Ch. 9 Lascoux POLYNOMIALS - Key polynomials for type $B, C, D$
9.1 $K^B, K^C, K^D$
Type $B$
Suppose that $x^v$ is a dominant monomial, i.e. $v_1 \ge \cdots \ge v_n \ge 0$. Then put
\[K_v^B \coloneqq x^v.\]Otherwise, we define
\[K_v^B \coloneqq K_{vs_i}^B \pi_i \text{ when } v_i < v_{i+1} \text{ and } i < n\]and
\[K_v^B \coloneqq K_{vs_n^B}^B \pi_n^B \text{ when } v_n < 0.\]Type $C$
Suppose that $x^v$ is a dominant monomial, i.e. $v_1 \ge \cdots \ge v_n \ge 0$. Then put
\[K_v^C \coloneqq x^v.\]Otherwise, we define
\[K_v^C \coloneqq K_{vs_i}^C \pi_i \text{ when } v_i < v_{i+1} \text{ and } i < n\]and
\[K_v^C \coloneqq K_{vs_n^C}^C \pi_n^C \text{ when } v_n < 0.\]Type $D$
Type $D$, in particular, requires some extra care. Unlike in types $A, B, C$, the set of dominant monomials is
\[x^v \text{ with } v = [v_1, \ldots, v_{n-1}, \pm v_n] \text{ where } v_1 \ge \cdots \ge v_n \ge 0.\]We then define
\[K_v^D \coloneqq x^v\]when $v$ is dominant in type $D$. Otherwise,
\[K_v^D \coloneqq K_{vs_i}^D \pi_i \text{ when } v_i < v_{i+1} \text{ and } i < n\]and
\[K_v^D \coloneqq K_{vs_n^D}^D \pi_n^D \text{ when } v_{n-1} + v_n < 0.\]