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直交多項式と表現論

 直交多項式と表現論

数理空間“τόπος” (トポス)新歓イベントhttps://peatix.com/event/643037 での発表資料です。

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Naoya Umezaki

May 11, 2019
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  1. ௚ަଟ߲ࣜͱදݱ࿦ ക࡚௚໵@unaoya 2019/5/11 ਺ཧۭؒ τ ´ oπoζʢτϙεʣ৽׻Πϕϯτ 1

  2. ࡾ֯ؔ਺ͷഒ֯ެࣜ sin 2θ = 2 sin θ cos θ sin

    3θ = sin 2θ cos θ + cos 2θ sin θ = 2 sin θ cos2 θ + (2 cos2 θ − 1) sin θ = sin θ(4 cos2 θ − 1) sin 4θ = sin 3θ cos θ + cos 3θ sin θ = sin θ(4 cos2 θ − 1) cos θ + (4 cos3 θ − 3 cos θ) sin θ = sin θ(8 cos3 θ − 4 cos θ) 2
  3. Chebyshev ଟ߲ࣜ sin θ = sin θ × 1 U0(x)

    = 1 sin 2θ = sin θ × 2 cos θ U1(x) = 2x sin 3θ = sin θ × (4 cos2 θ − 1) U2(x) = 4x2 − 1 sin 4θ = sin θ × (8 cos3 θ − 4 cos θ) U3(x) = 8x3 − 4x ͱ͢Δͱ sin(n + 1)θ = sin θ × Un(cos θ) ͱͳΔɻ͜ͷ Un(x) Λୈ 2 छ Chebyshev ଟ߲ࣜͱ͍͏ɻ 3
  4. ઴Խࣜ Ճ๏ఆཧ͔Β sin(n + 1)θ = cos θ sin nθ

    + cos nθ sin θ sin(n − 1)θ = cos θ sin nθ − cos nθ sin θ sin(n + 1)θ + sin(n − 1)θ = 2 sin nθ cos θ ͜͜Ͱ sin(n + 1)θ = sin θ × Un(cos θ) Λࢥ͍ग़͢ɻ྆ลΛ sin θ Ͱ ׂͬͯ x = cos θ ͱ͢Δͱɺ Un+1(x) + Un−1(x) = 2xUn(x) ͕੒Γཱͭɻ 4
  5. ฼ؔ਺ ઴Խࣜ Un(x) − 2xUn−1(x) + Un−2(x) = 0, U0(x)

    = 1, U1(x) = 2x Λ࢖ͬͯ ∞ n=0 Un(x)tn = U0(x) + U1(x)t + U2(x)t2 + U3(x)t3 + · · · Λมܗ͢Δɻt ഒ͢Δͱ antn ͸ antn+1 ͱͳΔ͔Βɺtn ͷ܎਺͸ an−1 ʹͳΔ͜ͱʹ஫ҙ͢Δͱɺ −2xt ∞ n=0 Un(x)tn = −2xU0(x)t − 2xU1(x)t2 − 2xU2(x)t3 − 2xU3(x)t4 − t2 ∞ n=0 Un(x)tn = U0(x)t2 + U1(x)t3 + U2(x)t4 + U3(x)t5 + · · · (1 − 2tx + t2) ∞ n=0 Un(x) = 1 5
  6. ฼ؔ਺ 1 1 − 2tx + t2 = U0(x) +

    U1(x)t + U2(x)t2 + U3(x)t3 + · · · ٯʹࠨลΛల։͢Δ͜ͱͰ 1 + (2tx − t2) + (2tx − t2)2 + (2tx − t2)3 + · · · = 1 + 2tx − t2 + 4t2x2 − 4t3x + t4 + 8t3x3 − 12t4x2 + · · · = 1 + (2x)t + (−1 + 4x2)t2 + (−4x + 8x3)t3 + · · · = U0(x) + U1(x)t + U2(x)t2 + U3(x)t3 + · · · ͱఆٛ͢Δ͜ͱ΋Ͱ͖Δɻ 6
  7. ঢ߱ԋࢉࢠ 1 1 − 2tx + t2 Λ t, x

