2025/08/25 by Dong, Xiangshuai, Wu, Tingzeng, Lai, HongJian
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2508.17743
Let χλ be an irreducible character of the symmetric group Sn. For an n × n matrix M = (mij), define the immanant of M corresponding to χλ by dλ(M) = ∑σ∈ Sn χλ(σ) ∏i=1n miσ(i). For λ= (k, 1n-k), the immanant d(k, 1n-k)(M) is called the hook immanant and denoted by dk(M). The hook immanant polynomial of matrix M is defined as dk(xIn - M), where In is the n × n identity matrix. Let G and \overrightarrowG be a graph and a digraph, respectively. Suppose that D(G) and A(G) (resp. D(\overrightarrowG) and A(\overrightarrowG)) are the degree matrix and adjacency matrix of G (resp. \overrightarrowG), respectively. In this paper, we characterize two hook immanantal equalities for the linear combination of matrices βD(G)+γA(G) and βD(\overrightarrowG)+γA(\overrightarrowG), where β and γ are real numbers. As applications, we derive recursive formulas for the hook immanantal polynomials and hook immanants of graph matrices.