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Primitive Central Idempotents in the Rational Group Algebras of Some Non-monomial Groups

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(1)國立臺灣師範大學理學院數學系 碩士論文 Department of Mathematics, College of Science. National Taiwan Normal University Master’s Thesis. Primitive Central Idempotents in the Rational Group Algebras of Some Non-monomial Groups. 林書愷 Lin, Shu-Kai. 指導教授:劉家新 博士 Advisor: Liu, Chia-Hsin, Ph.D.. 中華民國109年7月 July 2020.

(2) Abstract It is well-known that every group algebra of a finite group over the field of rational numbers is isomorphic to a direct sum of finitely many matrix rings over division rings. This is the so-called Wedderburn-Artin decomposition. It follows that there are finitely primitive central idempotents in the rational group algebra. However, it is not easy to write down an explicit form for each primitive central idempotent when an arbitrary group is given. It is known that primitive central idempotents have a nice description for finite monomial groups and nilpotent groups. Such description is investigated by E. Jespers, A. Olivieri and Á. del Río. In this thesis, we focus on some non-monomial groups and give an explicit form for primitive central idempotents. Keywords: rational group ring; primitive central idempotent; monomial group; nonmonomial group..

(3) Contents 1 Introduction. 1. 2 Preliminaries 2.1 Notations and definitions. . . . . . . . . . . . . . . . . . . . . . . . . . .. 4 4. 3 Construction by linear characters 3.1 The cyclic group from a character . . . . . . . . . . . . . . . . . . . . . . 3.2 The primitive central idempotents from linear characters . . . . . . . . .. 7 7 8. 4 Construction by Shoda pairs 4.1 Shoda pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Strong Shoda pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9 9 11. 5 The smallest non-monomial group 5.1 Primitive central idempotents of QA4 . . . . . . . . . . . . . . . . . . . . 5.2 Primitive central idempotents of QSL(2, 3) . . . . . . . . . . . . . . . . 5.2.1 GAP code for primitive central idempotents of QSL(2, 3) . . . .. 13 13 15 17. 6 Main results 6.1 Primitive central idempotents of QA5 . . . . . . . . 6.1.2 GAP code for primitive central idempotents 6.2 Primitive central idempotents of QSL(2, 5) . . . . 6.2.2 GAP code for primitive central idempotents 6.3 Primitive central idempotents of QS5 . . . . . . . . 6.3.2 GAP code for primitive central idempotents. 19 19 22 24 27 29 31. References. . . . . . . . . of QA5 . . . . . . . . . . . of QSL(2, 5) . . . . . . . . of QS5 . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 33. I.

(4) Chapter 1. Introduction Let F be a field with characteristic 0 and let G be a finite group. Then the algebra F G is semisimple, namely the regular F G-module is completely reducible. The WedderburnArtin theorem states that F G is isomorphic to a direct sum of finitely many matrix rings over division rings : F G ' Mn1 (D1 ) ⊕ · · · ⊕ Mns (Ds ). It is uniquely expressed up to isomorphism. Namely, if F G ' Mn1 (D1 ) ⊕ · · · ⊕ Mns (Ds ) ' Mm1 (D10 ) ⊕ · · · ⊕ Mmr (Dr0 ) where D1 , . . . , Ds , D10 , . . . , Dr0 are division rings. Then s = r and after a suitable permutation of indices, we have ni = mi and Di ' Di0 for all i. We want to know how this isomorphism actually sends the elements of F G to each component. Consider the elements ei in F G, which are corresponding to the units in each Mni (Di ) for i = 1, . . . , s. Note that ei has the following properties • (idempotent) e2i = ei . • (orthogonal) ei ej = 0 = ej ei for i 6= j. • (central) ei is in the center of F G. Moreover, a central idempotent ei of F G is called a primitive central idempotent if ei cannot be written as a sum f1 + f2 where f1 and f2 are nonzero orthogonal central idempotents. The following proposition shows the importance of primitive central idempotents. Proposition 1.0.1. Consider the regular F G-module F G as an F -algebra and let e ∈ F G be a central idempotent. Then e is primitive if and only if the ring (F G)e is indecomposable as a direct sum of two non-trivial two-sided ideals.. 1.

(5) Proof. Suppose e is not primitive, and we show that (F G)e is decomposable. By assumption there exist nonzero central idempotents e1 , e2 with e = e1 + e2 . Note that e1 , e2 and e are idempotents and hence e1 and e2 are orthogonal. Since e1 = e1 + 0 = e1 (e1 + e2 ) = e1 e, we have (F G)e1 = (F G)e1 e ⊆ (F G)e. Similarly, (F G)e2 = (F G)e2 e ⊆ (F G)e. On the other hand, if a ∈ F G then ae = a(e1 + e2 ) = ae1 + ae2 ∈ (F G)e1 + (F G)e2 and so (F G)e ⊆ (F G)e1 + (F G)e2 . We show that (F G)e = (F G)e1 + (F G)e2 . And since e1 ∈ (F G)e1 and e2 ∈ (F G)e2 , both of (F G)e1 and (F G)e2 are nonzero. Once we show that (F G)e1 ∩ (F G)e2 = {0} then it follows that (F G)e = (F G)e1 ⊕ (F G)e2 , and hence (F G)e is decomposable. For any x ∈ (F G)e1 , we have x = ae1 for some a ∈ F G, then x = ae1 = ae21 = (ae1 )e1 = xe1 . Therefore, for any x ∈ (F G)e1 ∩ (F G)e2 we have x = xe1 and x = xe2 . It follows that x = (xe1 )e2 = x(e1 e2 ) = x · 0 = 0. Finally, (F G)e1 ∩ (F G)e2 = {0}. Conversely, suppose that (F G)e is decomposable. Then (F G)e = I1 ⊕ I2 for some nonzero two-sided ideals I1 and I2 of (F G)e. Since e ∈ (F G)e, there exist unique x ∈ I1 and y ∈ I2 with e = x + y. It is immediate that x and y are central elements. If x = 0 then e = y ∈ I2 , which implies that I1 ⊕ I2 = (F G)e = (F G)y ⊆ I2 , contradicting with the assumption that I1 is nonzero. Therefore, x 6= 0. Similarly, y 6= 0. Next we show that both x and y are central idempotents and orthogonal. Since x ∈ (F G)e, we have x = re for some r ∈ F G. Therefore, x = re = re2 = xe. Thus, (1 − x)x = x − x2 = xe − x2 = x(x + y) − x2 = xy Since (1 − x)x ∈ I1 , xy ∈ I2 , and I1 ∩ I2 = {0}, we have x − x2 = 0. Therefore, x = x2 and xy = 0. Similarly, y = y 2 and yx = 0. Thus, e can be expressed as a sum of nonzero orthogonal central idempotents, and so e is not primitive. If χ is an irreducible character of G over F then the primitive central idempotent of F G associated to χ is denoted by eF (χ), that is, e = eF (χ) is the unique primitive central idempotent of F G such that χ(e) 6= 0. If F = C, we simply denote e(χ). In this thesis, a character of a group is assumed to be a complex character unless otherwise stated. For F = C, it is well known that the primitive central idempotents of CG are all elements of the form χ(1) X χ(g −1 )g, |G| g∈G. where χ is an irreducible character of G. Hence, using Galois descent, one obtains all primitive central idempotents of the rational group algebra QG from χ [JLP03], we denote it as eQ (χ). [JdR16, Theorem 3.3.1] gives an explicit form of eQ (χ) as the following theorem. 2.

