Equivariant Kazhdan--Lusztig Polynomials of Thagomizer Matroids with a Hyperoctahedral Group Action

The thagomizer matroid, realized as the graphic matroid of the complete tripartite graph $K_{1,1,n}$, has full automorphism group isomorphic to the hyperoctahedral group whenever $n \ge 2$. In the equivariant setting for this action, we compute both the Kazhdan--Lusztig polynomial and the inverse Kazhdan--Lusztig polynomial in the sense of Proudfoot's Kazhdan--Lusztig--Stanley theory, and we show that each coefficient is an honest representation with a multiplicity-free irreducible decomposition. Our main idea is to exploit the palindromicity of the equivariant $Z$-polynomial, reducing the computation to the already established symmetric-group equivariant Kazhdan--Lusztig theory for the graphic matroids of cycle graphs, and then to apply Proudfoot's equivariant Kazhdan--Lusztig--Stanley inversion identity to obtain the inverse polynomial. Passing to dimensions recovers the previously known nonequivariant thagomizer polynomials, while the coefficient formulas admit a natural expression in terms of the wreath product Frobenius characteristic for the hyperoctahedral group.