Polynomial Elements in Canonical Basis B with Two-dimensional Support for Type A₄ (I)

Abstract

All 62 monomial elements and 144 polynomial elements with one-dimensional support in the canonical basis B of the quantum group for type A4 have been determined in [5] and [4], respectively. It is conjectured in [4] that there are other polynomial elements in B with two- or three-dimensional support. In this paper, we 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 B of the quantum group for type A4.

Canonical Bases of Modified Quantum Algebras for TypeA2-monomial Elements

“…By now, the basis is only computed for type A 1 , A 2 , A 3 , B 2 (see [7,12,13]). For type A 4 , part of the basis is computed in [2,3,6].…”

Tight Monomials in Quantum Group for Type <i>A</i><sub>5</sub> with <i>t</i> ≤ 6

“…Based on Lusztig's work, Xi [5] found explicitly all 14 canonical basis elements of type A 3 (consisting of 8 longest monomials and 6 polynomials with one-dimensional support). For type A 4 , Hu, Ye and Yue [6] determined all 62 longest monomials in canonical basis, Hu and Ye [7] gave all 144 polynomials with one-dimensional support in canonical basis, and Li and Hu [8] got 112 polynomials with two-dimensional support in canonical basis. For type A n (n ≥ 5), Marsh [9] carried out thorough investigation.…”

Tightness of certain monomials

“…By now, the basis is only computed for types A 2 , A 3 and B 2 (see [10,20,21]). For type A 4 , part of the basis is computed in [4,5,7]. Lusztig [14] investigated the tightness of monomials, i.e., when a monomial is in the canonical basis or is a Z-linear combination of elements in the canonical basis.…”