Quasi-immanants

Abstract

For an integer partition λ of n and an $n\times n$ matrix A, consider the expansion of the immanant ${\text{Imm}}^{\lambda }\left(A\right)$ as a sum indexed by permutations σ of order n, with coefficients given by the irreducible characters ${\chi }^{\lambda }\left(\text{ctype}\right(\sigma \left)\right)$ of the symmetric group $S_n$, for the cycle type $\text{ctype}\left(\sigma \right)⊢n$ of σ. Skandera et al. have introduced combinatorial interpretations of a generalization of immanants given by replacing the coefficient ${\chi }^{\lambda }\left(\text{ctype}\right(\sigma \left)\right)$ with preimages with respect to the Frobenius morphism of elements among the distinguished bases of the algebra $\text{Sym}$ of symmetric functions. Since $\text{Sym}$ is contained in the algebra $\text{QSym}$ of quasisymmetric functions, this leads us to further generalize immanants with the use of quasisymmetric functions. Since bases of $\text{QSym}$ are indexed by integer compositions, we make use of cycle compositions in place of cycle types to define the family of quasi-immanants introduced in this paper. This is achieved through the use of the quasisymmetric power sum bases due to Ballantine et al., and we prove a combinatorial formula for the coefficients arising in an analogue, given by a special case of quasi-immanants associated with quasisymmetric Schur functions, of second immanants.