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F. Aliniaeifard and S. van Willigenburg, The extra basis in noncommuting variables, Adv. in Appl. Math. 168 (2025) [20 pages]
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Abstract:We answer a question of Bergeron, Hohlweg, Rosas, and Zabrocki from 2006 to give a combinatorial description for the coproduct of the x-basis in the Hopf algebra of symmetric functions in noncommuting variables, NCSym, which arises in the theory of Grothendieck bialgebras. We achieve this by applying the theory of Hopf monoids and the Fock functor. We also determine combinatorial expansions of this basis in terms of the monomial and power sum symmetric functions in NCSym, and by taking the commutative image of the x-basis we discover a new multiplicative basis for the algebra of symmetric functions.
https://doi.org/10.1016/j.aam.2025.102887
arXiv:2409.00177
Indexed by:Journal paper
Translation or Not:no
Pre One:Farid Aliniaeifard, Shamil Asgarli, Maria Esipova, Ethan Shelburne, Stephanie van Willigenburg and Tamsen Whitehead McGinley, Chromatic Quasisymmetric Functions of the Path Graph, Annals of Combinatorics, 2025
Next One:F. Aliniaeifard and S. Li, Peak algebra in noncommuting variables, The 37th Conference on Formal Power Series and Algebraic Combinatorics (FPSAC), Sapporo, Japan, 2025.
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