# Generalized Frobenius Formula

I would like to ask a very concrete question. First I recap what I know

1. One can label a representation $$\mathcal{R}$$ of $$\mathrm{U}(N)$$ with a sequence of ordered integers (positive, negative or null): $$\rho_1 \geq \rho_2 \geq \dots \geq \rho_N \;.$$
2. Characters of $$\mathrm{U}(N)$$ are $$\chi_{\rho}(x_1, \dots, x_N) = \frac{a_{(\rho_1+N-1, \rho_2+N-2, \dots, \rho_N )}(x_1,\dots,x_N)}{a_{(N-1, N-2, \dots, 0 )}(x_1,\dots,x_N)} \;,$$ where $$a_{(k_1, \dots k_N)}(x_1,\dots x_N) = \det \left(x_i^{k_j} \right)_{i,j=1}^N \;.$$
3. In particular, when all $$\rho_i \geq 0$$ we have that the character coincides with the corresponding Schur polynomial $$χ_\rho(x_1,\dots,x_N) = s_\rho(x_1,\dots,x_N)\;.$$
4. We can introduce power sums associated to a partition $$\lambda = (\lambda_1, \dots, \lambda_\ell)$$. $$p_k(x_1,\dots,x_N) = \sum_{i=1}^Nx_i^k \;, \qquad p_\lambda(x_1,\dots,x_N)= \prod_{j=1}^\ell p_{\lambda_\ell(x_1,\dots,x_N)} \;.$$ Then it is possible to express Schur polynomial in terms of these power sums and viceversa by means of $$s_\lambda(x_1,\dots,x_N) = \sum_\eta z_\eta^{-1} \chi_\lambda^\eta p_\eta(x_1,\dots,x_N)\;, \tag{1}$$ $$p_\lambda(x_1,\dots,x_N) = \sum_\eta \chi_\eta^\lambda s_\eta(x_1,\dots,x_N) \;, \tag{2}$$ where $$z_\lambda = \prod_n n^{a_n}a_n!$$ where $$n$$ is a number appearing $$a_n$$ times in the partition $$\lambda$$, and $$\chi_\lambda^\eta$$ is the coefficient of $$x_1^{\lambda_1+m-1}x_2^{\lambda_2+m-2}\cdot\dots\cdot x_m^{\lambda_m}$$ in the expansion of $$\prod_{1\leq i < j \leq m}(x_i-x_j)p_\eta(x_1,\dots,x_m)$$. In particular, formula $$(1)$$ is dubbed "Frobenius formula".

I noticed, however that even if some of $$\rho_i$$ are negative it is possible to expand $$χ_\rho$$, which, in this case is a rational function, in terms of power sums, in this case including also negative value of $$k$$. For example $$χ_{(1,-1)}(x_1,x_2) = 2 + x_1x_2^{-1} + x_1^{-1}x_2 = -\frac{1}{2}p_{\emptyset}(x_1,x_2) + p_{(1,-1)}(x_1,x_2) \\= -\frac{1}{2}p_0(x_1,x_2) + p_1(x_1,x_2)p_{-1}(x_1,x_2) \;,$$

## QUESTION

Is there a way to generalize eqs. $$(1)$$ and $$(2)$$ for a generic character of $$\mathrm{U}(N)$$ (not only those corresponding to $$\rho_i$$ all positive), that is to write any of these characters as combination of power sums and viceversa?

If yes what is the concrete experssion for $$\chi_\lambda^\eta$$ and $$z_\lambda$$?