The modified quantized enveloping algebraU has a remarkable canonical basis, which was introduced by Lusztig. In this paper, all these monomial elements of the canonical basis ofU for type A 2 are determined.
The modified quantized enveloping algebraU has a remarkable canonical basis, which was introduced by Lusztig. In this paper, all these monomial elements of the canonical basis ofU for type A 2 are determined.
“…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.…”
“…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.…”
Section: Introductionmentioning
confidence: 99%
“…. , a 5 1, we can easily see that z 32 , z 33 are not unique to make sure ξ = 0, so that (X λ , X λ ) 0 2. For example, one can easily check that (X λ , X λ ) 0 = 2 if λ = (1, 1, 1, 1, 1).…”
For a quantized enveloping algebra of finite type, one can associate a natural monomial to a dominant weight. We show that these monomials for types A 5 and D 4 are semitight (i.e., a Z-linear combination of elements in the canonical basis) by a direct calculation.
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