Petrie symmetric functions

Source abstract

For any positive integer k and nonnegative integer m , we consider the symmetric function G k , m defined as the sum of all monomials of degree m that involve only exponents smaller than k . We call G 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 G k , 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 G k , 1 , G k , 2 , G 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 G k , 2 k - 1 in the Schur basis.