The Character Tables of Centralizers in Weyl Group of E8 Iii

To classify the finite dimensional pointed Hopf algebras with Weyl group G of E8, we obtain the representatives of conjugacy classes of G and all character tables of centralizers of these representatives by means of software GAP. In this paper we only list character table 47–64. 2000 Mathematics Subject Classification: 16W30, 68M07 keywords: GAP, Hopf algebra, Weyl group, character. 0. Introduction This article is to contribute to the classification of finite-dimensional complex pointed Hopf algebras with Weyl groups of E8. Many papers are about the classification of finite dimensional pointed Hopf algebras, for example, [AS98, AS02, AS00, AS05, He06, AHS08, AG03, AFZ, AZ07, Gr00, Fa07, AF06, AF07, ZZC, ZC]. In these research ones need the centralizers and character tables of groups. In this paper we obtain the representatives of conjugacy classes of Weyl groups of E8 and all character tables of centralizers of these representatives by means of software GAP. In this paper we only list character table 47–64. By the Cartan-Killing classification of simple Lie algebras over complex field the Weyl groups to be considered are W (Al), (l ≥ 1); W (Bl), (l ≥ 2); W (Cl), (l ≥ 2); W (Dl), (l ≥ 4); W (El), (8 ≥ l ≥ 6); W (Fl), (l = 4); W (Gl), (l = 2). It is otherwise desirable to do this in view of the importance of Weyl groups in the theories of Lie groups, Lie algebras and algebraic groups. For example, the irreducible representations of Weyl groups were obtained by Frobenius, Schur and Young. The conjugace classes of W (F4) were obtained by Wall [Wa63] and its character tables were obtained by Kondo [Ko65]. The conjugace classes and character tables of W (E6), W (E7) and W (E8) were obtained by Frame [Fr51]. Carter gave a unified description of the conjugace classes of Weyl groups of simple Lie algebras [Ca72]. 1. Program By using the following program in GAP, we obtain the representatives of conjugacy classes of Weyl groups of E6 and all character tables of centralizers of these representatives. gap> L:=SimpleLieAlgebra(”E”,6,Rationals);; gap> R:=RootSystem(L);; gap> W:=WeylGroup(R);Display(Order(W)); gap > ccl:=ConjugacyClasses(W);; gap> q:=NrConjugacyClasses(W);; Display (q); gap> for i in [1..q] do > r:=Order(Representative(ccl[i]));Display(r);; > od; gap > s1:=Representative(ccl[1]);;cen1:=Centralizer(W,s1);; gap> cl1:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[2]);;cen1:=Centralizer(W,s1);; 1 2 SHOUCHUAN ZHANG, PENG WANG, JING CHENG, HUI YANG gap> cl2:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[3]);;cen1:=Centralizer(W,s1);; gap> cl3:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[4]);;cen1:=Centralizer(W,s1);; gap> cl4:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[5]);;cen1:=Centralizer(W,s1);; gap> cl5:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[6]);;cen1:=Centralizer(W,s1);; gap> cl6:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[7]);;cen1:=Centralizer(W,s1);; gap> cl7:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[8]);;cen1:=Centralizer(W,s1);; gap> cl8:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[9]);;cen1:=Centralizer(W,s1);; gap> cl9:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[10]);;cen1:=Centralizer(W,s1);; gap> cl10:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[11]);;cen1:=Centralizer(W,s1);; gap> cl11:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[12]);;cen1:=Centralizer(W,s1);; gap> cl2:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[13]);;cen1:=Centralizer(W,s1);; gap> cl13:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[14]);;cen1:=Centralizer(W,s1);; gap> cl14:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[15]);;cen1:=Centralizer(W,s1);; gap> cl15:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[16]);;cen1:=Centralizer(W,s1);; gap> cl16:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[17]);;cen1:=Centralizer(W,s1);; gap> cl17:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[18]);;cen1:=Centralizer(W,s1);; gap> cl18:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[19]);;cen1:=Centralizer(W,s1);; gap> cl19:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[20]);;cen1:=Centralizer(W,s1);; gap> cl20:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[21]);;cen1:=Centralizer(W,s1);; gap> cl21:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[22]);;cen1:=Centralizer(W,s1);; gap> cl22:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[23]);;cen1:=Centralizer(W,s1);; gap> cl23:=ConjugacyClasses(cen1); gap> s1:=Representative(ccl[24]);;cen1:=Centralizer(W,s1);; > cl24:=ConjugacyClasses(cen1); THE CHARACTER TABLES OF CENTRALIZERS IN WEYL GROUP OF E8 III 3 gap> s1:=Representative(ccl[25]);;cen1:=Centralizer(W,s1); gap> cl25:=ConjugacyClasses(cen1); gap> for i in [1..q] do > s:=Representative(ccl[i]);;cen:=Centralizer(W,s);; > char:=CharacterTable(cen);;Display (cen);Display(char); > od; The programs for Weyl groups of E7, E8, F4 and G2 are similar. 2. E8 The generators of Gs47 are: 0 B B B B B B B B B B B B B @ −2 −1 1 0 1 −1 1 0 −2 −2 2 0 1 −2 2 0 −3 −3 2 0 2 −2 2 0 −4 −4 4 −1 3 −3 3 0 −3 −3 3 −1 2 −2 3 0 −2 −3 2 0 1 −1 2 0 −2 −2 2 0 0 0 1 0 −1 −1 1 0 0 0 0 1 1 C C C C C C C C C C C C C