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I need the right answers for Q7 with explanation. (equation 1.4 on page 12)
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I need the right answers for Q7 with explanation. (equation 1.4 on page 12)
I need the right answers for Q7 with explanation. (equation 1.4on page 12)
CHAPTER 1. GROUP REPRESENTATIONS We now introduce the irreducible representations that will be the building blocks of all the others. 12 Definition 1.4.5 A nonzero G-module V is reducible if it contains a non- trivial submodule W. Otherwise, V is said to be irreducible. Equivalently, V is reducible if it has a basis B in which every g E G is assigned a block matrix of the formi Alg) | B(g) (1.4) X(g) = where the A(g) are square matrices, all of the same size, and 0 is a nonempty matrix of zeros. To see the equivalence, suppose V of dimension d has a submodule W of dimension f. 0</<d. Then let B (W₁, W₂, Wf, Vf+1 Vf+2₁ Va} where the first f vectors are a basis for W. Now we can compute the matrix of any g eG with respect to the basis B. Since W is a submodule, gw, e W for all i, 1 ≤ i ≤f. Thus the last d-f coordinates of gw, will all be zero. That accounts for the zero matrix in the lower left corner of X(g). Note that we have also shown that the A(g), g = G, are the matrices of the restriction of G to W. Hence they must all be square and of the same size. Conversely, suppose each X(g) has the given form with every A(9) being fx f. Let V=Cd and consider w = Clee.....es). where e, is the column vector with a 1 in the ith row and zeros elsewhere. (the standard basis for C"). Then the placement of the zeros in X(g) assures us that X(g)e, W for 1 ≤i</ and all ge G. Thus W G-module, and it is nontrivial because the matrix of zeros is nonempty. Clearly, any epresentation of degree 1 is irreducible. It seems hard to de- termine when a representation of greater degree will be irreducible. Certainly, checking all possible subspaces to find out which ones are submodules is out of the question. This unsatisfactory state of affairs will be remedied after we discuss inner products of group characters in Section 1.9. From the preceding examples, both the defining representation for S, and the group algebra for an arbitrary G are reducible if n 22 and [G] ≥ 2, respec- tively. After all, we produced nontrivial submodules. Let us now illustrate the alternative approach via matrices using the defining representation of Ss. We must extend the basis (1+2+3) for W to a basis for V = C(1,2,3)- Let us pick B=(1+2+3, 2, 3) Of course, X(e) remains the 3 x 3 identity matrix. To compute X( (1,2)), we look at (1, 2)'s action on our basis: (1,2)(1+2+3)=1+2+3, (1, 2)2=1=(1+2+3)-2-3, (1,2)3= 3. x(g) = the number of fixed points of g acting on S. 4. Let G be an abelian group. Find all inequivalent irreducible represen- tations of G. Hint: Use the fundamental theorem of abelian groups. 5. If X is a matrix representation of a group G, then its kernel is the set N = {g €G: X(g) = 1}. A representation is faithful if it is one-to-one. (a) Show that N is a normal subgroup of G and find a condition on N equivalent to the representation being faithful. (b) Suppose X has character x and degree d. Prove that g € N if and only if x(g) = d. Hint: Show that x(g) is a sum of roots of unity. (c) Show that for the coset representation, Nig, Hg,¹, where the 9 are the transversal. (d) For each of the following representations, under what conditions are they faithful: trivial, regular, coset, sign for Sn, defining for Sn, degree 1 for C₁? (e) Define a function Y on the group G/N by Y(gN) = X(g) for gN € G/N. i. Prove that Y is a well-defined faithful representation of G/N. ii. Show that Y is irreducible if and only if X is. iii. If X is the coset representation for a normal subgroup H of G, what is the corresponding representation Y? 6. It is possible to reverse the process of part (e) in the previous exercise. Let N be any normal subgroup of G and let Y be a representation of G/N. Define a function on G by X(g) = Y(gN). (a) Prove that X is a representation of G. We say that X has been lifted from the representation Y of G/N. (b) Show that if Y is faithful, then X has kernel N. (c) Show that X is irreducible if and only if Y is. Let X be a reducible matrix representation with block form given by equation (1.4). Let V be a module for X with submodule W correspond- ing to A. Consider the quotient vector space V/W = {v+W: VEV}. Show that V/W is a G-module with corresponding matrices C(g). Fur- thermore, show that we have Ve We (V/W).
CHAPTER 1. GROUP REPRESENTATIONS We now introduce the irreducible representations that will be the building blocks of all the others. 12 Definition 1.4.5 A nonzero G-module V is reducible if it contains a non- trivial submodule W. Otherwise, V is said to be irreducible. Equivalently, V is reducible if it has a basis B in which every g E G is assigned a block matrix of the formi Alg) | B(g) (1.4) X(g) = where the A(g) are square matrices, all of the same size, and 0 is a nonempty matrix of zeros. To see the equivalence, suppose V of dimension d has a submodule W of dimension f. 0</<d. Then let B (W₁, W₂, Wf, Vf+1 Vf+2₁ Va} where the first f vectors are a basis for W. Now we can compute the matrix of any g eG with respect to the basis B. Since W is a submodule, gw, e W for all i, 1 ≤ i ≤f. Thus the last d-f coordinates of gw, will all be zero. That accounts for the zero matrix in the lower left corner of X(g). Note that we have also shown that the A(g), g = G, are the matrices of the restriction of G to W. Hence they must all be square and of the same size. Conversely, suppose each X(g) has the given form with every A(9) being fx f. Let V=Cd and consider w = Clee.....es). where e, is the column vector with a 1 in the ith row and zeros elsewhere. (the standard basis for C"). Then the placement of the zeros in X(g) assures us that X(g)e, W for 1 ≤i</ and all ge G. Thus W G-module, and it is nontrivial because the matrix of zeros is nonempty. Clearly, any epresentation of degree 1 is irreducible. It seems hard to de- termine when a representation of greater degree will be irreducible. Certainly, checking all possible subspaces to find out which ones are submodules is out of the question. This unsatisfactory state of affairs will be remedied after we discuss inner products of group characters in Section 1.9. From the preceding examples, both the defining representation for S, and the group algebra for an arbitrary G are reducible if n 22 and [G] ≥ 2, respec- tively. After all, we produced nontrivial submodules. Let us now illustrate the alternative approach via matrices using the defining representation of Ss. We must extend the basis (1+2+3) for W to a basis for V = C(1,2,3)- Let us pick B=(1+2+3, 2, 3) Of course, X(e) remains the 3 x 3 identity matrix. To compute X( (1,2)), we look at (1, 2)'s action on our basis: (1,2)(1+2+3)=1+2+3, (1, 2)2=1=(1+2+3)-2-3, (1,2)3= 3. x(g) = the number of fixed points of g acting on S. 4. Let G be an abelian group. Find all inequivalent irreducible represen- tations of G. Hint: Use the fundamental theorem of abelian groups. 5. If X is a matrix representation of a group G, then its kernel is the set N = {g €G: X(g) = 1}. A representation is faithful if it is one-to-one. (a) Show that N is a normal subgroup of G and find a condition on N equivalent to the representation being faithful. (b) Suppose X has character x and degree d. Prove that g € N if and only if x(g) = d. Hint: Show that x(g) is a sum of roots of unity. (c) Show that for the coset representation, Nig, Hg,¹, where the 9 are the transversal. (d) For each of the following representations, under what conditions are they faithful: trivial, regular, coset, sign for Sn, defining for Sn, degree 1 for C₁? (e) Define a function Y on the group G/N by Y(gN) = X(g) for gN € G/N. i. Prove that Y is a well-defined faithful representation of G/N. ii. Show that Y is irreducible if and only if X is. iii. If X is the coset representation for a normal subgroup H of G, what is the corresponding representation Y? 6. It is possible to reverse the process of part (e) in the previous exercise. Let N be any normal subgroup of G and let Y be a representation of G/N. Define a function on G by X(g) = Y(gN). (a) Prove that X is a representation of G. We say that X has been lifted from the representation Y of G/N. (b) Show that if Y is faithful, then X has kernel N. (c) Show that X is irreducible if and only if Y is. Let X be a reducible matrix representation with block form given by equation (1.4). Let V be a module for X with submodule W correspond- ing to A. Consider the quotient vector space V/W = {v+W: VEV}. Show that V/W is a G-module with corresponding matrices C(g). Fur- thermore, show that we have Ve We (V/W).