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The center of the symmetric group Sn is trivial for n > 2. For n ≠ 6, S n ≃Aut(Sn). When n = 6, [Aut( S6) : Inn(S6] = 2. To understand the structure of A=Aut(S6), we study its linear (group) representations over C. A linear representation will later be defined as a module over the group ring CA. Maschke's Theorem implies that every finite dimensional linear representation decomposes into a finite direct sum of irreducible representations. Our goal is to find a character table for A. The character of a group element is the trace of the corresponding matrix under the representation. A character table is an array of characters. Start with irreducible representations of S6 and invoke the Induced Character Theorem to find representations of A. Representations of A may or may not be irreducible after induction. Since conjugacy classes of S n are in one-to-one correspondence with partitions of n, there is a beautiful combinatorial theory that enables us to explicitly construct all irreducible representations of Sn.

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AN EXPLICIT CONSTRUCTION OF THE CHARACTER TABLE FOR AUT( ) REPRESENTATIONS OF AUT( ) A THESIS Presented to the Department of Mathematics and Statistics California State University, Long Beach In Partial Fulfillment of the Requirements for the Degree Master of Science in Mathematics Committee Members: William L. Murray, Ph.D. (Chair) John O. Brevik, Ph.D. Robert C. Valentini, Ph.D. College Designee: Tangan Gao, Ph.D. By Jared R. Sutton B.S., 2014, California State University, Long Beach December 2016 ProQuest Number: 10240421 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. ProQuest 10240421 Published by ProQuest LLC ( 2016 ). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 ABSTRACT AN EXPLICIT CONSTRUCTION OF THE CHARACTER TABLE FOR AUT(S6 q REPRESENTATIONS OF AUT(S6 q By Jared R. Sutton December 2016 The center of the symmetric group Sn is trivial for n ° 2. For n ‰ 6, Sn » AutpSn q. When n “ 6, rAutpS6 q : InnpS6 qs “ 2. To understand the structure of A “ AutpS6 q, we study its linear (group) representations over C. A linear representation will later be defined as a module over the group ring CA. Maschke’s Theorem implies that every finite dimensional linear representation decomposes into a finite direct sum of irreducible representations. Our goal is to find a character table for A. The character of a group element is the trace of the corresponding matrix under the representation. A character table is an array of characters. Start with irreducible representations of S6 and invoke the Induced Character Theorem to find representations of A. Representations of A may or may not be irreducible after induction. Since conjugacy classes of Sn are in one-to-one correspondence with partitions of n, there is a beautiful combinatorial theory that enables us to explicitly construct all irreducible representations of Sn . ii ACKNOWLEDGEMENTS First, I would like to thank my parents Ellen and James Sutton for their support. Without them, I would not be where I am today. Second, I would like to thank my thesis advisor Dr. William Murray for allowing me the privilege to work side-by-side. Third, I would like to thank Esther Martinez and Michelle Butkivich for the many conversations regarding representation theory. Finally, I would like to thank all of my friends and family who supported me every step of the way. iii TABLE OF CONTENTS ABSTRACT.................................................................................................................................. ii ACKNOWLEDGEMENTS .......................................................................................................... iii 1. GROUP REPRESENTATIONS ....................................................................................... 