(6) Theorem 1.0.2. Let G be a finite group and let χ be a irreducible (complex) character of G. Then the primitive central idempotent of QG from χ has the following form eQ (χ) =. X. e(σ ◦ χ). σ∈Gal(Q(χ)/Q). where Q(χ) = Q(χ(g) : g ∈ G). However, the known methods to compute the character table of a finite group are tasks of exponential growth with respect to the order of the group. Therefore, we aim to find a character free expression of the primitive central idempotents of QG. In [JLP03], the expression in case G is nilpotent has been given. In [JdR16], it gives the expression in case G is monomial. In this thesis, we find the expressions for some non-monomial groups. In chapter 2, we give some notations and definitions. In chapter 3, we recall the construction of primitive central idempotents of F G from the linear characters of F G. In chapter 4, we introduce concepts of Shoda pairs and strong Shoda pairs which can be used to construct all primitive central idempotents of QG when G is monomial. In chapter 5, we illustrate that how to compute the primitive central idempotents of QG with G = SL(2, 3), which is the smallest non-monomial group. We will show that it is very complicated to find primitive central idempotents when G is non-monomial. In chapter 6, we compute the primitive central idempotents of QG with G = A5 , SL(2, 5) and S5 , non-monomial groups. The fact that these groups are non-monomial groups mentioned in [vdW73a], [vdW73b] and [vdW74] which shows the monomiality of groups of order less than 200.. 3.

(7) Chapter 2. Preliminaries 2.1. Notations and definitions. Definition 2.1.1. Let G be a finite group and F be a field. Let H be a subset of G. We e =P denote H h∈H h to be an element in F G. Moreover, if charF - |H|, we also denote P 1 b = H h∈H h to be an element in F G. |H| b is an idempotent of QG. Moreover, let N Remark. Let H be a subgroup of G, then H b is a central idempotent of QG. be a normal subgroup of G, then N b 0 = Proof. For h0 ∈ H, we have Hh bH b =H b is an idempotent of QG. H. 1 |H|. P. h∈H. hh0 =. 1 |H|. P. g bg = 1 P Moreover, if N E G then we have N n∈N n = |N | b is an central idempotent of QG. g ∈ G. Hence N. h∈Hh0 1 |N |. b Therefore, h = H.. P. n∈N g. b for all n=N. b of QG. We show that a normal subgroup N of G can induce a central idempotent N Conversely, let e be a central idempotent of QG. Consider the right group action of G on QG by the multiplication of QG, then the stabilizer subgroup of G with respect to e defined by Ge = {g ∈ G | eg = e} is a normal subgroup of G. b is also a central idempotent of QG If N is non-trivial, it is easy to check that 1 − N b and 1 − N b are mutually orthogonal. Since a finite product of central idempotents and N is still a central idempotent, one can consider the following elements in QG. Definition 2.1.2. Let G be a finite group. Define the element ε(G) in QG by. ε(G) =.   1,. if G = {1}; Y.  . c), (1 − M. M ∈M(G). 4. if G 6= {1};.

(8) where M(G) is the set of all minimal non-trivial normal subgroups of G. Let N be a normal subgroup of G. Define the element ε(G, N ) in QG by. ε(G, N ) =.  b  G,. if N = G; Y. b −M c) = N b (N.  . Y. c), (1 − M. if N 6= G.. M/N ∈M(G/N ). M/N ∈M(G/N ). Note that ε(G) and ε(G, N ) are central idempotents of QG. Remark. In fact, we need not to find M(G), the set of all minimal non-trivial normal bB b = B b and so subgroups of G. If A and B are subgroups of G with A ⊆ B, then A Q b b = (1 − A). b Hence, for G 6= {1}, we have ε(G) = b (1 − A)(1 − B) N ∈N (G) (1 − N ) where N (G) is the set of all non-trivial normal subgroup of G. In particular, if N is a b )ε(G) = ε(G). non-trivial normal subgroup of G, then (1 − N b and Q(G/N ). The following proposition gives us the connection between (QG)N Proposition 2.1.3. For any normal subgroup N of G, we have the natural isomorphism b ' Q(G/N ). (QG)N In particular, ε(G, N ) is the unique pre-image of ε(G/N ). Proof. Let φ be a homomorphism from QG to Q(G/N ), defined by QG −→ Q(G/N ) P P αg g 7−→ αg g¯.. φ:. b ) ⊆ ker φ. Since N b and 1 − N b are mutually It is clear that φ is onto and QG(1 − N b ⊕ QG(1 − N b ). Thus, we only need to show that orthogonal, we have QG = (QG)N b ). Namely, for any r ∈ ker φ write r = rN b + r(1 − N b ), then we claim ker φ ⊆ QG(1 − N b = 0. that rN P P For any r ∈ ker φ write r = g∈G αg g. Then we have φ(r) = g∈G αg g¯ = 0. P That is, g∈N x αg = 0 for any right coset N x in G. Write G = N x1 ∪ N x2 ∪ · · · ∪ N xn . Then we may write r=. X. αg g =. g∈G. =. n X X. αg g i=1 g∈N xi n  X X. X. i=1. g∈N. −. =−. αgxi (1 − g)xi +. g∈N. n X X. αgxi (1 − g)xi .. i=1 g∈N. 5. αgxi xi. .

(9) b =N b −N b = 0. Thus, Note that for any g ∈ N , we have (1 − g)N b= rN. X. . b= αg g N. . g∈G. =−. −. n X X.  b αgxi (1 − g)xi N. i=1 g∈N n XX. b xi = 0. αgxi (1 − g)N. i=1 g∈N. b ) and so (QG)N b ' Q(G/N ). Therefore, ker φ = QG(1 − N. 6.

(10) Chapter 3. Construction by linear characters In this chapter, we construct primitive central idempotents of QG by considering linear characters of G.. 3.1. The cyclic group from a character. Definition 3.1.1. Let χ be a character of G. Define the center of the character by Z(χ) = {g ∈ G | |χ(g)| = χ(1)} and define the kernel of the character by ker χ = {g ∈ G | χ(g) = χ(1)}. The following two lemmas lead us to find a cyclic group by a character of G. Lemma 3.1.2. [Isa94, Lemma 2.15] Let X be a C-representation of G affording the character χ with degree f and let g ∈ G. Then there exist ε1 , . . . , εf ∈ C with |εi | = 1 for all i such that (a) X(g) is similar to a diagonal matrix diag(ε1 , . . . , εf ). (b) χ(g) =. P. εi and so |χ(g)| ≤ χ(1).. Lemma 3.1.3. [Isa94, Lemma 2.27] Let X be a C-representation of G affording the character χ with degree f and let H = Z(χ) and K = ker χ. Then (a) H = {g ∈ G | X(g) = g I for some g ∈ F }. (b) H = Z(χ) is a subgroup of G. (c) χ|H = f λ where λ is a linear character of H. (d) K is a normal subgroup of H and H/K is a cyclic group. 7.

