Algebres d'Operateurs by Pierre de la Harpe

By Pierre de la Harpe

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2. Cartan data for 5 0 e ( 2 , 2 n - 2) W/H0\ mc,H)\ H dim A P Mss Hm 2 Pm SO(2n - 4) 2 Hn 1 Pn SL2M x SO(2n - 4) 1 2n~2(n-2)! 2n(n-l) \ Hc 0 G G 1 2 n " 2 (n - 1)! 2 n (n - 2)! | \VP\ n(n- 2n 1) 22 BRIAN D. BOE AND DAVID H. 10c). 4. Define another labeled graph, denoted T b , by replacing all occurrences of jj by b. 10f) VPm =T*VT*\ the ordering imposed above gives us all of the complex type weak order relations on T>pm. 4. 4 describes the ordered set V\ (together with part of P 0 , as will be explained below).

Fix /? = e* - e j G 6 = ft * . 9)(1), every element of ft~ occurs in this way. If /? , so we may assume 0 is not simple. 9)(5). Furthermore every Q G ft/5 must have parity e. 9)(3), the roots ek - et e ftp (i < k < I < j) must be determined by the following algorithm. (1) For any A; such that fce, (k + l)~ c G 8, ek - ek{ \ G ft/3. (2) Remove from the sequence 8 all numbers k\ V such that ek> — e/> has already been determined to belong to ftp. If k€t I e are adjacent in this shortened sequence, then ek — ei G ftp.

Assume that b > a-f 1. Then for every ft e £3, 8*rn+ consists of exactly one element ofV0.

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