Invariant inner product: hu ⊗ v; u0 ⊗ v 0i := hu; u0ihv; v 0i If we multiply matrix coefficients, 2m 2n ⊗ φu;u0 (k) φv;v 0 (k) = φu⊗v;u0⊗v 0 (k) Linearization: 2m 2n X 2p φ 0 (k) φ 0 (k) = c(2m; 2n; 2p; i; j) φ (k) u;u v;v ui ;uj p Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 16 / 43 Zonal spherical functions 2m Now choose unit spherical vectors v0 in V (2m) (resp. 2n) If we multiply, 2m 2n ⊗ φ 2m 2m (k) φ 2n 2n (k) = φ 2m 2n 2m 2n (k) v0 ;v0 v0 ,φ0 v0 ⊗v0 ;v0 ⊗v0 If 2m 2n X 2p v0 ⊗ v0 = c(2m; 2n; 2p) v0 then 2m 2n X 2 2p φv 2m;v 2m (k) φv 2n,φ2n (k) = jc(2m; 2n; 2p)j φ 2p 2p (k) 0 0 0 0 v0 ;v0 Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 17 / 43 Zonal spherical functions as Matrix coefficients 2 2 1 Restrict to one-parameter subgroup: Lcont (SO(3)) ! Lcont (S ) 01 0 0 1 cos(θ=2) i sin(θ=2) ! 0 cos(θ) − sin(θ) i sin(θ=2) cos(θ=2) @ A 0 sin(θ) cos(θ) 2m (0;0) φ 2m 2m (θ) = Pm (cos(θ)) v0 ;v0 Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 18 / 43 Physics Background Hydrogen Atom: Bound Electron, Discrete Binding Energy En (x; y; z) probability wave function of electron Time-Independent Schr¨odingerWave Equation @2 @2 @2 2m + + + (En − V ) = 0 @x2 @y 2 @z2 ~2 where e2 me4 E V = − and E = − = 1 n 2 2 2 2 2 4π0r 32π 0~ n n Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 19 / 43 Solution Space Solve using 1) rotation invariance (spherical coordinates) and 2) separation of variables.
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