(11) 3.2. The primitive central idempotents from linear characters. For a cyclic group A, the next proposition states that ε(A) is a primitive central idempotent of QA. The following proposition is mentioned in [JdR16, Section 3.3]. Proposition 3.2.1. Let A = hai be a cyclic group and ζ|A| is a primitive |A|-th root of unity. Then φ :  QAε(A) −→ Q(ζ|A| )  P P k k αk a ε(A) 7−→ αk ζ|A| is an isomorphism. In particular, Q(ζ|A| ) is indecomposable and hence ε(A) is a primitive central idempotent of QA. The following theorem is mentioned in [JdR16, Lemma 3.3.2]. Theorem 3.2.2. Let χ be a character of G, H = Z(χ) and K = ker χ, then ε(H, K) is a primitive central idempotent of QH. In particular, if χ is a linear character of G then ε(G, K) is a primitive central idempotent of QG. Proof. By Lemma 3.1.3 (d), we have H/K is a cyclic group. Denote A = H/K. By Proposition 3.2.1, ε(H/K) is a primitive central idempotent of Q(H/K). Therefore, from the natural isomorphism in Proposition 2.1.3, we have ε(H, K) is a primitive central idempotent of QH. In particular, if χ is a linear character, then H = Z(χ) = G. We have ε(G, K) is a primitive central idempotent of QG. For any finite group G, we always have the trivial character where its kernel is the b in QG. whole group G. Thus, we always have a primitive central idempotent e = G One may have interests in how many primitive central idempotents of QG are induced by linear characters of G. The following proposition tells us the number of linear characters of G. Proposition 3.2.3. [Isa94, Corollary 2.23] Let G be a finite group with the commutator subgroup G0 . Then (a) G0 =. T. {ker λ | λ is linear character of G}.. (b) |G : G0 | = the number of linear characters of G. In particular, we have at most |G : G0 | primitive central idempotents of QG which can be found via linear characters of G by Theorem 3.2.2.. 8.

(12) Chapter 4. Construction by Shoda pairs From the above discussion we know a linear character of G can induce a primitive central idempotent of QG. For a linear character χ of a subgroup H of G, the induced character χG of G given by X 1 χ(g t ) χG (g) = |H| t t∈G, g ∈H. may give us some information for primitive central idempotents of QG. Namely we have the proposition in the next section which can be deduced from Mackey’s Theorem [JdR16, Proposition 3.2.1].. 4.1. Shoda pairs. Proposition 4.1.1. [JdR16, Corollary 3.2.3] Let χ be a linear character of a subgroup H of G. Then the induced character χG is irreducible if and only if for every g ∈ G \ H there exists h ∈ H ∩ H g such that χ(hg ) 6= χ(h). Recall that the commutator of h, g in G is denoted by (h, g) = h−1 g −1 hg. Definition 4.1.2. Let G be a finite group and let (H, K) be a pair of subgroups of G. Then (H, K) is called a Shoda pair of G if it satisfies the following conditions. (S1) K is a normal subgroup of H. (S2) H/K is cyclic. (S3) For every g ∈ G \ H there exists h ∈ H such that (h, g) ∈ H \ K. In the view of Shoda pair, the Proposition 4.1.1 can be rewritten as the following statement. Corollary 4.1.3. If χ is a linear character of a subgroup of H of G with kernel K then the induced character χG is irreducible if and only if (H, K) is a Shoda pair of G. 9.

(13) Recall that, if H is a subgroup of G then a right transversal T is a subset of G which contains exactly one element of each right coset of H. Next, we consider the central elements of QG associated to linear characters of subgroups of G. Definition 4.1.4. Let H and K be subgroups of G with K E H and let e(G, H, K) denote the sum of all G-conjugates of ε(H, K), that is if T is a right transversal of CenG (ε(H, K)) in G then e(G, H, K) =. X. ε(H, K)t .. t∈T. In particular, e(G, H, K) is a central element of QG. Note that, the definition of e(G, H, K) is independent of the choice of T . In fact, if we have a linear character χ of a subgroup H of G such that the induced character χG is irreducible, then we may construct a primitive central idempotent of QG associated with χ. First, we need the following definition. Definition 4.1.5. Let χ be a character of G. Then the field of characters of χ over F is F (χ) = F (χ(g) : g ∈ G) Next, we may consider the following theorem. Theorem 4.1.6. [JdR16, Theorem 3.4.2] Let G be a finite group and let H be a subgroup of G. Suppose χ is a linear character of H with K = ker χ and χG is the induced character of χ on G. If χG is irreducible then the primitive central idempotent eQ (χG ) of QG associated to χG has the following expression eQ (χG ) = αe(G, H, K) where α =. |CenG (ε(H, K)) : H| . |Q(χ) : Q(χG )|. A group G is called monomial if all of irreducible characters of G are monomial, namely, induced from linear characters. Therefore, the theorem above gives the following corollary. Corollary 4.1.7. Let G be a monomial group. Then all primitive central idempotents of QG are precisely of the form |CenG (ε(H, K)) : H| e(G, H, K) where (H, K) is a Shoda pair of G. |Q(χ) : Q(χG )| For a nilpotent group G, we can find all primitive central idempotents of QG from the following proposition and Corollary 4.1.7. Proposition 4.1.8. [Isa94, Corollary 6.14] Every nilpotent group is a monomial group.. 10.

(14) 4.2. Strong Shoda pairs. It is easy to check if all distinct G-conjugates of ε(H, K) are orthogonal, then e(G, H, K) is a central idempotent of QG. Thus we consider the following stronger concept of Shoda pair. Definition 4.2.1. A strong Shoda pair of G is a pair (H, K) of subgroups of G satisfying the following conditions (SS1) H E NG (K). (SS2) H/K is cyclic and is also a maximal abelian subgroup of NG (K)/K. (SS3) For every g ∈ G \ NG (K), we have ε(H, K)ε(H, K)g = 0. We also define strongly monomial characters and strongly monomial groups as the following. Definition 4.2.2. Let ψ be a monomial character of G and write ψ = χG where χ is a linear character of a subgroup H of G with K = ker χ. Then ψ is called a strongly monomial character of G if (H, K) is a strong Shoda pair of G. A group G is said to be strongly monomial if every irreducible character of G is strongly monomial. Strong Shoda pairs play important roles for finding primitive central idempotents in some specific group algebras. For instance, if G is abelian-by-supersolvable (which we define below) then every primitive central idempotents of QG is of the form e(G, H, K) for some strong Shoda pair (H, K) of G. Recall that a group G is supersolvable if there is a series of normal subgroups of G {1} = H0 E H1 E · · · E Hs−1 E Hs = G such that each quotient group Hi+1 /Hi is cyclic and each Hi is normal in G. A group G is said to be abelian-by-supersolvable if G has an abelian normal subgroup A such that G/A is supersolvable. The following theorem is from [JdR16, Proposition 3.5.3, Corollary 3.5.4, Theorem 3.5.10]. Theorem 4.2.3. Let G be a finite abelian-by-supersolvable group and e ∈ QG. Then the following conditions are equivalent. (a) e is a primitive central idempotent of QG. (b) e = e(G, H, K) for some strong Shoda pair (H, K) of G.. 11.

(15) (c) e = e(G, H, K) for some pair (H, K) of subgroups of G satisfying the following conditions (SS10 ) K E H E CenG (ε(H, K)). (SS20 ) H/K is cyclic and is also a maximal abelian subgroup of CenG (ε(H, K))/K. (SS30 ) For every g ∈ G \ CenG (ε(H, K)), we have ε(H, K)ε(H, K)g = 0. Therefore, if G is a finite abelian-by-supersolvable group then G is strongly monomial. Note that the conditions (SS10 ), (SS20 ), (SS30 ) in (c) are an alternative definition of strong Shoda pair. In other words, we have Definition 4.2.4. A strong Shoda pair of G is a pair (H, K) of subgroups of G satisfying the following conditions (SS10 ) K E H E CenG (ε(H, K)). (SS20 ) H/K is cyclic and is also a maximal abelian subgroup of CenG (ε(H, K))/K. (SS30 ) For every g ∈ G \ CenG (ε(H, K)), we have ε(H, K)ε(H, K)g = 0.. 12.