1 2. REPRESENTATION DECOMPOSITION ...................................................................... 14 3. CHARACTER THEORY ................................................................................................. 21 4. THE AUTOMORPHISM GROUP OF ........................................................................ 41 5. REPRESENTATIONS OF .......................................................................................... 55 6. REPRESENTATIONS OF AUT( )................................................................................ 68 REFERENCES ............................................................................................................................. 85 iv CHAPTER 1 GROUP REPRESENTATIONS 1.1. Definition of Group Representation Throughout this work, we let G denote a finite group and V denote a finite dimensional vector space over a field F . There are several ways to define a group representation of G on V . For now, we define a group representation of G on V using either of the following equivalent definitions: 1. A binary operation ˚ : G ˆ V Ñ V that respects the group operation as well as the vector space structure: (a) ˚pgh, vq “ ˚pg, ˚ph, vqq (b) ˚pg, u ` vq “ ˚pg, uq ` ˚pg, vq (c) ˚pg, ↵vq “ ↵ ˚ pg, vq (d) ˚pe, vq “ v for all g, h P G, u, v P V and ↵ P F , where e P G is the group identity. 2. A group homomorphism ⇢ : G Ñ AutpV q » GLn pF q where n “ dim V . For “1 ùñ 2”, start with a binary operation ˚ : G ˆ V Ñ V satisfying (a) through (d). Define ⇢ : G Ñ AutpV q by ⇢pgqpvq “ ˚pg, vq. Then ⇢ is a homomorphism since ⇢pghqpvq “ ˚pgh, vq “ ˚pg, ˚ph, vqq ´ ¯ “ ⇢pgq ˚ ph, vq ´ ¯ “ ⇢pgq ⇢phqpvq “ p⇢pgq ˝ ⇢phqqpvq. Let us show that these definitions are indeed equivalent. For “2 ùñ 1”, start with a group homomorphism ⇢ : G Ñ AutpV q and define a binary operation ˚ : G ˆ V Ñ V by 1 ˚pg, vq “ ⇢pgqpvq. We must check that (a) through (d) hold. For (a), ˚pgh, vq “ ⇢pghqpvq “ ⇢pgq⇢phqpvq ” ı “ ⇢pgq ˚ ph, vq “ ˚pg, ˚ph, vqq. For (b), ˚pg, u ` vq “ ⇢pgqpu ` vq “ ⇢pgqpuq ` ⇢pgqpvq “ ˚pg, uq ` ˚pg, vq. For (c), ˚pg, ↵vq “ ⇢pgqp↵vq “ ↵⇢pgqpvq ” ı “ ↵ ˚ pg, vq . For (d), ˚pe, vq “ ⇢peqpvq “ In pvq “ v. ⌅ We will study representations as homomorphisms into matrix groups as opposed to 2 homomorphisms into abstract groups of automorphisms of V . Once we fix a basis of V , the definition(s) above imply that each g P G defines an invertible linear transformation of V and thus can be associated with an element of GLn pF q. Let us begin with a few concrete examples. Example 1.1.1. Let G “ S4 act by permutation on ta, b, c, du and let V “ spanta, b, c, du over Q. Extend the G-action linearly to all of V . Then ⇢ : G Ñ AutpV q is called the permutation representation. For example, consider pabcq P G and 2a ` 3b ´ 5c ` 7d P V . Linearity allows us to write: pabcqp2a ` 3b ´ 5c ` 7dq “ 2b ` 3c ´ 5a ` 7d “ ´5a ` 2b ` 3c ` 7d. Example 1.1.2. Let G be a group and let V “ spantg : g P Gu over an arbitrary field F . That is, each v P V is a