Abstract
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 Λ = aΛ0 + aΛ1, 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 ≤ k ≤ a.
| Original language | English |
|---|---|
| Pages (from-to) | 1929-1955 |
| Number of pages | 27 |
| Journal | Rocky Mountain Journal of Mathematics |
| Volume | 52 |
| Issue number | 6 |
| DOIs | |
| State | Published - Dec 2022 |
Bibliographical note
Publisher Copyright:© Rocky Mountain Mathematics Consortium.
Funding
This research was partially supported by a fellowship from the Israel Ministry of Science and Technology. Some of the results also appear in the Ph.D. thesis of Amara-Omari. 2020 AMS Mathematics subject classification: 17B10, 17B37. Keywords and phrases: canonical basis, multipartitions, Kashiwara crystals. Received by the editors on July 29, 2020, and in revised form on January 24, 2022.
| Funders | Funder number |
|---|---|
| Ministry of science and technology, Israel |
Keywords
- Kashiwara crystals
- canonical basis
- multipartitions