(16) Chapter 5. The smallest non-monomial group In this chapter, we provide an example to construct the primitive central idempotents of QG with G = SL(2, 3), the smallest non-monomial group, which is mentioned in [OdRS04]. Since SL(2, 3)/C2 ' A4 , we shall find the primitive central idempotents of QA4 first.. 5.1. Primitive central idempotents of QA4. Consider G = A4 , the alternating group of degree 4. Since V4 , the Klein four-group, is an abelian normal subgroup of A4 and A4 /V4 ' C3 is cyclic, it is supersolvable. Therefore, A4 is abelian-by-supersolvable and thus A4 is strongly monomial. We see that the primitive central idempotents of QA4 are precisely of the form e(G, H, K) for some strong Shoda pairs (H, K) of G. Consider the lattice of subgroups of the group A4 (For a subgroup H of G, the number of distinct conjugate subgroups of H is denoted in the lower left corner of H.) A4 V4 4 3. C3. C2 C1. Figure 5.1: The lattice of subgroups of A4 where Cn is cyclic group of order n. The strong Shoda pairs in A4 are precisely of the form (A4 , A4 ), (A4 , V4 ), and (V4 , C2 ). Note that there are 3 conjugates of C2 , so there are 3 Shoda pairs of the form (V4 , C2 ). As an obvious property that e(A4 , V4g , C2g ) = e(A4 , V4 , C2 ) for any g ∈ G, we have QA4 has precisely 3 primitive central idempotents by Theorem 4.2.3, namely, 13.

(17) c4 , e2 = e(A4 , A4 , V4 ) = ε(A4 , V4 ), and e3 = e(A4 , V4 , C2 ). e1 = e(A4 , A4 , A4 ) = A Next, we focus on the association between Shoda pairs and the irreducible characters of A4 . Consider the character table of A4 [Isa94, Appendix] and let 1, 2, 3A, 3B denote the conjugacy classes with representatives 1, (1 2)(3 4), (1 2 3), (1 3 2), respectively. class 1 size 1 χ1 χ2 χ3 χ4. 2 3A 3B 3 4 4. 1 1 1 1 1 1 3 −1. 1 ζ3 ζ32 0. 1 ζ32 ζ3 0. Table 5.1: Character table of A4 Recall that eQ (χ) denote the unique primitive central idempotent e of QG such that χ(e) 6= 0. It is easy to check that e1 = ε(A4 , A4 ) = eQ (χ1 ), e2 = ε(A4 , V4 ) = eQ (χ2 ) = eQ (χ3 ). To see the association between primitive central idempotent e3 and the irreducible character χ4 of A4 , we let λ be a non-trivial linear character of V4 . By the formula of induced character, λA4 is given by λA4 (s) =. 1 |V4 |. X. λ(st ) for s ∈ A4 .. t∈A4 ,st ∈V4. We have λA4 = χ4 , hence e3 = eQ (λA4 ) = eQ (χ4 ). Finally, we can describe the Wedderburn-Artin decomposition by the primitive central idempotents of QA4 . Using the techniques in [JdR16, Section 2.6 and Section 3.5], the structure of indecomposable algebras (QG)eQ (χi ) can be described. One has (QA4 )eQ (χ1 ) ' Q, (QA4 )eQ (χ2 ) ' Q(ζ3 ), (QA4 )eQ (χ4 ) ' M3 (Q). Therefore, we obtain the Wedderburn-Artin decomposition of QA4 . That is QA4 = (QA4 )e1 ⊕ (QA4 )e2 ⊕ (QA4 )e3 ' Q ⊕ Q(ζ3 ) ⊕ M3 (Q). This gives an example that for a monomial group G, we can obtain all the primitive central idempotents of QG by the construction above. 14.

(18) 5.2. Primitive central idempotents of QSL(2, 3). Next, we treat the smallest non-monomial case. This section is referred to [OdRS04, Example 5.7]. Consider G = SL(2, 3), the special linear group of degree 2 over F3 , which is the smallest non-monomial group. To describe SL(2, 3), let ! ! ! ! 1 1 −1 1 0 −1 1 −1 a= , b= , c= , t= . 1 −1 1 1 1 0 0 1 Then we have Q8 = ha, b, c | a2 = b2 = c2 = abci and C3 = ht | t3 = 1i SL(2, 3) = Q8 o C3 = ha, b, c, t | a4 = b4 = t3 = 1, b2 = a2 , ab = a−1 , at = b, bt = ci. Moreover, we consider the lattice of subgroups of SL(2, 3). SL(2, 3) Q8 3. 4. C6. 4. C3. C4 C2 C1. Figure 5.2: The lattice of subgroups of SL(2, 3) Note that SL(2, 3)/C2 ' A4 and the primitive central idempotents of QA4 are e(A4 , A4 , A4 ), e(A4 , A4 , V4 ), and e(A4 , V4 , C2 ). (To avoid confusion, we write subgroups of A4 in calligraphic fonts.) By the natural isomorphism which we mentioned in Proposition 2.1.3, c2 ' Q(SL(2, 3)/C2 ) ' QA4 . (QSL(2, 3))C b We obtain 3 primitive central idempotents of QG from QA4 , e1 = e(G, G, G) = G, e2 = e(G, G, Q8 ) = ε(G, Q8 ), and e3 = e(G, Q8 , C4 ). Consider a central idempotent e = 1 − (e1 + e2 + e3 ) = e(G, C2 , {1}). Although the pair (C2 , {1}) is a strong Shoda pair of SL(2, 3), the element e = e(G, C2 , {1}) is still not a primitive central idempotent of  QG. In fact, e is the sum of e4 = 21 e(G, C6 , C3 ) and e5 = 14 e(G, C6 , {1}) − e(G, C6 , C3 ) in QG. More precisely, we have  1 (1 − a2 ) 2 − (1 + (a + b + c))t − (1 − (a + b + c))t2 , 12  1 e5 = (1 − a2 ) 4 + (1 + (a + b + c))t + (1 − (a + b + c))t2 , 12 c2 = 1 − 1 (1 + a2 ) = e4 + e5 . e = e(G, C2 , {1}) = 1 − C 2 e4 =. 15.

(19) Note that the expression of primitive central idempotents may not unique. One has 1 e5 = e − e4 = e(G, C2 , {1}) − e(G, C6 , C3 ) 2 to be another expression of e5 . Consider the character table of SL(2, 3) [Isa94, Appendix] and let 1, 2, 3A, 3B, 4, 6A, 6B denote the conjugacy classes with representatives 1, a2 , t, t2 , c, a2 t, a2 t2 , respectively. class 1 size 1 χ1 χ2 χ3 χ4 χ5 χ6 χ7. 2 1. 3A 4. 3B 4. 4 6A 6B 6 4 4. 1 1 1 1 1 2 1 1 ζ3 ζ3 1 2 1 1 ζ3 ζ3 1 2 −2 −1 −1 0 2 2 −2 −ζ3 −ζ3 0 2 2 −2 −ζ3 −ζ3 0 3 3 0 0 −1. 1 ζ32 ζ3 1 ζ3 ζ32 0. 1 ζ3 ζ32 1 ζ32 ζ3 0. Table 5.2: Character table of SL(2, 3) We have the association between primitive central idempotents of QSL(2, 3) and the irreducible characters of SL(2, 3) as the following : e1 = ε(G, G) = eQ (χ1 ), e2 = ε(G, Q8 ) = eQ (χ2 ) = eQ (χ3 ), e3 = e(G, Q8 , C4 ) = eQ (χ7 ), 1 e4 = e(G, C6 , C3 ) = eQ (χ5 ) = eQ (χ6 ), 2 1 1 e5 = e(G, C6 , {1}) − e(G, C6 , C3 ) = eQ (χ4 ). 4 4 Since e4 = eQ (χ5 ) = eQ (χ6 ) and e5 = eQ (χ4 ), by Theorem 1.0.2, we can conclude that e4 and e5 are primitive central idempotents. To show that irreducible characters χ4 , χ5 , χ6 are non-monomial, we consider the following statement obtained immediately from the formula of induced character : if χ is a character of a subgroup H of G, then χG (1) = |G : H|χ(1). It is easy to check that there is no subgroup of index 2 in SL(2, 3), therefore all irreducible characters of degree 2 of SL(2, 3) are non-monomial. Hence, 1 e4 = e(G, C6 , C3 ) = eQ (χ5 ) = eQ (χ6 ) and 2 1 1 e5 = e(G, C6 , {1}) − e(G, C6 , C3 ) = eQ (χ4 ) 4 4 can not be written as the form αe(G, H, K) where (H, K) is a Shoda pair of G. 16.