formal linear combination of elements from G: ÿ v“ ↵g g gPG with ↵g P F . Define an action of G on V as ˆÿ ˙ ÿ h ↵g g “ ↵g hg gPG gPG for each h P G. This is called the regular representation and has dimension |G|. Example 1.1.3. Let G be a group and let V be a one-dimensional vector space. Elementary group theory enables us to compute all one-dimensional representations of G on V in an elegant way. Let’s make this precise; a one-dimensional representation of G is a homomorphism of the form ⇢ : G Ñ AutpV q “ GL1 pF q “ F ˚ . Since F ˚ is abelian, for such a homomorphism to exist, ⇢ must factor through the abelianization of G (the maximal abelian quotient): G Ñ G{rG, Gs Ñ F ˚ where rG, Gs “ xx´1 y ´1 xy | x, y P Gy. 3 In other words, we only have to consider homomorphisms of the form: : G{rG, Gs Ñ F ˚ . For example, rSn , Sn s “ An for n ° 2 (Dummit and Foote, 2004, Exercise 5.4.5). That is, Sn {rSn , Sn s » Z2 . Hence there are only two homomorphisms ⇢1 , ⇢2 : Z2 Ñ F ˚ and thus there are only two one-dimensional representations of Sn : 1. The Trivial Representation: ⇢1 : g ﬁÑ 1 for all g P Sn . 2. The Sign Representation: ⇢2 : g ﬁÑ 1 for all g P An and g ﬁÑ ´1 for all g P Sn zAn . Remark. If char F “ 2, then there is only the trivial representation. We now have an intuitive idea of what a group representation is. The next section introduces a third definition of group representation which will allow us to study representations as modules. 1.2. Representations as Modules Here we give a new definition of group representation that we will use throughout the remainder of the text. First we introduce some new notation. As usual, let G denote a finite group and let F be a field (later we require F “ C). Define the group ring or group algebra: # ˇ + ÿ ˇ ˇ R “ FG “ ↵g g ˇ ↵g P F . gPG ˇ Previously we defined the regular representation as F G but as a vector space. By extending F G multiplicatively we get a ring. For example, take G “ S3 and F “ C. We calculate the product as p1 ` iqp123q ¨ pip12q ` p13qq “ p´1 ` iqp123qp12q ` p1 ` iqp123qp13q “ p´1 ` iqp13q ` p1 ` iqp23q. The construction of group rings is essential for studying representations as modules. Recall that a (left) R-module is just an abelian group pM, `q equipped with a (left) R-action. A representation 4 of G on V defines an F G-module structure on V . Hence we have a one-to-one correspondence between representations of G and F G-modules: Theorem 1.2.1 (Representation Equivalence). There is a one-to-one correspondence between representations of G over F and F G-modules. Proof. Let ⇢ : G Ñ AutpV q be a representation of G on V over F . By defining the action of F G on V to be: ˆÿ ˙ ÿ ↵g g pvq “ ↵g ⇢pgqpvq, gPG gPG we will show that V is an F G-module. Conversely, if we start with an F G-module V , we will show that V is a representation of G. Since F e Ä F G is a subring, where we identify F with F e, we must have that V is also an F -module. Since an F -module is just an F -vector space and G is acting on it in a way that is compatible with the F -action, we have a representation of G on V . To make this precise, let M denote the collection of all F G-modules and R denote the collection of all representations of G over F . Let V P M be an F G-module. Define ⇢ “ f pV q as follows: ⇢pgqpvq “ 1 ¨ gpvq. To show that ⇢ P R, we must show that ⇢ satisfies conditions (a) through (d) in the definition of group representation. For (a), ˚pgh, vq “ ⇢pghqpvq “ 1 ¨ ghpvq “ 1 ¨ gphvq “ ⇢pgqphvq “ ˚pg, ˚ph, vqq. 5 For (b), ˚pg, u ` vq “ ⇢pgqpu ` vq “ 1 ¨ gpu ` vq “ 1 ¨ gpuq ` 1 ¨ gpvq “ ⇢pgqpuq ` ⇢pgqpvq “ ˚pg, uq ` ˚pg, vq. For (c), ˚pg, ↵vq “ ⇢pgqp↵vq “ 1 ¨ gp↵vq “ ↵ ¨ 1 ¨ gpvq “ ↵⇢pgqpvq ` ˘ “ ↵ ˚ pg, vq . For (d), ˚pe, vq “ ⇢peqpvq “ 1 ¨ epvq “ 1¨v “ v. This establishes a function f : M Ñ R. Similarly, let P R be a representation of G over F . Define W “ hp q by ÿ ÿ ↵g gpvq “ ↵g pgqpvq. gPG gPG 6 To see that W is an F G-module, we verify the three axioms for modules below. For the first, ÿ ÿ ↵g gpu ` vq “ ↵g pgqpu ` vq gPG gPG ÿ ÿ “ ↵g pgqpuq ` ↵g pgqpvq gPG gPG ÿ ÿ “ ↵g gpuq ` ↵g gpvq. gPG gPG For the second, ˆÿ ÿ ˙ ÿ ↵g g ` g g pvq “ p↵g ` g qgpvq gPG gPG gPG ÿ “ p↵g ` gq pgqpvq gPG ÿ ÿ “ ↵g pgqpvq ` g pgqpvq gPG gPG ÿ ÿ “ ↵g gpvq ` g gpvq. gPG gPG For the third, ˆÿ ÿ ˙ ÿ ↵g g h h pvq “ ↵g h pghqpvq gPG hPG g,h P G ÿ “ ↵g h pghqpvq g,h P G ÿ “ ↵g h pgq phqpvq g,h P G ÿ “ ↵g hg phqpvq g,h P G ˆÿ ˙« ÿ ff “ ↵g g h phqpvq gPG hPG ˆÿ ˙«ˆ ÿ ˙ ff “ ↵g g h h pvq . gPG hPG 7 For the fourth, 1 ¨ epvq “ 1 ¨ peqpvq “ 1 ¨ Ipvq “ 1 ¨ pvq. This establishes a function h : R Ñ M . To show that there is a one-to-one correspondence between M and R, we need to show that h ˝ f “ IdM and f ˝ h “ IdR . To show h ˝ f “ IdM , we need to show that the F G-action on ph ˝ f qpV q agrees with the F G-action on V . That is, we must ÿ ÿ ÿ take ↵g g P F G and v P V and show that ↵g gpvq defined via h ˝ f is equal to ↵g gpvq gPG gPG gPG when defined directly. Suppose f pV q “ ⇢. Then „ ˆÿ ˙⇢ „ ˆ ˆÿ ˙˙⇢ ph ˝ f q ↵g g pvq “ h f ↵g g pvq gPG gPG „ ˆÿ ˙⇢ “ h ↵g ⇢pgq pvq gPG ˆÿ ˙ “ ↵g g pvq. gPG Finally, ” ı ” ` ˘ ı pf ˝ hqp qpgq pvq “ f hp q pgq pvq “ 1 ¨ gpvq pwhere 1 ¨ g acts as hp qq ˆÿ ˙ “ ↵g g pvq pwhere ↵g “ 1, ↵g1 “ 0 for g 1 ‰ gq gPG “ 1 ¨ pgqpvq “ pgqpvq. ⌅ Definition 1.2.2. Let ⇢1 : G Ñ AutpV q and ⇢2 : G Ñ AutpV q be two n-dimensional representations of G. An isomorphism of representations is a vector space isomorphism „ Ñ W that respects the G-action. That is, f p⇢1 pgqvq “ ⇢2 pgqf pvq for all g P G and v P V . f :V › 1.3. Subrepresentations When studying group representations, we want to understand the action of G on V 8 completely, that is, how does G act on subspaces W Ä V ? More specifically, which subspaces are invariant under the action of G? Definition 1.3.1. Let T : V Ñ V be a linear map. A subspace W Ä V is said to be invariant if T pW q Ñ W . Definition 1.3.2. Let ⇢ : G Ñ AutpV q » GLn pF q be a representation. A subrepresentation of G over F is a subspace W Ä V that is invariant under the action of G. That is, : G Ñ AutpW q » GLm pF q where m “ dim W † dim V . Remark. A subrepresentation of G on W Ñ V gives W the structure of an F G-submodule of V and vice versa. Definition 1.3.3. A representation of G is said to be irreducible if it contains no proper nontrivial subrepresentations. Example 1.3.4. Let G “ S3 act by permutation on the coordinate components of vectors in V “ F 3 . Consider the one-dimensional subspace W “ tpx, y, zq | x “ y “ zu. If px, y, zq P W , then px, y, zq P W for all P Sn . Hence W is G-invariant and thus yields a subrepresentation for this action. A less obvious subrepresentation of G is the subspace U “ tpx, y, zq | x ` y ` z “ 0u. Claim. W Ä U if and only if char F “ 3, and U is irreducible otherwise. Proof. A basis for W is tp1, 1, 1qu and a basis for U is tp1, ´1, 0q, p0, 1, ´1qu. If char F “ 3, then W Ä U