- I Need The Right Answer For Q13 With Explanation 1 (121.83 KiB) Viewed 39 times
I need the right answer for Q13 with explanation.
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answerhappygod
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I need the right answer for Q13 with explanation.
I need the right answer for Q13 with explanation.
50 CHAPTER 1. GROUP REPRESENTATIONS 8. Let V be a vector space. Show that the following properties will hold in V if and only if a related property holds for a basis of V. (a) V is a G-module. (b) The map 0: VW is a G-homomorphism. (c) The inner product (,) on V is G-invariant. 9. Why won't replacing the inner products in the proof of Maschke's the- orem by bilinear forms give a demonstration valid over any field? Give a correct proof over an arbitrary field as follows. Assume that X is a reducible matrix representation of the form (1.4). (a) Write out the equations obtained by equating blocks in X (gh) = X(g) X (h). What interpretation can you give to the equations obtained from the upper left and lower right blocks? (b) Use part (a) to show that where T = 0 TX (9)T-1 = (A(9) C)). 0 C(g) I D 0 I and D= EgeG A(g-¹)B(g). 10. Verify that the map X: R+ → GL2 given in the example at the end of Section 1.5 is a representation and that the subspace W is invariant. 11. Find H≤ S, and a set of tabloids S such that CH CS C{1, 2,..., n}. 12. Let X be an irreducible matrix representation of G. Show that if g € ZG (the center of G), then X(g) = cI for some scalar c. 13. Let (X1, X2,..., Xn) CGLd be a subgroup of commuting matrices. Show that these matrices are simultaneously diagonalizable using rep- resentation theory. 14. Prove the following converse of Schur's lemma. Let X be a represen- tation of G over C with the property that only scalar multiples cl commute with X (g) for all g E G. Prove that X is irreducible. 15. Let X and Y be representations of G. The inner tensor product, XÂY, assigns to each g E G the matrix (X&Y)(g) = X(g) & Y(g). (a) Verify that X&Y is a representation of G. (b) Show that if X, Y, and XÂY have characters denoted by X, 4, and xô, respectively, then (xv) (g) = x(g)(g). (c) Find a group with irreducible representations X and Y such that XÂY is not irreducible.
50 CHAPTER 1. GROUP REPRESENTATIONS 8. Let V be a vector space. Show that the following properties will hold in V if and only if a related property holds for a basis of V. (a) V is a G-module. (b) The map 0: VW is a G-homomorphism. (c) The inner product (,) on V is G-invariant. 9. Why won't replacing the inner products in the proof of Maschke's the- orem by bilinear forms give a demonstration valid over any field? Give a correct proof over an arbitrary field as follows. Assume that X is a reducible matrix representation of the form (1.4). (a) Write out the equations obtained by equating blocks in X (gh) = X(g) X (h). What interpretation can you give to the equations obtained from the upper left and lower right blocks? (b) Use part (a) to show that where T = 0 TX (9)T-1 = (A(9) C)). 0 C(g) I D 0 I and D= EgeG A(g-¹)B(g). 10. Verify that the map X: R+ → GL2 given in the example at the end of Section 1.5 is a representation and that the subspace W is invariant. 11. Find H≤ S, and a set of tabloids S such that CH CS C{1, 2,..., n}. 12. Let X be an irreducible matrix representation of G. Show that if g € ZG (the center of G), then X(g) = cI for some scalar c. 13. Let (X1, X2,..., Xn) CGLd be a subgroup of commuting matrices. Show that these matrices are simultaneously diagonalizable using rep- resentation theory. 14. Prove the following converse of Schur's lemma. Let X be a represen- tation of G over C with the property that only scalar multiples cl commute with X (g) for all g E G. Prove that X is irreducible. 15. Let X and Y be representations of G. The inner tensor product, XÂY, assigns to each g E G the matrix (X&Y)(g) = X(g) & Y(g). (a) Verify that X&Y is a representation of G. (b) Show that if X, Y, and XÂY have characters denoted by X, 4, and xô, respectively, then (xv) (g) = x(g)(g). (c) Find a group with irreducible representations X and Y such that XÂY is not irreducible.