Bases for Coinvariants and Cyclic Permutations
G. Lusztig · Birkhäuser Boston eBooks · 2010
28.1.1. In this chapter we assume that (I, ·) is of finite type. Let $$\lambda \in X^ + $$ . For any sequence i = (i1,i 2 ,…, i N ) in I such that $${\rm{s}}_{i_1 } {\rm{s}}_{i_2 } \cdots {\rm{s}}_{i_N } $$ is a reduced expression of an element w $$ \in $$ W, we consider the element $$\theta {\rm{(i,}}\,\lambda {\rm{)}}\,{\rm{ = }}\theta _{i_1 }^{(a_1 )} \theta _{i_2 }^{(a_2 )} \cdots \theta _{i_N }^{(a_N )} \in {\rm{f}}$$ where $$a_1 = \left\langle {s_{i_N } \cdots s_{i_2 } (i_1 ),\lambda } \right\rangle , \cdots ,a_{N - 1} = \left\langle {s_{i_N } (i_{n - 1} ),\lambda } \right\rangle ,a_N = \left\langle {i_N ,\lambda } \right\rangle ;$$ Note that $$a_1 ,a_2 , \cdots ,a_N \in {\rm{N}}$$ by 2.2.7