since 1 ` 1 ` 1 “ 3 “ 0. Conversely, if W Ä U , then we can write c1 p1, ´1, 0q ` c2 p0, 1, ´1q “ p1, 1, 1q which says that c1 “ 1, c2 “ ´1, and c2 ´ c1 “ 1. The last equation says 0 “ 3 which implies that char F “ 3. Suppose now char F ‰ 3 and suppose U0 Ä U is a one-dimensional subrepresentation. Let us write U0 “ spantpa, b, cqu where a ` b ` c “ 0, thus c “ ´a ´ b. Since 9 U0 “ spantpa, b, ´a ´ bqu is a subrepresentation, the action of p12q P S3 on U0 tells us that pb, a, ´a ´ bq P U0 . Since U0 is a 1-dimensional subspace of U , there exists a scalar k such that pa, b, ´a ´ bq “ kpb, a, ´a ´ bq. There are two possible solutions: either k “ 1 with a “ b or we have ´a ´ b “ 0. If ´a ´ b “ 0, then pa, ´a, 0q “ kp´a, a, 0q implies that a “ ´ak. We cannot have a “ 0 for otherwise dim U0 “ 0. So we must have k “ ´1. Then tp1, ´1, 0qu is a basis for U0 . But the action of p23q on p1, ´1, 0q would imply dim U0 • 2. Suppose now that k “ 1 with a “ b. Then U0 “ spantpa, a, ´2aqu “ spantp1, 1, ´2qu. The action of p23q on U0 says that p1, ´2, 1q P U0 . So there must exist a scalar m such that p1, ´2, 1q “ mp1, 1, ´2q. The second coordinate component tells us that m “ ´2. But p1, ´2, 1q “ p´2, ´2, 4q if and only if char F “ 3, which we assumed to be false. ⌅ 1.4. Direct Sums In this section we introduce the notion of “adding” two representations. Definition 1.4.1. Let ⇢1 : G Ñ AutpV q and ⇢2 : G Ñ AutpW q be representations of G with dim V “ n and dim W “ m. We define the direct sum of ⇢1 and ⇢2 to be the representation p⇢1 ‘ ⇢2 q : G Ñ AutpV ‘ W q » GLn`m pF q where p⇢1 ‘ ⇢2 qpgqpv, wq “ p⇢1 pgqpvq, ⇢2 pgqpwqq. For short, we write gpv, wq “ pgv, gwq. Let ⇢1 pgq and ⇢2 pgq be the associated linear transformations. Then ¨ ˛ ˚ ⇢pgq 0 ‹ p⇢1 ‘ ⇢2 qpgq “ ˝ ‚. 0 pgq 10 Note that V ‘ W is a direct sum of F G-modules by Representation Equivalence. Proposition 1.4.2. Let V be a vector space with a G-representation, and suppose that U and W are subspaces of V such that V “ U ‘ W as vector spaces and such that U and W are subrepresentations of V . Then V “ U ‘ W as representations. Remark. If ⇢ “ ⇢ 1 ‘ ⇢1 ‘ ¨ ¨ ¨ ‘ ⇢1 ‘ ⇢ loooooooooomoooooooooon 2 ‘ ⇢2 ‘ ¨ ¨ ¨ ‘ ⇢2 , then we write ⇢ “ n⇢1 ‘ m⇢2 . loooooooooomoooooooooon n m Example 1.4.3. In Example 1.3.4, we found two subrepresentations W “ tpx, y, zq | x “ y “ zu and U “ tpx, y, zq | x ` y ` z “ 0u. We showed that if char F ‰ 3, then U is irreducible. In this case, we have V “ U ‘ W . 1.5. Tensor Products The notion of a tensor product may be applied to the context of a variety of mathematical objects and structures. Tensor products will be necessary when defining induced representations. Since representations can be defined as F G-modules, we will want to study tensor products of modules. Definition 1.5.1. Let R be a ring and let M, N be right and left R-modules respectively. Then the tensor product of M and N denoted M bR N is defined to be # +O ÿ pmi , ni q : mi P M, ni P N „ finite where „ is the set of relations that define bilinearity: ` ˘ ` m, pn1 ` n2 q „ m, n1 q ` pm, n2 q ` ˘ pm1 ` m2 q, n „ pm1 , nq ` pm2 , nq pmr, nq „ pm, rnq for all m, m1 , m2 P M , n, n1 , n2 P N and r P R. A simple tensor m b n is defined to be the image of pm, nq in the quotient. The Universal Mapping Property says that b is a module through which all bilinear maps factor uniquely. Elements of M bR N are finite sums of simple tensors. 