Symmetric Functions as Characters of Hyperoctahedral Group

In Symmetric Group Characters as Symmetric Functions, the authors Orellana and Zabrocki showed that there is a non-homogenous basis of the ring of symmetric functions such that when these elements are evaluated at the eigenvalues of a permutation matrix, they are the values of the irreducible characters of the symmetric group. They found a formula by developing a slightly simpler basis of the symmetric functions called the induced trivial character basis which represent the trivial characters induced from a subgroup to the symmetric group.

We will be introducing an analogous idea of the irreducible character basis of symmetric group to the hyperoctahedral group, namely, hyperoctahedral group - irreducible character basis. To do this, we have defined an intermediate basis, hyperoctahedral hyperoctahedral group - induced trivial character basis that is constructed in tensor square of ring of symmetric functions in type A. We will provide an algebraic proof that these bases have the analogous properties when evaluated at pairs of eigenvalues of permutation matrices. Moreover, a combinatorial interpretation of this calculation will be provided using fillings of pairs of diagrams in two alphabets.

We will define hyperoctahedral group - irreducible character basis by change of basis using Kostka coefficients and show that these bases are families of symmetric functions that are the values of the irreducible characters of the hyperoctahedral group when evaluated at pair of multi-sets of eigenvalues of permutation matrices. We will also define a computational tool, the Frobenius map on tensor square of ring of symmetric functions such that the image is the character of hyper octahedral group that is plethysm of Schur functions in two alphabets.

The Kronecker product of two irreducible representations of hyperoctahedral group has the stability property, that is, there exists the reduced Kronecker coefficients that stabilizes in expansion of this Kronecker product for n sufficiently large. We will show that the structure coefficients in this expansion is the same as the coefficients in expansion of the regular product of any two hyperoctahedral group - irreducible character basis.