Journal article
Petrie symmetric functions
Algebraic combinatorics, v 5(5), pp 947-1013
01 Jan 2022
Abstract
For any positive integer k and nonnegative integer m, we consider the symmetric function C (k, m) defined as the sum of all monomials of degree m that involve only exponents smaller than k. We call C (k, 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 C (k, m) <middle dot> s mu in the Schur basis whenever is a partition; all coefficients in this expansion belong to {0, 1, -1}. We also show that C (k, 1) , C (k, 2) , C (k, 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 C (k, 2k - 1) in the Schur basis.
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Details
- Title
- Petrie symmetric functions
- Creators
- Darij Grinberg - Drexel University
- Publication Details
- Algebraic combinatorics, v 5(5), pp 947-1013
- Publisher
- Combinatorics Consortium
- Number of pages
- 68
- Grant note
- 2016-06596 / Swedish Research Council
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:001343960500007
- Scopus ID
- 2-s2.0-85143634644
- Other Identifier
- 991022237375904721