11 If V and W are finite-dimensional F -vector spaces with bases tvi u and twj u respectively, then tvi b wj u is a basis for V bF W . Let : V Ñ V and : W Ñ W be linear transformations with bases tvi u and twj u and associated matrices A and B respectively where 1 § i § n and 1 § j § m. Then the matrix of the linear transformation b : V b W Ñ V b W with respect to the basis tvi b vj u is given by ¨ ˛ ¨ ˛ A B A1,2 B ¨ ¨ ¨ A1,n B B A B1,2 A ¨¨¨ B1,m A ˚ 1,1 ‹ ˚ 1,1 ‹ ˚ ‹ ˚ ‹ ˚ A2,1 B A2,2 B ¨ ¨ ¨ A2,n B ‹ ˚ B2,1 A B2,2 A ¨ ¨ ¨ B2,m A ‹ ˚ ‹ ˚ ‹ AbB “˚ .. .. .. ‹ or ˚ .. .. .. ‹. ˚ ... ‹ ˚ ... ‹ ˚ . . . ‹ ˚ . . . ‹ ˝ ‚ ˝ ‚ An,1 B An,2 B ¨ ¨ ¨ An,n B Bm,1 A Bm,2 A ¨ ¨ ¨ Bm,m A depending on how we order tvi b wj u (i.e., horizontally or vertically). Example 1.5.2. Let R “ Z, MR “ Zm , and R N “ Zn where pm, nq “ 1. Since Z is a commutative ring, define a left-action of Z on Zm by rm “ mr so that Zm is a Z-bimodule. Then the tensor product Zm bZ Zn is a left Z-module defined by rpm b nq “ prmq b n. Now, what is Zm bZ Zn ? In other words, what is Zm bZ Zn isomorphic to? Consider a b b P Zm bZ Zn for some a P Zm and b P Zn . Since pm, nq “ 1, we may write mr ` ns “ 1 for some r, s P Z. Then a b b “ 1 ¨ pa b bq “ pmr ` nsqpa b bq “ pmrq ¨ pa b bq ` pnsq ¨ pa b bq “ pmraq b b ` pnsaq b b “ pmraq b b ` a b pnsbq “ 0bb`ab0 “ 0. Therefore, Zm bZ Zn » t0u. 12 Definition 1.5.3. Let ⇢1 : G Ñ AutpV q and ⇢2 : G Ñ AutpW q be representations of G over F with n “ dim V and m “ dim W . We define the tensor representation of ⇢1 and ⇢2 to be p⇢1 b ⇢2 q : G Ñ AutpV b W q » GLnm pF q where p⇢1 b ⇢2 qpgqpv b wq “ ⇢1 pgqpvq b ⇢2 pgqpwq. For short, write gpv b wq “ gpvq b gpwq. Note that V b W is a tensor product of F G-modules by Representation Equivalence. Example 1.5.4. Suppose A “ ⇢1 pgq is a 2 ˆ 2 matrix and B “ ⇢2 pgq is a 3 ˆ 3 matrix. Then ¨ ˛ ¨ ˛ ˚ B1,1 A B1,2 A B1,3 A ‹ A ˚ 1,1 B A 1,2 B ‹ ˚ ‹ p⇢1 b ⇢2 qpgq “ A b B “ ˝ ‚ or ˚ ˚ B2,1 A B2,2 A B2,3 A ‹. ‹ A2,1 B A2,2 B ˝ ‚ B3,1 A B3,2 A B3,3 A 13 CHAPTER 2 REPRESENTATION DECOMPOSITION 2.1. Indecomposable vs. Irreducible Now that we have a fundamental grasp on what representations are and how to study them, we would like to know if all representations decompose into subrepresentations. The next couple of sections provide us with enough background to prove Maschke’s Theorem, which says that all representations decompose if char F - |G|. Recall that V is irreducible if it has no proper nontrivial subrepresentations of G. Equivalently, V is simple as an F G-module (V contains no proper nontrivial submodules). The notion of irreducibility raises the notion of representation decomposition, that is, decomposing a representation into irreducible subrepresentations. Definition 2.1.1. V is indecomposable if cannot be written as a direct sum of nontrivial subrepresentations. Remark. Note that irreducible implies indecomposable. Example 2.1.2. Let W “ tpx, y, zq | x “ y “ zu and U “ tpx, y, zq | x ` y ` z “ 0u be the subrepresentations from Example 1.3.4. Earlier we saw that char F ‰ 3 implied that V could be written as a direct sum of U and W . But char F “ 3 implied that W à U . In the first case, U is irreducible (and hence indecomposable). If char F “ 3, U is not irreducible, but is indecomposable since U “ W ‘ W 1 would imply W 1 à U , which is impossible since W is the only one-dimensional subrepresentation of U . Assume by way of contradiction

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