Let $b_\lambda = \sum_{g \in Q_\lambda} (-1)^{sgn(g)}g$, where $Q_\lambda$ is the subgroup of elements that permute the numbers in columns, $\lambda$ a Young diagram corresponding to partition $\lambda$, filled with consecutive numbers going left to right, top-down. I have to calculate the character of $\mathbb{C}[S_n]b_\lambda$.
I know that $\mathbb{C}[S_n]b_\lambda$ is a direct sum of $V_\mu$, $V_\mu = \mathbb{C}[S_n]c_\mu$, where $c_\mu$ is the product $a_\mu b_\mu$, and $a_\mu = \sum_{g \in P_\mu}g$, $P_\mu$ the subgroup permuting the numbers in rows, and $\mu \leq_{lex} \lambda$. What are the coefficients with which $V_\mu$ enter this sum? Are they Kostka numbers? Is there a neat formula this character, or is it just the sum of the characters computed via Frobenius formula with coefficients being Kostka numbers?