(20) Finally, we can describe the Wedderburn-Artin decomposition by the primitive central idempotents of QSL(2, 3) by the isomorphism between indecomposable algebra (QG)e and matrix ring Mn (D). One has (QSL(2, 3))eQ (χ1 ) ' Q, (QSL(2, 3))eQ (χ2 ) ' Q(ζ3 ), (QSL(2, 3))eQ (χ7 ) ' M3 (Q), (QSL(2, 3))eQ (χ5 ) ' M2 (Q(ζ3 )), (QSL(2, 3))eQ (χ4 ) ' H(Q), where H(Q) is the Hamilton quaternion algebra over the rationals, defined as the following H(Q) = {a + bi + cj + dk | a, b, c, d ∈ Q}, where {1, i, j, k} is a Q-basis of H(Q), and the multiplication is determined by the rules i2 = j 2 = −1,. ij = −ji = k.. Note that the ring H(Q) is a division ring since it is a subring of the division ring H(R). (See also [JdR16, Section 1.4].) Therefore, we obtain the Wedderburn-Artin decomposition of QSL(2, 3). That is QSL(2, 3) = (QSL(2, 3))e1 ⊕ (QSL(2, 3))e2 ⊕ · · · ⊕ (QSL(2, 3))e5 ' Q ⊕ Q(ζ3 ) ⊕ M3 (Q) ⊕ M2 (Q(ζ3 )) ⊕ H(Q). Although SL(2, 3) has order only 24, it is still very complicated to find primitive central idempotents of QSL(2, 3).. 5.2.1. GAP code for primitive central idempotents of QSL(2, 3). We provide a GAP code below for the readers to compute above primitive central idempotents by using GAP [GAP20] and the Wedderga package [BCH+ 18]. LoadPackage("wedderga"); #define e(G,H,K) eGHK := function(R,G,H,K) local IS, S, ep, cen, g, T, e, i; IS:=IntermediateSubgroups(H,K).subgroups;; Add(IS,H); ep:=Product(IS, S-> AverageSum(R,K) - AverageSum(R,S));; #cen:=Centralizer(G,ep);; #note "Centralizer" will compute too much time cen:=[];; for g in G do. 17.

(21) if g* ep=ep*g then AddSet(cen,g); fi; od; cen:=Group(cen);; T:=List(RightTransversal(G,cen),i->CanonicalRightCosetElement(cen,i));; e:=Sum([1..Size(T)], i -> (T[i]∧ -1* One(R))* ep* (T[i]* One(R)));; return e; end; # compute primitive central idempotents t:=Z(3);; 1t:=t∧ 0;; 0t:=0* t;; a:=[[1t, 1t],[1t,t]];; b:=[[1t,t],[0t,1t]];; sl:=Group(a,b);; #id=[24,3] Q8:=Group(a,b* a* b∧ 2);; #id=[8,4] C4:=Group(a);; C6:=Group(a* b∧ 2);; C3:=Group((a* b∧ 2)∧ 2);; Tr:=TrivialSubgroup(sl);; R:=GroupRing(Rationals, sl);; e1:=AverageSum(R,sl);; e2:=eGHK(R,sl,sl,Q8);; e3:=eGHK(R,sl,Q8,C4);; e4:=eGHK(R,sl,C6,C3)/2;; e5:=eGHK(R,sl,C6,Tr)/4 - eGHK(R,sl,C6,C3)/4;;. 18.

(22) Chapter 6. Main results In this chapter, we focus on these three specific non-monomial group A5 , SL(2, 5) and S5 which we are more familiar with. That is, we will find all primitive central idempotents of QA5 , QSL(2, 5) and QS5 .. 6.1. Primitive central idempotents of QA5. For G = A5 , the alternating group of degree 5. It is the non-monomial group of order 60. To see A5 is non-monomial, it is easy to check that A5 has an irreducible character of degree 3, but A5 has no subgroup of index 3. Since A5 is simple, the natural isomorphism b to Q(G/N ) cannot help us to find primitive central idempotents of QG. from (QG)N Since the commutator subgroup A05 of A5 is still A5 , we have |A5 : A05 | = 1. Therefore A5 has only one linear character, namely, the trivial character, and hence the primitive c5 . central idempotent of QA5 constructed from the trivial character of A5 is e1 = A Note that e1 corresponds to the Shoda pair (A5 , A5 ) in A5 . To find other Shoda pairs of A5 , we consider the lattice of subgroups of A5 . For convenience, the common node does not necessarily mean to have the common subgroup. A5. 6. D10 6. 5. A4. C5 5. V4. 15. C2. 10. S3. 10. C3. C1 Figure 6.1: The lattice of subgroups of A5. 19.

(23) There are 5 Shoda pairs of the form (A4 , V4 ) of A5 . Note that (A4 , V4 ) is not a strong Shoda pair of A5 . To find the coefficient α, observe that CenG (ε(A4 , V4 )) = A4 , and therefore 4 X i e(A5 , A4 , V4 ) = ε(A4 , V4 )σ where σ = (1 2 3 4 5). i=0. Let λ be one of the two characters of A4 with ker λ = V4 (see Table 5.1). Then λA5 is the following character 3 5A 5B class 1 2 λA5. 5 1 −1. 0. 0. where the representatives of conjugacy classes are mentioned in Table 6.1 below. Therefore, we obtain [Q(λ) : Q(λA5 )] = [Q(ζ3 ) : Q] = 2 and we deduce from Theorem 4.1.6 that the primitive central idempotent of QA5 corresponds with the Shoda pair (A4 , V4 ) in A5 is 1 e2 = e(A5 , A4 , V4 ) 2 . Next, we consider the central element e3 in QA5 where 1 1 e3 = e(A5 , C5 , {1}) − e(A5 , C3 , {1}). 3 5 To show e3 is a primitive central idempotent of QA5 , it suffices to find an irreducible character of A5 such that e3 = eQ (χ). Consider the character table of A5 [Isa94, Appendix] where 1, 2, 3, 5A, 5B denote the conjugacy classes with representatives (1), (1 2)(3 4), (1 2 3), (1 2 3 4 5), (1 3 5 4 2), respectively. class 1 size 1. 2 15. 3 20. 1. 1. χ1. 1. χ2. 3 −1. χ3 χ4 χ5. 3 −1 0 4 0 1 5 1 −1. 0. 5A 12 1. 5B 12 1. √ 1+ 5 2√ 1− 5 2. √ 1− 5 2√ 1+ 5 2. −1 0. −1 0. Table 6.1: Character table of A5 Let Kn denote the set of all elements of order n in A5 . Therefore, e3 can be expressed as.  1 1 1 f f3 − K f5 = eQ (χ4 ), e3 = e(A5 , C5 , {1}) − e(A5 , C3 , {1}) = 4K1 + K 3 5 15 20.

