Nonrecursive canonical basis computations for low rank Kashiwara crystals of type \(A\). (English) Zbl 1526.17012

Amara-Omari, Ola; Schaps, Mary

Rocky Mt. J. Math. 52, No. 6, 1929-1955 (2022).

Summary: We give an improved algorithm for the action of the divided power of a Chevalley basis element of an affine Lie algebra of type \(A\) on canonical basis elements satisfying an easily checked uniformity condition and compare calculation times for our algorithm against the standard algorithm. For symmetric Kashiwara crystals of affine type \(A\) and rank \(e=2\), and for the canonical basis elements that we call external, corresponding to weights on the outer skin of the Kashiwara crystal, we construct the canonical basis elements in a nonrecursive manner. In particular, for a symmetric crystal with \(\Lambda=a\Lambda_0+a\Lambda_1\), we give formulae for the canonical basis elements for all the \(e\)-regular multipartitions with defects either \(k(a-k)\) or \(k(a-k)+2a\), for \(0\leq k\leq a\).

MSC:

17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B37	Quantum groups (quantized enveloping algebras) and related deformations

Keywords:

canonical basis; multipartitions; Kashiwara crystals

Software:

SageMath

References:

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