2025/01/26 by John M. Campbell, Campbell, John M.
Computer Science · Mathematics · #15A15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2501.15667
openalex publication_date 2025/01/26 · openalex created_date 2025/01/29 · openalex updated_date 2026/07/28
For an integer partition λ of n and an n × n matrix A, consider the expansion of the immanant Immλ(A) as a sum indexed by permutations σ of order n, with coefficients given by the irreducible characters χλ(ctype(σ)) of the symmetric group Sn, for the cycle type ctype(σ) \vdash n of σ. Skandera et al. have introduced combinatorial interpretations of a generalization of immanants given by replacing the coefficient χλ(ctype(σ)) with preimages with respect to the Frobenius morphism of elements among the distinguished bases of the algebra \textsfSym of symmetric functions. Since \textsfSym is contained in the algebra \textsfQSym of quasisymmetric functions, this leads us to further generalize immanants with the use of quasisymmetric functions. Since bases of \textsfQSym 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.