(24) and hence e3 is a primitive central idempotent of QA5 . Let e4 = 1 − (e1 + e2 + e3 ), we will show that e4 is a primitive central idempotent of QA5 . In fact, e4 has the following expression 1 e4 = e(A5 , D10 , C5 ). 2 Also, by using the techniques in [JdR16, Section 2.6 and Section 3.5], the structure of indecomposable algebras (QG)eQ (χi ) can be described. One has (QA5 )e1 = (QA5 )eQ (χ1 ) ' Q, (QA5 )e2 = (QA5 )eQ (χ5 ) ' M5 (Q), (QA5 )e3 = (QA5 )eQ (χ4 ) ' M4 (Q). Therefore, we have QA5 ' Q ⊕ M5 (Q) ⊕ M4 (Q) ⊕ S where S is the product of the remaining indecomposable components with the dimension dimQ S = 60 − (12 + 52 + 42 ) = 18. First notice that S will not have a division ring as a component. If there is such a division ring R, then we have a map θ : A5 −→ R. Since A5 is simple, ker θ = {1} or A5 . If ker θ = A5 , then θ is the restriction of the augmentation map which is corresponding to e1 . Hence, ker θ = {1}, and so A5 will be embedded into a division ring, by [Ami55, Theorem 2], it is impossible. Hence any component of S will be to Mn (D) with n ≥ 2. Since dimQ S = 18, S is equal to M3 (D) or equal to M3 (Q) ⊕ M3 (Q). However, if S = M3 (Q) ⊕ M3 (Q), it will follow that the irreducible characters of degree 3 of G will be rational-valued characters, which yields a contradiction. Therefore, we have (QA5 )e4 ' S = M3 (D) is indecomposable. By proposition 1.0.1 we know e4 is a primitive central idempotent of QA5 and the corresponding characters of e4 are as the following. 1 e4 = e(A5 , D10 , C5 ) = eQ (χ2 ) = eQ (χ3 ). 2 √ Also, since (QA5 )eQ (χ2 ) ' M3 (Q( 5)), we obtain the Wedderburn-Artin decomposition of QA5 . That is √ QA5 ' Q ⊕ M5 (Q) ⊕ M4 (Q) ⊕ M3 (Q( 5)).. 21.

(25) We have the following theorem as a result. Theorem 6.1.1. For the non-monomial group A5 , the primitive central idempotents of QA5 have the following expression e1 = e(A5 , A5 , A5 ), 1 e2 = e(A5 , A4 , V4 ), 2 1 1 e3 = e(A5 , C5 , {1}) − e(A5 , C3 , {1}), 3 5 1 e4 = e(A5 , D10 , C5 ). 2 We give specific subgroups below for the readers to compute above primitive central idempotents. A5 = h(1, 2, 3, 4, 5), (3, 4, 5)i, A4 = h(1, 2)(3, 5), (1, 3)(2, 5), (1, 2, 3)i, V4 = h(1, 2)(3, 5), (1, 3)(2, 5)i, C5 = h(1, 2, 3, 4, 5)i, C3 = h(1, 2, 4)i, D10 = h(1, 2, 3, 4, 5), (1, 2)(3, 5)i.. 6.1.2. GAP code for primitive central idempotents of QA5. We provide a GAP code below for the readers to compute above primitive central idempotents by using GAP [GAP20] and the Wedderga package [BCH+ 18]. LoadPackage("wedderga"); #define e(G,H,K) eGHK := function(R,G,H,K) local IS, S, ep, cen, g, T, e, i; IS:=IntermediateSubgroups(H,K).subgroups;; Add(IS,H); ep:=Product(IS, S-> AverageSum(R,K) - AverageSum(R,S));; #cen:=Centralizer(G,ep);; #note "Centralizer" will compute too much time cen:=[];; for g in G do if g* ep=ep*g then AddSet(cen,g); fi; od; cen:=Group(cen);; T:=List(RightTransversal(G,cen),i->CanonicalRightCosetElement(cen,i));; e:=Sum([1..Size(T)], i -> (T[i]∧ -1* One(R))* ep* (T[i]* One(R)));; return e;. 22.

(26) end; # compute primitive central idempotents a:=(1,2,3,4,5);; b:=(3,4,5);; c:=(1,2)* (3,5);; A5:=Group(a,b);; #id=[60,5] D:=Group(a,c);; #id=[10,1] C5:=Group(a);; V:=Group(c, (1,3)* (2,5));; #id=[4,2] A4:=Group(c, (1,3)* (2,5),(1,2,3));; #id=[12,3] C3:=Group((1,2,4));; S3:=Group((1,2,4),c);; #id=[6,1] Tr:=TrivialSubgroup(A5);; R:=GroupRing(Rationals, A5);; e1:=AverageSum(R,A5);; e2:=eGHK(R,A5,A4,V)/2;; e3:=eGHK(R,A5,C5,Tr)/3 - eGHK(R,A5,C3,Tr)/5;; e4:=eGHK(R,A5,D,C5)/2;;. 23.

(27) 6.2. Primitive central idempotents of QSL(2, 5). For G = SL(2, 5), the special linear group of degree 2 over F5 . It is a non-monomial group of order 120. To see SL(2, 5) is non-monomial, it is easy to check that SL(2, 5) has an irreducible character of degree 2, but SL(2, 5) has no subgroup of index 2. Consider the lattice of subgroups of SL(2, 5). For convenience, the common node does not necessarily mean to have the common subgroup. SL(2, 5) 5. SL(2, 3). 6 Dic20 10 6. C10 5. 6 C5 15. Dic12. Q8 10. C6. 10. C3. C4 C2 C1. Figure 6.2: The lattice of subgroups of SL(2, 5) Here Dic4n = ha, x | a2n = 1, x2 = an , ax = a−1 i is dicyclic (or binary dihedral) group of order 4n. Note that Z(SL(2, 5)) = C2 is the only nontrivial proper normal subgroup of SL(2, 5). In particular, SL(2, 5)/C2 = P SL(2, 5) ' A5 . By the natural isomorphism we mentioned in Proposition 2.1.3 c2 ' Q(SL(2, 5)/C2 ) ' QA5 . QSL(2, 5)C We obtain 4 primitive central idempotents of QSL(2, 5) from QA5 , namely e1 = e(G, G, G), 1 e2 = e(G, SL(2, 3), Q8 ), 2 1 1 e3 = e(G, C10 , C2 ) − e(G, C6 , C2 ), 3 5 1 e4 = e(G, Dic20 , C10 ). 2 Consider the character table of SL(2, 5) [JL01, page 445] where 1, 2, 3, 4, 5A, 5B,   0 2 2 6, 10A, 10B denote the conjugacy classes with representatives ( 10 01 ) , −1 0 −1 , −1 2 ,     2 0 , ( 1 1 ) , ( 1 2 ) , −2 2 , −1 2 , −1 1 , respectively. 0 −2 01 01 −1 −2 0 −1 0 −1. 24.

