Polynomial elements in canonical basis \(\mathbf B\) with two-dimensional support for type \(A_4\). I (Q2880068)

This paper continues the second author's former work. All \(62\) monomial elements and \(144\) polynomial elements with one-dimensional support in the canonical basis \(\mathbb{B}\) of the quantum group for type \(A_{4}\) have been determined in [\textit{Y. Hu, J. Ye} and \textit{X. Yue}, J. Algebra 263, No. 2, 228--245 (2003; Zbl 1125.17309)] and [\textit{Y. Hu, J. Ye}, Commun. Algebra 33, No. 11, 3855--3877 (2005; Zbl 1130.17007)], respectively. It is conjectured that there are other polynomial elements in \(\mathbb{B}\) with two- or three-dimensional support. In this paper, the authors compute \(50\) polynomial elements with two-dimensional support of different-exponent type and \(62\) polynomial elements with two-dimensional support of same-exponent type in the canonical basis \(\mathbb{B}\) of the quantum group for type \(A_{4}\).