Abacus-histories and the combinatorics of creation operators
| Autor: | Loehr, Nicholas A., Warrington, Gregory S. |
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| Rok vydání: | 2020 |
| Předmět: | |
| Zdroj: | Journal of Combinatorial Theory, Series A, Volume 177, January 2021 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.jcta.2020.105330 |
| Popis: | Creation operators act on symmetric functions to build Schur functions, Hall--Littlewood polynomials, and related symmetric functions one row at a time. Haglund, Morse, Zabrocki, and others have studied more general symmetric functions $H_{\alpha}$, $C_{\alpha}$, and $B_{\alpha}$ obtained by applying any sequence of creation operators to $1$. We develop new combinatorial models for the Schur expansions of these and related symmetric functions using objects called abacus-histories. These formulas arise by chaining together smaller abacus-histories that encode the effect of an individual creation operator on a given Schur function. We give a similar treatment for operators such as multiplication by $h_m$, $h_m^{\perp}$, $\omega$, etc., which serve as building blocks to construct the creation operators. We use involutions on abacus-histories to give bijective proofs of properties of the Bernstein creation operator and Hall-Littlewood polynomials indexed by three-row partitions. Comment: 23 pages, 11 figures |
| Databáze: | arXiv |
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