(28) class 1 size 1. 2 1. 3 20. 4 30. 1. 1. 1. χ1. 1. χ2. 2 −2 −1. 0. χ3. 2 −2 −1. 0. χ4. 3. χ5 χ6 χ7 χ8 χ9. 3 3 0 −1 4 4 1 0 4 −4 1 0 5 5 −1 1 6 −6 0 0. 3. 0. −1. 5A 12 1. 5B 12. 6 20. 1. 1. √ −1+ 5 2√ −1− 5 2√ 1− 5 2√ 1+ 5 2. √ −1− 5 2√ −1+ 5 2√ 1+ 5 2√ 1− 5 2. −1 −1 0 1. −1 −1 0 1. 1 1 0 0 1 −1 −1 0. 10A 12 1. 10B 12 1. √ 1+ 5 2√ 1− 5 2√ 1+ 5 2√ 1− 5 2. √ 1− 5 2√ 1+ 5 2√ 1− 5 2√ 1+ 5 2. −1 1 0 −1. −1 1 0 −1. Table 6.3: Character table of SL(2, 5) By directly computing, the association between primitive central idempotents e1 , e2 , e3 , e4 of QSL(2, 5) and the irreducible characters of SL(2, 5) is as the following e1 = e(G, G, G) = eQ (χ1 ), 1 e2 = e(G, SL(2, 3), Q8 ) = eQ (χ8 ), 2 1 1 e3 = e(G, C10 , C2 ) − e(G, C6 , C2 ) = eQ (χ6 ), 3 5 1 e4 = e(G, Dic20 , C10 ) = eQ (χ4 ) = eQ (χ5 ). 2 c2 = e(G, C2 , {1}) is a central idempoAlthough e = 1 − (e1 + e2 + e3 + e4 ) = 1 − C tent of QSL(2, 5), it is still not primitive. Note that there are 4 remaining characters χ2 , χ3 , χ7 , χ9 have not corresponded to primitive central idempotents. Hence, we have to find out at most 4 primitive central idempotents of QSL(2, 5). Consider the elements e5 , e6 in QSL(2, 5) with the expression 1 1 e5 = e(G, C10 , {1}) − e(G, C6 , {1}), 3 5 1 e6 = e(G, C10 , C5 ). 2 Let Kn denote the subset of all elements of order n in SL(2, 5). We have  1 f f2 + K f3 − K f5 − K f6 + K g e5 = 4K1 − 4K 10 = eQ (χ7 ), 30  1 f g f2 + K f5 − K e6 = 6K1 − 6K 10 = eQ (χ9 ). 20 Therefore, e5 , e6 are primitive central idempotents of QSL(2, 5). Consider e7 = 1 − (e1 + · · · + e6 ), that is 1 1 e7 = e(G, C10 , {1}) − e(G, C6 , C3 ). 6 5 25.

(29) To show that e7 = eQ (χ2 ) = eQ (χ3 ), we have X e(σ ◦ χ2 ) eQ (χ2 ) = σ∈Gal(Q(χ2 )/Q). X. =. √ σ∈Gal(Q( 5)/Q). e(σ ◦ χ2 ). = e(χ2 ) + e(χ3 )  1 f f2 − 2K f3 − K f5 + 2K f6 + K g = 4K1 − 4K 10 , 60 and    12  f f2 − 1 K f5 − K g K1 − K 10 , 5 10  1  5f f f3 − K f6 . e(G, C6 , C3 ) = K1 − K2 + K 3 6. e(G, C10 , {1}) =. Therefore, by Theorem 1.0.2, 1 1 e7 = e(G, C10 , {1}) − e(G, C6 , C3 ) = eQ (χ2 ) = eQ (χ3 ) 6 5 is a primitive central idempotent of QSL(2, 5). We have the following theorem as a result. Theorem 6.2.1. For the non-monomial group SL(2, 5), the primitive central idempotents of QSL(2, 5) have the following expression  e1 = e SL(2, 5), SL(2, 5), SL(2, 5) ,  1 e2 = e SL(2, 5), SL(2, 3), Q8 , 2  1  1 e3 = e SL(2, 5), C10 , C2 − e SL(2, 5), C6 , C2 , 3 5  1 e4 = e SL(2, 5), Dic20 , C10 , 2  1  1 e5 = e SL(2, 5), C10 , {1} − e SL(2, 5), C6 , {1} , 3 5  1 e6 = e SL(2, 5), C10 , C5 ) , 2  1  1 e7 = e SL(2, 5), C10 , {1} − e SL(2, 5), C6 , C3 . 6 5 We give specific subgroups below for the readers to compute above primitive central idempotents.. 26.

(30) a = ( 11 12 ) , b = ( 21 11 ) , c = ab, d = a2 b3 , x = ab2 , y = aba, z = xy −1 , SL(2, 5) = ha, bi, SL(2, 3) = hc, di, Q8 = hc2 d, cdci, Dic20 = hx, yi, C10 = hzi, C6 = hci, C5 = hz 2 i, C3 = hc2 i, C2 = hz 5 i = hc3 i.. GAP code for primitive central idempotents of QSL(2, 5). 6.2.2. We provide a GAP code below for the readers to compute above primitive central idempotents by using GAP [GAP20] and the Wedderga package [BCH+ 18]. LoadPackage("wedderga"); #define e(G,H,K) eGHK := function(R,G,H,K) local IS, S, ep, cen, g, T, e, i; IS:=IntermediateSubgroups(H,K).subgroups;; Add(IS,H); ep:=Product(IS, S-> AverageSum(R,K) - AverageSum(R,S));; #cen:=Centralizer(G,ep);; #note "Centralizer" will compute too much time cen:=[];; for g in G do if g* ep=ep*g then AddSet(cen,g); fi; od; cen:=Group(cen);; T:=List(RightTransversal(G,cen),i->CanonicalRightCosetElement(cen,i));; e:=Sum([1..Size(T)], i -> (T[i]∧ -1* One(R))* ep* (T[i]* One(R)));; return e; end; # compute primitive central idempotents t:=Z(5);; 1t:=t∧ 0;; 0t:=0* t;; a:=[[1t, 1t],[1t,t]];; b:=[[t, 1t],[1t,1t]];;. 27.

(31) sl:=Group(a,b);; #id=[120,5] c:=a* b;; d:=a∧ 2* b∧ 3;; sl23:= Group(c,d);; #id=[24,3] #note:. Order(a)=Order(b)=10, Order(a* b)=6, Order(a∧ 2* b∧ 3)=3. Q8:=Group(c∧ 2* d, c* d* c);; #id=[8,4] Dic20:=Group(a* b∧ 2, a* b* a);; #id=[20,1] #note Order(a* b∧ 2)=Order(a* b* a)=4 C10:=Group(a* b∧ 2* (a* b* a)∧ -1);; C6:=Group(a* b);; C5:=Group(a∧ 2);; C3:=Group(a* b)∧ 2;; R:=GroupRing(Rationals,sl);; e1:=AverageSum(R,sl);; e2:=eGHK(R,sl,sl23,Q8)/2;; e3:=eGHK(R,sl,Group(a),Group(a∧ 5))/3 - eGHK(R,sl,C6,Group((a* b)∧ 3))/5;; e4:=eGHK(R,sl,Dic20,C10)/2;; e5:=eGHK(R,sl,Group(a),[One(sl)])/3 - eGHK(R,sl,C6,[One(sl)])/5;; e6:=eGHK(R,sl,Group(a),C5)/2;; e7:=eGHK(R,sl,Group(a),[One(sl)])/6 - eGHK(R,sl,C6,C3)/5;;. 28.

