Math 299 - Independent Study
The main text will be Representations and Characters of Groups, 2nd Edition by Gordon James and Martin Liebeck. An online version of the book is available through the Sacramento State library. Additional resources are listed below.
| Day | Topics Covered | For Next Time |
|---|---|---|
| Mon/W01 | Syllabus and Planning | Skim: Ch. 1 and 2. Maybe start by reading the chapter summaries at the end of each chapter. Problems to Try: 1.4(b), 1.7, 1.9, 2.2, 2.8, 2.9 |
| Thu/W01 | Chatted about Ch. 1. Focused on problems 1.7 and 1.9 | Read: Ch. 2 and 3. Problems to Try: 2.2, 2.8, 2.9, 3.2, 3.5, 3.7 Note: it may be good to review Example 1.4 when thinking about problem 3.5 |
| Mon/W02 | Labor day–No Meeting | |
| Thu/W02 | Chatted about Ch. 2. Focused on problems 2.8 and 2.9 | Read: Ch. 3 and start on 4. To supplement Chapter 4, you could also look at the Notes from the ANTC Seminar on Representation Theory of the Symmetric Group. The notes talk specifically about $\operatorname{Sym}(n)$-modules, but you can easily generalize them to $FG$-modules for an arbitrary field $F$ and arbitrary group $G$. Problems to Try: 2.2, 3.2, 3.5, 3.7 Portfolio Problem: try to write up a nice clean solution to 2.9 for the portfolio. Next time I can start an Overleaf document for the portfolio, and we can add this problem it then. |
| Mon/W03 | Ch. 3. | Read: Ch. 4 and start Ch.5 (just the section on submodules). Problems to Try: 4.1, 4.2 Additional Problem 1: Let $V$ be the permutation module for $\operatorname{Sym}(n)$ over a field $F$ with basis $v_1,\ldots, v_n$ (as in Defintion 4.10). Let $W_1 = \operatorname{span}(v_1+\cdots + v_n)$ and $W_2 = \operatorname{span}(v_i- v_j \mid 1\le i < j \le n)$. Prove that $W_1$ and $W_2$ are $F\operatorname{Sym}(n)$-submodules of $V$. Can you find a basis for $W_2$ and determine its dimension? |
| Thu/W03 | Ch. 4. | Read: Review Ch. 4 and read Ch.5. Problems to Try (same as last time): 4.1, 4.2 Additional Problem 1: Let $V$ be the permutation module for $\operatorname{Sym}(n)$ over a field $F$ with basis $v_1,\ldots, v_n$ (as in Defintion 4.10). Let $W_1 = \operatorname{span}(v_1+\cdots + v_n)$ and $W_2 = \operatorname{span}(v_i- v_j \mid 1\le i < j \le n)$. Prove that $W_1$ and $W_2$ are $F\operatorname{Sym}(n)$-submodules of $V$. Can you find a basis for $W_2$ and determine its dimension? |
| Mon/W04 | Modules & Submodules | Read: Review Ch. 5 and read Ch.6. Problems to Try: 6.1 Additional Problem 2: Look at AP 1 from last time. Assume that $n=3$ and that $F$ does not have characteristic $3$. Prove that $W_2$ is irreducible. (Possible approach: towards a contradiction, assume $W_2$ is reducible, so there is some proper nonzero submodule $\{0\} < U < W_2$. In AP 1, we learned $\dim W_2 = n-1 = 2$, so $\dim U = 1$. Consider how the transpositions act on $U$; Problem 9 from Chapter 2 may help. Try to show that $\operatorname{Alt}(3)$ is in the kernel of the action on $U$, and then use this to show that $U = W_1$. To get a contradiction, separately show that $W_1 \cap W_2 = \{0\}$). Portfolio Problem: try to write up AP 1 for the portfolio. |
| Thu/W04 | Irreducibility & the Group Algebra | Read: Review Ch.6 Problems to Try: 6.2, 6.3, 6.4 Additional Problem 2 (just the end): In our meeting we showed that $U\subseteq W_1 \cap W_2$. Finish the proof of AP 2 by showing that $W_1 \cap W_2 = \{0\}$. (Thus $U$ does not exist and we may conclude that $W_1$ is irreducible.) Portfolio Problem (if not already done last time): try to write up AP 1 for the portfolio. |
| Mon/W05 | The Group Algebra | Read: Ch.7 Problems to Try: 7.3, 7.5, 7.6 Problems to Revise: start writing up AP 2 for the portfolio. (Aim to have it done by next Monday.) |
