MATH 689-601 Random Matrix Theory II, spring 2018

Course information

Course Notes

The notes will be updated cumulatively.

Matlab demonstrations

Homework Assignments

  1. Due Apr 30, target submission 5*.
    1. (*) Show that for a permutation \(\alpha \in S_n\), \[\#\{i \in [N]^n : i = i_\alpha\} = N^{|\alpha|} = N^{|\alpha^{-1}|}.\]
    2. (*) Show that for two permutations \(\sigma, \rho \in S_n\) and \(i, i': [n] \rightarrow [N]\), \[i_\rho = i'_\sigma \quad \Leftrightarrow \quad i_{\rho \sigma^{-1}} = i'\] and \[i_{\sigma \rho} = i'_\sigma \quad \Leftrightarrow \quad i_{\sigma \rho \sigma^{-1}} = i'.\]
    3. (*) Let \(G\) be a finite group and \(f,g : G \rightarrow \mathbb{C}\). Show that \[\widehat{f \ast g} = \hat{f} \cdot \hat{g},\] where the Fourier transform \(\hat{f}\) and convolution \(\ast\) were defined in class.
    4. (*) Take \(\mathbb{Z}_3 = \{0, 1, 2\}\). This finite group has irreducible representations \[R_q: \mathbb{Z}_3 \rightarrow \mathbb{C}, \quad R_q(p) = e^{\frac{2 \pi i}{3} p q}.\] Compute the direct and inverse Fourier transforms for this group.
    5. (*) Use the Grand Orthogonality Theorem to prove that for any two irreducible representations \(\rho, \rho'\), \[\frac{1}{|G|} \sum_{g \in G} \chi_{\rho}(g^{-1}) \rho'(g) = \delta_{\rho = \rho'} \frac{1}{d_\rho} I_{d_\rho}.\]
    6. (**) Prove the relation \[(\chi_{\rho} \ast \chi_{\rho'})(\pi) = \frac{|S_t|}{d_\rho} \delta_{\rho = \rho'} \chi_\rho(\pi).\]
    7. (**) Use the relation in the preceding exercise to prove the Samuel-Beukers Theorem.
  1. Due Mar 22, target submission 7*.
    1. (*) We have defined Hermite polynomials as \[h_n(x) = (-1)^n e^{x^2/2} \frac{d^n}{dx^n} \left[e^{-x^2/2}\right].\] From this definition, prove properties (1)-(3), namely
      • \(h_0(x) = 1, \quad h_1(x)=x, \quad h_2(x) = x^2-1, \quad h_{n+1}(x) = xh_n(x) - h'_n(x),\)
      • \(h_n(x)\) is a monic polynomial of degree \(n\).
      • \(h_n(x)\) is even (corresp. odd) when \(n\) is even (corresp. odd).
    2. (**) Calculate numerically and plot the histogram of zeros of Hermite polynomial of a large order \(N\). Submit your code printout and a picture. Hint: it is nicer to have \(N\) fairly large, about 500. But the straightforward way (defining the polynomial, finding its roots) will not work because the coefficients grow too fast. Recast the problem as looking for eigenvalues of a matrix instead (check the typed lecture notes!).
    3. (**) Exercise 5.12 from the notes.
    4. (*) Exercise 5.16 from the notes.
    5. Exercise 6.12 from the notes (optional).
    6. (*) Exercise 6.15 from the notes.
    7. (*) Exercise 6.17 from the notes.
    8. (*) Exercise 6.18 from the notes.
    9. (*) Exercise 6.19 from the notes.
    10. Exercise 6.24 from the notes (optional).
    11. (**) Exercise 6.26 from the notes.
    12. (*) Exercise 6.36 from the notes.
    13. (**) Exercise 6.38 from the notes. An outline is sufficient.
    14. (*) Exercise 6.39 from the notes. An outline is sufficient.
    15. (*) Exercise 6.43 from the notes.

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