(32) 6.3. Primitive central idempotents of QS5. For G = S5 , the symmetric group of degree 5. It is another non-monomial group of order 120. To see S5 is non-monomial, it is easy to check that S5 has an irreducible character of degree 4, but S5 has no subgroup of index 4. Consider the lattice of subgroups of S5 . For convenience, the common node does not necessarily mean to have the common subgroup. S5. 6. 15. F5,4. D8. A5. 6. 15. D10. C4. 10. 5. 15. V4. 5. S3. 5. 6. C5. S4. 10. A4. 10. 10. C2B. D12. S3. C3. 10. 15. 10. C6. C2 × C2. C2A. C1 Figure 6.3: The lattice of subgroups of S5 Here F5,4 = C5 o C4 = ha, b | a5 = b4 = 1, ab = a3 i is the Frobenius group of order 20. Since S5 has 2 linear characters, the trivial character and the sign character (which takes the value 1 for even permutations and the value −1 for odd permutations), we obtain 2 primitive central idempotents of QS5 by Theorem 3.2.2 that e1 = e(S5 , S5 , S5 ), e2 = e(S5 , S5 , A5 ). Consider the character table of S5 [JL01, page 429] where 1, 2A, 2B, 3, 4, 5, 6 denote the conjugacy classes with representatives (1), (1 2), (1 2)(3 4), (1 2 3), (1 2 3 4), (1 2 3 4 5), (1 2 3 4 5 6), respectively.. 29.

(33) class 1 2A 2B size 1 10 15 χ1 χ2 χ3 χ4 χ5 χ6 χ7. 3 20. 4 30. 5 24. 6 20. 1 1 1 1 1 1 1 1 −1 1 1 −1 1 −1 4 −2 0 1 0 −1 1 4 2 0 1 0 −1 −1 5 1 1 −1 −1 0 1 5 −1 1 −1 1 0 −1 6 0 −2 0 0 1 0. Table 6.5: Character table of S5 Since all irreducible characters χ are rational-valued, we have eQ (χ) = e(χ). Hence, there are 7 primitive central idempotents of QS5 . For each ei in the following theorem, by directly computing, one has ei = eQ (χi ) and by Theorem 1.0.2, ei will be a primitive central idempotent of QS5 . Theorem 6.3.1. For the non-monomial group S5 , the primitive central idempotents of QS5 have the following expression e1 = e(S5 , S5 , S5 ), e2 = e(S5 , S5 , A5 ), 1 1 e3 = − e(S5 , C6 , C2A ) + e(S5 , C5 , {1}), 5 6 1 1 e4 = − e(S5 , C6 , {1}) + e(S5 , C5 , {1}), 5 6 1 1 e5 = − e(S5 , C6 , C3 ) + e(S5 , C4 , C2B ), 4 6 1 1 e6 = − e(S5 , C6 , C6 ) + e(S5 , C4 , C4 ), 4 6 1 e7 = e(S5 , D10 , C5 ). 2 where C2A is a cyclic group generated by a 2-cycle and C2B is a cyclic group generated by a (2, 2)-cycle. We give specific subgroups below for the readers to compute above primitive central idempotents.. 30.

(34) S5 = h(1, 2, 3, 4, 5), (1, 2)i, A5 = h(1, 2, 3, 4, 5), (3, 4, 5)i, C6 = h(1, 2), (3, 4, 5)i, C3 = h(3, 4, 5)i, C2A = h(1, 2)i, C5 = h(1, 2, 3, 4, 5)i, C4 = h(1, 3, 2, 5)i, C2B = h(1, 2)(3, 5)i, D10 = h(1, 2, 3, 4, 5), (1, 2)(3, 5)i.. 6.3.2. GAP code for primitive central idempotents of QS5. We provide a GAP code below for the readers to compute above primitive central idempotents by using GAP [GAP20] and the Wedderga package [BCH+ 18]. LoadPackage("wedderga"); #define e(G,H,K) eGHK := function(R,G,H,K) local IS, S, ep, cen, g, T, e, i; IS:=IntermediateSubgroups(H,K).subgroups;; Add(IS,H); ep:=Product(IS, S-> AverageSum(R,K) - AverageSum(R,S));; #cen:=Centralizer(G,ep);; #note "Centralizer" will compute too much time cen:=[];; for g in G do if g* ep=ep*g then AddSet(cen,g); fi; od; cen:=Group(cen);; T:=List(RightTransversal(G,cen),i->CanonicalRightCosetElement(cen,i));; e:=Sum([1..Size(T)], i -> (T[i]∧ -1* One(R))* ep* (T[i]* One(R)));; return e; end; # compute primitive central idempotents a:=(1,2,3,4,5);; b:=(1,2);; c:=(3,4,5);; S5:=Group(a,b);; #id=[120,34] A5:=Group(a,c);; #id=[60,5] C6:=Group(b,c);; C3:=Group(c);;. 31.

(35) C2A:=Group(b);; C5:=Group(a);; C4:=Group((1,3,2,5));; C2B:=Group((1,2)* (3,5));; D10:=Group(a,(1,2)* (3,5));; #id=[10,1] tr:=[One(S5)];; R:=GroupRing(Rationals,S5);; e1:=AverageSum(R,S5);; e2:=eGHK(R,S5,S5,A5);; e3:=-eGHK(R,S5,C6,C2A)/5 + eGHK(R,S5,C5,tr)/6;; e4:=-eGHK(R,S5,C6,tr)/5 + eGHK(R,S5,C5,tr)/6;; e5:=-eGHK(R,S5,C6,C3)/4 + eGHK(R,S5,C4,C2B)/6;; e6:=-eGHK(R,S5,C6,C6)/4 + eGHK(R,S5,C4,C4)/6;; e7:=eGHK(R,S5,D10,C5)/2;;. 32.

(36) Bibliography [Ami55]. S. A. Amitsur. Finite subgroups of division rings. Trans. Amer. Math. Soc., 80:361–386, 1955.. [BCH+ 18] G. Kaur Bakshi, O. Broche Cristo, A. Herman, A. Konovalov, S. Maheshwary, A. Olivieri, G. Olteanu, Á. del Río, and I. Van Gelder. Wedderga wedderburn decomposition of group algebras, version 4.9.5, November 2018. [GAP20]. The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.11.0, 2020.. [Isa94]. I. Martin Isaacs. Character theory of finite groups. Dover Publications, Inc., New York, 1994. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423 (57 #417)].. [JdR16]. Eric Jespers and Ángel del Río. Group ring groups. Vol. 1. Orders and generic constructions of units. De Gruyter Graduate. De Gruyter, Berlin, 2016.. [JL01]. Gordon James and Martin Liebeck. Representations and characters of groups. Cambridge University Press, New York, second edition, 2001.. [JLP03]. Eric Jespers, Guilherme Leal, and Antonio Paques. Central idempotents in the rational group algebra of a finite nilpotent group. J. Algebra Appl., 2(1):57–62, 2003.. [OdRS04] Aurora Olivieri, Ángel del Río, and Juan Jacobo Simón. On monomial characters and central idempotents of rational group algebras. Comm. Algebra, 32(4):1531–1550, 2004. [vdW73a] Robert W. van der Waall. On monomial groups. J. Reine Angew. Math., 264:103–134, 1973. [vdW73b] Robert W. van der Waall. On the monomiality of groups of order between 100 and 200. I. J. Reine Angew. Math., 262(263):82–92, 1973. [vdW74]. Robert W. van der Waall. On the monomiality of groups of order between 100 and 200. II. J. Reine Angew. Math., 270:184–194, 1974.. 33.

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