On signed multiplicities of Schur expansions surrounding Petrie symmetric functions
HTML articles powered by AMS MathViewer
- by Yen-Jen Cheng, Meng-Chien Chou, Sen-Peng Eu, Tung-Shan Fu and Jyun-Cheng Yao
- Proc. Amer. Math. Soc. 151 (2023), 1839-1854
- DOI: https://doi.org/10.1090/proc/16263
- Published electronically: February 2, 2023
- HTML | PDF | Request permission
Abstract:
For $k\ge 1$, the homogeneous symmetric functions $G(k,m)$ of degree $m$ defined by $\sum _{m\ge 0} G(k,m) z^m=\prod _{i\ge 1} \big (1+x_iz+x^2_iz^2+\cdots +x^{k-1}_iz^{k-1}\big )$ are called Petrie symmetric functions. As derived by Grinberg and Fu–Mei independently, the expansion of $G(k,m)$ in the basis of Schur functions $s_{\lambda }$ turns out to be signed multiplicity free, i.e., the coefficients are $-1$, $0$ and $1$. In this paper we give a combinatorial interpretation of the coefficient of $s_{\lambda }$ in terms of the $k$-core of $\lambda$ and a sequence of rim hooks of size $k$ removed from $\lambda$. We further study the product of $G(k,m)$ with a power sum symmetric function $p_n$. For all $n\ge 1$, we give necessary and sufficient conditions on the parameters $k$ and $m$ in order for the expansion of $G(k,m)\cdot p_n$ in the basis of Schur functions to be signed multiplicity free. This settles affirmatively a conjecture of Alexandersson as the special case $n=2$.References
- Stephen Doty and Grant Walker, Modular symmetric functions and irreducible modular representations of general linear groups, J. Pure Appl. Algebra 82 (1992), no. 1, 1–26. MR 1181090, DOI 10.1016/0022-4049(92)90007-3
- Houshan Fu and Zhousheng Mei, Truncated homogeneous symmetric functions, Linear Multilinear Algebra 70 (2022), no. 3, 438–448. MR 4388836, DOI 10.1080/03081087.2020.1733460
- Manfred Gordon and E. Martin Wilkinson, Determinants of Petrie matrices, Pacific J. Math. 51 (1974), 451–453. MR 354411, DOI 10.2140/pjm.1974.51.451
- Darij Grinberg, Petrie symmetric functions, Algebr. Comb. 5 (2022), no. 5, 947–1013. MR 4511157
- Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, MA, 1981. With a foreword by P. M. Cohn; With an introduction by Gilbert de B. Robinson. MR 644144
- Linyuan Liu and Patrick Polo, On the cohomology of line bundles over certain flag schemes II, J. Combin. Theory Ser. A 178 (2021), Paper No. 105352, 11. MR 4169831, DOI 10.1016/j.jcta.2020.105352
- I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 2015. With contribution by A. V. Zelevinsky and a foreword by Richard Stanley; Reprint of the 2008 paperback edition [ MR1354144]. MR 3443860
- R.P. Stanley, Enumerative Combinatorics, vol. 2, Cambridge University Press, New York/Cambridge, 1996.
- Grant Walker, Modular Schur functions, Trans. Amer. Math. Soc. 346 (1994), no. 2, 569–604. MR 1273543, DOI 10.1090/S0002-9947-1994-1273543-0
Bibliographic Information
- Yen-Jen Cheng
- Affiliation: Department of Applied Mathematics, National Yang Ming Chiao Tung University, Hsinchu 300093, Taiwan, Republc of China
- MR Author ID: 1261410
- Email: yjc7755@nycu.edu.tw
- Meng-Chien Chou
- Affiliation: Department of Mathematics, National Taiwan Normal University, Taipei 116325, Taiwan, Republc of China
- Email: aoliver466@gmail.com
- Sen-Peng Eu
- Affiliation: Department of Mathematics, National Taiwan Normal University, Taipei 116325, Taiwan, Republc of China
- MR Author ID: 705346
- ORCID: 0000-0002-2645-2625
- Email: speu@math.ntnu.edu.tw
- Tung-Shan Fu
- Affiliation: Department of Applied Mathematics, National Pingtung University, Pingtung 900391, Taiwan, Republc of China
- MR Author ID: 350587
- Email: tsfu@mail.nptu.edu.tw
- Jyun-Cheng Yao
- Affiliation: Department of Mathematics, National Taiwan Normal University, Taipei 116325, Taiwan, Republc of China
- Email: 0955526287er@gmail.com
- Received by editor(s): June 28, 2022
- Received by editor(s) in revised form: July 21, 2022, August 7, 2022, August 17, 2022, August 18, 2022, and August 20, 2022
- Published electronically: February 2, 2023
- Additional Notes: This research was supported in part by Ministry of Science and Technology (MOST), Taiwan, grants 110-2115-M-003-011-MY3 (the third author), 111-2115-M-153-004-MY2 (the fourth author), and MOST postdoctoral fellowship 111-2811-M-A49-537-MY2 (the first author).
- Communicated by: Isabella Novik
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 1839-1854
- MSC (2020): Primary 05E05; Secondary 05A17
- DOI: https://doi.org/10.1090/proc/16263
- MathSciNet review: 4556182