Petrie symmetric functions
Algebraic Combinatorics, Volume 5 (2022) no. 5, pp. 947-1013.

For any positive integer k and nonnegative integer m, we consider the symmetric function Gk,m defined as the sum of all monomials of degree m that involve only exponents smaller than k. We call Gk,m a Petrie symmetric function in honor of Flinders Petrie, as the coefficients in its expansion in the Schur basis are determinants of Petrie matrices (and thus belong to 0,1,-1 by a classical result of Gordon and Wilkinson). More generally, we prove a Pieri-like rule for expanding a product of the form Gk,m·s μ in the Schur basis whenever μ is a partition; all coefficients in this expansion belong to 0,1,-1. We also show that Gk,1,Gk,2,Gk,3,... form an algebraically independent generating set for the symmetric functions when 1-k is invertible in the base ring, and we prove a conjecture of Liu and Polo about the expansion of Gk,2k-1 in the Schur basis.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.232
Classification: 05E05
Keywords: symmetric functions, Schur functions, Schur polynomials, combinatorial Hopf algebras, Petrie matrices, Pieri rules, Murnaghan–Nakayama rule
Grinberg, Darij 1

1 Drexel University Korman Center 15 S 33rd Street Office #263 Philadelphia, PA 19104 (USA)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{ALCO_2022__5_5_947_0,
     author = {Grinberg, Darij},
     title = {Petrie symmetric functions},
     journal = {Algebraic Combinatorics},
     pages = {947--1013},
     publisher = {The Combinatorics Consortium},
     volume = {5},
     number = {5},
     year = {2022},
     doi = {10.5802/alco.232},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.232/}
}
TY  - JOUR
AU  - Grinberg, Darij
TI  - Petrie symmetric functions
JO  - Algebraic Combinatorics
PY  - 2022
SP  - 947
EP  - 1013
VL  - 5
IS  - 5
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.232/
DO  - 10.5802/alco.232
LA  - en
ID  - ALCO_2022__5_5_947_0
ER  - 
%0 Journal Article
%A Grinberg, Darij
%T Petrie symmetric functions
%J Algebraic Combinatorics
%D 2022
%P 947-1013
%V 5
%N 5
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.232/
%R 10.5802/alco.232
%G en
%F ALCO_2022__5_5_947_0
Grinberg, Darij. Petrie symmetric functions. Algebraic Combinatorics, Volume 5 (2022) no. 5, pp. 947-1013. doi : 10.5802/alco.232. https://alco.centre-mersenne.org/articles/10.5802/alco.232/

[1] Ahmia, Moussa; Merca, Mircea A generalization of complete and elementary symmetric functions (2020) | arXiv

[2] Atiyah, M.F.; Tall, D.O. Group representations, λ-rings and the J-homomorphism, Topology, Volume 8 (1969) no. 3, pp. 253-297 | DOI | MR

[3] Bazeniar, Abdelghafour; Ahmia, Moussa; Belbachir, Hacène Connection between bi s nomial coefficients and their analogs and symmetric functions, Turkish J. Math., Volume 42 (2018), pp. 807-818 | DOI | MR | Zbl

[4] Berg, Chris; Bergeron, Nantel; Saliola, Franco; Serrano, Luis; Zabrocki, Mike A Lift of the Schur and Hall–Littlewood Bases to Non-commutative Symmetric Functions, Canad. J. Math., Volume 66 (2014) no. 3, pp. 525-565 | DOI | MR | Zbl

[5] Bergeron, François Symmetric Functions and Rectangular Catalan Combinatorics http://bergeron.math.uqam.ca/wp-content/uploads/2019/09/Symmetric-Functions.pdf (Accessed 2019-09-07)

[6] Cameron, Peter J. Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, 1994 (http://www.maths.qmul.ac.uk/~pjc/comb/)

[7] Crane, Rixon Plane Partitions in Number Theory and Algebra, Masters thesis, University of Queensland (1998) (honors’ thesis; https://rixonc.github.io/files/Rixon_Crane_Honours_Thesis.pdf)

[8] Doty, Stephen; Walker, Grant Modular symmetric functions and irreducible modular representations of general linear groups, J. Pure Appl. Algebra, Volume 82 (1992) no. 1, pp. 1-26 | DOI | MR | Zbl

[9] Egge, Eric S. An Introduction to Symmetric Functions and Their Combinatorics, Student Mathematical Library, 91, AMS, 2019 | DOI | MR

[10] Fu, Houshan; Mei, Zhousheng Truncated homogeneous symmetric functions, Linear Multilinear Algebra (2020), pp. 1-11 | DOI | Zbl

[11] Fulkerson, D. R.; Gross, O. A. Incidence matrices and interval graphs, Pacific J. Math., Volume 15 (1965), pp. 835-855 | DOI | MR | Zbl

[12] Fulton, William; Lang, Serge Riemann–Roch algebra, Springer, 1985 | DOI

[13] Gordon, Manfred; Wilkinson, E. Martin Determinants of Petrie matrices, Pacific J. Math., Volume 51 (1974), pp. 451-453 | DOI | MR | Zbl

[14] Grinberg, Darij, FPSAC 2020 Proceedings | Zbl

[15] Grinberg, Darij Petrie symmetric functions (2021) (arXiv version of the present paper) | arXiv

[16] Grinberg, Darij Petrie symmetric functions [detailed version] (2021) detailed version of the present paper (ancillary file) | arXiv

[17] Grinberg, Darij; Reiner, Victor Hopf algebras in Combinatorics (2020) | arXiv

[18] Hazewinkel, Michiel Witt vectors. Part 1 (2008) | arXiv

[19] Liu, Linyuan; Polo, Patrick On the cohomology of line bundles over certain flag schemes II, J. Combin. Theory Ser. A, Volume 178 (2021), p. 105352 | DOI | MR | Zbl

[20] Macdonald, Ian G. Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs, Oxford Science Publications, 1995

[21] Mendes, Anthony; Remmel, Jeffrey Counting with Symmetric Functions, Developments in Mathematics, 43, Springer, 2015 | DOI

[22] Olsson, Jørn Combinatorics and representations of finite groups, Vorlesungen aus dem FB Mathematik der Univ. Essen, 20, Universität Essen, 1993 (http://web.math.ku.dk/~olsson/manus/comb_rep_all.pdf)

[23] Sagan, Bruce Combinatorics: The Art of Counting, Graduate Studies in Mathematics, 210, AMS, 2015

[24] Sam, Steven V. Notes for Math 740 (Symmetric Functions) https://mathweb.ucsd.edu/~ssam/old/17S-740/notes.pdf (Accessed 2017-04-27)

[25] Stanley, Richard Enumerative Combinatorics, volume 2, Cambridge University Press, 2001

[26] The Sage Developers SageMath, the Sage Mathematics Software System (Version 9.0) (2020) (https://www.sagemath.org)

[27] Walker, Grant Modular Schur functions, Trans. Amer. Math. Soc., Volume 346 (1994) no. 2, pp. 569-604 | DOI | MR | Zbl

[28] Wildon, Mark A Combinatorial Proof of a Plethystic Murnaghan–Nakayama Rule, SIAM J. Discrete Math., Volume 30 (2016) no. 3, pp. 1526-1533 | DOI | MR | Zbl

[29] Zelevinsky, Andrey V. Representations of Finite Classical Groups: A Hopf Algebra Approach, Lecture Notes in Mathematics, 869, Springer, 1981 | DOI | MR

Cited by Sources: