Math 100C: Homomorphism Rings of G-modules Supplement 1. Homomorphism rings Let V and W be G-modules (finite dimensional, over field C of complex numbers, where we assume the group G is finite). The aim of this blurb is to `determine' the C-vector space HomG(V; W ) of G-module endomorphisms ' : V ! W . Our basic tool is Schur's Lemma: Lemma 1.1. (Schur's Lemma) Let V be an irreducible G-module and let ' : V ! V be a G-module endomorphism. There exists α 2 C such that '(v) = αv for all v 2 V . Furthermore, if V and W are nonisomorphic irreducibles and if ' : V ! V is a G-module endomorphism, then ' ≡ 0. If V and W are both irreducible, Schur's Lemma gives us our answer { we have ( V ∼= W Hom (V; W ) ∼= C G G 0 otherwise. But what if V and W are not irreducible? The first step is to apply Maschke's Theorem to write V and W as direct sums of irreducible modules: ∼ ⊕n1 ⊕nr V =G (U1) ⊕ · · · ⊕ (Ur) ∼ ⊕m1 ⊕mr W =G (U1) ⊕ · · · ⊕ (Ur) ; where U1;:::;Ur are pairwise nonisomorphic irreducible G-modules and n1; : : : ; nr and m1; : : : ; mr are nonnegative integers (by allowing some of the ni; mi to be zero, we can always write down such a decomposition). We know ( i = j Hom (U ;U ) ∼= C G i j 0 i 6= j: The idea is to boost this result to get a reasonable picture of HomG(V; W ). To transfer between V; W; and the Ui modules, we will make use of inclusion and projection maps. For 1 ≤ i ≤ r and 1 ≤ j ≤ ni, let ⊕n1 ⊕nr ιi;j : Ui ,! (U1) ⊕ · · · ⊕ (Ur) = V th ⊕ni be the inclusion of Ui into the j factor in the direct sum Ui . The map ιi;j is an injection of G-modules. Similarly, for 1 ≤ i ≤ r and 1 ≤ j ≤ mi, let ⊕m1 ⊕mr πi;j : W = (U1) ⊕ · · · ⊕ (Ur) Ui th ⊕mi be projection of W onto the j factor in the direct sum Ui . 1 2 How do the ιi;j and πi;j maps help us? Well, if ' : V ! W is any G-module homomorphism, then for any i; i0; j; j0 we get a G-module homomorphism πi0;j0 ◦ ' ◦ ιi;j : Ui ! Ui0 between the irreducible G-modules Ui and Ui0 . These `little maps' characterize ': Problem 1.2. Let ' : V ! W and '0 : V ! W be two G-module homomor- 0 0 phisms. Prove that ' = ' if and only if πi0;j0 ◦ ' ◦ ιi;j = πi0;j0 ◦ ' ◦ ιi;j for all 0 0 1 ≤ i; i ≤ r and 1 ≤ j ≤ ni and 1 ≤ j ≤ mi. 0 Schur's Lemma tells us that πi0;j0 ◦ ' ◦ ιi;j ≡ 0 unless i = i , in which case there exists α 2 C such that πi;j ◦ ' ◦ ιi;j scales by α. Encoding the numbers 0 0 α = αi;j;i0;j0 for the various choices of i; j; i ; j gives rise to an injective C-linear map Φ : HomG(V; W ) ! Matm1×n1 (C) ⊕ · · · ⊕ Matmr×nr (C): Problem 1.3. Prove that Φ is a surjection. (Hint: Suppose you have an m × n complex matrix A. If U is an irreducible G-module, how can you build a G-module homomorphism U ⊕n ! U ⊕m using the matrix A?) By the above problem, the map Φ is an isomorphism of C-vector spaces, so that dimC HomG(V; W ) = m1n1 + ··· + mrnr: Problem 1.4. Let V be a G-module. Prove that the decomposition of V into irreducibles is unique. That is, suppose we have ∼ ⊕n1 ⊕nr V =G (U1) ⊕ · · · ⊕ (Ur) ; ∼ ⊕m1 ⊕mr V =G (U1) ⊕ · · · ⊕ (Ur) : for irreducible modules U1;:::;Ur and nonnegative integers n1; : : : ; nr and m1; : : : ; mr. Prove that ni = mi for all i. (Hint: Consider the C-vector space Hom(Ui;V ) { what is its dimension? This problem could also be solved by appeal to the Jordan-H¨olderTheorem.) As an example, suppose that V; U; and W are irreducible G-modules. Con- sider the direct sums V ⊕3 ⊕U ⊕2 and V ⊕4 ⊕U ⊕3 ⊕W ⊕3; what does the C-vector ⊕3 ⊕2 ⊕4 ⊕3 ⊕3 space HomG(V ⊕ U ;V ⊕ U ⊕ W ) look like? Answer: It looks like a direct sum of matrix vector spaces, namely Mat3×4(C) ⊕ Mat2×3(C) ⊕ Mat0×3(C); where we identify the space of `0 × 3' matrices Mat0×3(C) with the zero space 0. In particular, the dimension of this space is 3 × 4 + 2 × 3 + 0 × 3 = 12 + 6 + 0 = 18: 3 2. Endomorphism rings What if V = W in the above discussion? The homomorphism vector space HomG(V; V ) gets a new name: the endomorphism algebra EndG(V ) := HomG(V; V ) = fall G-module homomorphisms ' : V ! V g: As the name suggests, EndG(V ) is not just a C-vector space, but also a C- algebra { the multiplication operation is the composition of homomorphisms: ('1 · '2)(v) := '1('2(v)): If V 6= W , there is no natural algebra structure on the vector space HomG(V; W ) { try not to confuse these situations. What does the endomorphism ring EndG(V ) look like as a C-algebra? Let us decompose V as in Maschke's Theorem: ∼ ⊕n1 ⊕nr V =G (U1) ⊕ · · · ⊕ (Ur) ; for some nonisomorphic irreducibles U1;:::;Ur and some positive integers n1; : : : ; nr. In the last section we constructed an isomorphism of C-vector spaces: Φ : EndG(V ) ! Matn1 (C) ⊕ · · · ⊕ Matnr (C): The right hand side may be regarded as a C-algebra (where the multiplication is matrix multiplication in each direct summand). Here is a boring problem. Problem 2.1. Prove that Φ is an isomorphism of C-algebras (i.e. it sends composition of endomorphisms to multiplication of matrices). To make this theory more interesting, let's consider an explicit example. Let n > 1 be an integer and consider the permutation action of the symmetric n n group Sn on V = C . In the standard basis of C , this corresponds to the group homomorphism P : Sn ! GLn(C); where for any σ 2 Sn the matrix Pσ is the permutation matrix of σ: ( 1 σ(i) = j (P ) = σ i;j 0 otherwise. Since n > 1, the module V is not irreducible { rather, it breaks up into ∼ two irreducible modules V =Sn U ⊕ W , where U is the trivial 1-dimensional n representation of Sn and W = f(α1; : : : ; αn) 2 C : α1 + ··· + αn = 0g. Since U and W are not isomorphic (why?) we know that ∼ ∼ 2 EndSn (V ) = Mat1(C) ⊕ Mat1(C) = C : 4 Let us see the results of the above paragraph, explicitly, using matrices. We can identify Sn-module endomorphisms V ! V with n × n complex matrices T 2 Matn(C) such that TPσ = PσT for all σ 2 Sn. Here is a more interesting problem. Problem 2.2. Let T 2 Matn(C). Prove that TPσ = PσT for all permutation matrices Pσ if and only if T has the form 0α β ··· β1 Bβ α ··· βC T = B . C @ . A β β ··· α for some complex numbers α; β 2 C. Use this fact to deduce an isomorphism of C-algebras ∼ 2 EndSn (V ) = C : (Hint: What happens when you multiply two matrices in the above form to- gether? Try to write down an isomorphism that respects this multiplication!).