Character table and blocks of finite simple triality groups
Character table and blocks of finite simple triality groups
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DOI：
10.1090/s0002-9947-1987-0896007-9
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发表时间：
1987
影响因子：
1.3
通讯作者：
D. Deriziotis;G. Michler
中科院分区：
文献类型：
--
作者：
D. Deriziotis;G. Michler
本文在Spalstein[14]近期工作的基础上，利用李型有限群的不可约特征标的Deligne-Lusztig理论，给出了有限单群ID4(Q)的特征标表。作为应用，我们得到了IDi(Q)的不可约特征标对于所有素数r>；0到r-块的分类。这使得我们能够验证Brauer的高度为零的猜想，他关于属于给定块的不可约特征标界的猜想，以及关于简单三重群的Alperin-McKay猜想}D4(Q)。它还得出，对于每个素数r，在}Di(Q)中都有零缺陷块。导言。设G“-3D4(Q)是定义在有限域GF(Q)上的具有q个p”元的简单三重群，其中p>；0是素数，？是正整数.在[14]中，N·Spalstein计算了G“的8个单幂不可约特征标的值。在定理4.3中，G“的非单位不可约特征标以虚Deligne-Lusztig特征标RT&的精确线性组合的形式给出，其中？是对应代数群G的a-稳定极大环面T的a-不动点的线性特征标。Deligne-Lusztig特征标值如表3.6所示。根据李型有限群的不可约特征标的Lusztig Jordan形式[11]，G“的每个不可约特征标x的形式为x=xi？m>；其中t是Ga的半单元，Xu LS是t的中心化子CC(T)的酉元不可约特征标.命题2.2给出了GA的半单元t的中心化子CC(T)的群论结构，以及7(直到GA-共轭)最大环面7)，0 X/，st>；X>；1 X.qsor X/.stst表4.4给出了Ga的不可约特征标及其度数的完整分类。在GA的半单元t的共轭类集合上，可以定义如下等价关系。两个这样的共轭类IP？和IP？等价的充要条件是它们的中心化子CC(I，)和CC(T2)是G0-共轭的。若q为奇数，则有15个等价类，其代表为S，1 0，G“的所有块B，缺陷群8(B)=Gd。G”的所有不可约特征标的个数由k(B)0有界，见推论5.1。关于符号和术语，我们参考了Carter[2]、Deriziotis[4]、Feit[7]和Lusztig[11]的著作。1.关于3D4(Q)的记号和已知结果。设G是素域GF(/>；)=fp，p>；0的代数闭包K上的D4型动态图型单连通代数群.设q=Pm为正整数m，GF(Hx，r(H2)=h2。图G的三重自同构a=Rq是由t乘以/C的场自同构z-？z‘得到的，单群3d4(Q)=Ga={gG G|a(G)=g}称为Steinberg-Tits三重.其序为Ga912(98+Q4+L)(QB L)(Q21)。环面T是a-稳定的。A=qj在T上的限制引出V的线性变换，同样记为a。对每个x G Hom(X，K*)，h(X)=t G T，其中x(^)=A(I)对所有X g X，则h是同构.设A，A2，A3和A4是X中的基本权重。每个元素h(X)g7可以唯一地写为A(X)-n*(x*，.“)，1-1其中Xa，z(A)=zmh，)for/g$，zg A‘*，其中x(^，)=z，对于1 r2’R3}·/4={-/()}y5=0 a，A7=iŒ/，=z2？y，=OI-3^2+4^2-4)=Z2≈Y4=X X(w3+4>；=(Z2)3a，=如果许可或版权限制可应用于再分配；见http://www.ams.org/journal-terms-of-use 42 D.I.deriziotis和G.O.Michler设^j是CC(X)的所有a-稳定G-共轭的集合，其中x是G的半单元素，对所有Re/，r(X)=1，则群Ga通过共轭作用于#y。如果7=0是空集，则Q0=W，而x是Ga的正则元素。G的a-稳定极大环面的G0轨道与h1(a，W)的类之间存在一一对应关系，见[1，第186页]。对于三元数Ga=3D4(Q)，已知\hl(a，W)\=7；[14]。设T是G的a-稳定极大环面，Weyl群W=NC(T)/T。如果7“是G的a-稳定极大环面，则存在唯一类[w]。G^a，if)，其中Je{0，1，…，6}使得T‘“是G-共轭到77。=T0={i g r|H>；ya(í)=i}。特别地，元素h(X)-nf=1h(x>；，，z，)g T属于T当且仅当4A(X)=Wjoh(X)=El a(xwt(a，)，z，i)/=i为了简单起见，每个元素h(X)Y4=xh(xh，z)&T由A(X)=(ZLT z2>；z3>；z4)表示，用这个符号我们可以将所有元素参数化
Based on recent work of Spaltenstein [14] and the Deligne-Lusztig theory of irreducible characters of finite groups of Lie type, in this paper the character table of the finite simple groups iD4(q) is given. As an application we obtain a classification of the irreducible characters of iDi(q) into r-blocks for all primes r > 0. This enables us to verify Brauer's height zero conjecture, his conjecture on the bound of irreducible characters belonging to a give block, and the Alperin-McKay conjecture for the simple triality groups }D4(q). It also follows that for every prime r there are blocks of defect zero in }Di(q). Introduction. Let G„ -3D4(q) be a simple triality group defined over a finite field GF(q) with q p" elements, where p > 0 is a prime number and « is a positive integer. In [14] N. Spaltenstein computed the values of the eight unipotent irreducible characters of G„. Using his results we determine the character table of G„ in §4. In Theorem 4.3 the nonunipotent irreducible characters of G„ are presented in the form of precise linear combinations of the virtual Deligne-Lusztig characters RT&, where © is a linear character of the a-fixed points of a a-stable maximal torus T of the corresponding algebraic group