A generating function associated with the alternating elements in the positive part of. Issue 4 (3rd April 2023)

• Record Type: Journal Article
• Authors: Ruan, Chenwei
• Is Part Of: Communications in algebra. Volume 51:Issue 4(2023)
• Page Start: 1707
• Page End: 1720
• Publication Date: 2023-04-03
• Subjects: Catalan word -- generating function -- q-shuffle algebra; Primary: 17B37 -- Secondary: 05E14 -- 81R50; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2022.2140350
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: The positive part U q + of U q ( s l ̂ 2 ) admits an embedding into a q -shuffle algebra. This embedding was introduced by M. Rosso in 1995. In 2019, Terwilliger introduced the alternating elements { W − n } n ∈ N, { W n + 1 } n ∈ N, { G n + 1 } n ∈ N, { G ˜ n + 1 } n ∈ N in U q + using the Rosso embedding. He showed that the alternating elements { W − n } n ∈ N, { W n + 1 } n ∈ N, { G ˜ n + 1 } n ∈ N form a PBW basis for U q +, and he expressed { G n + 1 } n ∈ N in this alternating PBW basis. In his calculation, Terwilliger used some elements { D n } n ∈ N with the following property: the generating function D ( t ) = ∑ n ∈ N D n t n is the multiplicative inverse of the generating function G ˜ ( t ) = ∑ n ∈ N G ˜ n t n where G ˜ 0 = 1 . Terwilliger defined { D n } n ∈ N recursively; in this paper, we will express { D n } n ∈ N in closed form. Communicated by Alberto Elduque