    ͰͦΕͧΕඍ෼͢Δͱɺ 2x − 2t (1 − 2tx + t2)2 = 2(x − t) (1 − 2tx + t2)2 , 2t (1 − 2tx + t2)2 ͱͳΔɻ t(x − t) 2(x − t) (1 − 2tx + t2)2 = 2t (x − t)2 (1 − 2tx + t2)2 = 2t x2 − 2xt + t2 (1 − 2tx + t2)2 (1 − x2) 2t (1 − 2tx + t2)2 = 2t 1 − x2 (1 − 2tx + t2)2 (t(x − t) d dt + (1 − x2) d dx ) 1 1 − 2tx + t2 = 2t 1 − 2xt + t2 (1 − 2tx + t2)2 = 2t 1 1 − 2tx + t2 ͱͳΔɻ 7
  8. ঢ߱ԋࢉࢠ ͭ·Γ (t(x − t) d dt + (1 −

    x2) d dx − 2t) 1 1 − 2tx + t2 = 0 ͕੒Γཱͭɻ 1 1 − 2tx + t2 = U0(x) + U1(x)t + U2(x)t2 + U3(x)t3 + · · · ͷӈลʹ΋ಉ͡ܭࢉΛ͢Δɻantn Λ t ഒͱ t Ͱͷඍ෼ʹΑͬͯ tn ͷ܎਺͕ an−1, (n + 1)an+1 ͱͳΔ͜ͱʹ஫ҙ͢Δͱɺ nxUn(x) − (n − 1)Un−1(x) + (1 − x2) d dx Un(x) − 2Un−1(x) = 0 {(1 − x2) d dx + nx}Un(x) = (n + 1)Un−1 8
  9. ঢ߱ԋࢉࢠ {(1 − x2) d dx + nx}Un(x) = (n

    + 1)Un−1(x) (n + 1)2xUn(x) = (n + 1)Un+1(x) + (n + 1)Un−1(x) ྆ลҾ͘ͱ {(1 − x2) d dx − (n + 2)x}Un(x) = −(n + 1)Un+1(x) ͜ͷೋ͕ͭঢ߱ԋࢉࢠͱݺ͹ΕΔɻ {(1 − x2) d dx + nx}Un(x) = (n + 1)Un−1 {(1 − x2) d dx − (n + 2)x}Un(x) = −(n + 1)Un+1(x) 9
  10. ඍ෼ํఔࣜ ঢ߱ԋࢉࢠΛଓ͚ͯ࡞༻ͤ͞Δͱɺ {(1 − x2) d dx − (n +

    1)x}{(1 − x2) d dx + nx}Un(x) = −n(n + 1)Un−1 ({(1 − x2) d dx − (n + 1)x}{(1 − x2) d dx + nx} + n(n + 1)})Un(x) = 0 ࠨลΛܭࢉ͢Δͱɺ {(1 − x2)2 d2 dx2 + (1 − x2)(−2x) d dx − (n + 1)x(1 − x2) d dx + nx(1 − x2) d dx + n(1 − x2) − n(n + 1)x2 + n(n + 1)}Un(x) = 0 (1 − x2) d2 dx Un(x) − 3x d dx Un(x) + n(n + 2)Un(x) = 0 ͕͑ΒΕΔɻ 10
  11. ඍ෼ํఔࣜ d2 dθ2 (sin θUn(cos θ)) = sin θ(−Un(cos θ)

    − 3 cos θ d dx Un(cos θ) + (1 − cos2 θ) d2 dx2 Un(cos θ)) ͱͳΔͷͰɺ (1 − x2) d2 dx Un(x) − 3x d dx Un(x) + n(n + 2)Un(x) = 0 ͱ߹ΘͤΔͱɺ d2 dθ2 sin θUn(cos θ) = −(n + 1)2 sin θUn(cos θ) ͜Ε͸྆୺͕ݻఆ͞Εͨ೾ͷํఔࣜʢແݶͷߴ͞ͷҪށܕϙςϯ γϟϧʣ 11
  12. ௚ަؔ܎ π 0 sin(n + 1)θ sin(m + 1)θdθ =

