Representation Theory

notes

• [Groups and linear algebra - 2024-01-17] [pdf]
• [Representations - 2024-01-22] [pdf]
• [Invariant subspaces and complete reducibilty - 2024-01-24] [pdf]
• [Schur’s Lemma & irreducibility - 2024-01-29] [pdf]
• [The character table of a finite group - 2024-01-31] [pdf]
• [Characters of irreducible representations - 2024-02-05] [pdf]
• [Functions on \(G\) via regular representation - 2024-02-07 pt1] [pdf]
• [Representations & the symmetric group - 2024-02-07 pt2] [pdf]
• [The number of irreducible representations - 2024-02-12] [pdf]

assignments

• due 2024-01-26 [problem set 01] [pdf] [solutions] [solutions pdf]
• due 2024-02-02 [problem set 02] [pdf] [solutions] [solutions pdf]
• due 2024-02-16 [problem set 03] [pdf] [solutions] [solutions pdf]

Error-Correcting codes

notes

• [Overview on linear codes - 2024-02-14] [pdf]
• [Check matrices; finite projective space; Hamming codes- 2024-02-21] [pdf]
• [Shannon’s Theorem; block codes - 2024-02-22] [pdf]
• [Bounds for block codes; decoding - 2024-02-26] [pdf]
• [Dual codes; McWilliams identity - 2024-02-28] [pdf]
• [Codes from points in projective space - 2024-03-04] [pdf]
• [Cyclic codes - 2024-03-06] [pdf]
• [Reed-Solomon codes - 2024-03-13] [pdf]

sage-math notebooks

• example of ternary code [via cocalc] [download notebook]
• Hamming code example [via cocalc] [download notebook]
• perfect codes [via cocalc] [download notebook]
• array decoding [via cocalc] [download notebook]
• syndrome decoding [via cocalc] [download notebook]
• dual-codes [via cocalc] [download notebook]
• ternary Golay codes [via cocalc] [download notebook]
• decoding Reed-Solomon codes [via cocalc] [download notebook]

assignments

• due 2024-03-01 [problem set 04] [pdf] [solutions] [solutions pdf]
  
• due 2024-03-29 [problem set 05] [pdf] [solutions] [solutions pdf]
  
• due 2024-03-29 [final project proposal] [pdf]