Quasi-immanants

TL;DR

This paper introduces a new family of quasi-immanants generalizing classical immanants by utilizing quasisymmetric functions and cycle compositions, with combinatorial formulas for their coefficients.

Contribution

It extends the concept of immanants to the algebra of quasisymmetric functions using cycle compositions and quasisymmetric power sum bases.

Findings

Defined quasi-immanants via quasisymmetric functions.

Derived combinatorial formulas for coefficients of quasi-immanants.

Connected quasi-immanants to quasisymmetric Schur functions.

Abstract

For an integer partition $ \lambda$ of $n$ and an $n \times n$ matrix $A$, consider the expansion of the immanant $\text{Imm}^{\lambda}(A)$ as a sum indexed by permutations $\sigma$ of order $n$, with coefficients given by the irreducible characters $\chi^{\lambda}(\text{ctype}(\sigma))$ of the symmetric group $S_{n}$, for the cycle type $\text{ctype}(\sigma) \vdash n$ of $\sigma$. Skandera et al. have introduced combinatorial interpretations of a generalization of immanants given by replacing the coefficient $\chi^{\lambda}(\text{ctype}(\sigma))$ with preimages with respect to the Frobenius morphism of elements among the distinguished bases of the algebra $\textsf{Sym}$ of symmetric functions. Since $ \textsf{Sym}$ is contained in the algebra $\textsf{QSym}$ of quasisymmetric functions, this leads us to further generalize immanants with the use of quasisymmetric functions. Since bases of $ \textsf{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.