A decomposition of the group algebra of a finite Coxeter group
A decomposition of the group algebra of a finite Coxeter group
复制标题
DOI：
10.1016/0021-8693(68)90022-7
复制
发表时间：
1968-06
影响因子：
0.9
通讯作者：
Louis Solomon
中科院分区：
文献类型：
--
作者：
Louis Solomon
设W是Z阶的有限Coxeter群(见[IO])。这意味着W是一个有限群，它有一组R={rl，…，rz}的对合生成元和一组形式为(r，>；nl)的定义关系。IG=1.对于每个子集Kofi={L，…，Z}，设W是由Rk生成的子群，K是K，在[8]中我证明了一个特征标公式，其中+K是由W的主标诱导的W的特征标，E是W的交错特征标，W到群{+L，-i}的同态使得E(？J=-1，避免了对W的性质的任何仔细考察，而是依赖于Hopf迹公式在W在一个合适的单纯复形--W的Coxeter复形上的作用(见[ZO，12 1).本文对这个公式提供了一个代数解释，并推广了[S]中的猜想.给出了关于W的群代数的一些新的信息，特别是将W的群代数分解成2z个左理想的直和。我们首先假定W是由集合R={ri：i Ei}生成的有限群，并且对于i Ei存在同态E：W-+{+1，-L}且e(Ri)=-1，这相当于说生成集R位于W中指标2的某个子群之外。定理1的证明不需要额外的假设，从而证明了对于更大类的有限群，(1)型公式是存在的。在
Let W be a finite Coxeter group of rank Z,(see [IO]). This means that W is a finite group with a set R={rl,..., rz} of involutory generators and a set of defining relations of the form (r,> nl. ig= 1. For each subset KofI={l,..., Z} let W, be the subgroup of W generated by the rk with k E K. In [8] I proved a character formula where+ K is the character of W induced by the principal character of W,, and E is the alternating character of W, the homomorphism of W into the group {+ l,-I} such that E (? J=-1 for i E 1. The argument avoided any close scrutiny of the characters of W and depended on application of the Hopf trace formula to the action of W on a suitable simplicial complex, the Coxeter complex of W (see [ZO, 121).The present paper provides an algebraic explanation for this formula and a generalization of it conjectured in [S]. It also gives some new information about the group algebra of W, in particular a decomposition of the group algebra of W into a direct sum of 2z left ideals. We begin by assuming only that W is a finite group generated by a set R={ri: i E I} and that there exists a homomorphism E: W-+{+ 1,-l} with e (ri)=-1 for i E I. This amounts to saying that the generating set R lies outside some subgroup of index two in W. The proof of Theorem 1 requires no additional hypothesis and hence shows that formulas of type (1) exist for a wide class of finite groups. In the