| Thu/W05 | $FG$-homs | Read: Ch.8 Problems to Try: 7.5, 8.1 Additional Problem 3: Rewrite the proof of Maschke’s Theorem in your own words (adding in detail and highlighting any questions you have). Portfolio Problem: finish writing up AP 2 for the portfolio. |
| Mon/W06 | Maschke | Read: Beginning of Ch.9 (pp. 78–80) Problems to Try: 8.2, 8.7 Additional Problem 4: Rewrite the proof of Schur’s Lemma in your own words (adding in detail and highlighting any questions you have). If you have the time and desire, do the same for Corollary 9.3. |
| Thu/W06 | Maschke | Read: Ch.9 Problems to Try: 9.1, 9.4 Additional Problem 4: Rewrite the proof of Schur’s Lemma in your own words Additional Problem 5: Rewrite Theorem 9.8 in your own words. (The proof in the book occurs before the statement of the theorem.) |
| Mon/W07 | Schur | Read: Ch.10 Problems to Try: 9.2, 9.4 (from last time), 10.1 (just the first part), 10.2 Portfolio Problem: Find all irreducible representations of $C_3$ and of $C_2 \times C_2$. |
| Thu/W07 | No Meeting | |
| Mon/W08 | Schur | Read: Ch.11: skim up through Proposition 11.3. Read: Ch.12: skim up through Example 12.16. Problems to Try (some repeats from last time): 10.1 (just the first part), 10.2, 12.3, 12.7 Portfolio Problem (if not already done): Find all irreducible representations of $C_3$ and of $C_2 \times C_2$. |
| Thu/W08 | No Meeting | |
| Mon/W09 | Conjugacy Classes | Read: Ch.11: skim up through Corollary 11.6. Read: Ch.12: skim through Example 12.20, then carefully read the rest. Problems to Try (some repeats from last time): 10.1 (just the first part), 10.2, 12.4, 12.6 Portfolio Problem (if not already done): Find all irreducible representations of $C_3$ and of $C_2 \times C_2$. |
| Thu/W09 | Conjugacy Classes | Read: Ch.13: up through Example 13.6. Problems to Try (some repeats from last time): 10.1 (just the first part), 10.2, 12.6, 13.3 |
| Mon/W10 | Characters | Read: Finish Ch.13 (but you can skip the section on the regular character for now) Problems to Try (some repeats from last time): 10.2, 13.1, 13.7 Additional Problem 5: Rewrite the proof of Proposition 13.9 in your own words (adding detail and highlighting any questions you have). |
| Thu/W10 | Characters | Read: Ch.11: from 11.9 through the end. Aim to understand the statements, but it’s fine if you just skim the proofs. Example 11.13 illustrates important ideas. Read: Ch.13: the section on the regular character Problems to Try: 10.2, 11.1, 11.2, 13.2 |
| Mon/W11 | Characters | Read: Ch.14: beginning – Example 14.6; Theorem 14.12 – Theorem 14.23. Problems to Try: 14.1 (Theorem 14.20 helps), 14.2 (follow Example 14.22) Additional Problem 6: For each of the groups $S_3$, $S_4$ and $S_5$, determine the character given in Proposition 13.24 by writing out the values of the character on each conjugacy class. Then use Theorem 14.20 to determine, in each case, if the character is irreducible or not. |
| Thu/W11 | Inner Product | Read: Ch.15 and Ch. 16 Problems to Try: 15.1 (follow Example 15.7), 16.2 (follow Example 16.5), 16.3 |
| Mon/W12 | Orthogonality | Read: Ch.17 and Sections 18.1 and 18.2 Problems to Try: 17.1, 17.3, 17.4 |
| Thu/W12 | No Meeting | |
| Mon/W13 | Lifting | Read: Ch.22: beginning through 22.12; you can skip the proof of 22.3 since we didn’t talk about tensor products Problems to Try: 17.3, 17.4, 22.1 Portfolio Problem: write up 17.1 for the portfolio. Frame the problem as simply finding the character table for $Q_8$ (but use the approach outlined in the problem by doing part (b) first). Also, you can just state what the conjugacy classes of $Q_8$ are without proof. |
| Thu/W13 | No Meeting |
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