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- Title
On Antipodes of Immaculate Functions.
- Authors
Campbell, John Maxwell
- Abstract
The immaculate basis of the Hopf algebra NSym of noncommutative symmetric functions is a Schur-like basis of NSym that has been applied in many areas in the field of algebraic combinatorics. The problem of determining a cancellation-free formula for the antipode of NSym evaluated at an arbitrary immaculate function S α remains open, letting α denote an integer composition. However, for the cases whereby we let α be a hook or consist of at most two rows, Benedetti and Sagan (J Combin Theory Ser A 148:275–315, 2017) have determined cancellation-free formulas for expanding S (S α) in the S -basis. According to a Jacobi–Trudi-like formula for expanding immaculate functions in the ribbon basis that we had previously proved bijectively (Discrete Math 340(7):1716–1726, 2017), by applying the antipode S of NSym to both sides of this formula, we obtain a cancellation-free formula for expressing S (S (m n)) in the R-basis, for an arbitrary rectangle (m n) . We explore the idea of using this R-expansion, together with sign-reversing involutions, to determine combinatorial interpretations of the S -coefficients of antipodes of rectangular immaculate functions. We then determine cancellation-free formulas for antipodes of immaculate functions much more generally, using a Jacobi–Trudi-like formula recently introduced by Allen and Mason that generalizes Campbell's formulas for expanding S -elements into the R-basis, and we further explore how new families of composition tableaux may be used to obtain combinatorial interpretations for expanding S (S α) into the S -basis.
- Subjects
SYMMETRIC functions; ALGEBRAIC fields; FAMILY structure; HOPF algebras; NONCOMMUTATIVE algebras; RECTANGLES
- Publication
Annals of Combinatorics, 2023, Vol 27, Issue 3, p579
- ISSN
0218-0006
- Publication type
Article
- DOI
10.1007/s00026-022-00632-0