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.