G. The values of the Deligne-Lusztig characters are given in Table 3.6. By Lusztig's Jordan form of the irreducible characters of a finite group of Lie type [11] each irreducible character x of G„ is of the form x = Xi?M> where t is a semisimple element of Ga and xu lS a unipotent irreducible character of the centralizer Cc(t) of t. The group theoretical structure of the centralizers Cc(t) of the semisimple elements t of Ga is given in Proposition 2.2, and of the 7 (up to Ga-conjugacy) maximal tori 7), 0 X/,st> X>1 Xi.qsor X/.ststA complete classification of the irreducible characters of Ga with their degrees is given in Table 4.4. On the set of conjugacy classes of semisimple elements t of Ga one can define an equivalence relation as follows. Two such conjugacy classes ip» and ip» are equivalent if and only if their centralizers Cc(i,) and Cc(t2) are G0-conjugate. If q is odd, there are 15 equivalence classes with representatives s¡, 1 0 and all /--blocks B of G„ with defect group 8(B) = GD the number of all irreducible characters of G„ belonging to B is bounded by k(B) 0; see Corollary 5.1. Concerning the notation and terminology we refer to the books by Carter [2], Deriziotis [4], Feit [7], and Lusztig [11]. 1. Notations and known results on 3D4(q). Let G be a simple simply connected algebraic group of Dynkin diagram type D4 over the algebraic closure K of the prime field GF(/>) = Fp, p > 0. Let q = pm for some positive integer m, and let GF( hx and r(h2) = h2. Then t induces an isometry on V which again is denoted by t. The triality automorphism a = rq of G is induced by t times the field automorphism z -» z' of /C The simple group 3D4(q) = Ga = {g G G|a(g) = g} is called the Steinberg-Tits triality. Its order \Ga\ = 912(98 + q4 + l)(qb l)(q2 1). The torus T is a-stable. The restriction of a = qj onto T induces a linear transformation of V, again denoted by a. Let h: Hom(X, K*) -> T be defined as follows. For every x G Hom(X,K*), h(x) = t G T, where x(^) = A(i) for all X g X Then h is an isomorphism. Let A,, A2, A3, and A4 be the fundamental weights in X. Each element h(x) g 7 can uniquely be written as A(x)-n*(x*,.„), 1-1 where Xa,,z(a) = zMh,) for /g $, z g A'*, and where x(^,) = z, for 1 r2' r3} •/4= {-/■()} y5 = 0 a, a7 = i Œ/, = = Z2 ßy, = Oi -3^2 + 4^2-4) = z2 Ûy4 = X X (w3 + 4> = (Z2)3 a, = if License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 42 D. I. deriziotis and g. o. michler Let ^j be the collection of all a-stable G-conjugates of Cc(x) where x is a semisimple element of G with r(x) = 1 for all re/, Then the group Ga acts on #y by conjugation. If 7 = 0 is the empty set, then Q0 = W, and x is a regular element of Ga. There is a one-to-one correspondence between the G0-orbits of a-stable maximal tori of G and the classes of Hl(a, W), see [1, p. 186]. It is known for the triality Ga =3D4(q) that \Hl(a, W)\ = 7; cf. [14]. Let T be a a-stable maximal torus of G, with Weyl group W = NC(T)/T. If 7" is a a-stable maximal torus of G, then there is a unique class [w.] g rY^a, IF) with je {0,1,..., 6} such that T'„ is G-conjugate to 77. = T 0 = {i g r | H>ya(í) = i}. In particular, the element h(x) — nf=1 h(x>,, z,) g T belongs to T¡ if and only if 4 A(x) = Wjoh(x) = El a(Xwt(a,),z,í)/ = i For the sake of simplicity, each element h(x) Y\4=xh(xh ,z) & T is denoted by A(x) = (zlt z2> z3> z4)With this notation we can parametrize all the elemen