Quasi-immanants
John M. Campbell
TL;DR
The paper introduces a quasisymmetric generalization of immanants by replacing symmetric-function coefficients with data from $\textsf{QSym}$ indexed by cycle compositions via a Frobenius-type map and Ballantine et al.'s quasisymmetric power sums. It defines quasi-immanants $\text{QImm}^{Q}_{\Psi}$ and $\text{QImm}^{Q}_{\Phi}$ for $Q\in \textsf{QSym}$ and proves recovery of the classical immanants in key specializations ($Q=s_{\lambda}$, $Q=e_{\lambda}$, $Q=m_{\lambda}$). The work further analyzes quasi-Schur second immanants, showing that $\mathcal{S}_{(2,1^{n})}\notin \textsf{Sym}$ and giving an explicit coefficient rule for $\text{QImm}^{\mathcal{S}_{(2,1^{n-2})}}_{\Psi}$ that depends on the cycle composition, thereby demonstrating that these quasi-immanants are not simply equivalents of the classical second immanant. It also discusses Toeplitz matrix cases and outlines directions for future study, including deeper connections with quasisymmetric function theory and potential Frobenius-type mappings in the quasisymmetric setting.
Abstract
For an integer partition $ λ$ of $n$ and an $n \times n$ matrix $A$, consider the expansion of the immanant $\text{Imm}^λ(A)$ as a sum indexed by permutations $σ$ of order $n$, with coefficients given by the irreducible characters $χ^λ(\text{ctype}(σ))$ of the symmetric group $S_{n}$, for the cycle type $\text{ctype}(σ) \vdash n$ of $σ$. Skandera et al. have introduced combinatorial interpretations of a generalization of immanants given by replacing the coefficient $χ^λ(\text{ctype}(σ))$ 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.