    ⎧ ⎨ ⎩ π (n = m) 0 (n ̸= m) ࠨลΛ π 0 sin(n + 1)θ sin θ sin(m + 1)θ sin θ sin θ sin θdθ ͱมܗ͔ͯ͠Β x = cos θ ͱஔ׵ੵ෼͢Δͱ 1 −1 Un(x)Um(x) 1 − x2dx = ⎧ ⎨ ⎩ π (n = m) 0 (n ̸= m) ͜ΕΛ࢖͏ͱؔ਺Λ Chebyshev ଟ߲ࣜͰల։Ͱ͖ɺ਺஋ܭࢉͳͲ ʹԠ༻͞ΕΔɻ 12
  13. SU(2) ͷදݱ SU(2) = a b −¯ b ¯ a

    , |a|2 + |b|2 = 1, a, b ∈ C ͷ 2 ม਺ଟ߲ࣜ ΁ͷ࡞༻Λ a b −¯ b ¯ a f (x, y) = f (¯ ax − by, ¯ bx + ay) ͰఆΊΔɻ eiθ 0 0 e−iθ f (x, y) = f (e−iθx, eiθy) Λܭࢉ͢Δͱ eiθ 0 0 e−iθ x = e−iθx, eiθ 0 0 e−iθ y = eiθy, eiθ 0 0 e−iθ x2 = e−2iθx2, eiθ 0 0 e−iθ xy = xy, eiθ 0 0 e−iθ y2 = e2iθy2 13
  14. දݱͷࢦඪ eiθ 0 0 e−iθ x3 = e−3iθx3, eiθ 0

    0 e−iθ x2y = e−iθx2y, eiθ 0 0 e−iθ xy2 = eiθxy2, eiθ 0 0 e−iθ y3 = e3iθy3 ಉ࣍͡਺ͷ෦෼ʹ͍ͭͯ܎਺͚ͩ଍͢ɻ౳ൺ਺ྻͷ࿨ͷެࣜͱ Euler ͷެࣜͰܭࢉ͢Δͱ e−iθ + eiθ = e−2iθ − e2iθ e−iθ − eiθ = sin 2θ sin θ e−2iθ + 1 + e2iθ = e−3iθ − e3iθ e−iθ − eiθ = sin 3θ sin θ e−3iθ + e−iθ + eiθ + e3iθ = e−4iθ − e4iθ e−iθ − eiθ = sin 4θ sin θ 14
  15. ࢦඪͷ௚ަੑ ܈ͷදݱͷࢦඪʹ͸௚ަੑ͕͋Δɻྫ͑͹ n ̸= m ͳΒ sin(n + m) 2π

    N + sin 2(n + m) 2π N + · · · + sin(N − 1)(n + m) 2π N = 0 ͕੒Γཱͭͱ͍͏͜ͱɻ ಉ͡Α͏ʹ n ̸= m ͷͱ͖ɺ π 0 sin(n + 1)θ sin θ sin(m + 1)θ sin θ sin θ sin θdθ = 0 0 −1 Un(x)Um(x) 1 − x2dx = 0 ͕੒Γཱͭɻ 15
  16. ௚ަଟ߲ࣜ ্ͰݟͨΑ͏ͳ௚ަؔ܎ɺ઴Խࣜɺඍ෼ํఔࣜͳͲͷੑ࣭Λຬͨ ͢ଟ߲ࣜͷܥྻ͕͍͔ͭ͋͘Δɻ ྫ͑͹ྔࢠྗֶʹग़ͯ͘Δ΋ͷͱͯ͠ • Hermite ଟ߲ࣜ͸ௐ࿨ৼಈࢠ • Legendre ଟ߲ࣜ͸֯ӡಈྔͷྔࢠԽ

    • Laguerre ଟ߲ࣜ͸ਫૉݪࢠͷಈܘํఔࣜ ͕͋Δɻ ௚ަଟ߲ࣜͷੑ࣭Λදݱ࿦ͰཧղͰ͖Δʢͱࢥ͍ͬͯΔͷͰ͜Ε ͔Βษڧ͢Δʣ ɻ 16