Homework Representation Theory, B.Math

Week 1

1. Let $R$ be a ring and $A$, $B$ and $C$ be $R$-modules. Show that $A\otimes_R (B \otimes_R C)$ is isomorphic to $(A \otimes_R B)\otimes_R C$ as $R$-modules.

2. Let $\{e_1,\ldots,e_n\}$ be a basis of a vector space V. Show that $\{(e_{i_1}\cdot\ldots\cdot e_{i_a}):1\le i_1\le i_2\le \ldots i_a\le n\}$ is a basis of $Sym^a(V)$. Compute its dimension.

3. Show that the subspaces $H=< v_1 \otimes\ldots\otimes v_a: v_i\in V$ for all $i$ and $v_i=v_j$ for some $i\ne j >$ and $H'=< v_1\otimes\ldots \otimes v_a - sgn(\sigma)v_{\sigma 1}\otimes\ldots\otimes v_{\sigma a}:\sigma\in S_a, v_i\in V >$ of $T^aV$ are the same.

4. Show that $Sym^{\bullet}V \cong T^{\bullet}V/I$ where $I$ is the two sided ideal generated by $\{v\otimes v'-v'\otimes v:v,v'\in V\}$. Also $Ext^{\bullet}V\cong T^{\bullet}V/J$ where $J$ is generated by $\{v\otimes v:v\in V\}$.

5. Show that $T^2V\cong Sym^2V\oplus Ext^2V$ for a finite dimensional vector space V.

   
   

   Week 2

6. Let $V$ be a one dimensional representation of a group $G$. Show that there is an injective $G$-equivariant map from $V$ to the regular representation of $G$.

7. Let $V$ be a representation of $G$ and $v\in V$ be such that $\{g.v| g\in G \}$ is a basis of $V$ then show that $V$ isomorphic to a regular representation.

8. Let $V$ and $W$ be $G$-representations given by $\rho_1$ and $\rho_2$ respectively. Compute the matrix of $\rho_1\otimes\rho_2(g)$ in terms of the matrix of $\rho_1(g)$ and $\rho_2(g)$ w.r.t. to a basis $B_1$ and $B_2$ of $V$, $W$ and the basis of $V\otimes W$ obtained using $B_1$ and $B_2$ as explained in class.

9. Let $V$ be a $G$-representation. Show that $Sym^n(V)$ and $Ext^n(V)$ are subrepresentation of $T^n(V)$. Show that for $n=2$, the isomorphism in problem (5) above is $G$-equivariant.

10. Let $V$ be a one dimensional representation of a group $G$. Show that $T^n(V)$ is the trivial representation if $|G|$ divides $n$.

    Week 3

11. Fulton-Harris 1.3

12. Fulton-Harris 1.4

13. Serre 2.1

    Week 4

14. Serre 2.6

15. Fulton-Harris 2.3

16. Fulton-Harris 2.4

17. Let $\rho:G\to GL(V)$ be a representation. Show that for $g\in G$ the endomorphism $\rho(g)$ of $V$ is $G$-equivariant iff $\rho(g)$ is in center of Image($\rho$). Show that every irreducible representation of an abelian group is one dimensional.

    Week 5

18. Fultoh-Harris 2.7

19. Serre 2.7

20. Serre 3.2

21. Serre 3.5

22. Serre 3.6

    Week 6

23. Serre 5.1

24. Serre 5.2

25. Serre 5.4

26. Let $V$ be the regular representation of $A_4$. Decompose $Sym^2V$ as direct sum of irreducible representation.

27. Fulton-Harris 2.25

    Week 7

28. Fulton-Harris 2.37

29. Serre 7.1

30. Seree 7.2

31. Fulton-Harris 3.23

    Week 8

32. Fulton-Harris 3.16

33. Fulton-Harris 4.4

34. Fulton-Harris 4.6

35. Fulton-Harris 4.16

36. Fulton-Harris 4.20

37. Fulton-Harris 5.2