Canonical Basis for Type $A_4$ (II)-Polynomial Elements in One Variable

All the 62 monomial elements in the canonical basis B of the quantized enveloping algebra for type $A_4$ have been determined in [2]. According to Lusztig’s idea [7], the elements in the canonical basis B consist of monomials and linear combinations of monomials (for convenience, we call them polynomials). In this note, we compute all the 144 polynomial elements in one variable in the canonical basis B of the quantized enveloping algebra for type $A_4$ based on our joint note [2]. We conjecture that there are other polynomial elements in two or three variables in the canonical basis B, which include independent variables and dependent variables. Moreover, it is conjectured that there are no polynomial elements in the canonical basis B with four or more variables