TY - JOUR

T1 - ON SIGNED MULTIPLICITIES OF SCHUR EXPANSIONS SURROUNDING PETRIE SYMMETRIC FUNCTIONS

AU - Cheng, Yen Jen

AU - Chou, Meng Chien

AU - Eu, Sen Peng

AU - Fu, Tung Shan

AU - Yao, Jyun Cheng

N1 - Funding Information:
Received by the editors June 28, 2022, and, in revised form, July 21, 2022, August 7, 2022, August 17, 2022, August 18, 2022, and August 20, 2022. 2020 Mathematics Subject Classification. Primary 05E05; Secondary 05A17. Key words and phrases. Petrie symmetric functions, truncated homogeneous symmetric functions, modular complete symmetric functions, signed multiplicity free. 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).
Publisher Copyright:
©2023 American Mathematical Society.

PY - 2023/5/1

Y1 - 2023/5/1

N2 - For k ≥ 1, the homogeneous symmetric functions G(k, m) of degree m defined by Formula Presented 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 sλ 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 sλ in terms of the k-core of λ and a sequence of rim hooks of size k removed from λ. We further study the product of G(k, m) with a power sum symmetric function pn. For all n ≥ 1, we give necessary and sufficient conditions on the parameters k and m in order for the expansion of G(k, m) · pn in the basis of Schur functions to be signed multiplicity free. This settles affirmatively a conjecture of Alexandersson as the special case n = 2.

AB - For k ≥ 1, the homogeneous symmetric functions G(k, m) of degree m defined by Formula Presented 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 sλ 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 sλ in terms of the k-core of λ and a sequence of rim hooks of size k removed from λ. We further study the product of G(k, m) with a power sum symmetric function pn. For all n ≥ 1, we give necessary and sufficient conditions on the parameters k and m in order for the expansion of G(k, m) · pn in the basis of Schur functions to be signed multiplicity free. This settles affirmatively a conjecture of Alexandersson as the special case n = 2.

KW - Petrie symmetric functions

KW - modular complete symmetric functions

KW - signed multiplicity free

KW - truncated homogeneous symmetric functions

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U2 - 10.1090/proc/16263

DO - 10.1090/proc/16263

M3 - Article

AN - SCOPUS:85150054321

SN - 0002-9939

VL - 151

SP - 1839

EP - 1854

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 5

ER -