Cells in the affine Weyl group of type \(\tilde D_ 4\)
Let \(W=W_ a(\tilde D_ 4)\) be the affine Weyl group of type \(\tilde D_ 4\) with \(S=\{s_ i|\) \(0\leq i\leq 4\}\) the set of its Coxeter generators such that the order of the products \(s_ is_ 2\) is equal to 3 for any \(i\neq 2\). For any proper subset I of S, let \(w_ I\) be the longest element of the subgroup of W generated by I. Let \(\ell(x)\) be the length function on W with respect to S. For \(x,y\in W\), the notation \(w=x\cdot y\) means that \(w=xy\) and \(\ell(w)=\ell(x)+\ell(y)\). In the paper under review, the author gives the following explicit description for the two-sided cells of W (in the sense of \textit{D. Kazhdan} and \textit{G. Lusztig} [see Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)]). Let \(S(i)=\{I\subset S|\ell(w_ I)=i\}\), for \(i\geq 0\). Let \(S(2)_ 1=\{\{s_ 0,s_ 1\},\{s_ 3,s_ 4\}\}\), \(S(2)_ 2=\{\{s_ 0,s_ 3\},\{s_ 1,s_ 4\}\}\), \(S(2)_ 3=\{\{s_ 0,s_ 4\},\{s_ 1,s_ 3\}\}\), \(S(6)_ 1=\{\{s_ 0,s_ 2,s_ 1\},\{s_ 3,s_ 2,s_ 4\}\}\), \(S(6)_ 2=\{\{s_ 0,s_ 2,s_ 3\},\{s_ 1,s_ 2,s_ 4\}\}\), \(S(6)_ 3=\{\{s_ 0,s_ 2,s_ 4\},\{s_ 1,s_ 2,s_ 3\}\}\). For each i with S(i)\(\neq \emptyset\), define by C(i) the set of all elements \(w\in W\) satisfying the following conditions: (a) \(w=x\cdot w_ I\cdot z\) for some \(I\in S(i)\) and x,z\(\in W;\) (b) \(w\neq x\cdot w_ J\cdot z\) for any \(J\not\in \cup_{0\leq j\leq i}S(j)\) and \(x,z\in W.\) For each j, \(1\leq j\leq 3\), and each \(i\in \{2,6\}\), define \(C(i)_ j=\{w\in C(i)|\) \(w\neq x\cdot w_ J\cdot z\) for any \(J\in S(i)- S(i)_ j\) and x,z\(\in W\}\) and define \(C(7)=C(6)-\cup_{1\leq j\leq 3}C(6)_ j\). Then the author shows that \[ W=C(0)\cup C(1)\cup (\cup_{1\leq j\leq 3}C(2)_ j)\cup C(3)\cup C(4)\cup (\cup_{1\leq j\leq 3}C(6)_ j)\cup C(7)\cup C(12) \] is the decomposition of W into 12 two- sided cells. Unfortunately, the author does not describe the left cells of W in this paper. But recently this has been done independently by Chen Chengdong and the reviewer in two different ways.
scientific article; zbMATH DE number 4214258
scientific article